SciPy Reference Guide

SciPy Reference Guide
SciPy Reference Guide
Release 0.8.0rc3
Written by the SciPy community
July 15, 2010
CONTENTS
1
SciPy Tutorial
1.1 Introduction . . . . . . . . . . . . . . . . . . .
1.2 Basic functions in Numpy (and top-level scipy) .
1.3 Special functions (scipy.special) . . . . .
1.4 Integration (scipy.integrate) . . . . . . .
1.5 Optimization (optimize) . . . . . . . . . . . . .
1.6 Interpolation (scipy.interpolate) . . . .
1.7 Fourier Transforms (scipy.fftpack) . . . .
1.8 Signal Processing (signal) . . . . . . . . . . . .
1.9 Linear Algebra . . . . . . . . . . . . . . . . . .
1.10 Statistics . . . . . . . . . . . . . . . . . . . . .
1.11 Multi-dimensional image processing (ndimage)
1.12 File IO (scipy.io) . . . . . . . . . . . . . . .
1.13 Weave . . . . . . . . . . . . . . . . . . . . . .
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3
3
6
10
10
14
26
34
36
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62
85
91
2
Release Notes
127
2.1 SciPy 0.8.0 Release Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3
Reference
3.1 Clustering package (scipy.cluster) . . . . . . . . . .
3.2 Constants (scipy.constants) . . . . . . . . . . . . . .
3.3 Fourier transforms (scipy.fftpack) . . . . . . . . . .
3.4 Integration and ODEs (scipy.integrate) . . . . . . .
3.5 Interpolation (scipy.interpolate) . . . . . . . . . .
3.6 Input and output (scipy.io) . . . . . . . . . . . . . . . .
3.7 Linear algebra (scipy.linalg) . . . . . . . . . . . . .
3.8 Maximum entropy models (scipy.maxentropy) . . . .
3.9 Miscellaneous routines (scipy.misc) . . . . . . . . . .
3.10 Multi-dimensional image processing (scipy.ndimage) .
3.11 Orthogonal distance regression (scipy.odr) . . . . . . .
3.12 Optimization and root finding (scipy.optimize) . . . .
3.13 Signal processing (scipy.signal) . . . . . . . . . . . .
3.14 Sparse matrices (scipy.sparse) . . . . . . . . . . . . .
3.15 Sparse linear algebra (scipy.sparse.linalg) . . . .
3.16 Spatial algorithms and data structures (scipy.spatial)
3.17 Special functions (scipy.special) . . . . . . . . . . .
3.18 Statistical functions (scipy.stats) . . . . . . . . . . .
3.19 C/C++ integration (scipy.weave) . . . . . . . . . . . .
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133
133
156
171
183
195
215
221
254
267
271
324
331
359
390
407
420
443
472
673
i
Bibliography
677
Index
683
ii
SciPy Reference Guide, Release 0.8.0rc3
Release
0.8
Date
July 15, 2010
SciPy (pronounced “Sigh Pie”) is open-source software for mathematics, science, and engineering.
CONTENTS
1
SciPy Reference Guide, Release 0.8.0rc3
2
CONTENTS
CHAPTER
ONE
SCIPY TUTORIAL
1.1 Introduction
Contents
• Introduction
– SciPy Organization
– Finding Documentation
SciPy is a collection of mathematical algorithms and convenience functions built on the Numpy extension for Python.
It adds significant power to the interactive Python session by exposing the user to high-level commands and classes
for the manipulation and visualization of data. With SciPy, an interactive Python session becomes a data-processing
and system-prototyping environment rivaling sytems such as Matlab, IDL, Octave, R-Lab, and SciLab.
The additional power of using SciPy within Python, however, is that a powerful programming language is also available
for use in developing sophisticated programs and specialized applications. Scientific applications written in SciPy
benefit from the development of additional modules in numerous niche’s of the software landscape by developers
across the world. Everything from parallel programming to web and data-base subroutines and classes have been
made available to the Python programmer. All of this power is available in addition to the mathematical libraries in
SciPy.
This document provides a tutorial for the first-time user of SciPy to help get started with some of the features available
in this powerful package. It is assumed that the user has already installed the package. Some general Python facility
is also assumed such as could be acquired by working through the Tutorial in the Python distribution. For further
introductory help the user is directed to the Numpy documentation.
For brevity and convenience, we will often assume that the main packages (numpy, scipy, and matplotlib) have been
imported as:
>>>
>>>
>>>
>>>
import
import
import
import
numpy as np
scipy as sp
matplotlib as mpl
matplotlib.pyplot as plt
These are the import conventions that our community has adopted after discussion on public mailing lists. You will
see these conventions used throughout NumPy and SciPy source code and documentation. While we obviously don’t
require you to follow these conventions in your own code, it is highly recommended.
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SciPy Reference Guide, Release 0.8.0rc3
1.1.1 SciPy Organization
SciPy is organized into subpackages covering different scientific computing domains. These are summarized in the
following table:
Subpackage
cluster
constants
fftpack
integrate
interpolate
io
linalg
maxentropy
ndimage
odr
optimize
signal
sparse
spatial
special
stats
weave
Description
Clustering algorithms
Physical and mathematical constants
Fast Fourier Transform routines
Integration and ordinary differential equation solvers
Interpolation and smoothing splines
Input and Output
Linear algebra
Maximum entropy methods
N-dimensional image processing
Orthogonal distance regression
Optimization and root-finding routines
Signal processing
Sparse matrices and associated routines
Spatial data structures and algorithms
Special functions
Statistical distributions and functions
C/C++ integration
Scipy sub-packages need to be imported separately, for example:
>>> from scipy import linalg, optimize
Because of their ubiquitousness, some of the functions in these subpackages are also made available in the scipy
namespace to ease their use in interactive sessions and programs. In addition, many basic array functions from numpy
are also available at the top-level of the scipy package. Before looking at the sub-packages individually, we will first
look at some of these common functions.
1.1.2 Finding Documentation
Scipy and Numpy have HTML and PDF versions of their documentation available at http://docs.scipy.org/, which
currently details nearly all available functionality. However, this documentation is still work-in-progress, and some
parts may be incomplete or sparse. As we are a volunteer organization and depend on the community for growth,
your participation - everything from providing feedback to improving the documentation and code - is welcome and
actively encouraged.
Python also provides the facility of documentation strings. The functions and classes available in SciPy use this method
for on-line documentation. There are two methods for reading these messages and getting help. Python provides the
command help in the pydoc module. Entering this command with no arguments (i.e. >>> help ) launches an
interactive help session that allows searching through the keywords and modules available to all of Python. Running
the command help with an object as the argument displays the calling signature, and the documentation string of the
object.
The pydoc method of help is sophisticated but uses a pager to display the text. Sometimes this can interfere with
the terminal you are running the interactive session within. A scipy-specific help system is also available under the
command sp.info. The signature and documentation string for the object passed to the help command are printed
to standard output (or to a writeable object passed as the third argument). The second keyword argument of sp.info
defines the maximum width of the line for printing. If a module is passed as the argument to help than a list of the
functions and classes defined in that module is printed. For example:
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SciPy Reference Guide, Release 0.8.0rc3
>>> sp.info(optimize.fmin)
fmin(func, x0, args=(), xtol=0.0001, ftol=0.0001, maxiter=None, maxfun=None,
full_output=0, disp=1, retall=0, callback=None)
Minimize a function using the downhill simplex algorithm.
:Parameters:
func : callable func(x,*args)
The objective function to be minimized.
x0 : ndarray
Initial guess.
args : tuple
Extra arguments passed to func, i.e. ‘‘f(x,*args)‘‘.
callback : callable
Called after each iteration, as callback(xk), where xk is the
current parameter vector.
:Returns: (xopt, {fopt, iter, funcalls, warnflag})
xopt : ndarray
Parameter that minimizes function.
fopt : float
Value of function at minimum: ‘‘fopt = func(xopt)‘‘.
iter : int
Number of iterations performed.
funcalls : int
Number of function calls made.
warnflag : int
1 : Maximum number of function evaluations made.
2 : Maximum number of iterations reached.
allvecs : list
Solution at each iteration.
*Other Parameters*:
xtol : float
Relative error
ftol : number
Relative error
maxiter : int
Maximum number
maxfun : number
Maximum number
full_output : bool
Set to True if
disp : bool
Set to True to
retall : bool
Set to True to
in xopt acceptable for convergence.
in func(xopt) acceptable for convergence.
of iterations to perform.
of function evaluations to make.
fval and warnflag outputs are desired.
print convergence messages.
return list of solutions at each iteration.
:Notes:
Uses a Nelder-Mead simplex algorithm to find the minimum of
function of one or more variables.
Another useful command is source. When given a function written in Python as an argument, it prints out a listing
of the source code for that function. This can be helpful in learning about an algorithm or understanding exactly what
1.1. Introduction
5
SciPy Reference Guide, Release 0.8.0rc3
a function is doing with its arguments. Also don’t forget about the Python command dir which can be used to look
at the namespace of a module or package.
1.2 Basic functions in Numpy (and top-level scipy)
Contents
• Basic functions in Numpy (and top-level scipy)
– Interaction with Numpy
– Top-level scipy routines
* Type handling
* Index Tricks
* Shape manipulation
* Polynomials
* Vectorizing functions (vectorize)
* Other useful functions
– Common functions
1.2.1 Interaction with Numpy
To begin with, all of the Numpy functions have been subsumed into the scipy namespace so that all of those functions are available without additionally importing Numpy. In addition, the universal functions (addition, subtraction,
division) have been altered to not raise exceptions if floating-point errors are encountered; instead, NaN’s and Inf’s
are returned in the arrays. To assist in detection of these events, several functions (sp.isnan, sp.isfinite,
sp.isinf) are available.
Finally, some of the basic functions like log, sqrt, and inverse trig functions have been modified to return complex
numbers instead of NaN’s where appropriate (i.e. sp.sqrt(-1) returns 1j).
1.2.2 Top-level scipy routines
The purpose of the top level of scipy is to collect general-purpose routines that the other sub-packages can use and to
provide a simple replacement for Numpy. Anytime you might think to import Numpy, you can import scipy instead
and remove yourself from direct dependence on Numpy. These routines are divided into several files for organizational
purposes, but they are all available under the numpy namespace (and the scipy namespace). There are routines for
type handling and type checking, shape and matrix manipulation, polynomial processing, and other useful functions.
Rather than giving a detailed description of each of these functions (which is available in the Numpy Reference Guide
or by using the help, info and source commands), this tutorial will discuss some of the more useful commands
which require a little introduction to use to their full potential.
Type handling
Note the difference between sp.iscomplex/sp.isreal and sp.iscomplexobj/sp.isrealobj. The former command is array based and returns byte arrays of ones and zeros providing the result of the element-wise test.
The latter command is object based and returns a scalar describing the result of the test on the entire object.
Often it is required to get just the real and/or imaginary part of a complex number. While complex numbers and arrays
have attributes that return those values, if one is not sure whether or not the object will be complex-valued, it is better
to use the functional forms sp.real and sp.imag . These functions succeed for anything that can be turned into
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SciPy Reference Guide, Release 0.8.0rc3
a Numpy array. Consider also the function sp.real_if_close which transforms a complex-valued number with
tiny imaginary part into a real number.
Occasionally the need to check whether or not a number is a scalar (Python (long)int, Python float, Python complex,
or rank-0 array) occurs in coding. This functionality is provided in the convenient function sp.isscalar which
returns a 1 or a 0.
Finally, ensuring that objects are a certain Numpy type occurs often enough that it has been given a convenient interface
in SciPy through the use of the sp.cast dictionary. The dictionary is keyed by the type it is desired to cast to and
the dictionary stores functions to perform the casting. Thus, sp.cast[’f’](d) returns an array of sp.float32
from d. This function is also useful as an easy way to get a scalar of a certain type:
>>> sp.cast[’f’](sp.pi)
array(3.1415927410125732, dtype=float32)
Index Tricks
There are some class instances that make special use of the slicing functionality to provide efficient means for array
construction. This part will discuss the operation of sp.mgrid , sp.ogrid , sp.r_ , and sp.c_ for quickly
constructing arrays.
One familiar with Matlab may complain that it is difficult to construct arrays from the interactive session with Python.
Suppose, for example that one wants to construct an array that begins with 3 followed by 5 zeros and then contains 10
numbers spanning the range -1 to 1 (inclusive on both ends). Before SciPy, you would need to enter something like
the following
>>> concatenate(([3],[0]*5,arange(-1,1.002,2/9.0)))
With the r_ command one can enter this as
>>> r_[3,[0]*5,-1:1:10j]
which can ease typing and make for more readable code. Notice how objects are concatenated, and the slicing syntax
is (ab)used to construct ranges. The other term that deserves a little explanation is the use of the complex number
10j as the step size in the slicing syntax. This non-standard use allows the number to be interpreted as the number of
points to produce in the range rather than as a step size (note we would have used the long integer notation, 10L, but
this notation may go away in Python as the integers become unified). This non-standard usage may be unsightly to
some, but it gives the user the ability to quickly construct complicated vectors in a very readable fashion. When the
number of points is specified in this way, the end- point is inclusive.
The “r” stands for row concatenation because if the objects between commas are 2 dimensional arrays, they are stacked
by rows (and thus must have commensurate columns). There is an equivalent command c_ that stacks 2d arrays by
columns but works identically to r_ for 1d arrays.
Another very useful class instance which makes use of extended slicing notation is the function mgrid. In the simplest
case, this function can be used to construct 1d ranges as a convenient substitute for arange. It also allows the use of
complex-numbers in the step-size to indicate the number of points to place between the (inclusive) end-points. The real
purpose of this function however is to produce N, N-d arrays which provide coordinate arrays for an N-dimensional
volume. The easiest way to understand this is with an example of its usage:
>>> mgrid[0:5,0:5]
array([[[0, 0, 0, 0,
[1, 1, 1, 1,
[2, 2, 2, 2,
[3, 3, 3, 3,
[4, 4, 4, 4,
0],
1],
2],
3],
4]],
1.2. Basic functions in Numpy (and top-level scipy)
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SciPy Reference Guide, Release 0.8.0rc3
[[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4]]])
>>> mgrid[0:5:4j,0:5:4j]
array([[[ 0.
, 0.
,
[ 1.6667, 1.6667,
[ 3.3333, 3.3333,
[ 5.
, 5.
,
[[ 0.
, 1.6667,
[ 0.
, 1.6667,
[ 0.
, 1.6667,
[ 0.
, 1.6667,
0.
,
1.6667,
3.3333,
5.
,
3.3333,
3.3333,
3.3333,
3.3333,
0.
],
1.6667],
3.3333],
5.
]],
5.
],
5.
],
5.
],
5.
]]])
Having meshed arrays like this is sometimes very useful. However, it is not always needed just to evaluate some
N-dimensional function over a grid due to the array-broadcasting rules of Numpy and SciPy. If this is the only purpose
for generating a meshgrid, you should instead use the function ogrid which generates an “open “grid using NewAxis
judiciously to create N, N-d arrays where only one dimension in each array has length greater than 1. This will save
memory and create the same result if the only purpose for the meshgrid is to generate sample points for evaluation of
an N-d function.
Shape manipulation
In this category of functions are routines for squeezing out length- one dimensions from N-dimensional arrays, ensuring that an array is at least 1-, 2-, or 3-dimensional, and stacking (concatenating) arrays by rows, columns, and “pages
“(in the third dimension). Routines for splitting arrays (roughly the opposite of stacking arrays) are also available.
Polynomials
There are two (interchangeable) ways to deal with 1-d polynomials in SciPy. The first is to use the poly1d class from
Numpy. This class accepts coefficients or polynomial roots to initialize a polynomial. The polynomial object can then
be manipulated in algebraic expressions, integrated, differentiated, and evaluated. It even prints like a polynomial:
>>> p = poly1d([3,4,5])
>>> print p
2
3 x + 4 x + 5
>>> print p*p
4
3
2
9 x + 24 x + 46 x + 40 x + 25
>>> print p.integ(k=6)
3
2
x + 2 x + 5 x + 6
>>> print p.deriv()
6 x + 4
>>> p([4,5])
array([ 69, 100])
The other way to handle polynomials is as an array of coefficients with the first element of the array giving the
coefficient of the highest power. There are explicit functions to add, subtract, multiply, divide, integrate, differentiate,
and evaluate polynomials represented as sequences of coefficients.
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Vectorizing functions (vectorize)
One of the features that NumPy provides is a class vectorize to convert an ordinary Python function which accepts
scalars and returns scalars into a “vectorized-function” with the same broadcasting rules as other Numpy functions
(i.e. the Universal functions, or ufuncs). For example, suppose you have a Python function named addsubtract
defined as:
>>> def addsubtract(a,b):
...
if a > b:
...
return a - b
...
else:
...
return a + b
which defines a function of two scalar variables and returns a scalar result. The class vectorize can be used to “vectorize
“this function so that
>>> vec_addsubtract = vectorize(addsubtract)
returns a function which takes array arguments and returns an array result:
>>> vec_addsubtract([0,3,6,9],[1,3,5,7])
array([1, 6, 1, 2])
This particular function could have been written in vector form without the use of vectorize . But, what if the
function you have written is the result of some optimization or integration routine. Such functions can likely only be
vectorized using vectorize.
Other useful functions
There are several other functions in the scipy_base package including most of the other functions that are also in the
Numpy package. The reason for duplicating these functions is to allow SciPy to potentially alter their original interface
and make it easier for users to know how to get access to functions
>>> from scipy import *
Functions which should be mentioned are mod(x,y) which can replace x % y when it is desired that the result take the sign of y instead of x . Also included is fix which always rounds to the nearest integer towards
zero. For doing phase processing, the functions angle, and unwrap are also useful. Also, the linspace
and logspace functions return equally spaced samples in a linear or log scale. Finally, it’s useful to be
aware of the indexing capabilities of Numpy. Mention should be made of the new function select which extends the functionality of where to include multiple conditions and multiple choices. The calling convention is
select(condlist,choicelist,default=0). select is a vectorized form of the multiple if-statement.
It allows rapid construction of a function which returns an array of results based on a list of conditions. Each element
of the return array is taken from the array in a choicelist corresponding to the first condition in condlist that
is true. For example
>>> x = r_[-2:3]
>>> x
array([-2, -1, 0, 1, 2])
>>> select([x > 3, x >= 0],[0,x+2])
array([0, 0, 2, 3, 4])
1.2. Basic functions in Numpy (and top-level scipy)
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1.2.3 Common functions
Some functions depend on sub-packages of SciPy but should be available from the top-level of SciPy due to their
common use. These are functions that might have been placed in scipy_base except for their dependence on other
sub-packages of SciPy. For example the factorial and comb functions compute n! and n!/k!(n − k)! using either
exact integer arithmetic (thanks to Python’s Long integer object), or by using floating-point precision and the gamma
function. The functions rand and randn are used so often that they warranted a place at the top level. There are
convenience functions for the interactive use: disp (similar to print), and who (returns a list of defined variables and
memory consumption–upper bounded). Another function returns a common image used in image processing: lena.
Finally, two functions are provided that are useful for approximating derivatives of functions using discrete-differences.
The function central_diff_weights returns weighting coefficients for an equally-spaced N -point approximation to the derivative of order o. These weights must be multiplied by the function corresponding to these points and
the results added to obtain the derivative approximation. This function is intended for use when only samples of the
function are avaiable. When the function is an object that can be handed to a routine and evaluated, the function
derivative can be used to automatically evaluate the object at the correct points to obtain an N-point approximation to the o-th derivative at a given point.
1.3 Special functions (scipy.special)
The main feature of the scipy.special package is the definition of numerous special functions of mathematical
physics. Available functions include airy, elliptic, bessel, gamma, beta, hypergeometric, parabolic cylinder, mathieu,
spheroidal wave, struve, and kelvin. There are also some low-level stats functions that are not intended for general
use as an easier interface to these functions is provided by the stats module. Most of these functions can take array
arguments and return array results following the same broadcasting rules as other math functions in Numerical Python.
Many of these functions also accept complex numbers as input. For a complete list of the available functions with a
one-line description type >>> help(special). Each function also has its own documentation accessible using
help. If you don’t see a function you need, consider writing it and contributing it to the library. You can write the
function in either C, Fortran, or Python. Look in the source code of the library for examples of each of these kinds of
functions.
1.4 Integration (scipy.integrate)
The scipy.integrate sub-package provides several integration techniques including an ordinary differential
equation integrator. An overview of the module is provided by the help command:
>>> help(integrate)
Methods for Integrating Functions given function object.
quad
dblquad
tplquad
fixed_quad
quadrature
romberg
-------
General purpose integration.
General purpose double integration.
General purpose triple integration.
Integrate func(x) using Gaussian quadrature of order n.
Integrate with given tolerance using Gaussian quadrature.
Integrate func using Romberg integration.
Methods for Integrating Functions given fixed samples.
trapz
cumtrapz
simps
romb
10
-----
Use
Use
Use
Use
trapezoidal rule to compute integral from samples.
trapezoidal rule to cumulatively compute integral.
Simpson’s rule to compute integral from samples.
Romberg Integration to compute integral from
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(2**k + 1) evenly-spaced samples.
See the special module’s orthogonal polynomials (special) for Gaussian
quadrature roots and weights for other weighting factors and regions.
Interface to numerical integrators of ODE systems.
odeint
ode
-- General integration of ordinary differential equations.
-- Integrate ODE using VODE and ZVODE routines.
1.4.1 General integration (quad)
The function quad is provided to integrate a function of one variable between two points. The points can be ±∞ (±
inf) to indicate infinite limits. For example, suppose you wish to integrate a bessel function jv(2.5,x) along the
interval [0, 4.5].
Z 4.5
I=
J2.5 (x) dx.
0
This could be computed using quad:
>>> result = integrate.quad(lambda x: special.jv(2.5,x), 0, 4.5)
>>> print result
(1.1178179380783249, 7.8663172481899801e-09)
>>> I = sqrt(2/pi)*(18.0/27*sqrt(2)*cos(4.5)-4.0/27*sqrt(2)*sin(4.5)+
sqrt(2*pi)*special.fresnel(3/sqrt(pi))[0])
>>> print I
1.117817938088701
>>> print abs(result[0]-I)
1.03761443881e-11
The first argument to quad is a “callable” Python object (i.e a function, method, or class instance). Notice the use of a
lambda- function in this case as the argument. The next two arguments are the limits of integration. The return value
is a tuple, with the first element holding the estimated value of the integral and the second element holding an upper
bound on the error. Notice, that in this case, the true value of this integral is
r √
2 18 √
4√
3
√
I=
2 cos (4.5) −
2 sin (4.5) + 2πSi
,
π 27
27
π
where
Z
Si (x) =
x
sin
0
π t2 dt.
2
is the Fresnel sine integral. Note that the numerically-computed integral is within 1.04 × 10−11 of the exact result —
well below the reported error bound.
Infinite inputs are also allowed in quad by using ± inf as one of the arguments. For example, suppose that a
numerical value for the exponential integral:
Z ∞ −xt
e
En (x) =
dt.
tn
1
is desired (and the fact that this integral can be computed as special.expn(n,x) is forgotten). The functionality
of the function special.expn can be replicated by defining a new function vec_expint based on the routine
quad:
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>>> from scipy.integrate import quad
>>> def integrand(t,n,x):
...
return exp(-x*t) / t**n
>>> def expint(n,x):
...
return quad(integrand, 1, Inf, args=(n, x))[0]
>>> vec_expint = vectorize(expint)
>>> vec_expint(3,arange(1.0,4.0,0.5))
array([ 0.1097, 0.0567, 0.0301, 0.0163,
>>> special.expn(3,arange(1.0,4.0,0.5))
array([ 0.1097, 0.0567, 0.0301, 0.0163,
0.0089,
0.0049])
0.0089,
0.0049])
The function which is integrated can even use the quad argument (though the error bound may underestimate the error
due to possible numerical error in the integrand from the use of quad ). The integral in this case is
Z ∞ Z ∞ −xt
1
e
dt dx = .
In =
n
t
n
0
1
>>> result = quad(lambda x: expint(3, x), 0, inf)
>>> print result
(0.33333333324560266, 2.8548934485373678e-09)
>>> I3 = 1.0/3.0
>>> print I3
0.333333333333
>>> print I3 - result[0]
8.77306560731e-11
This last example shows that multiple integration can be handled using repeated calls to quad. The mechanics of this
for double and triple integration have been wrapped up into the functions dblquad and tplquad. The function,
dblquad performs double integration. Use the help function to be sure that the arguments are defined in the correct
order. In addition, the limits on all inner integrals are actually functions which can be constant functions. An example
of using double integration to compute several values of In is shown below:
>>> from scipy.integrate import quad, dblquad
>>> def I(n):
...
return dblquad(lambda t, x: exp(-x*t)/t**n, 0, Inf, lambda x: 1, lambda x: Inf)
>>> print I(4)
(0.25000000000435768, 1.0518245707751597e-09)
>>> print I(3)
(0.33333333325010883, 2.8604069919261191e-09)
>>> print I(2)
(0.49999999999857514, 1.8855523253868967e-09)
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1.4.2 Gaussian quadrature (integrate.gauss_quadtol)
A few functions are also provided in order to perform simple Gaussian quadrature over a fixed interval. The first
is fixed_quad which performs fixed-order Gaussian quadrature. The second function is quadrature which
performs Gaussian quadrature of multiple orders until the difference in the integral estimate is beneath some tolerance
supplied by the user. These functions both use the module special.orthogonal which can calculate the roots
and quadrature weights of a large variety of orthogonal polynomials (the polynomials themselves are available as
special functions returning instances of the polynomial class — e.g. special.legendre).
1.4.3 Integrating using samples
There are three functions for computing integrals given only samples: trapz , simps, and romb . The first two
functions use Newton-Coates formulas of order 1 and 2 respectively to perform integration. These two functions
can handle, non-equally-spaced samples. The trapezoidal rule approximates the function as a straight line between
adjacent points, while Simpson’s rule approximates the function between three adjacent points as a parabola.
If the samples are equally-spaced and the number of samples available is 2k + 1 for some integer k, then Romberg
integration can be used to obtain high-precision estimates of the integral using the available samples. Romberg integration uses the trapezoid rule at step-sizes related by a power of two and then performs Richardson extrapolation
on these estimates to approximate the integral with a higher-degree of accuracy. (A different interface to Romberg
integration useful when the function can be provided is also available as romberg).
1.4.4 Ordinary differential equations (odeint)
Integrating a set of ordinary differential equations (ODEs) given initial conditions is another useful example. The
function odeint is available in SciPy for integrating a first-order vector differential equation:
dy
= f (y, t) ,
dt
given initial conditions y (0) = y0 , where y is a length N vector and f is a mapping from RN to RN . A higher-order
ordinary differential equation can always be reduced to a differential equation of this type by introducing intermediate
derivatives into the y vector.
For example suppose it is desired to find the solution to the following second-order differential equation:
d2 w
− zw(z) = 0
dz 2
1
√ 1
and dw
with initial conditions w (0) = √
3 2
dz z=0 = − 3 3Γ( 1 ) . It is known that the solution to this differential
3 Γ( 23 )
3
equation with these boundary conditions is the Airy function
w = Ai (z) ,
which gives a means to check the integrator using special.airy.
First, convert this ODE into standard form by setting y = dw
dz , w and t = z. Thus, the differential equation becomes
dy
ty1
0 t
y0
0 t
=
=
=
y.
y0
1 0
y1
1 0
dt
In other words,
f (y, t) = A (t) y.
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Rt
As an interesting reminder, if A (t) commutes with 0 A (τ ) dτ under matrix multiplication, then this linear differential equation has an exact solution using the matrix exponential:
Z t
A (τ ) dτ y (0) ,
y (t) = exp
0
However, in this case, A (t) and its integral do not commute.
There are many optional inputs and outputs available when using odeint which can help tune the solver. These additional inputs and outputs are not needed much of the time, however, and the three required input arguments and
the output solution suffice. The required inputs are the function defining the derivative, fprime, the initial conditions
vector, y0, and the time points to obtain a solution, t, (with the initial value point as the first element of this sequence).
The output to odeint is a matrix where each row contains the solution vector at each requested time point (thus, the
initial conditions are given in the first output row).
The following example illustrates the use of odeint including the usage of the Dfun option which allows the user to
specify a gradient (with respect to y ) of the function, f (y, t).
>>>
>>>
>>>
>>>
>>>
>>>
...
from scipy.integrate import odeint
from scipy.special import gamma, airy
y1_0 = 1.0/3**(2.0/3.0)/gamma(2.0/3.0)
y0_0 = -1.0/3**(1.0/3.0)/gamma(1.0/3.0)
y0 = [y0_0, y1_0]
def func(y, t):
return [t*y[1],y[0]]
>>> def gradient(y,t):
...
return [[0,t],[1,0]]
>>>
>>>
>>>
>>>
>>>
x = arange(0,4.0, 0.01)
t = x
ychk = airy(x)[0]
y = odeint(func, y0, t)
y2 = odeint(func, y0, t, Dfun=gradient)
>>> print ychk[:36:6]
[ 0.355028 0.339511 0.324068
0.308763
0.293658
0.278806]
>>> print y[:36:6,1]
[ 0.355028 0.339511
0.324067
0.308763
0.293658
0.278806]
>>> print y2[:36:6,1]
[ 0.355028 0.339511 0.324067
0.308763
0.293658
0.278806]
1.5 Optimization (optimize)
There are several classical optimization algorithms provided by SciPy in the scipy.optimize package. An
overview of the module is available using help (or pydoc.help):
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from scipy import optimize
>>> info(optimize)
Optimization Tools
==================
A collection of general-purpose optimization routines.
fmin
--
fmin_powell -fmin_cg
--
fmin_bfgs
--
fmin_ncg
--
leastsq
--
Nelder-Mead Simplex algorithm
(uses only function calls)
Powell’s (modified) level set method (uses only
function calls)
Non-linear (Polak-Ribiere) conjugate gradient algorithm
(can use function and gradient).
Quasi-Newton method (Broydon-Fletcher-Goldfarb-Shanno);
(can use function and gradient)
Line-search Newton Conjugate Gradient (can use
function, gradient and Hessian).
Minimize the sum of squares of M equations in
N unknowns given a starting estimate.
Constrained Optimizers (multivariate)
fmin_l_bfgs_b -- Zhu, Byrd, and Nocedal’s L-BFGS-B constrained optimizer
(if you use this please quote their papers -- see help)
fmin_tnc
-- Truncated Newton Code originally written by Stephen Nash and
adapted to C by Jean-Sebastien Roy.
fmin_cobyla
-- Constrained Optimization BY Linear Approximation
Global Optimizers
anneal
brute
---
Simulated Annealing
Brute force searching optimizer
Scalar function minimizers
fminbound
brent
golden
bracket
-----
Bounded minimization of a scalar function.
1-D function minimization using Brent method.
1-D function minimization using Golden Section method
Bracket a minimum (given two starting points)
Also a collection of general-purpose root-finding routines.
fsolve
--
Non-linear multi-variable equation solver.
Scalar function solvers
brentq
brenth
---
ridder
bisect
---
quadratic interpolation Brent method
Brent method (modified by Harris with hyperbolic
extrapolation)
Ridder’s method
Bisection method
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newton
--
fixed_point --
Secant method or Newton’s method
Single-variable fixed-point solver.
A collection of general-purpose nonlinear multidimensional solvers.
broyden1
--
broyden2
--
broyden3
--
broyden_generalized --
anderson
--
anderson2
--
Broyden’s first method - is a quasi-Newton-Raphson
method for updating an approximate Jacobian and then
inverting it
Broyden’s second method - the same as broyden1, but
updates the inverse Jacobian directly
Broyden’s second method - the same as broyden2, but
instead of directly computing the inverse Jacobian,
it remembers how to construct it using vectors, and
when computing inv(J)*F, it uses those vectors to
compute this product, thus avoding the expensive NxN
matrix multiplication.
Generalized Broyden’s method, the same as broyden2,
but instead of approximating the full NxN Jacobian,
it construct it at every iteration in a way that
avoids the NxN matrix multiplication. This is not
as precise as broyden3.
extended Anderson method, the same as the
broyden_generalized, but added w_0^2*I to before
taking inversion to improve the stability
the Anderson method, the same as anderson, but
formulated differently
Utility Functions
line_search -check_grad --
Return a step that satisfies the strong Wolfe conditions.
Check the supplied derivative using finite difference
techniques.
The first four algorithms are unconstrained minimization algorithms (fmin: Nelder-Mead simplex, fmin_bfgs:
BFGS, fmin_ncg: Newton Conjugate Gradient, and leastsq: Levenburg-Marquardt). The last algorithm actually
finds the roots of a general function of possibly many variables. It is included in the optimization package because at
the (non-boundary) extreme points of a function, the gradient is equal to zero.
1.5.1 Nelder-Mead Simplex algorithm (fmin)
The simplex algorithm is probably the simplest way to minimize a fairly well-behaved function. The simplex algorithm
requires only function evaluations and is a good choice for simple minimization problems. However, because it does
not use any gradient evaluations, it may take longer to find the minimum. To demonstrate the minimization function
consider the problem of minimizing the Rosenbrock function of N variables:
f (x) =
N
−1
X
100 xi − x2i−1
2
2
+ (1 − xi−1 ) .
i=1
The minimum value of this function is 0 which is achieved when xi = 1. This minimum can be found using the fmin
routine as shown in the example below:
>>> from scipy.optimize import fmin
>>> def rosen(x):
...
"""The Rosenbrock function"""
...
return sum(100.0*(x[1:]-x[:-1]**2.0)**2.0 + (1-x[:-1])**2.0)
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>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = fmin(rosen, x0, xtol=1e-8)
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 339
Function evaluations: 571
>>> print xopt
[ 1. 1. 1. 1.
1.]
Another optimization algorithm that needs only function calls to find the minimum is Powell’s method available as
fmin_powell.
1.5.2 Broyden-Fletcher-Goldfarb-Shanno algorithm (fmin_bfgs)
In order to converge more quickly to the solution, this routine uses the gradient of the objective function. If the gradient
is not given by the user, then it is estimated using first-differences. The Broyden-Fletcher-Goldfarb-Shanno (BFGS)
method typically requires fewer function calls than the simplex algorithm even when the gradient must be estimated.
To demonstrate this algorithm, the Rosenbrock function is again used. The gradient of the Rosenbrock function is the
vector:
∂f
∂xj
=
N
X
200 xi − x2i−1 (δi,j − 2xi−1 δi−1,j ) − 2 (1 − xi−1 ) δi−1,j .
i=1
=
200 xj − x2j−1 − 400xj xj+1 − x2j − 2 (1 − xj ) .
This expression is valid for the interior derivatives. Special cases are
∂f
∂x0
∂f
∂xN −1
=
−400x0 x1 − x20 − 2 (1 − x0 ) ,
=
200 xN −1 − x2N −2 .
A Python function which computes this gradient is constructed by the code-segment:
>>> def rosen_der(x):
...
xm = x[1:-1]
...
xm_m1 = x[:-2]
...
xm_p1 = x[2:]
...
der = zeros_like(x)
...
der[1:-1] = 200*(xm-xm_m1**2) - 400*(xm_p1 - xm**2)*xm - 2*(1-xm)
...
der[0] = -400*x[0]*(x[1]-x[0]**2) - 2*(1-x[0])
...
der[-1] = 200*(x[-1]-x[-2]**2)
...
return der
The calling signature for the BFGS minimization algorithm is similar to fmin with the addition of the fprime argument. An example usage of fmin_bfgs is shown in the following example which minimizes the Rosenbrock
function.
>>> from scipy.optimize import fmin_bfgs
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>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = fmin_bfgs(rosen, x0, fprime=rosen_der)
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 53
Function evaluations: 65
Gradient evaluations: 65
>>> print xopt
[ 1. 1. 1. 1. 1.]
1.5.3 Newton-Conjugate-Gradient (fmin_ncg)
The method which requires the fewest function calls and is therefore often the fastest method to minimize functions of
many variables is fmin_ncg. This method is a modified Newton’s method and uses a conjugate gradient algorithm
to (approximately) invert the local Hessian. Newton’s method is based on fitting the function locally to a quadratic
form:
1
T
f (x) ≈ f (x0 ) + ∇f (x0 ) · (x − x0 ) + (x − x0 ) H (x0 ) (x − x0 ) .
2
where H (x0 ) is a matrix of second-derivatives (the Hessian). If the Hessian is positive definite then the local minimum
of this function can be found by setting the gradient of the quadratic form to zero, resulting in
xopt = x0 − H−1 ∇f.
The inverse of the Hessian is evaluted using the conjugate-gradient method. An example of employing this method
to minimizing the Rosenbrock function is given below. To take full advantage of the NewtonCG method, a function
which computes the Hessian must be provided. The Hessian matrix itself does not need to be constructed, only a
vector which is the product of the Hessian with an arbitrary vector needs to be available to the minimization routine.
As a result, the user can provide either a function to compute the Hessian matrix, or a function to compute the product
of the Hessian with an arbitrary vector.
Full Hessian example:
The Hessian of the Rosenbrock function is
Hij =
∂2f
∂xi ∂xj
200 (δi,j − 2xi−1 δi−1,j ) − 400xi (δi+1,j − 2xi δi,j ) − 400δi,j xi+1 − x2i + 2δi,j ,
= 202 + 1200x2i − 400xi+1 δi,j − 400xi δi+1,j − 400xi−1 δi−1,j ,
=
if i, j ∈ [1, N − 2] with i, j ∈ [0, N − 1] defining the N × N matrix. Other non-zero entries of the matrix are
∂2f
∂x20
∂2f
∂2f
=
∂x0 ∂x1
∂x1 ∂x0
2
∂ f
∂2f
=
∂xN −1 ∂xN −2
∂xN −2 ∂xN −1
∂2f
∂x2N −1
18
=
1200x20 − 400x1 + 2,
= −400x0 ,
= −400xN −2 ,
=
200.
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For example, the Hessian when N = 5 is

1200x20 − 400x1 + 2
−400x0
2

−400x
202
+
1200x
0
1 − 400x2

0
−400x
H=
1


0
0
0
0
−400x1
202 + 1200x22 − 400x3
−400x2
0
0
0
−400x2
202 + 1200x23 − 400x4
−400x3
0
0
0
−400x3
200
The code which computes this Hessian along with the code to minimize the function using fmin_ncg is shown in
the following example:
>>> from scipy.optimize import fmin_ncg
>>> def rosen_hess(x):
...
x = asarray(x)
...
H = diag(-400*x[:-1],1) - diag(400*x[:-1],-1)
...
diagonal = zeros_like(x)
...
diagonal[0] = 1200*x[0]-400*x[1]+2
...
diagonal[-1] = 200
...
diagonal[1:-1] = 202 + 1200*x[1:-1]**2 - 400*x[2:]
...
H = H + diag(diagonal)
...
return H
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = fmin_ncg(rosen, x0, rosen_der, fhess=rosen_hess, avextol=1e-8)
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 23
Function evaluations: 26
Gradient evaluations: 23
Hessian evaluations: 23
>>> print xopt
[ 1. 1. 1. 1. 1.]
Hessian product example:
For larger minimization problems, storing the entire Hessian matrix can consume considerable time and memory. The
Newton-CG algorithm only needs the product of the Hessian times an arbitrary vector. As a result, the user can supply
code to compute this product rather than the full Hessian by setting the fhess_p keyword to the desired function.
The fhess_p function should take the minimization vector as the first argument and the arbitrary vector as the second
argument. Any extra arguments passed to the function to be minimized will also be passed to this function. If possible,
using Newton-CG with the hessian product option is probably the fastest way to minimize the function.
In this case, the product of the Rosenbrock Hessian with an arbitrary vector is not difficult to
arbitrary vector, then H (x) p has elements:

1200x20 − 400x1 + 2 p0 − 400x0 p1

..

.

2
H (x) p = 
 −400xi−1 pi−1 + 202 + 1200xi − 400xi+1 pi − 400xi pi+1

..

.
−400xN −2 pN −2 + 200pN −1
compute. If p is the




.



Code which makes use of the fhess_p keyword to minimize the Rosenbrock function using fmin_ncg follows:
1.5. Optimization (optimize)
19



.


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>>> from scipy.optimize import fmin_ncg
>>> def rosen_hess_p(x,p):
...
x = asarray(x)
...
Hp = zeros_like(x)
...
Hp[0] = (1200*x[0]**2 - 400*x[1] + 2)*p[0] - 400*x[0]*p[1]
...
Hp[1:-1] = -400*x[:-2]*p[:-2]+(202+1200*x[1:-1]**2-400*x[2:])*p[1:-1] \
...
-400*x[1:-1]*p[2:]
...
Hp[-1] = -400*x[-2]*p[-2] + 200*p[-1]
...
return Hp
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = fmin_ncg(rosen, x0, rosen_der, fhess_p=rosen_hess_p, avextol=1e-8)
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 22
Function evaluations: 25
Gradient evaluations: 22
Hessian evaluations: 54
>>> print xopt
[ 1. 1. 1. 1. 1.]
1.5.4 Least-square fitting (leastsq)
All of the previously-explained minimization procedures can be used to solve a least-squares problem provided the
appropriate objective function is constructed. For example, suppose it is desired to fit a set of data {xi , yi } to a known
model, y = f (x, p) where p is a vector of parameters for the model that need to be found. A common method for
determining which parameter vector gives the best fit to the data is to minimize the sum of squares of the residuals.
The residual is usually defined for each observed data-point as
ei (p, yi , xi ) = kyi − f (xi , p)k .
An objective function to pass to any of the previous minization algorithms to obtain a least-squares fit is.
J (p) =
N
−1
X
e2i (p) .
i=0
The leastsq algorithm performs this squaring and summing of the residuals automatically. It takes as an input
argument the vector function e (p) and returns the value of p which minimizes J (p) = eT e directly. The user is also
encouraged to provide the Jacobian matrix of the function (with derivatives down the columns or across the rows). If
the Jacobian is not provided, it is estimated.
An example should clarify the usage. Suppose it is believed some measured data follow a sinusoidal pattern
yi = A sin (2πkxi + θ)
where the parameters A, k , and θ are unknown. The residual vector is
ei = |yi − A sin (2πkxi + θ)| .
By defining a function to compute the residuals and (selecting an appropriate starting position), the least-squares fit
routine can be used to find the best-fit parameters Â, k̂, θ̂. This is shown in the following example:
>>>
>>>
>>>
>>>
>>>
20
from numpy import *
x = arange(0,6e-2,6e-2/30)
A,k,theta = 10, 1.0/3e-2, pi/6
y_true = A*sin(2*pi*k*x+theta)
y_meas = y_true + 2*random.randn(len(x))
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>>> def residuals(p, y, x):
...
A,k,theta = p
...
err = y-A*sin(2*pi*k*x+theta)
...
return err
>>> def peval(x, p):
...
return p[0]*sin(2*pi*p[1]*x+p[2])
>>> p0 = [8, 1/2.3e-2, pi/3]
>>> print array(p0)
[ 8.
43.4783
1.0472]
>>> from scipy.optimize import leastsq
>>> plsq = leastsq(residuals, p0, args=(y_meas, x))
>>> print plsq[0]
[ 10.9437 33.3605
0.5834]
>>> print array([A, k, theta])
[ 10.
33.3333
0.5236]
>>>
>>>
>>>
>>>
>>>
import matplotlib.pyplot as plt
plt.plot(x,peval(x,plsq[0]),x,y_meas,’o’,x,y_true)
plt.title(’Least-squares fit to noisy data’)
plt.legend([’Fit’, ’Noisy’, ’True’])
plt.show()
Least-squares fit to noisy data
15
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Noisy
True
10
5
0
5
10
15
0.00
0.01
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0.02
0.03
0.04
0.05
0.06
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1.5.5 Sequential Least-square fitting with constraints (fmin_slsqp)
This module implements the Sequential Least SQuares Programming optimization algorithm (SLSQP).
min F (x)
subject to
Cj (X) = 0,
j = 1, ..., MEQ
Cj (x) ≥ 0,
j = MEQ + 1, ..., M
XL ≤ x ≤ XU,
I = 1, ..., N.
The following script shows examples for how constraints can be specified.
"""
This script tests fmin_slsqp using Example 14.4 from Numerical Methods for
Engineers by Steven Chapra and Raymond Canale. This example maximizes the
function f(x) = 2*x*y + 2*x - x**2 - 2*y**2, which has a maximum at x=2,y=1.
"""
from scipy.optimize import fmin_slsqp
from numpy import array, asfarray, finfo,ones, sqrt, zeros
def testfunc(d,*args):
"""
Arguments:
d
- A list of two elements, where d[0] represents x and
d[1] represents y in the following equation.
sign - A multiplier for f. Since we want to optimize it, and the scipy
optimizers can only minimize functions, we need to multiply it by
-1 to achieve the desired solution
Returns:
2*x*y + 2*x - x**2 - 2*y**2
"""
try:
sign = args[0]
except:
sign = 1.0
x = d[0]
y = d[1]
return sign*(2*x*y + 2*x - x**2 - 2*y**2)
def testfunc_deriv(d,*args):
""" This is the derivative of testfunc, returning a numpy array
representing df/dx and df/dy
"""
try:
sign = args[0]
except:
sign = 1.0
x = d[0]
y = d[1]
dfdx = sign*(-2*x + 2*y + 2)
dfdy = sign*(2*x - 4*y)
return array([ dfdx, dfdy ],float)
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from time import time
print ’\n\n’
print "Unbounded optimization. Derivatives approximated."
t0 = time()
x = fmin_slsqp(testfunc, [-1.0,1.0], args=(-1.0,), iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results",x
print "\n\n"
print "Unbounded optimization. Derivatives provided."
t0 = time()
x = fmin_slsqp(testfunc, [-1.0,1.0], args=(-1.0,), iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results",x
print "\n\n"
print "Bound optimization. Derivatives approximated."
t0 = time()
x = fmin_slsqp(testfunc, [-1.0,1.0], args=(-1.0,),
eqcons=[lambda x, y: x[0]-x[1] ], iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results",x
print "\n\n"
print "Bound optimization (equality constraints). Derivatives provided."
t0 = time()
x = fmin_slsqp(testfunc, [-1.0,1.0], fprime=testfunc_deriv, args=(-1.0,),
eqcons=[lambda x, y: x[0]-x[1] ], iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results",x
print "\n\n"
print "Bound optimization (equality and inequality constraints)."
print "Derivatives provided."
t0 = time()
x = fmin_slsqp(testfunc,[-1.0,1.0], fprime=testfunc_deriv, args=(-1.0,),
eqcons=[lambda x, y: x[0]-x[1] ],
ieqcons=[lambda x, y: x[0]-.5], iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results",x
print "\n\n"
def test_eqcons(d,*args):
try:
sign = args[0]
except:
sign = 1.0
x = d[0]
y = d[1]
return array([ x**3-y ])
def test_ieqcons(d,*args):
try:
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sign = args[0]
except:
sign = 1.0
x = d[0]
y = d[1]
return array([ y-1 ])
print "Bound optimization (equality and inequality constraints)."
print "Derivatives provided via functions."
t0 = time()
x = fmin_slsqp(testfunc, [-1.0,1.0], fprime=testfunc_deriv, args=(-1.0,),
f_eqcons=test_eqcons, f_ieqcons=test_ieqcons,
iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results",x
print "\n\n"
def test_fprime_eqcons(d,*args):
try:
sign = args[0]
except:
sign = 1.0
x = d[0]
y = d[1]
return array([ 3.0*(x**2.0), -1.0 ])
def test_fprime_ieqcons(d,*args):
try:
sign = args[0]
except:
sign = 1.0
x = d[0]
y = d[1]
return array([ 0.0, 1.0 ])
print "Bound optimization (equality and inequality constraints)."
print "Derivatives provided via functions."
print "Constraint jacobians provided via functions"
t0 = time()
x = fmin_slsqp(testfunc,[-1.0,1.0], fprime=testfunc_deriv, args=(-1.0,),
f_eqcons=test_eqcons, f_ieqcons=test_ieqcons,
fprime_eqcons=test_fprime_eqcons,
fprime_ieqcons=test_fprime_ieqcons, iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results",x
print "\n\n"
1.5.6 Scalar function minimizers
Often only the minimum of a scalar function is needed (a scalar function is one that takes a scalar as input and returns
a scalar output). In these circumstances, other optimization techniques have been developed that can work faster.
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Unconstrained minimization (brent)
There are actually two methods that can be used to minimize a scalar function (brent and golden), but golden is
included only for academic purposes and should rarely be used. The brent method uses Brent’s algorithm for locating
a minimum. Optimally a bracket should be given which contains the minimum desired. A bracket is a triple (a, b, c)
such that f (a) > f (b) < f (c) and a < b < c . If this is not given, then alternatively two starting points can be
chosen and a bracket will be found from these points using a simple marching algorithm. If these two starting points
are not provided 0 and 1 will be used (this may not be the right choice for your function and result in an unexpected
minimum being returned).
Bounded minimization (fminbound)
Thus far all of the minimization routines described have been unconstrained minimization routines. Very often, however, there are constraints that can be placed on the solution space before minimization occurs. The fminbound
function is an example of a constrained minimization procedure that provides a rudimentary interval constraint for
scalar functions. The interval constraint allows the minimization to occur only between two fixed endpoints.
For example, to find the minimum of J1 (x) near x = 5 , fminbound can be called using the interval [4, 7] as a
constraint. The result is xmin = 5.3314 :
>>> from scipy.special import j1
>>> from scipy.optimize import fminbound
>>> xmin = fminbound(j1, 4, 7)
>>> print xmin
5.33144184241
1.5.7 Root finding
Sets of equations
To find the roots of a polynomial, the command roots is useful. To find a root of a set of non-linear equations, the
command fsolve is needed. For example, the following example finds the roots of the single-variable transcendental
equation
x + 2 cos (x) = 0,
and the set of non-linear equations
x0 cos (x1 )
=
4,
x0 x1 − x1
=
5.
The results are x = −1.0299 and x0 = 6.5041, x1 = 0.9084 .
>>> def func(x):
...
return x + 2*cos(x)
>>> def func2(x):
...
out = [x[0]*cos(x[1]) - 4]
...
out.append(x[1]*x[0] - x[1] - 5)
...
return out
>>> from scipy.optimize import fsolve
>>> x0 = fsolve(func, 0.3)
>>> print x0
-1.02986652932
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>>> x02 = fsolve(func2, [1, 1])
>>> print x02
[ 6.50409711 0.90841421]
Scalar function root finding
If one has a single-variable equation, there are four different root finder algorithms that can be tried. Each of these
root finding algorithms requires the endpoints of an interval where a root is suspected (because the function changes
signs). In general brentq is the best choice, but the other methods may be useful in certain circumstances or for
academic purposes.
Fixed-point solving
A problem closely related to finding the zeros of a function is the problem of finding a fixed-point of a function. A
fixed point of a function is the point at which evaluation of the function returns the point: g (x) = x. Clearly the fixed
point of g is the root of f (x) = g (x) − x. Equivalently, the root of f is the fixed_point of g (x) = f (x) + x. The
routine fixed_point provides a simple iterative method using Aitkens sequence acceleration to estimate the fixed
point of g given a starting point.
1.6 Interpolation (scipy.interpolate)
Contents
• Interpolation (scipy.interpolate)
– Linear 1-d interpolation (interp1d)
– Spline interpolation in 1-d: Procedural (interpolate.splXXX)
– Spline interpolation in 1-d: Object-oriented (UnivariateSpline)
– Two-dimensional spline representation: Procedural (bisplrep)
– Two-dimensional spline representation: Object-oriented (BivariateSpline)
– Using radial basis functions for smoothing/interpolation
* 1-d Example
* 2-d Example
There are two general interpolation facilities available in SciPy. The first facility is an interpolation class which
performs linear 1-dimensional interpolation. The second facility is based on the FORTRAN library FITPACK and
provides functions for 1- and 2-dimensional (smoothed) cubic-spline interpolation. There are both procedural and
object-oriented interfaces for the FITPACK library.
1.6.1 Linear 1-d interpolation (interp1d)
The interp1d class in scipy.interpolate is a convenient method to create a function based on fixed data points which can
be evaluated anywhere within the domain defined by the given data using linear interpolation. An instance of this class
is created by passing the 1-d vectors comprising the data. The instance of this class defines a __call__ method and
can therefore by treated like a function which interpolates between known data values to obtain unknown values (it
also has a docstring for help). Behavior at the boundary can be specified at instantiation time. The following example
demonstrates it’s use.
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>>> import numpy as np
>>> from scipy import interpolate
>>> x = np.arange(0,10)
>>> y = np.exp(-x/3.0)
>>> f = interpolate.interp1d(x, y)
>>> xnew = np.arange(0,9,0.1)
>>> import matplotlib.pyplot as plt
>>> plt.plot(x,y,’o’,xnew,f(xnew),’-’)
1.0
0.8
0.6
0.4
0.2
0.0
0
1
2
3
4
5
6
7
8
9
1.6.2 Spline interpolation in 1-d: Procedural (interpolate.splXXX)
Spline interpolation requires two essential steps: (1) a spline representation of the curve is computed, and (2) the spline
is evaluated at the desired points. In order to find the spline representation, there are two different ways to represent
a curve and obtain (smoothing) spline coefficients: directly and parametrically. The direct method finds the spline
representation of a curve in a two- dimensional plane using the function splrep. The first two arguments are the
only ones required, and these provide the x and y components of the curve. The normal output is a 3-tuple, (t, c, k) ,
containing the knot-points, t , the coefficients c and the order k of the spline. The default spline order is cubic, but this
can be changed with the input keyword, k.
For curves in N -dimensional space the function splprep allows defining the curve parametrically. For this function
only 1 input argument is required. This input is a list of N -arrays representing the curve in N -dimensional space. The
length of each array is the number of curve points, and each array provides one component of the N -dimensional data
point. The parameter variable is given with the keword argument, u, which defaults to an equally-spaced monotonic
sequence between 0 and 1 . The default output consists of two objects: a 3-tuple, (t, c, k) , containing the spline
representation and the parameter variable u.
The keyword argument,
√s , is used to specify the amount of smoothing to perform during the spline fit. The default
value of s is s = m − 2m where m is the number of data-points being fit. Therefore, if no smoothing is desired a
value of s = 0 should be passed to the routines.
Once the spline representation of the data has been determined, functions are available for evaluating the spline
(splev) and its derivatives (splev, spalde) at any point and the integral of the spline between any two points
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( splint). In addition, for cubic splines ( k = 3 ) with 8 or more knots, the roots of the spline can be estimated (
sproot). These functions are demonstrated in the example that follows.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy import interpolate
Cubic-spline
>>>
>>>
>>>
>>>
>>>
x = np.arange(0,2*np.pi+np.pi/4,2*np.pi/8)
y = np.sin(x)
tck = interpolate.splrep(x,y,s=0)
xnew = np.arange(0,2*np.pi,np.pi/50)
ynew = interpolate.splev(xnew,tck,der=0)
>>>
>>>
>>>
>>>
>>>
>>>
plt.figure()
plt.plot(x,y,’x’,xnew,ynew,xnew,np.sin(xnew),x,y,’b’)
plt.legend([’Linear’,’Cubic Spline’, ’True’])
plt.axis([-0.05,6.33,-1.05,1.05])
plt.title(’Cubic-spline interpolation’)
plt.show()
Derivative of spline
>>>
>>>
>>>
>>>
>>>
>>>
>>>
yder = interpolate.splev(xnew,tck,der=1)
plt.figure()
plt.plot(xnew,yder,xnew,np.cos(xnew),’--’)
plt.legend([’Cubic Spline’, ’True’])
plt.axis([-0.05,6.33,-1.05,1.05])
plt.title(’Derivative estimation from spline’)
plt.show()
Integral of spline
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
def integ(x,tck,constant=-1):
x = np.atleast_1d(x)
out = np.zeros(x.shape, dtype=x.dtype)
for n in xrange(len(out)):
out[n] = interpolate.splint(0,x[n],tck)
out += constant
return out
yint = integ(xnew,tck)
plt.figure()
plt.plot(xnew,yint,xnew,-np.cos(xnew),’--’)
plt.legend([’Cubic Spline’, ’True’])
plt.axis([-0.05,6.33,-1.05,1.05])
plt.title(’Integral estimation from spline’)
plt.show()
Roots of spline
>>> print interpolate.sproot(tck)
[ 0.
3.1416]
Parametric spline
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>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
t = np.arange(0,1.1,.1)
x = np.sin(2*np.pi*t)
y = np.cos(2*np.pi*t)
tck,u = interpolate.splprep([x,y],s=0)
unew = np.arange(0,1.01,0.01)
out = interpolate.splev(unew,tck)
plt.figure()
plt.plot(x,y,’x’,out[0],out[1],np.sin(2*np.pi*unew),np.cos(2*np.pi*unew),x,y,’b’)
plt.legend([’Linear’,’Cubic Spline’, ’True’])
plt.axis([-1.05,1.05,-1.05,1.05])
plt.title(’Spline of parametrically-defined curve’)
plt.show()
Cubic-spline interpolation
1.0
Linear
Cubic Spline
True
0.5
0.0
0.5
1.0
0
1
2
3
4
5
6
Derivative estimation from spline
1.0
Cubic Spline
True
0.5
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1.0
0
1
2
3
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6
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Integral estimation from spline
1.0
Cubic Spline
True
0.5
0.0
0.5
1.0
0
1
2
3
4
5
6
Spline of parametrically-defined curve
1.0
Linear
Cubic Spline
True
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0.0
0.5
1.0
1.0
0.5
0.0
0.5
1.0
1.6.3 Spline interpolation in 1-d: Object-oriented (UnivariateSpline)
The spline-fitting capabilities described above are also available via an objected-oriented interface. The one dimensional splines are objects of the UnivariateSpline class, and are created with the x and y components of the curve
provided as arguments to the constructor. The class defines __call__, allowing the object to be called with the x-axis
values at which the spline should be evaluated, returning the interpolated y-values. This is shown in the example below
for the subclass InterpolatedUnivariateSpline. The methods integral, derivatives, and roots methods are
also available on UnivariateSpline objects, allowing definite integrals, derivatives, and roots to be computed for the
spline.
The UnivariateSpline class can also be used to smooth data by providing a non-zero value of the smoothing parameter
s, with the same meaning as the s keyword of the splrep function described above. This results in a spline that
has fewer knots than the number of data points, and hence is no longer strictly an interpolating spline, but rather
a smoothing spline. If this is not desired, the InterpolatedUnivariateSpline class is available. It is a subclass of
UnivariateSpline that always passes through all points (equivalent to forcing the smoothing parameter to 0). This class
is demonstrated in the example below.
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The LSQUnivarateSpline is the other subclass of UnivarateSpline. It allows the user to specify the number and location
of internal knots as explicitly with the parameter t. This allows creation of customized splines with non-linear spacing,
to interpolate in some domains and smooth in others, or change the character of the spline.
1.6.4 Two-dimensional spline representation: Procedural (bisplrep)
For (smooth) spline-fitting to a two dimensional surface, the function bisplrep is available. This function takes as
required inputs the 1-D arrays x, y, and z which represent points on the surface z = f (x, y) . The default output is a
list [tx, ty, c, kx, ky] whose entries represent respectively, the components of the knot positions, the coefficients of the
spline, and the order of the spline in each coordinate. It is convenient to hold this list in a single object, tck, so that
it can be passed easily to the function bisplev. The keyword, s , can be used to change the amount
of smoothing
√
performed on the data while determining the appropriate spline. The default value is s = m − 2m where m is the
number of data points in the x, y, and z vectors. As a result, if no smoothing is desired, then s = 0 should be passed to
bisplrep .
To evaluate the two-dimensional spline and it’s partial derivatives (up to the order of the spline), the function bisplev
is required. This function takes as the first two arguments two 1-D arrays whose cross-product specifies the domain
over which to evaluate the spline. The third argument is the tck list returned from bisplrep. If desired, the fourth
and fifth arguments provide the orders of the partial derivative in the x and y direction respectively.
It is important to note that two dimensional interpolation should not be used to find the spline representation of
images. The algorithm used is not amenable to large numbers of input points. The signal processing toolbox contains
more appropriate algorithms for finding the spline representation of an image. The two dimensional interpolation
commands are intended for use when interpolating a two dimensional function as shown in the example that follows.
This example uses the mgrid command in SciPy which is useful for defining a “mesh-grid “in many dimensions.
(See also the ogrid command if the full-mesh is not needed). The number of output arguments and the number of
dimensions of each argument is determined by the number of indexing objects passed in mgrid.
>>> import numpy as np
>>> from scipy import interpolate
>>> import matplotlib.pyplot as plt
Define function over sparse 20x20 grid
>>> x,y = np.mgrid[-1:1:20j,-1:1:20j]
>>> z = (x+y)*np.exp(-6.0*(x*x+y*y))
>>>
>>>
>>>
>>>
>>>
plt.figure()
plt.pcolor(x,y,z)
plt.colorbar()
plt.title("Sparsely sampled function.")
plt.show()
Interpolate function over new 70x70 grid
>>> xnew,ynew = np.mgrid[-1:1:70j,-1:1:70j]
>>> tck = interpolate.bisplrep(x,y,z,s=0)
>>> znew = interpolate.bisplev(xnew[:,0],ynew[0,:],tck)
>>>
>>>
>>>
>>>
>>>
plt.figure()
plt.pcolor(xnew,ynew,znew)
plt.colorbar()
plt.title("Interpolated function.")
plt.show()
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Sparsely sampled function.
1.0
0.20
0.15
0.10
0.05
0.00
0.05
0.10
0.15
0.20
0.5
0.0
0.5
1.0
1.0
0.5
0.0
0.5
1.0
Interpolated function.
1.0
0.20
0.15
0.10
0.05
0.00
0.05
0.10
0.15
0.20
0.5
0.0
0.5
1.0
1.0
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1.6.5 Two-dimensional spline representation: Object-oriented (BivariateSpline)
The BivariateSpline class is the 2-dimensional analog of the UnivariateSpline class. It and its subclasses
implement the FITPACK functions described above in an object oriented fashion, allowing objects to be instantiated
that can be called to compute the spline value by passing in the two coordinates as the two arguments.
1.6.6 Using radial basis functions for smoothing/interpolation
Radial basis functions can be used for smoothing/interpolating scattered data in n-dimensions, but should be used with
caution for extrapolation outside of the observed data range.
1-d Example
This example compares the usage of the Rbf and UnivariateSpline classes from the scipy.interpolate module.
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>>> import numpy as np
>>> from scipy.interpolate import Rbf, InterpolatedUnivariateSpline
>>> import matplotlib.pyplot as plt
>>>
>>>
>>>
>>>
# setup data
x = np.linspace(0, 10, 9)
y = np.sin(x)
xi = np.linspace(0, 10, 101)
>>> # use fitpack2 method
>>> ius = InterpolatedUnivariateSpline(x, y)
>>> yi = ius(xi)
>>>
>>>
>>>
>>>
>>>
plt.subplot(2, 1, 1)
plt.plot(x, y, ’bo’)
plt.plot(xi, yi, ’g’)
plt.plot(xi, np.sin(xi), ’r’)
plt.title(’Interpolation using univariate spline’)
>>> # use RBF method
>>> rbf = Rbf(x, y)
>>> fi = rbf(xi)
>>>
>>>
>>>
>>>
>>>
>>>
plt.subplot(2, 1, 2)
plt.plot(x, y, ’bo’)
plt.plot(xi, fi, ’g’)
plt.plot(xi, np.sin(xi), ’r’)
plt.title(’Interpolation using RBF - multiquadrics’)
plt.show()
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2-d Example
This example shows how to interpolate scattered 2d data.
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>>>
>>>
>>>
>>>
import numpy as np
from scipy.interpolate import Rbf
import matplotlib.pyplot as plt
from matplotlib import cm
>>>
>>>
>>>
>>>
>>>
>>>
# 2-d tests - setup scattered data
x = np.random.rand(100)*4.0-2.0
y = np.random.rand(100)*4.0-2.0
z = x*np.exp(-x**2-y**2)
ti = np.linspace(-2.0, 2.0, 100)
XI, YI = np.meshgrid(ti, ti)
>>> # use RBF
>>> rbf = Rbf(x, y, z, epsilon=2)
>>> ZI = rbf(XI, YI)
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
# plot the result
n = plt.normalize(-2., 2.)
plt.subplot(1, 1, 1)
plt.pcolor(XI, YI, ZI, cmap=cm.jet)
plt.scatter(x, y, 100, z, cmap=cm.jet)
plt.title(’RBF interpolation - multiquadrics’)
plt.xlim(-2, 2)
plt.ylim(-2, 2)
plt.colorbar()
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1.7 Fourier Transforms (scipy.fftpack)
Warning: This is currently a stub page
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Contents
• Fourier Transforms (scipy.fftpack)
– Fast Fourier transforms
– One dimensional discrete Fourier transforms
– Two and n dimensional discrete Fourier transforms
– Discrete Cosine Transforms
* type I
* type II
* type III
* References
– FFT convolution
Fourier analysis is fundamentally a method for expressing a function as a sum of periodic components, and for recovering the signal from those components. When both the function and its Fourier transform are replaced with discretized
counterparts, it is called the discrete Fourier transform (DFT). The DFT has become a mainstay of numerical computing in part because of a very fast algorithm for computing it, called the Fast Fourier Transform (FFT), which was
known to Gauss (1805) and was brought to light in its current form by Cooley and Tukey [CT]. Press et al. [NR]
provide an accessible introduction to Fourier analysis and its applications.
1.7.1 Fast Fourier transforms
1.7.2 One dimensional discrete Fourier transforms
fft, ifft, rfft, irfft
1.7.3 Two and n dimensional discrete Fourier transforms
fft in more than one dimension
1.7.4 Discrete Cosine Transforms
Return the Discrete Cosine Transform [Mak] of arbitrary type sequence x.
For a single dimension array x, dct(x, norm=’ortho’) is equal to matlab dct(x).
There are theoretically 8 types of the DCT [WP], only the first 3 types are implemented in scipy. ‘The’ DCT generally
refers to DCT type 2, and ‘the’ Inverse DCT generally refers to DCT type 3.
type I
There are several definitions of the DCT-I; we use the following (for norm=None):
k
yk = x0 + (−1) xN −1 + 2
N
−2
X
n=1
xn cos
πnk
N −1
,
0 ≤ k < N.
Only None is supported as normalization mode for DCT-I. Note also that the DCT-I is only supported for input size >
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type II
There are several definitions of the DCT-II; we use the following (for norm=None):
yk = 2
N
−1
X
xn cos
n=0
π(2n + 1)k
2N
0 ≤ k < N.
If norm=’ortho’, yk is multiplied by a scaling factor f :
(p
1/(4N ), if k = 0
f= p
1/(2N ), otherwise
Which makes the corresponding matrix of coefficients orthonormal (OO’ = Id).
type III
There are several definitions, we use the following (for norm=None):
yk = x0 + 2
N
−1
X
xn cos
n=1
πn(2k + 1)
2N
0 ≤ k < N,
or, for norm=’ortho’:
N −1
1 X
πn(2k + 1)
x0
xn cos
yk = √ + √
2N
N
N n=1
0 ≤ k < N.
The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up to a factor 2N. The orthonormalized
DCT-III is exactly the inverse of the orthonormalized DCT-II.
References
1.7.5 FFT convolution
scipy.fftpack.convolve performs a convolution of two one-dimensional arrays in frequency domain.
1.8 Signal Processing (signal)
The signal processing toolbox currently contains some filtering functions, a limited set of filter design tools, and a few
B-spline interpolation algorithms for one- and two-dimensional data. While the B-spline algorithms could technically
be placed under the interpolation category, they are included here because they only work with equally-spaced data and
make heavy use of filter-theory and transfer-function formalism to provide a fast B-spline transform. To understand
this section you will need to understand that a signal in SciPy is an array of real or complex numbers.
1.8.1 B-splines
A B-spline is an approximation of a continuous function over a finite- domain in terms of B-spline coefficients and knot
points. If the knot- points are equally spaced with spacing ∆x , then the B-spline approximation to a 1-dimensional
function is the finite-basis expansion.
x
X
−j .
y (x) ≈
cj β o
∆x
j
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In two dimensions with knot-spacing ∆x and ∆y , the function representation is
x
y
XX
z (x, y) ≈
cjk β o
− j βo
−k .
∆x
∆y
j
k
In these expressions, β o (·) is the space-limited B-spline basis function of order, o . The requirement of equallyspaced knot-points and equally-spaced data points, allows the development of fast (inverse-filtering) algorithms for
determining the coefficients, cj , from sample-values, yn . Unlike the general spline interpolation algorithms, these
algorithms can quickly find the spline coefficients for large images.
The advantage of representing a set of samples via B-spline basis functions is that continuous-domain operators
(derivatives, re- sampling, integral, etc.) which assume that the data samples are drawn from an underlying continuous function can be computed with relative ease from the spline coefficients. For example, the second-derivative
of a spline is
x
1 X
o00
c
β
−
j
.
y 00 (x) =
j
∆x2 j
∆x
Using the property of B-splines that
d2 β o (w)
= β o−2 (w + 1) − 2β o−2 (w) + β o−2 (w − 1)
dw2
it can be seen that
y 00 (x) =
i
x
x
1 X h o−2 x
o−2
o−2
c
−
j
+
1
−
2β
−
j
+
β
−
j
−
1
.
β
j
∆x2 j
∆x
∆x
∆x
If o = 3 , then at the sample points,
∆x2 y 0 (x)|x=n∆x
=
X
cj δn−j+1 − 2cj δn−j + cj δn−j−1 ,
j
=
cn+1 − 2cn + cn−1 .
Thus, the second-derivative signal can be easily calculated from the spline fit. if desired, smoothing splines can be
found to make the second-derivative less sensitive to random-errors.
The savvy reader will have already noticed that the data samples are related to the knot coefficients via a convolution
operator, so that simple convolution with the sampled B-spline function recovers the original data from the spline coefficients. The output of convolutions can change depending on how boundaries are handled (this becomes increasingly
more important as the number of dimensions in the data- set increases). The algorithms relating to B-splines in the
signal- processing sub package assume mirror-symmetric boundary conditions. Thus, spline coefficients are computed
based on that assumption, and data-samples can be recovered exactly from the spline coefficients by assuming them
to be mirror-symmetric also.
Currently the package provides functions for determining second- and third-order cubic spline coefficients
from equally spaced samples in one- and two-dimensions (signal.qspline1d, signal.qspline2d,
signal.cspline1d, signal.cspline2d). The package also supplies a function ( signal.bspline ) for
evaluating the bspline basis function, β o (x) for arbitrary order and x. For large o , the B-spline basis function can be
approximated well by a zero-mean Gaussian function with standard-deviation equal to σo = (o + 1) /12 :
1
x2
.
β o (x) ≈ p
exp −
2σo
2πσo2
A function to compute this Gaussian for arbitrary x and o is also available ( signal.gauss_spline ). The
following code and Figure uses spline-filtering to compute an edge-image (the second-derivative of a smoothed
spline) of Lena’s face which is an array returned by the command lena. The command signal.sepfir2d
was used to apply a separable two-dimensional FIR filter with mirror- symmetric boundary conditions to the spline
coefficients. This function is ideally suited for reconstructing samples from spline coefficients and is faster than
signal.convolve2d which convolves arbitrary two-dimensional filters and allows for choosing mirror-symmetric
boundary conditions.
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>>> from numpy import *
>>> from scipy import signal, misc
>>> import matplotlib.pyplot as plt
>>>
>>>
>>>
>>>
>>>
image = misc.lena().astype(float32)
derfilt = array([1.0,-2,1.0],float32)
ck = signal.cspline2d(image,8.0)
deriv = signal.sepfir2d(ck, derfilt, [1]) + \
signal.sepfir2d(ck, [1], derfilt)
Alternatively we could have done:
laplacian = array([[0,1,0],[1,-4,1],[0,1,0]],float32)
deriv2 = signal.convolve2d(ck,laplacian,mode=’same’,boundary=’symm’)
>>>
>>>
>>>
>>>
>>>
plt.figure()
plt.imshow(image)
plt.gray()
plt.title(’Original image’)
plt.show()
>>>
>>>
>>>
>>>
>>>
plt.figure()
plt.imshow(deriv)
plt.gray()
plt.title(’Output of spline edge filter’)
plt.show()
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1.8.2 Filtering
Filtering is a generic name for any system that modifies an input signal in some way. In SciPy a signal can be thought
of as a Numpy array. There are different kinds of filters for different kinds of operations. There are two broad kinds
of filtering operations: linear and non-linear. Linear filters can always be reduced to multiplication of the flattened
Numpy array by an appropriate matrix resulting in another flattened Numpy array. Of course, this is not usually the
best way to compute the filter as the matrices and vectors involved may be huge. For example filtering a 512 × 512
image with this method would require multiplication of a 5122 ×5122 matrix with a 5122 vector. Just trying to store the
5122 × 5122 matrix using a standard Numpy array would require 68, 719, 476, 736 elements. At 4 bytes per element
this would require 256GB of memory. In most applications most of the elements of this matrix are zero and a different
method for computing the output of the filter is employed.
Convolution/Correlation
Many linear filters also have the property of shift-invariance. This means that the filtering operation is the same at
different locations in the signal and it implies that the filtering matrix can be constructed from knowledge of one row
(or column) of the matrix alone. In this case, the matrix multiplication can be accomplished using Fourier transforms.
Let x [n] define a one-dimensional signal indexed by the integer n. Full convolution of two one-dimensional signals
can be expressed as
∞
X
y [n] =
x [k] h [n − k] .
k=−∞
This equation can only be implemented directly if we limit the sequences to finite support sequences that can be stored
in a computer, choose n = 0 to be the starting point of both sequences, let K + 1 be that value for which y [n] = 0
for all n > K + 1 and M + 1 be that value for which x [n] = 0 for all n > M + 1 , then the discrete convolution
expression is
min(n,K)
X
y [n] =
x [k] h [n − k] .
k=max(n−M,0)
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For convenience assume K ≥ M. Then, more explicitly the output of this operation is
y [0]
= x [0] h [0]
y [1]
= x [0] h [1] + x [1] h [0]
y [2]
..
.
=
..
.
x [0] h [2] + x [1] h [1] + x [2] h [0]
..
.
y [M ]
=
x [0] h [M ] + x [1] h [M − 1] + · · · + x [M ] h [0]
y [M + 1] =
.. ..
. .
y [K] =
x [1] h [M ] + x [2] h [M − 1] + · · · + x [M + 1] h [0]
..
.
y [K + 1] =
.. ..
. .
x [K + 1 − M ] h [M ] + · · · + x [K] h [1]
..
.
y [K + M − 1]
y [K + M ]
x [K − M ] h [M ] + · · · + x [K] h [0]
= x [K − 1] h [M ] + x [K] h [M − 1]
=
x [K] h [M ] .
Thus, the full discrete convolution of two finite sequences of lengths K + 1 and M + 1 respectively results in a finite
sequence of length K + M + 1 = (K + 1) + (M + 1) − 1.
One dimensional convolution is implemented in SciPy with the function signal.convolve . This function takes
as inputs the signals x, h , and an optional flag and returns the signal y. The optional flag allows for specification of
which part of the output signal to return. The default value of ‘full’
returns
the entire signal. If the flag has a value of
‘same’ then only the middle K values are returned starting at y M2−1 so that the output has the same length as the
largest input. If the flag has a value of ‘valid’ then only the middle K − M + 1 = (K + 1) − (M + 1) + 1 output
values are returned where z depends on all of the values of the smallest input from h [0] to h [M ] . In other words only
the values y [M ] to y [K] inclusive are returned.
This same function signal.convolve can actually take N -dimensional arrays as inputs and will return the N
-dimensional convolution of the two arrays. The same input flags are available for that case as well.
Correlation is very similar to convolution except for the minus sign becomes a plus sign. Thus
w [n] =
∞
X
y [k] x [n + k]
k=−∞
is the (cross) correlation of the signals y and x. For finite-length signals with y [n] = 0 outside of the range [0, K] and
x [n] = 0 outside of the range [0, M ] , the summation can simplify to
min(K,M −n)
w [n] =
X
y [k] x [n + k] .
k=max(0,−n)
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Assuming again that K ≥ M this is
w [−K]
=
y [K] x [0]
w [−K + 1] =
.. ..
. .
y [K − 1] x [0] + y [K] x [1]
..
.
w [M − K]
y [K − M ] x [0] + y [K − M + 1] x [1] + · · · + y [K] x [M ]
=
w [M − K + 1] =
.. ..
. .
w [−1] =
w [0]
y [K − M − 1] x [0] + · · · + y [K − 1] x [M ]
..
.
y [1] x [0] + y [2] x [1] + · · · + y [M + 1] x [M ]
= y [0] x [0] + y [1] x [1] + · · · + y [M ] x [M ]
w [1]
= y [0] x [1] + y [1] x [2] + · · · + y [M − 1] x [M ]
w [2]
..
.
=
..
.
w [M − 1]
w [M ]
y [0] x [2] + y [1] x [3] + · · · + y [M − 2] x [M ]
..
.
= y [0] x [M − 1] + y [1] x [M ]
=
y [0] x [M ] .
The SciPy function signal.correlate implements this operation. Equivalent flags are available for this operation
to return the
full K
+M +1 length sequence (‘full’) or a sequence with the same size as the largest sequence starting at
w −K + M2−1 (‘same’) or a sequence where the values depend on all the values of the smallest sequence (‘valid’).
This final option returns the K − M + 1 values w [M − K] to w [0] inclusive.
The function signal.correlate can also take arbitrary N -dimensional arrays as input and return the N dimensional convolution of the two arrays on output.
When N = 2, signal.correlate and/or signal.convolve can be used to construct arbitrary image filters
to perform actions such as blurring, enhancing, and edge-detection for an image.
Convolution is mainly used for filtering when one of the signals is much smaller than the other ( K M ), otherwise
linear filtering is more easily accomplished in the frequency domain (see Fourier Transforms).
Difference-equation filtering
A general class of linear one-dimensional filters (that includes convolution filters) are filters described by the difference
equation
N
M
X
X
ak y [n − k] =
bk x [n − k]
k=0
k=0
where x [n] is the input sequence and y [n] is the output sequence. If we assume initial rest so that y [n] = 0 for n < 0
, then this kind of filter can be implemented using convolution. However, the convolution filter sequence h [n] could
be infinite if ak 6= 0 for k ≥ 1. In addition, this general class of linear filter allows initial conditions to be placed on
y [n] for n < 0 resulting in a filter that cannot be expressed using convolution.
The difference equation filter can be thought of as finding y [n] recursively in terms of it’s previous values
a0 y [n] = −a1 y [n − 1] − · · · − aN y [n − N ] + · · · + b0 x [n] + · · · + bM x [n − M ] .
Often a0 = 1 is chosen for normalization. The implementation in SciPy of this general difference equation filter is
a little more complicated then would be implied by the previous equation. It is implemented so that only one signal
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needs to be delayed. The actual implementation equations are (assuming a0 = 1 ).
y [n]
=
b0 x [n] + z0 [n − 1]
z0 [n]
=
b1 x [n] + z1 [n − 1] − a1 y [n]
z1 [n] =
.. ..
. .
b2 x [n] + z2 [n − 1] − a2 y [n]
..
.
zK−2 [n]
= bK−1 x [n] + zK−1 [n − 1] − aK−1 y [n]
zK−1 [n]
=
bK x [n] − aK y [n] ,
where K = max (N, M ) . Note that bK = 0 if K > M and aK = 0 if K > N. In this way, the output at time n
depends only on the input at time n and the value of z0 at the previous time. This can always be calculated as long as
the K values z0 [n − 1] . . . zK−1 [n − 1] are computed and stored at each time step.
The difference-equation filter is called using the command signal.lfilter in SciPy. This command takes as
inputs the vector b, the vector, a, a signal x and returns the vector y (the same length as x ) computed using the
equation given above. If x is N -dimensional, then the filter is computed along the axis provided. If, desired, initial
conditions providing the values of z0 [−1] to zK−1 [−1] can be provided or else it will be assumed that they are all
zero. If initial conditions are provided, then the final conditions on the intermediate variables are also returned. These
could be used, for example, to restart the calculation in the same state.
Sometimes it is more convenient to express the initial conditions in terms of the signals x [n] and y [n] . In other words,
perhaps you have the values of x [−M ] to x [−1] and the values of y [−N ] to y [−1] and would like to determine what
values of zm [−1] should be delivered as initial conditions to the difference-equation filter. It is not difficult to show
that for 0 ≤ m < K,
K−m−1
X
(bm+p+1 x [n − p] − am+p+1 y [n − p]) .
zm [n] =
p=0
Using this formula we can find the intial condition vector z0 [−1] to zK−1 [−1] given initial conditions on y (and x ).
The command signal.lfiltic performs this function.
Other filters
The signal processing package provides many more filters as well.
Median Filter
A median filter is commonly applied when noise is markedly non- Gaussian or when it is desired to preserve edges. The
median filter works by sorting all of the array pixel values in a rectangular region surrounding the point of interest.
The sample median of this list of neighborhood pixel values is used as the value for the output array. The sample
median is the middle array value in a sorted list of neighborhood values. If there are an even number of elements in the
neighborhood, then the average of the middle two values is used as the median. A general purpose median filter that
works on N-dimensional arrays is signal.medfilt . A specialized version that works only for two-dimensional
arrays is available as signal.medfilt2d .
Order Filter
A median filter is a specific example of a more general class of filters called order filters. To compute the output
at a particular pixel, all order filters use the array values in a region surrounding that pixel. These array values are
sorted and then one of them is selected as the output value. For the median filter, the sample median of the list of
array values is used as the output. A general order filter allows the user to select which of the sorted values will be
used as the output. So, for example one could choose to pick the maximum in the list or the minimum. The order
filter takes an additional argument besides the input array and the region mask that specifies which of the elements
in the sorted list of neighbor array values should be used as the output. The command to perform an order filter is
signal.order_filter .
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Wiener filter
The Wiener filter is a simple deblurring filter for denoising images. This is not the Wiener filter commonly described
in image reconstruction problems but instead it is a simple, local-mean filter. Let x be the input signal, then the output
is
( 2
2
σ
1 − σσ2 x σx2 ≥ σ 2 ,
2 mx +
σ
x
x
y=
mx
σx2 < σ 2 .
Where mx is the local estimate of the mean and σx2 is the local estimate of the variance. The window for these estimates
is an optional input parameter (default is 3 × 3 ). The parameter σ 2 is a threshold noise parameter. If σ is not given
then it is estimated as the average of the local variances.
Hilbert filter
The Hilbert transform constructs the complex-valued analytic signal from a real signal. For example if x = cos ωn
then y = hilbert (x) would return (except near the edges) y = exp (jωn) . In the frequency domain, the hilbert
transform performs
Y =X ·H
where H is 2 for positive frequencies, 0 for negative frequencies and 1 for zero-frequencies.
1.9 Linear Algebra
When SciPy is built using the optimized ATLAS LAPACK and BLAS libraries, it has very fast linear algebra capabilities. If you dig deep enough, all of the raw lapack and blas libraries are available for your use for even more speed.
In this section, some easier-to-use interfaces to these routines are described.
All of these linear algebra routines expect an object that can be converted into a 2-dimensional array. The output of
these routines is also a two-dimensional array. There is a matrix class defined in Numpy, which you can initialize with
an appropriate Numpy array in order to get objects for which multiplication is matrix-multiplication instead of the
default, element-by-element multiplication.
1.9.1 Matrix Class
The matrix class is initialized with the SciPy command mat which is just convenient short-hand for matrix. If you
are going to be doing a lot of matrix-math, it is convenient to convert arrays into matrices using this command. One
advantage of using the mat command is that you can enter two-dimensional matrices using MATLAB-like syntax
with commas or spaces separating columns and semicolons separting rows as long as the matrix is placed in a string
passed to mat .
1.9.2 Basic routines
Finding Inverse
The inverse of a matrix A is the matrix B such that AB = I where I is the identity matrix consisting of ones down
the main diagonal. Usually B is denoted B = A−1 . In SciPy, the matrix inverse of the Numpy array, A, is obtained
using linalg.inv (A) , or using A.I if A is a Matrix. For example, let


1 3 5
A = 2 5 1 
2 3 8
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then

A−1
−37 9
1 
14
2
=
25
4
−3
 

22
−1.48 0.36
0.88
−9  =  0.56
0.08 −0.36  .
1
0.16 −0.12 0.04
The following example demonstrates this computation in SciPy
>>> A = mat(’[1 3 5; 2 5 1; 2 3 8]’)
>>> A
matrix([[1, 3, 5],
[2, 5, 1],
[2, 3, 8]])
>>> A.I
matrix([[-1.48, 0.36, 0.88],
[ 0.56, 0.08, -0.36],
[ 0.16, -0.12, 0.04]])
>>> from scipy import linalg
>>> linalg.inv(A)
array([[-1.48, 0.36, 0.88],
[ 0.56, 0.08, -0.36],
[ 0.16, -0.12, 0.04]])
Solving linear system
Solving linear systems of equations is straightforward using the scipy command linalg.solve. This command
expects an input matrix and a right-hand-side vector. The solution vector is then computed. An option for entering a
symmetrix matrix is offered which can speed up the processing when applicable. As an example, suppose it is desired
to solve the following simultaneous equations:
x + 3y + 5z
=
10
2x + 5y + z
=
8
2x + 3y + 8z
=
3
We could find the solution vector using a matrix inverse:

 
x
1
 y = 2
z
2
3
5
3
−1 


 

5
10
−232
−9.28
1
 129  =  5.16  .
1   8 =
25
8
3
19
0.76
However, it is better to use the linalg.solve command which can be faster and more numerically stable. In this case it
however gives the same answer as shown in the following example:
>>> A = mat(’[1 3 5; 2 5 1; 2 3 8]’)
>>> b = mat(’[10;8;3]’)
>>> A.I*b
matrix([[-9.28],
[ 5.16],
[ 0.76]])
>>> linalg.solve(A,b)
array([[-9.28],
[ 5.16],
[ 0.76]])
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Finding Determinant
The determinant of a square matrix A is often denoted |A| and is a quantity often used in linear algebra. Suppose aij
are the elements of the matrix A and let Mij = |Aij | be the determinant of the matrix left by removing the ith row
and j th column from A . Then for any row i,
X
i+j
|A| =
(−1) aij Mij .
j
This is a recursive way to define the determinant where the base case is defined by accepting that the determinant of a
1 × 1 matrix is the only matrix element. In SciPy the determinant can be calculated with linalg.det . For example,
the determinant of


1 3 5
A = 2 5 1 
2 3 8
is
|A|
2
1 − 3 2
8 2
1 + 5 2
8 5 3 =
5
1 3
=
1 (5 · 8 − 3 · 1) − 3 (2 · 8 − 2 · 1) + 5 (2 · 3 − 2 · 5) = −25.
In SciPy this is computed as shown in this example:
>>> A = mat(’[1 3 5; 2 5 1; 2 3 8]’)
>>> linalg.det(A)
-25.000000000000004
Computing norms
Matrix and vector norms can also be computed with SciPy. A wide range of norm definitions are available using
different parameters to the order argument of linalg.norm . This function takes a rank-1 (vectors) or a rank-2
(matrices) array and an optional order argument (default is 2). Based on these inputs a vector or matrix norm of the
requested order is computed.
For vector x , the order parameter can be any real number including inf or -inf. The computed norm is

max |xi |
ord = inf


min
|x
|
ord
= −inf
i
kxk =
1/ord

 P |x |ord
|ord| < ∞.
i
i
For matrix A the only valid values for norm are ±2, ±1, ± inf, and ‘fro’ (or ‘f’) Thus,
P

maxi j |aij |
ord = inf


P


min
|a
|
ord
= −inf

i
ij

Pj


ord = 1
 maxj P i |aij |
kAk =
minj i |aij |
ord = −1


max
σ
ord = 2

i



min
σ
ord
= −2

i

 p
trace (AH A) ord = ’fro’
where σi are the singular values of A .
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Solving linear least-squares problems and pseudo-inverses
Linear least-squares problems occur in many branches of applied mathematics. In this problem a set of linear scaling
coefficients is sought that allow a model to fit data. In particular it is assumed that data yi is related to data xi through
a set of coefficients cj and model functions fj (xi ) via the model
X
yi =
cj fj (xi ) + i
j
where i represents uncertainty in the data. The strategy of least squares is to pick the coefficients cj to minimize
2
X
X yi −
c
f
(x
)
J (c) =
j j
i .
j
i Theoretically, a global minimum will occur when

∂J
=0=
∂c∗n
X

yi −
X
i
cj fj (xi ) (−fn∗ (xi ))
j
or
X
j
cj
X
fj (xi ) fn∗ (xi )
=
X
i
yi fn∗ (xi )
i
H
A Ac
= AH y
where
{A}ij = fj (xi ) .
When AH A is invertible, then
c = AH A
−1
AH y = A† y
where A† is called the pseudo-inverse of A. Notice that using this definition of A the model can be written
y = Ac + .
The command linalg.lstsq will solve the linear least squares problem for c given A and y . In addition
linalg.pinv or linalg.pinv2 (uses a different method based on singular value decomposition) will find A†
given A.
The following example and figure demonstrate the use of linalg.lstsq and linalg.pinv for solving a datafitting problem. The data shown below were generated using the model:
yi = c1 e−xi + c2 xi
where xi = 0.1i for i = 1 . . . 10 , c1 = 5 , and c2 = 4. Noise is added to yi and the coefficients c1 and c2 are estimated
using linear least squares.
>>> from numpy import *
>>> from scipy import linalg
>>> import matplotlib.pyplot as plt
>>>
>>>
>>>
>>>
>>>
46
c1,c2= 5.0,2.0
i = r_[1:11]
xi = 0.1*i
yi = c1*exp(-xi)+c2*xi
zi = yi + 0.05*max(yi)*random.randn(len(yi))
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>>> A = c_[exp(-xi)[:,newaxis],xi[:,newaxis]]
>>> c,resid,rank,sigma = linalg.lstsq(A,zi)
>>> xi2 = r_[0.1:1.0:100j]
>>> yi2 = c[0]*exp(-xi2) + c[1]*xi2
>>>
>>>
>>>
>>>
>>>
plt.plot(xi,zi,’x’,xi2,yi2)
plt.axis([0,1.1,3.0,5.5])
plt.xlabel(’$x_i$’)
plt.title(’Data fitting with linalg.lstsq’)
plt.show()
Data fitting with linalg.lstsq
5.5
5.0
4.5
4.0
3.5
3.0
0.0
0.2
0.4
xi
0.6
0.8
1.0
Generalized inverse
The generalized inverse is calculated using the command linalg.pinv or linalg.pinv2. These two commands
differ in how they compute the generalized inverse. The first uses the linalg.lstsq algorithm while the second uses
singular value decomposition. Let A be an M × N matrix, then if M > N the generalized inverse is
−1
AH
A# = AH AAH
−1
A† = AH A
while if M < N matrix the generalized inverse is
.
In both cases for M = N , then
A† = A# = A−1
as long as A is invertible.
1.9.3 Decompositions
In many applications it is useful to decompose a matrix using other representations. There are several decompositions
supported by SciPy.
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Eigenvalues and eigenvectors
The eigenvalue-eigenvector problem is one of the most commonly employed linear algebra operations. In one popular
form, the eigenvalue-eigenvector problem is to find for some square matrix A scalars λ and corresponding vectors v
such that
Av = λv.
For an N × N matrix, there are N (not necessarily distinct) eigenvalues — roots of the (characteristic) polynomial
|A − λI| = 0.
The eigenvectors, v , are also sometimes called right eigenvectors to distinguish them from another set of left eigenvectors that satisfy
H
H
vL
A = λvL
or
AH v L = λ ∗ v L .
With it’s default optional arguments, the command linalg.eig returns λ and v. However, it can also return vL and
just λ by itself ( linalg.eigvals returns just λ as well).
In addtion, linalg.eig can also solve the more general eigenvalue problem
Av
= λBv
= λ∗ BH vL
H
A vL
for square matrices A and B. The standard eigenvalue problem is an example of the general eigenvalue problem for
B = I. When a generalized eigenvalue problem can be solved, then it provides a decomposition of A as
A = BVΛV−1
where V is the collection of eigenvectors into columns and Λ is a diagonal matrix of eigenvalues.
By definition, eigenvectors areP
only defined up to a constant scale factor. In SciPy, the scaling factor for the eigenvec2
tors is chosen so that kvk = i vi2 = 1.
As an example, consider finding the eigenvalues and eigenvectors of the matrix


1 5 2
A =  2 4 1 .
3 6 2
The characteristic polynomial is
|A − λI| =
(1 − λ) [(4 − λ) (2 − λ) − 6] −
5 [2 (2 − λ) − 3] + 2 [12 − 3 (4 − λ)]
=
−λ3 + 7λ2 + 8λ − 3.
The roots of this polynomial are the eigenvalues of A :
λ1
=
7.9579
λ2
= −1.2577
λ3
=
0.2997.
The eigenvectors corresponding to each eigenvalue can be found using the original equation. The eigenvectors associated with these eigenvalues can then be found.
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>>> from scipy import linalg
>>> A = mat(’[1 5 2; 2 4 1; 3 6 2]’)
>>> la,v = linalg.eig(A)
>>> l1,l2,l3 = la
>>> print l1, l2, l3
(7.95791620491+0j) (-1.25766470568+0j) (0.299748500767+0j)
>>> print v[:,0]
[-0.5297175 -0.44941741 -0.71932146]
>>> print v[:,1]
[-0.90730751 0.28662547 0.30763439]
>>> print v[:,2]
[ 0.28380519 -0.39012063 0.87593408]
>>> print sum(abs(v**2),axis=0)
[ 1. 1. 1.]
>>> v1 = mat(v[:,0]).T
>>> print max(ravel(abs(A*v1-l1*v1)))
8.881784197e-16
Singular value decomposition
Singular Value Decompostion (SVD) can be thought of as an extension of the eigenvalue problem to matrices that are
not square. Let A be an M × N matrix with M and N arbitrary. The matrices AH A and AAH are square hermitian
matrices 1 of size N × N and M × M respectively. It is known that the eigenvalues of square hermitian matrices are
real and non-negative. In addtion, there are at most min (M, N ) identical non-zero eigenvalues of AH A and AAH .
Define these positive eigenvalues as σi2 . The square-root of these are called singular values of A. The eigenvectors of
AH A are collected by columns into an N × N unitary 2 matrix V while the eigenvectors of AAH are collected by
columns in the unitary matrix U , the singular values are collected in an M × N zero matrix Σ with main diagonal
entries set to the singular values. Then
A = UΣVH
is the singular-value decomposition of A. Every matrix has a singular value decomposition. Sometimes, the singular
values are called the spectrum of A. The command linalg.svd will return U , VH , and σi as an array of the
singular values. To obtain the matrix Σ use linalg.diagsvd. The following example illustrates the use of
linalg.svd .
>>> A = mat(’[1 3 2; 1 2 3]’)
>>> M,N = A.shape
>>> U,s,Vh = linalg.svd(A)
>>> Sig = mat(linalg.diagsvd(s,M,N))
>>> U, Vh = mat(U), mat(Vh)
>>> print U
[[-0.70710678 -0.70710678]
[-0.70710678 0.70710678]]
>>> print Sig
[[ 5.19615242 0.
0.
]
[ 0.
1.
0.
]]
>>> print Vh
[[ -2.72165527e-01 -6.80413817e-01 -6.80413817e-01]
[ -6.18652536e-16 -7.07106781e-01
7.07106781e-01]
[ -9.62250449e-01
1.92450090e-01
1.92450090e-01]]
1
2
A hermitian matrix D satisfies DH = D.
A unitary matrix D satisfies DH D = I = DDH so that D−1 = DH .
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>>> print A
[[1 3 2]
[1 2 3]]
>>> print U*Sig*Vh
[[ 1. 3. 2.]
[ 1. 2. 3.]]
LU decomposition
The LU decompostion finds a representation for the M × N matrix A as
A = PLU
where P is an M × M permutation matrix (a permutation of the rows of the identity matrix), L is in M × K lower
triangular or trapezoidal matrix ( K = min (M, N ) ) with unit-diagonal, and U is an upper triangular or trapezoidal
matrix. The SciPy command for this decomposition is linalg.lu .
Such a decomposition is often useful for solving many simultaneous equations where the left-hand-side does not
change but the right hand side does. For example, suppose we are going to solve
Axi = bi
for many different bi . The LU decomposition allows this to be written as
PLUxi = bi .
Because L is lower-triangular, the equation can be solved for Uxi and finally xi very rapidly using forward- and
back-substitution. An initial time spent factoring A allows for very rapid solution of similar systems of equations in the future. If the intent for performing LU decomposition is for solving linear systems then the command
linalg.lu_factor should be used followed by repeated applications of the command linalg.lu_solve to
solve the system for each new right-hand-side.
Cholesky decomposition
Cholesky decomposition is a special case of LU decomposition applicable to Hermitian positive definite matrices.
When A = AH and xH Ax ≥ 0 for all x , then decompositions of A can be found so that
A =
UH U
A =
LLH
where L is lower-triangular and U is upper triangular. Notice that L = UH . The command linagl.cholesky
computes the cholesky factorization. For using cholesky factorization to solve systems of equations there are also
linalg.cho_factor and linalg.cho_solve routines that work similarly to their LU decomposition counterparts.
QR decomposition
The QR decomposition (sometimes called a polar decomposition) works for any M × N array and finds an M × M
unitary matrix Q and an M × N upper-trapezoidal matrix R such that
A = QR.
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Notice that if the SVD of A is known then the QR decomposition can be found
A = UΣVH = QR
implies that Q = U and R = ΣVH . Note, however, that in SciPy independent algorithms are used to find QR and
SVD decompositions. The command for QR decomposition is linalg.qr .
Schur decomposition
For a square N × N matrix, A , the Schur decomposition finds (not-necessarily unique) matrices T and Z such that
A = ZTZH
where Z is a unitary matrix and T is either upper-triangular or quasi-upper triangular depending on whether or not a
real schur form or complex schur form is requested. For a real schur form both T and Z are real-valued when A is
real-valued. When A is a real-valued matrix the real schur form is only quasi-upper triangular because 2 × 2 blocks
extrude from the main diagonal corresponding to any complex- valued eigenvalues. The command linalg.schur
finds the Schur decomposition while the command linalg.rsf2csf converts T and Z from a real Schur form to
a complex Schur form. The Schur form is especially useful in calculating functions of matrices.
The following example illustrates the schur decomposition:
>>> from scipy import linalg
>>> A = mat(’[1 3 2; 1 4 5; 2 3 6]’)
>>> T,Z = linalg.schur(A)
>>> T1,Z1 = linalg.schur(A,’complex’)
>>> T2,Z2 = linalg.rsf2csf(T,Z)
>>> print T
[[ 9.90012467 1.78947961 -0.65498528]
[ 0.
0.54993766 -1.57754789]
[ 0.
0.51260928 0.54993766]]
>>> print T2
[[ 9.90012467 +0.00000000e+00j -0.32436598 +1.55463542e+00j
-0.88619748 +5.69027615e-01j]
[ 0.00000000 +0.00000000e+00j 0.54993766 +8.99258408e-01j
1.06493862 +1.37016050e-17j]
[ 0.00000000 +0.00000000e+00j 0.00000000 +0.00000000e+00j
0.54993766 -8.99258408e-01j]]
>>> print abs(T1-T2) # different
[[ 1.24357637e-14
2.09205364e+00
6.56028192e-01]
[ 0.00000000e+00
4.00296604e-16
1.83223097e+00]
[ 0.00000000e+00
0.00000000e+00
4.57756680e-16]]
>>> print abs(Z1-Z2) # different
[[ 0.06833781 1.10591375 0.23662249]
[ 0.11857169 0.5585604
0.29617525]
[ 0.12624999 0.75656818 0.22975038]]
>>> T,Z,T1,Z1,T2,Z2 = map(mat,(T,Z,T1,Z1,T2,Z2))
>>> print abs(A-Z*T*Z.H) # same
[[ 1.11022302e-16
4.44089210e-16
4.44089210e-16]
[ 4.44089210e-16
1.33226763e-15
8.88178420e-16]
[ 8.88178420e-16
4.44089210e-16
2.66453526e-15]]
>>> print abs(A-Z1*T1*Z1.H) # same
[[ 1.00043248e-15
2.22301403e-15
5.55749485e-15]
[ 2.88899660e-15
8.44927041e-15
9.77322008e-15]
[ 3.11291538e-15
1.15463228e-14
1.15464861e-14]]
>>> print abs(A-Z2*T2*Z2.H) # same
[[ 3.34058710e-16
8.88611201e-16
4.18773089e-18]
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[
[
1.48694940e-16
1.33228956e-15
8.95109973e-16
1.33582317e-15
8.92966151e-16]
3.55373104e-15]]
1.9.4 Matrix Functions
Consider the function f (x) with Taylor series expansion
f (x) =
∞
X
f (k) (0)
k!
k=0
xk .
A matrix function can be defined using this Taylor series for the square matrix A as
f (A) =
∞
X
f (k) (0)
k!
k=0
Ak .
While, this serves as a useful representation of a matrix function, it is rarely the best way to calculate a matrix function.
Exponential and logarithm functions
The matrix exponential is one of the more common matrix functions. It can be defined for square matrices as
eA =
∞
X
1 k
A .
k!
k=0
The command linalg.expm3 uses this Taylor series definition to compute the matrix exponential. Due to poor
convergence properties it is not often used.
Another method to compute the matrix exponential is to find an eigenvalue decomposition of A :
A = VΛV−1
and note that
eA = VeΛ V−1
where the matrix exponential of the diagonal matrix Λ is just the exponential of its elements. This method is implemented in linalg.expm2 .
The preferred method for implementing the matrix exponential is to use scaling and a Padé approximation for ex .
This algorithm is implemented as linalg.expm .
The inverse of the matrix exponential is the matrix logarithm defined as the inverse of the matrix exponential.
A ≡ exp (log (A)) .
The matrix logarithm can be obtained with linalg.logm .
Trigonometric functions
The trigonometric functions sin , cos , and tan are implemented for matrices in linalg.sinm, linalg.cosm,
and linalg.tanm respectively. The matrix sin and cosine can be defined using Euler’s identity as
52
sin (A)
=
cos (A)
=
ejA − e−jA
2j
ejA + e−jA
.
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The tangent is
tan (x) =
sin (x)
−1
= [cos (x)] sin (x)
cos (x)
and so the matrix tangent is defined as
−1
[cos (A)]
sin (A) .
Hyperbolic trigonometric functions
The hyperbolic trigonemetric functions sinh , cosh , and tanh can also be defined for matrices using the familiar
definitions:
sinh (A)
=
cosh (A)
=
tanh (A)
=
eA − e−A
2
eA + e−A
2
−1
[cosh (A)] sinh (A) .
These matrix functions can be found using linalg.sinhm, linalg.coshm , and linalg.tanhm.
Arbitrary function
Finally, any arbitrary function that takes one complex number and returns a complex number can be called as a matrix
function using the command linalg.funm. This command takes the matrix and an arbitrary Python function. It
then implements an algorithm from Golub and Van Loan’s book “Matrix Computations “to compute function applied
to the matrix using a Schur decomposition. Note that the function needs to accept complex numbers as input in order
to work with this algorithm. For example the following code computes the zeroth-order Bessel function applied to a
matrix.
>>> from scipy import special, random, linalg
>>> A = random.rand(3,3)
>>> B = linalg.funm(A,lambda x: special.jv(0,x))
>>> print A
[[ 0.72578091 0.34105276 0.79570345]
[ 0.65767207 0.73855618 0.541453 ]
[ 0.78397086 0.68043507 0.4837898 ]]
>>> print B
[[ 0.72599893 -0.20545711 -0.22721101]
[-0.27426769 0.77255139 -0.23422637]
[-0.27612103 -0.21754832 0.7556849 ]]
>>> print linalg.eigvals(A)
[ 1.91262611+0.j 0.21846476+0.j -0.18296399+0.j]
>>> print special.jv(0, linalg.eigvals(A))
[ 0.27448286+0.j 0.98810383+0.j 0.99164854+0.j]
>>> print linalg.eigvals(B)
[ 0.27448286+0.j 0.98810383+0.j 0.99164854+0.j]
Note how, by virtue of how matrix analytic functions are defined, the Bessel function has acted on the matrix eigenvalues.
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1.10 Statistics
1.10.1 Introduction
SciPy has a tremendous number of basic statistics routines with more easily added by the end user (if you create
one please contribute it). All of the statistics functions are located in the sub-package scipy.stats and a fairly
complete listing of these functions can be had using info(stats).
Random Variables
There are two general distribution classes that have been implemented for encapsulating continuous random variables
and discrete random variables . Over 80 continuous random variables and 10 discrete random variables have been
implemented using these classes. The list of the random variables available is in the docstring for the stats subpackage.
Note: The following is work in progress
1.10.2 Distributions
First some imports
>>>
>>>
>>>
>>>
import numpy as np
from scipy import stats
import warnings
warnings.simplefilter(’ignore’, DeprecationWarning)
We can obtain the list of available distribution through introspection:
>>> dist_continu = [d for d in dir(stats) if
...
isinstance(getattr(stats,d), stats.rv_continuous)]
>>> dist_discrete = [d for d in dir(stats) if
...
isinstance(getattr(stats,d), stats.rv_discrete)]
>>> print ’number of continuous distributions:’, len(dist_continu)
number of continuous distributions: 84
>>> print ’number of discrete distributions: ’, len(dist_discrete)
number of discrete distributions:
12
Distributions can be used in one of two ways, either by passing all distribution parameters to each method call or by
freezing the parameters for the instance of the distribution. As an example, we can get the median of the distribution
by using the percent point function, ppf, which is the inverse of the cdf:
>>> print stats.nct.ppf(0.5, 10, 2.5)
2.56880722561
>>> my_nct = stats.nct(10, 2.5)
>>> print my_nct.ppf(0.5)
2.56880722561
help(stats.nct) prints the complete docstring of the distribution. Instead we can print just some basic information:
>>> print stats.nct.extradoc #contains the distribution specific docs
Non-central Student T distribution
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df**(df/2) * gamma(df+1)
nct.pdf(x,df,nc) = -------------------------------------------------2**df*exp(nc**2/2)*(df+x**2)**(df/2) * gamma(df/2)
for df > 0, nc > 0.
>>> print
...
number of
>>> print
...
bounds of
’number of arguments: %d, shape parameters: %s’% (stats.nct.numargs,
stats.nct.shapes)
arguments: 2, shape parameters: df,nc
’bounds of distribution lower: %s, upper: %s’ % (stats.nct.a,
stats.nct.b)
distribution lower: -1.#INF, upper: 1.#INF
We can list all methods and properties of the distribution with dir(stats.nct). Some of the methods are private
methods, that are not named as such, i.e. no leading underscore, for example veccdf or xa and xb are for internal
calculation. The main methods we can see when we list the methods of the frozen distribution:
>>> print dir(my_nct) #reformatted
[’__class__’, ’__delattr__’, ’__dict__’, ’__doc__’, ’__getattribute__’,
’__hash__’, ’__init__’, ’__module__’, ’__new__’, ’__reduce__’, ’__reduce_ex__’,
’__repr__’, ’__setattr__’, ’__str__’, ’__weakref__’, ’args’, ’cdf’, ’dist’,
’entropy’, ’isf’, ’kwds’, ’moment’, ’pdf’, ’pmf’, ’ppf’, ’rvs’, ’sf’, ’stats’]
The main public methods are:
• rvs: Random Variates
• pdf: Probability Density Function
• cdf: Cumulative Distribution Function
• sf: Survival Function (1-CDF)
• ppf: Percent Point Function (Inverse of CDF)
• isf: Inverse Survival Function (Inverse of SF)
• stats: Return mean, variance, (Fisher’s) skew, or (Fisher’s) kurtosis
• moment: non-central moments of the distribution
The main additional methods of the not frozen distribution are related to the estimation of distrition parameters:
• fit: maximum likelihood estimation of distribution parameters, including location
and scale
• fit_loc_scale: estimation of location and scale when shape parameters are given
• nnlf: negative log likelihood function
• expect: Calculate the expectation of a function against the pdf or pmf
All continuous distributions take loc and scale as keyword parameters to adjust the location and scale of the distribution, e.g. for the standard normal distribution location is the mean and scale is the standard deviation. The standardized
distribution for a random variable x is obtained through (x - loc) / scale.
Discrete distribution have most of the same basic methods, however pdf is replaced the probability mass function pmf,
no estimation methods, such as fit, are available, and scale is not a valid keyword parameter. The location parameter,
keyword loc can be used to shift the distribution.
The basic methods, pdf, cdf, sf, ppf, and isf are vectorized with np.vectorize, and the usual numpy broadcasting is
applied. For example, we can calculate the critical values for the upper tail of the t distribution for different probabilites
and degrees of freedom.
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>>> stats.t.isf([0.1, 0.05, 0.01], [[10], [11]])
array([[ 1.37218364, 1.81246112, 2.76376946],
[ 1.36343032, 1.79588482, 2.71807918]])
Here, the first row are the critical values for 10 degrees of freedom and the second row is for 11 d.o.f., i.e. this is the
same as
>>> stats.t.isf([0.1, 0.05, 0.01], 10)
array([ 1.37218364, 1.81246112, 2.76376946])
>>> stats.t.isf([0.1, 0.05, 0.01], 11)
array([ 1.36343032, 1.79588482, 2.71807918])
If both, probabilities and degrees of freedom have the same array shape, then element wise matching is used. As an
example, we can obtain the 10% tail for 10 d.o.f., the 5% tail for 11 d.o.f. and the 1% tail for 12 d.o.f. by
>>> stats.t.isf([0.1, 0.05, 0.01], [10, 11, 12])
array([ 1.37218364, 1.79588482, 2.68099799])
Performance and Remaining Issues
The performance of the individual methods, in terms of speed, varies widely by distribution and method. The results of a method are obtained in one of two ways, either by explicit calculation or by a generic algorithm that is
independent of the specific distribution. Explicit calculation, requires that the method is directly specified for the
given distribution, either through analytic formulas or through special functions in scipy.special or numpy.random for
rvs. These are usually relatively fast calculations. The generic methods are used if the distribution does not specify any explicit calculation. To define a distribution, only one of pdf or cdf is necessary, all other methods can be
derived using numeric integration and root finding. These indirect methods can be very slow. As an example, rgh
= stats.gausshyper.rvs(0.5, 2, 2, 2, size=100) creates random variables in a very indirect way
and takes about 19 seconds for 100 random variables on my computer, while one million random variables from the
standard normal or from the t distribution take just above one second.
The distributions in scipy.stats have recently been corrected and improved and gained a considerable test suite, however
a few issues remain:
• skew and kurtosis, 3rd and 4th moments and entropy are not thoroughly tested and some coarse testing indicates
that there are still some incorrect results left.
• the distributions have been tested over some range of parameters, however in some corner ranges, a few incorrect
results may remain.
• the maximum likelihood estimation in fit does not work with default starting parameters for all distributions
and the user needs to supply good starting parameters. Also, for some distribution using a maximum likelihood
estimator might inherently not be the best choice.
The next example shows how to build our own discrete distribution, and more examples for the usage of the distributions are shown below together with the statistical tests.
Example: discrete distribution rv_discrete
In the following we use stats.rv_discrete to generate a discrete distribution that has the probabilites of the truncated
normal for the intervalls centered around the integers.
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>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
...
npoints = 20 # number of integer support points of the distribution minus 1
npointsh = npoints / 2
npointsf = float(npoints)
nbound = 4 # bounds for the truncated normal
normbound = (1+1/npointsf) * nbound # actual bounds of truncated normal
grid = np.arange(-npointsh, npointsh+2, 1) # integer grid
gridlimitsnorm = (grid-0.5) / npointsh * nbound # bin limits for the truncnorm
gridlimits = grid - 0.5
grid = grid[:-1]
probs = np.diff(stats.truncnorm.cdf(gridlimitsnorm, -normbound, normbound))
gridint = grid
normdiscrete = stats.rv_discrete(values = (gridint,
np.round(probs, decimals=7)), name=’normdiscrete’)
From the docstring of rv_discrete:
“You can construct an aribtrary discrete rv where P{X=xk} = pk by passing to the rv_discrete initialization
method (through the values= keyword) a tuple of sequences (xk, pk) which describes only those values of X
(xk) that occur with nonzero probability (pk).”
There are some requirements for this distribution to work. The keyword name is required. The support points of the
distribution xk have to be integers. Also, I needed to limit the number of decimals. If the last two requirements are not
satisfied an exception may be raised or the resulting numbers may be incorrect.
After defining the distribution, we obtain access to all methods of discrete distributions.
>>> print ’mean = %6.4f, variance = %6.4f, skew = %6.4f, kurtosis = %6.4f’% \
...
normdiscrete.stats(moments = ’mvsk’)
mean = -0.0000, variance = 6.3302, skew = 0.0000, kurtosis = -0.0076
>>> nd_std = np.sqrt(normdiscrete.stats(moments =
’v’))
Generate a random sample and compare observed frequencies with probabilities
>>> n_sample = 500
>>> np.random.seed(87655678) # fix the seed for replicability
>>> rvs = normdiscrete.rvs(size=n_sample)
>>> rvsnd = rvs
>>> f, l = np.histogram(rvs, bins=gridlimits)
>>> sfreq = np.vstack([gridint, f, probs*n_sample]).T
>>> print sfreq
[[ -1.00000000e+01
0.00000000e+00
2.95019349e-02]
[ -9.00000000e+00
0.00000000e+00
1.32294142e-01]
[ -8.00000000e+00
0.00000000e+00
5.06497902e-01]
[ -7.00000000e+00
2.00000000e+00
1.65568919e+00]
[ -6.00000000e+00
1.00000000e+00
4.62125309e+00]
[ -5.00000000e+00
9.00000000e+00
1.10137298e+01]
[ -4.00000000e+00
2.60000000e+01
2.24137683e+01]
[ -3.00000000e+00
3.70000000e+01
3.89503370e+01]
[ -2.00000000e+00
5.10000000e+01
5.78004747e+01]
[ -1.00000000e+00
7.10000000e+01
7.32455414e+01]
[ 0.00000000e+00
7.40000000e+01
7.92618251e+01]
[ 1.00000000e+00
8.90000000e+01
7.32455414e+01]
[ 2.00000000e+00
5.50000000e+01
5.78004747e+01]
[ 3.00000000e+00
5.00000000e+01
3.89503370e+01]
[ 4.00000000e+00
1.70000000e+01
2.24137683e+01]
[ 5.00000000e+00
1.10000000e+01
1.10137298e+01]
[ 6.00000000e+00
4.00000000e+00
4.62125309e+00]
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[
[
[
[
7.00000000e+00
8.00000000e+00
9.00000000e+00
1.00000000e+01
3.00000000e+00
0.00000000e+00
0.00000000e+00
0.00000000e+00
1.65568919e+00]
5.06497902e-01]
1.32294142e-01]
2.95019349e-02]]
Next, we can test, whether our sample was generated by our normdiscrete distribution. This also verifies, whether the
random numbers are generated correctly
The chisquare test requires that there are a minimum number of observations in each bin. We combine the tail bins
into larger bins so that they contain enough observations.
>>> f2 = np.hstack([f[:5].sum(), f[5:-5], f[-5:].sum()])
>>> p2 = np.hstack([probs[:5].sum(), probs[5:-5], probs[-5:].sum()])
>>> ch2, pval = stats.chisquare(f2, p2*n_sample)
>>> print ’chisquare for normdiscrete: chi2 = %6.3f pvalue = %6.4f’ % (ch2, pval)
chisquare for normdiscrete: chi2 = 12.466 pvalue = 0.4090
The pvalue in this case is high, so we can be quite confident that our random sample was actually generated by the
distribution.
1.10.3 Analysing One Sample
First, we create some random variables. We set a seed so that in each run we get identical results to look at. As an
example we take a sample from the Student t distribution:
>>> np.random.seed(282629734)
>>> x = stats.t.rvs(10, size=1000)
Here, we set the required shape parameter of the t distribution, which in statistics corresponds to the degrees of freedom, to 10. Using size=100 means that our sample consists of 1000 independently drawn (pseudo) random numbers.
Since we did not specify the keyword arguments loc and scale, those are set to their default values zero and one.
Descriptive Statistics
x is a numpy array, and we have direct access to all array methods, e.g.
>>> print x.max(), x.min() # equivalent to np.max(x), np.min(x)
5.26327732981 -3.78975572422
>>> print x.mean(), x.var() # equivalent to np.mean(x), np.var(x)
0.0140610663985 1.28899386208
How do the some sample properties compare to their theoretical counterparts?
>>> m, v, s, k = stats.t.stats(10, moments=’mvsk’)
>>> n, (smin, smax), sm, sv, ss, sk = stats.describe(x)
>>> print ’distribution:’,
distribution:
>>> sstr = ’mean = %6.4f, variance = %6.4f, skew = %6.4f, kurtosis = %6.4f’
>>> print sstr %(m, v, s ,k)
mean = 0.0000, variance = 1.2500, skew = 0.0000, kurtosis = 1.0000
>>> print ’sample:
’,
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sample:
>>> print sstr %(sm, sv, ss, sk)
mean = 0.0141, variance = 1.2903, skew = 0.2165, kurtosis = 1.0556
Note: stats.describe uses the unbiased estimator for the variance, while np.var is the biased estimator.
For our sample the sample statistics differ a by a small amount from their theoretical counterparts.
T-test and KS-test
We can use the t-test to test whether the mean of our sample differs in a statistcally significant way from the theoretical
expectation.
>>> print ’t-statistic = %6.3f pvalue = %6.4f’ %
t-statistic = 0.391 pvalue = 0.6955
stats.ttest_1samp(x, m)
The pvalue is 0.7, this means that with an alpha error of, for example, 10%, we cannot reject the hypothesis that the
sample mean is equal to zero, the expectation of the standard t-distribution.
As an exercise, we can calculate our ttest also directly without using the provided function, which should give us the
same answer, and so it does:
>>> tt = (sm-m)/np.sqrt(sv/float(n)) # t-statistic for mean
>>> pval = stats.t.sf(np.abs(tt), n-1)*2 # two-sided pvalue = Prob(abs(t)>tt)
>>> print ’t-statistic = %6.3f pvalue = %6.4f’ % (tt, pval)
t-statistic = 0.391 pvalue = 0.6955
The Kolmogorov-Smirnov test can be used to test the hypothesis that the sample comes from the standard t-distribution
>>> print ’KS-statistic D = %6.3f pvalue = %6.4f’ % stats.kstest(x, ’t’, (10,))
KS-statistic D = 0.016 pvalue = 0.9606
Again the p-value is high enough that we cannot reject the hypothesis that the random sample really is distributed
according to the t-distribution. In real applications, we don’t know what the underlying distribution is. If we perform
the Kolmogorov-Smirnov test of our sample against the standard normal distribution, then we also cannot reject the
hypothesis that our sample was generated by the normal distribution given that in this example the p-value is almost
40%.
>>> print ’KS-statistic D = %6.3f pvalue = %6.4f’ % stats.kstest(x,’norm’)
KS-statistic D = 0.028 pvalue = 0.3949
However, the standard normal distribution has a variance of 1, while our sample has a variance of 1.29. If we standardize our sample and test it against the normal distribution, then the p-value is again large enough that we cannot
reject the hypothesis that the sample came form the normal distribution.
>>> d, pval = stats.kstest((x-x.mean())/x.std(), ’norm’)
>>> print ’KS-statistic D = %6.3f pvalue = %6.4f’ % (d, pval)
KS-statistic D = 0.032 pvalue = 0.2402
Note: The Kolmogorov-Smirnov test assumes that we test against a distribution with given parameters, since in the
last case we estimated mean and variance, this assumption is violated, and the distribution of the test statistic on which
the p-value is based, is not correct.
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Tails of the distribution
Finally, we can check the upper tail of the distribution. We can use the percent point function ppf, which is the inverse
of the cdf function, to obtain the critical values, or, more directly, we can use the inverse of the survival function
>>> crit01, crit05, crit10 = stats.t.ppf([1-0.01, 1-0.05, 1-0.10], 10)
>>> print ’critical values from ppf at 1%%, 5%% and 10%% %8.4f %8.4f %8.4f’% (crit01, crit05, cr
critical values from ppf at 1%, 5% and 10%
2.7638
1.8125
1.3722
>>> print ’critical values from isf at 1%%, 5%% and 10%% %8.4f %8.4f %8.4f’% tuple(stats.t.isf([
critical values from isf at 1%, 5% and 10%
2.7638
1.8125
1.3722
>>> freq01 = np.sum(x>crit01) / float(n) *
>>> freq05 = np.sum(x>crit05) / float(n) *
>>> freq10 = np.sum(x>crit10) / float(n) *
>>> print ’sample %%-frequency at 1%%, 5%%
sample %-frequency at 1%, 5% and 10% tail
100
100
100
and 10%% tail %8.4f %8.4f %8.4f’% (freq01, freq05, fr
1.4000
5.8000 10.5000
In all three cases, our sample has more weight in the top tail than the underlying distribution. We can briefly check
a larger sample to see if we get a closer match. In this case the empirical frequency is quite close to the theoretical
probability, but if we repeat this several times the fluctuations are still pretty large.
>>> freq05l = np.sum(stats.t.rvs(10, size=10000) > crit05) / 10000.0 * 100
>>> print ’larger sample %%-frequency at 5%% tail %8.4f’% freq05l
larger sample %-frequency at 5% tail
4.8000
We can also compare it with the tail of the normal distribution, which has less weight in the tails:
>>> print ’tail prob. of normal at 1%%, 5%% and 10%% %8.4f %8.4f %8.4f’% \
...
tuple(stats.norm.sf([crit01, crit05, crit10])*100)
tail prob. of normal at 1%, 5% and 10%
0.2857
3.4957
8.5003
The chisquare test can be used to test, whether for a finite number of bins, the observed frequencies differ significantly
from the probabilites of the hypothesized distribution.
>>> quantiles = [0.0, 0.01, 0.05, 0.1, 1-0.10, 1-0.05, 1-0.01, 1.0]
>>> crit = stats.t.ppf(quantiles, 10)
>>> print crit
[
-Inf -2.76376946 -1.81246112 -1.37218364 1.37218364 1.81246112
2.76376946
Inf]
>>> n_sample = x.size
>>> freqcount = np.histogram(x, bins=crit)[0]
>>> tprob = np.diff(quantiles)
>>> nprob = np.diff(stats.norm.cdf(crit))
>>> tch, tpval = stats.chisquare(freqcount, tprob*n_sample)
>>> nch, npval = stats.chisquare(freqcount, nprob*n_sample)
>>> print ’chisquare for t:
chi2 = %6.3f pvalue = %6.4f’ % (tch, tpval)
chisquare for t:
chi2 = 2.300 pvalue = 0.8901
>>> print ’chisquare for normal: chi2 = %6.3f pvalue = %6.4f’ % (nch, npval)
chisquare for normal: chi2 = 64.605 pvalue = 0.0000
We see that the standard normal distribution is clearly rejected while the standard t-distribution cannot be rejected.
Since the variance of our sample differs from both standard distribution, we can again redo the test taking the estimate
for scale and location into account.
The fit method of the distributions can be used to estimate the parameters of the distribution, and the test is repeated
using probabilites of the estimated distribution.
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>>> tdof, tloc, tscale = stats.t.fit(x)
>>> nloc, nscale = stats.norm.fit(x)
>>> tprob = np.diff(stats.t.cdf(crit, tdof, loc=tloc, scale=tscale))
>>> nprob = np.diff(stats.norm.cdf(crit, loc=nloc, scale=nscale))
>>> tch, tpval = stats.chisquare(freqcount, tprob*n_sample)
>>> nch, npval = stats.chisquare(freqcount, nprob*n_sample)
>>> print ’chisquare for t:
chi2 = %6.3f pvalue = %6.4f’ % (tch, tpval)
chisquare for t:
chi2 = 1.577 pvalue = 0.9542
>>> print ’chisquare for normal: chi2 = %6.3f pvalue = %6.4f’ % (nch, npval)
chisquare for normal: chi2 = 11.084 pvalue = 0.0858
Taking account of the estimated parameters, we can still reject the hypothesis that our sample came from a normal
distribution (at the 5% level), but again, with a p-value of 0.95, we cannot reject the t distribution.
Special tests for normal distributions
Since the normal distribution is the most common distribution in statistics, there are several additional functions
available to test whether a sample could have been drawn from a normal distribution
First we can test if skew and kurtosis of our sample differ significantly from those of a normal distribution:
>>> print ’normal skewtest teststat = %6.3f pvalue = %6.4f’ % stats.skewtest(x)
normal skewtest teststat = 2.785 pvalue = 0.0054
>>> print ’normal kurtosistest teststat = %6.3f pvalue = %6.4f’ % stats.kurtosistest(x)
normal kurtosistest teststat = 4.757 pvalue = 0.0000
These two tests are combined in the normality test
>>> print ’normaltest teststat = %6.3f pvalue = %6.4f’ % stats.normaltest(x)
normaltest teststat = 30.379 pvalue = 0.0000
In all three tests the p-values are very low and we can reject the hypothesis that the our sample has skew and kurtosis
of the normal distribution.
Since skew and kurtosis of our sample are based on central moments, we get exactly the same results if we test the
standardized sample:
>>> print ’normaltest teststat = %6.3f pvalue = %6.4f’ % \
...
stats.normaltest((x-x.mean())/x.std())
normaltest teststat = 30.379 pvalue = 0.0000
Because normality is rejected so strongly, we can check whether the normaltest gives reasonable results for other
cases:
>>> print ’normaltest teststat = %6.3f pvalue = %6.4f’ % stats.normaltest(stats.t.rvs(10, size=100))
normaltest teststat = 4.698 pvalue = 0.0955
>>> print ’normaltest teststat = %6.3f pvalue = %6.4f’ % stats.normaltest(stats.norm.rvs(size=1000))
normaltest teststat = 0.613 pvalue = 0.7361
When testing for normality of a small sample of t-distributed observations and a large sample of normal distributed
observation, then in neither case can we reject the null hypothesis that the sample comes from a normal distribution.
In the first case this is because the test is not powerful enough to distinguish a t and a normally distributed random
variable in a small sample.
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1.10.4 Comparing two samples
In the following, we are given two samples, which can come either from the same or from different distribution, and
we want to test whether these samples have the same statistical properties.
Comparing means
Test with sample with identical means:
>>> rvs1 = stats.norm.rvs(loc=5, scale=10, size=500)
>>> rvs2 = stats.norm.rvs(loc=5, scale=10, size=500)
>>> stats.ttest_ind(rvs1, rvs2)
(-0.54890361750888583, 0.5831943748663857)
Test with sample with different means:
>>> rvs3 = stats.norm.rvs(loc=8, scale=10, size=500)
>>> stats.ttest_ind(rvs1, rvs3)
(-4.5334142901750321, 6.507128186505895e-006)
Kolmogorov-Smirnov test for two samples ks_2samp
For the example where both samples are drawn from the same distribution, we cannot reject the null hypothesis since
the pvalue is high
>>> stats.ks_2samp(rvs1, rvs2)
(0.025999999999999995, 0.99541195173064878)
In the second example, with different location, i.e. means, we can reject the null hypothesis since the pvalue is below
1%
>>> stats.ks_2samp(rvs1, rvs3)
(0.11399999999999999, 0.0027132103661283141)
1.11 Multi-dimensional image processing (ndimage)
1.11.1 Introduction
Image processing and analysis are generally seen as operations on two-dimensional arrays of values. There are however a number of fields where images of higher dimensionality must be analyzed. Good examples of these are medical
imaging and biological imaging. numpy is suited very well for this type of applications due its inherent multidimensional nature. The scipy.ndimage packages provides a number of general image processing and analysis
functions that are designed to operate with arrays of arbitrary dimensionality. The packages currently includes functions for linear and non-linear filtering, binary morphology, B-spline interpolation, and object measurements.
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1.11.2 Properties shared by all functions
All functions share some common properties. Notably, all functions allow the specification of an output array with the
output argument. With this argument you can specify an array that will be changed in-place with the result with the
operation. In this case the result is not returned. Usually, using the output argument is more efficient, since an existing
array is used to store the result.
The type of arrays returned is dependent on the type of operation, but it is in most cases equal to the type of the input.
If, however, the output argument is used, the type of the result is equal to the type of the specified output argument.
If no output argument is given, it is still possible to specify what the result of the output should be. This is done by
simply assigning the desired numpy type object to the output argument. For example:
>>> print correlate(arange(10), [1, 2.5])
[ 0 2 6 9 13 16 20 23 27 30]
>>> print correlate(arange(10), [1, 2.5], output = Float64)
[ 0.
2.5
6.
9.5 13.
16.5 20.
23.5 27.
30.5]
Note: In previous versions of scipy.ndimage, some functions accepted the output_type argument to achieve the
same effect. This argument is still supported, but its use will generate an deprecation warning. In a future version
all instances of this argument will be removed. The preferred way to specify an output type, is by using the output
argument, either by specifying an output array of the desired type, or by specifying the type of the output that is to be
returned.
1.11.3 Filter functions
The functions described in this section all perform some type of spatial filtering of the the input array: the elements
in the output are some function of the values in the neighborhood of the corresponding input element. We refer to
this neighborhood of elements as the filter kernel, which is often rectangular in shape but may also have an arbitrary
footprint. Many of the functions described below allow you to define the footprint of the kernel, by passing a mask
through the footprint parameter. For example a cross shaped kernel can be defined as follows:
>>>
>>>
[[0
[1
[0
footprint = array([[0,1,0],[1,1,1],[0,1,0]])
print footprint
1 0]
1 1]
1 0]]
Usually the origin of the kernel is at the center calculated by dividing the dimensions of the kernel shape by two.
For instance, the origin of a one-dimensional kernel of length three is at the second element. Take for example the
correlation of a one-dimensional array with a filter of length 3 consisting of ones:
>>> a = [0, 0, 0, 1, 0, 0, 0]
>>> correlate1d(a, [1, 1, 1])
[0 0 1 1 1 0 0]
Sometimes it is convenient to choose a different origin for the kernel. For this reason most functions support the origin
parameter which gives the origin of the filter relative to its center. For example:
>>> a = [0, 0, 0, 1, 0, 0, 0]
>>> print correlate1d(a, [1, 1, 1], origin = -1)
[0 1 1 1 0 0 0]
The effect is a shift of the result towards the left. This feature will not be needed very often, but it may be useful
especially for filters that have an even size. A good example is the calculation of backward and forward differences:
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>>> a = [0, 0, 1, 1, 1, 0, 0]
>>> print correlate1d(a, [-1, 1])
## backward difference
[ 0 0 1 0 0 -1 0]
>>> print correlate1d(a, [-1, 1], origin = -1) ## forward difference
[ 0 1 0 0 -1 0 0]
We could also have calculated the forward difference as follows:
>>> print correlate1d(a, [0, -1, 1])
[ 0 1 0 0 -1 0 0]
however, using the origin parameter instead of a larger kernel is more efficient. For multi-dimensional kernels origin
can be a number, in which case the origin is assumed to be equal along all axes, or a sequence giving the origin along
each axis.
Since the output elements are a function of elements in the neighborhood of the input elements, the borders of the
array need to be dealt with appropriately by providing the values outside the borders. This is done by assuming that
the arrays are extended beyond their boundaries according certain boundary conditions. In the functions described
below, the boundary conditions can be selected using the mode parameter which must be a string with the name of the
boundary condition. Following boundary conditions are currently supported:
“nearest”
“wrap”
“reflect”
“constant”
Use the value at the boundary
Periodically replicate the array
Reflect the array at the boundary
Use a constant value, default is 0.0
[1 2 3]->[1 1 2 3 3]
[1 2 3]->[3 1 2 3 1]
[1 2 3]->[1 1 2 3 3]
[1 2 3]->[0 1 2 3 0]
The “constant” mode is special since it needs an additional parameter to specify the constant value that should be used.
Note: The easiest way to implement such boundary conditions would be to copy the data to a larger array and extend
the data at the borders according to the boundary conditions. For large arrays and large filter kernels, this would be
very memory consuming, and the functions described below therefore use a different approach that does not require
allocating large temporary buffers.
Correlation and convolution
The correlate1d function calculates a one-dimensional correlation along the given axis. The lines of
the array along the given axis are correlated with the given weights. The weights parameter must be a
one-dimensional sequences of numbers.
The function correlate implements multi-dimensional correlation of the input array with a given
kernel.
The convolve1d function calculates a one-dimensional convolution along the given axis. The lines of
the array along the given axis are convoluted with the given weights. The weights parameter must be a
one-dimensional sequences of numbers.
Note: A convolution is essentially a correlation after mirroring the kernel. As a result, the origin parameter behaves differently than in the case of a correlation: the result is shifted in the opposite directions.
The function convolve implements multi-dimensional convolution of the input array with a given kernel.
Note: A convolution is essentially a correlation after mirroring the kernel. As a result, the origin parameter behaves differently than in the case of a correlation: the results is shifted in the opposite direction.
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Smoothing filters
The gaussian_filter1d function implements a one-dimensional Gaussian filter. The standarddeviation of the Gaussian filter is passed through the parameter sigma. Setting order = 0 corresponds
to convolution with a Gaussian kernel. An order of 1, 2, or 3 corresponds to convolution with the first,
second or third derivatives of a Gaussian. Higher order derivatives are not implemented.
The gaussian_filter function implements a multi-dimensional Gaussian filter. The standarddeviations of the Gaussian filter along each axis are passed through the parameter sigma as a sequence or
numbers. If sigma is not a sequence but a single number, the standard deviation of the filter is equal along
all directions. The order of the filter can be specified separately for each axis. An order of 0 corresponds
to convolution with a Gaussian kernel. An order of 1, 2, or 3 corresponds to convolution with the first,
second or third derivatives of a Gaussian. Higher order derivatives are not implemented. The order parameter must be a number, to specify the same order for all axes, or a sequence of numbers to specify a
different order for each axis.
Note: The multi-dimensional filter is implemented as a sequence of one-dimensional Gaussian filters.
The intermediate arrays are stored in the same data type as the output. Therefore, for output types with a
lower precision, the results may be imprecise because intermediate results may be stored with insufficient
precision. This can be prevented by specifying a more precise output type.
The uniform_filter1d function calculates a one-dimensional uniform filter of the given size along
the given axis.
The uniform_filter implements a multi-dimensional uniform filter. The sizes of the uniform filter
are given for each axis as a sequence of integers by the size parameter. If size is not a sequence, but a
single number, the sizes along all axis are assumed to be equal.
Note: The multi-dimensional filter is implemented as a sequence of one-dimensional uniform filters.
The intermediate arrays are stored in the same data type as the output. Therefore, for output types with a
lower precision, the results may be imprecise because intermediate results may be stored with insufficient
precision. This can be prevented by specifying a more precise output type.
Filters based on order statistics
The minimum_filter1d function calculates a one-dimensional minimum filter of given size along the
given axis.
The maximum_filter1d function calculates a one-dimensional maximum filter of given size along
the given axis.
The minimum_filter function calculates a multi-dimensional minimum filter. Either the sizes of a
rectangular kernel or the footprint of the kernel must be provided. The size parameter, if provided, must
be a sequence of sizes or a single number in which case the size of the filter is assumed to be equal along
each axis. The footprint, if provided, must be an array that defines the shape of the kernel by its non-zero
elements.
The maximum_filter function calculates a multi-dimensional maximum filter. Either the sizes of a
rectangular kernel or the footprint of the kernel must be provided. The size parameter, if provided, must
be a sequence of sizes or a single number in which case the size of the filter is assumed to be equal along
each axis. The footprint, if provided, must be an array that defines the shape of the kernel by its non-zero
elements.
The rank_filter function calculates a multi-dimensional rank filter. The rank may be less then zero,
i.e., rank = -1 indicates the largest element. Either the sizes of a rectangular kernel or the footprint of the
kernel must be provided. The size parameter, if provided, must be a sequence of sizes or a single number
in which case the size of the filter is assumed to be equal along each axis. The footprint, if provided, must
be an array that defines the shape of the kernel by its non-zero elements.
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The percentile_filter function calculates a multi-dimensional percentile filter. The percentile
may be less then zero, i.e., percentile = -20 equals percentile = 80. Either the sizes of a rectangular kernel
or the footprint of the kernel must be provided. The size parameter, if provided, must be a sequence of
sizes or a single number in which case the size of the filter is assumed to be equal along each axis. The
footprint, if provided, must be an array that defines the shape of the kernel by its non-zero elements.
The median_filter function calculates a multi-dimensional median filter. Either the sizes of a rectangular kernel or the footprint of the kernel must be provided. The size parameter, if provided, must be
a sequence of sizes or a single number in which case the size of the filter is assumed to be equal along
each axis. The footprint if provided, must be an array that defines the shape of the kernel by its non-zero
elements.
Derivatives
Derivative filters can be constructed in several ways. The function gaussian_filter1d described in Smoothing
filters can be used to calculate derivatives along a given axis using the order parameter. Other derivative filters are the
Prewitt and Sobel filters:
The prewitt function calculates a derivative along the given axis.
The sobel function calculates a derivative along the given axis.
The Laplace filter is calculated by the sum of the second derivatives along all axes. Thus, different Laplace filters
can be constructed using different second derivative functions. Therefore we provide a general function that takes a
function argument to calculate the second derivative along a given direction and to construct the Laplace filter:
The function generic_laplace calculates a laplace filter using the function passed through
derivative2 to calculate second derivatives. The function derivative2 should have the following
signature:
derivative2(input, axis, output, mode, cval, *extra_arguments, **extra_keywords)
It should calculate the second derivative along the dimension axis. If output is not None it should use that
for the output and return None, otherwise it should return the result. mode, cval have the usual meaning.
The extra_arguments and extra_keywords arguments can be used to pass a tuple of extra arguments and a
dictionary of named arguments that are passed to derivative2 at each call.
For example:
>>> def d2(input, axis, output, mode, cval):
...
return correlate1d(input, [1, -2, 1], axis, output, mode, cval, 0)
...
>>> a = zeros((5, 5))
>>> a[2, 2] = 1
>>> print generic_laplace(a, d2)
[[ 0 0 0 0 0]
[ 0 0 1 0 0]
[ 0 1 -4 1 0]
[ 0 0 1 0 0]
[ 0 0 0 0 0]]
To demonstrate the use of the extra_arguments argument we could do:
>>> def d2(input, axis, output, mode, cval, weights):
...
return correlate1d(input, weights, axis, output, mode, cval, 0,)
...
>>> a = zeros((5, 5))
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>>> a[2, 2] = 1
>>> print generic_laplace(a, d2, extra_arguments = ([1, -2, 1],))
[[ 0 0 0 0 0]
[ 0 0 1 0 0]
[ 0 1 -4 1 0]
[ 0 0 1 0 0]
[ 0 0 0 0 0]]
or:
>>> print generic_laplace(a, d2, extra_keywords = {’weights’: [1, -2, 1]})
[[ 0 0 0 0 0]
[ 0 0 1 0 0]
[ 0 1 -4 1 0]
[ 0 0 1 0 0]
[ 0 0 0 0 0]]
The following two functions are implemented using generic_laplace by providing appropriate functions for the
second derivative function:
The function laplace calculates the Laplace using discrete differentiation for the second derivative (i.e.
convolution with [1, -2, 1]).
The function gaussian_laplace calculates the Laplace using gaussian_filter to calculate the
second derivatives. The standard-deviations of the Gaussian filter along each axis are passed through the
parameter sigma as a sequence or numbers. If sigma is not a sequence but a single number, the standard
deviation of the filter is equal along all directions.
The gradient magnitude is defined as the square root of the sum of the squares of the gradients in all directions. Similar
to the generic Laplace function there is a generic_gradient_magnitude function that calculated the gradient
magnitude of an array:
The function generic_gradient_magnitude calculates a gradient magnitude using the function
passed through derivative to calculate first derivatives. The function derivative should have the
following signature:
derivative(input, axis, output, mode, cval, *extra_arguments, **extra_keywords)
It should calculate the derivative along the dimension axis. If output is not None it should use that for the
output and return None, otherwise it should return the result. mode, cval have the usual meaning.
The extra_arguments and extra_keywords arguments can be used to pass a tuple of extra arguments and a
dictionary of named arguments that are passed to derivative at each call.
For example, the sobel function fits the required signature:
>>>
>>>
>>>
[[0
[0
[0
[0
[0
a = zeros((5, 5))
a[2, 2] = 1
print generic_gradient_magnitude(a, sobel)
0 0 0 0]
1 2 1 0]
2 0 2 0]
1 2 1 0]
0 0 0 0]]
See the documentation of generic_laplace for examples of using the extra_arguments and extra_keywords arguments.
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The sobel and prewitt functions fit the required signature and can therefore directly be used with
generic_gradient_magnitude. The following function implements the gradient magnitude using Gaussian
derivatives:
The function gaussian_gradient_magnitude calculates the gradient magnitude using
gaussian_filter to calculate the first derivatives. The standard-deviations of the Gaussian filter
along each axis are passed through the parameter sigma as a sequence or numbers. If sigma is not a
sequence but a single number, the standard deviation of the filter is equal along all directions.
Generic filter functions
To implement filter functions, generic functions can be used that accept a callable object that implements the filtering
operation. The iteration over the input and output arrays is handled by these generic functions, along with such
details as the implementation of the boundary conditions. Only a callable object implementing a callback function
that does the actual filtering work must be provided. The callback function can also be written in C and passed using
a PyCObject (see Extending ndimage in C for more information).
The generic_filter1d function implements a generic one-dimensional filter function, where the
actual filtering operation must be supplied as a python function (or other callable object). The
generic_filter1d function iterates over the lines of an array and calls function at each line.
The arguments that are passed to function are one-dimensional arrays of the tFloat64 type. The
first contains the values of the current line. It is extended at the beginning end the end, according to
the filter_size and origin arguments. The second array should be modified in-place to provide the output
values of the line. For example consider a correlation along one dimension:
>>> a = arange(12, shape = (3,4))
>>> print correlate1d(a, [1, 2, 3])
[[ 3 8 14 17]
[27 32 38 41]
[51 56 62 65]]
The same operation can be implemented using generic_filter1d as follows:
>>> def fnc(iline, oline):
...
oline[...] = iline[:-2] + 2 * iline[1:-1] + 3 * iline[2:]
...
>>> print generic_filter1d(a, fnc, 3)
[[ 3 8 14 17]
[27 32 38 41]
[51 56 62 65]]
Here the origin of the kernel was (by default) assumed to be in the middle of the filter of length 3.
Therefore, each input line was extended by one value at the beginning and at the end, before the function
was called.
Optionally extra arguments can be defined and passed to the filter function. The extra_arguments and
extra_keywords arguments can be used to pass a tuple of extra arguments and/or a dictionary of named
arguments that are passed to derivative at each call. For example, we can pass the parameters of our filter
as an argument:
>>> def fnc(iline, oline, a, b):
...
oline[...] = iline[:-2] + a * iline[1:-1] + b * iline[2:]
...
>>> print generic_filter1d(a, fnc, 3, extra_arguments = (2, 3))
[[ 3 8 14 17]
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[27 32 38 41]
[51 56 62 65]]
or
>>> print generic_filter1d(a, fnc, 3, extra_keywords = {’a’:2, ’b’:3})
[[ 3 8 14 17]
[27 32 38 41]
[51 56 62 65]]
The generic_filter function implements a generic filter function, where the actual filtering operation must be supplied as a python function (or other callable object). The generic_filter function
iterates over the array and calls function at each element. The argument of function is a onedimensional array of the tFloat64 type, that contains the values around the current element that are
within the footprint of the filter. The function should return a single value that can be converted to a
double precision number. For example consider a correlation:
>>> a = arange(12, shape = (3,4))
>>> print correlate(a, [[1, 0], [0, 3]])
[[ 0 3 7 11]
[12 15 19 23]
[28 31 35 39]]
The same operation can be implemented using generic_filter as follows:
>>> def fnc(buffer):
...
return (buffer * array([1, 3])).sum()
...
>>> print generic_filter(a, fnc, footprint = [[1, 0], [0, 1]])
[[ 0 3 7 11]
[12 15 19 23]
[28 31 35 39]]
Here a kernel footprint was specified that contains only two elements. Therefore the filter function receives
a buffer of length equal to two, which was multiplied with the proper weights and the result summed.
When calling generic_filter, either the sizes of a rectangular kernel or the footprint of the kernel
must be provided. The size parameter, if provided, must be a sequence of sizes or a single number in
which case the size of the filter is assumed to be equal along each axis. The footprint, if provided, must
be an array that defines the shape of the kernel by its non-zero elements.
Optionally extra arguments can be defined and passed to the filter function. The extra_arguments and
extra_keywords arguments can be used to pass a tuple of extra arguments and/or a dictionary of named
arguments that are passed to derivative at each call. For example, we can pass the parameters of our filter
as an argument:
>>> def fnc(buffer, weights):
...
weights = asarray(weights)
...
return (buffer * weights).sum()
...
>>> print generic_filter(a, fnc, footprint = [[1, 0], [0, 1]], extra_arguments = ([1, 3],))
[[ 0 3 7 11]
[12 15 19 23]
[28 31 35 39]]
or
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>>> print generic_filter(a, fnc, footprint = [[1, 0], [0, 1]], extra_keywords= {’weights’: [1, 3
[[ 0 3 7 11]
[12 15 19 23]
[28 31 35 39]]
These functions iterate over the lines or elements starting at the last axis, i.e. the last index changes the fastest. This
order of iteration is guaranteed for the case that it is important to adapt the filter depending on spatial location. Here
is an example of using a class that implements the filter and keeps track of the current coordinates while iterating.
It performs the same filter operation as described above for generic_filter, but additionally prints the current
coordinates:
>>> a = arange(12, shape = (3,4))
>>>
>>> class fnc_class:
...
def __init__(self, shape):
...
# store the shape:
...
self.shape = shape
...
# initialize the coordinates:
...
self.coordinates = [0] * len(shape)
...
...
def filter(self, buffer):
...
result = (buffer * array([1, 3])).sum()
...
print self.coordinates
...
# calculate the next coordinates:
...
axes = range(len(self.shape))
...
axes.reverse()
...
for jj in axes:
...
if self.coordinates[jj] < self.shape[jj] - 1:
...
self.coordinates[jj] += 1
...
break
...
else:
...
self.coordinates[jj] = 0
...
return result
...
>>> fnc = fnc_class(shape = (3,4))
>>> print generic_filter(a, fnc.filter, footprint = [[1, 0], [0, 1]])
[0, 0]
[0, 1]
[0, 2]
[0, 3]
[1, 0]
[1, 1]
[1, 2]
[1, 3]
[2, 0]
[2, 1]
[2, 2]
[2, 3]
[[ 0 3 7 11]
[12 15 19 23]
[28 31 35 39]]
For the generic_filter1d function the same approach works, except that this function does not iterate over the
axis that is being filtered. The example for generic_filter1d then becomes this:
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>>> a = arange(12, shape = (3,4))
>>>
>>> class fnc1d_class:
...
def __init__(self, shape, axis = -1):
...
# store the filter axis:
...
self.axis = axis
...
# store the shape:
...
self.shape = shape
...
# initialize the coordinates:
...
self.coordinates = [0] * len(shape)
...
...
def filter(self, iline, oline):
...
oline[...] = iline[:-2] + 2 * iline[1:-1] + 3 * iline[2:]
...
print self.coordinates
...
# calculate the next coordinates:
...
axes = range(len(self.shape))
...
# skip the filter axis:
...
del axes[self.axis]
...
axes.reverse()
...
for jj in axes:
...
if self.coordinates[jj] < self.shape[jj] - 1:
...
self.coordinates[jj] += 1
...
break
...
else:
...
self.coordinates[jj] = 0
...
>>> fnc = fnc1d_class(shape = (3,4))
>>> print generic_filter1d(a, fnc.filter, 3)
[0, 0]
[1, 0]
[2, 0]
[[ 3 8 14 17]
[27 32 38 41]
[51 56 62 65]]
Fourier domain filters
The functions described in this section perform filtering operations in the Fourier domain. Thus, the input array
of such a function should be compatible with an inverse Fourier transform function, such as the functions from the
numpy.fft module. We therefore have to deal with arrays that may be the result of a real or a complex Fourier
transform. In the case of a real Fourier transform only half of the of the symmetric complex transform is stored.
Additionally, it needs to be known what the length of the axis was that was transformed by the real fft. The functions
described here provide a parameter n that in the case of a real transform must be equal to the length of the real
transform axis before transformation. If this parameter is less than zero, it is assumed that the input array was the
result of a complex Fourier transform. The parameter axis can be used to indicate along which axis the real transform
was executed.
The fourier_shift function multiplies the input array with the multi-dimensional Fourier transform
of a shift operation for the given shift. The shift parameter is a sequences of shifts for each dimension, or
a single value for all dimensions.
The fourier_gaussian function multiplies the input array with the multi-dimensional Fourier transform of a Gaussian filter with given standard-deviations sigma. The sigma parameter is a sequences of
values for each dimension, or a single value for all dimensions.
The fourier_uniform function multiplies the input array with the multi-dimensional Fourier transform of a uniform filter with given sizes size. The size parameter is a sequences of values for each
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dimension, or a single value for all dimensions.
The fourier_ellipsoid function multiplies the input array with the multi-dimensional Fourier
transform of a elliptically shaped filter with given sizes size. The size parameter is a sequences of values
for each dimension, or a single value for all dimensions. This function is only implemented for dimensions
1, 2, and 3.
1.11.4 Interpolation functions
This section describes various interpolation functions that are based on B-spline theory. A good introduction to Bsplines can be found in: M. Unser, “Splines: A Perfect Fit for Signal and Image Processing,” IEEE Signal Processing
Magazine, vol. 16, no. 6, pp. 22-38, November 1999.
Spline pre-filters
Interpolation using splines of an order larger than 1 requires a pre- filtering step. The interpolation functions described
in section Interpolation functions apply pre-filtering by calling spline_filter, but they can be instructed not to
do this by setting the prefilter keyword equal to False. This is useful if more than one interpolation operation is done
on the same array. In this case it is more efficient to do the pre-filtering only once and use a prefiltered array as the
input of the interpolation functions. The following two functions implement the pre-filtering:
The spline_filter1d function calculates a one-dimensional spline filter along the given axis. An
output array can optionally be provided. The order of the spline must be larger then 1 and less than 6.
The spline_filter function calculates a multi-dimensional spline filter.
Note: The multi-dimensional filter is implemented as a sequence of one-dimensional spline filters. The
intermediate arrays are stored in the same data type as the output. Therefore, if an output with a limited
precision is requested, the results may be imprecise because intermediate results may be stored with
insufficient precision. This can be prevented by specifying a output type of high precision.
Interpolation functions
Following functions all employ spline interpolation to effect some type of geometric transformation of the input array.
This requires a mapping of the output coordinates to the input coordinates, and therefore the possibility arises that input
values outside the boundaries are needed. This problem is solved in the same way as described in Filter functions for
the multi-dimensional filter functions. Therefore these functions all support a mode parameter that determines how the
boundaries are handled, and a cval parameter that gives a constant value in case that the ‘constant’ mode is used.
The geometric_transform function applies an arbitrary geometric transform to the input. The given
mapping function is called at each point in the output to find the corresponding coordinates in the input.
mapping must be a callable object that accepts a tuple of length equal to the output array rank and returns
the corresponding input coordinates as a tuple of length equal to the input array rank. The output shape
and output type can optionally be provided. If not given they are equal to the input shape and type.
For example:
>>> a = arange(12, shape=(4,3), type = Float64)
>>> def shift_func(output_coordinates):
...
return (output_coordinates[0] - 0.5, output_coordinates[1] - 0.5)
...
>>> print geometric_transform(a, shift_func)
[[ 0.
0.
0.
]
[ 0.
1.3625 2.7375]
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[ 0.
[ 0.
4.8125
8.2625
6.1875]
9.6375]]
Optionally extra arguments can be defined and passed to the filter function. The extra_arguments and
extra_keywords arguments can be used to pass a tuple of extra arguments and/or a dictionary of named
arguments that are passed to derivative at each call. For example, we can pass the shifts in our example as
arguments:
>>> def shift_func(output_coordinates, s0, s1):
...
return (output_coordinates[0] - s0, output_coordinates[1] - s1)
...
>>> print geometric_transform(a, shift_func, extra_arguments = (0.5, 0.5))
[[ 0.
0.
0.
]
[ 0.
1.3625 2.7375]
[ 0.
4.8125 6.1875]
[ 0.
8.2625 9.6375]]
or
>>> print geometric_transform(a, shift_func, extra_keywords = {’s0’: 0.5, ’s1’: 0.5})
[[ 0.
0.
0.
]
[ 0.
1.3625 2.7375]
[ 0.
4.8125 6.1875]
[ 0.
8.2625 9.6375]]
Note: The mapping function can also be written in C and passed using a PyCObject. See Extending
ndimage in C for more information.
The function map_coordinates applies an arbitrary coordinate transformation using the given array
of coordinates. The shape of the output is derived from that of the coordinate array by dropping the first
axis. The parameter coordinates is used to find for each point in the output the corresponding coordinates
in the input. The values of coordinates along the first axis are the coordinates in the input array at which
the output value is found. (See also the numarray coordinates function.) Since the coordinates may be
non- integer coordinates, the value of the input at these coordinates is determined by spline interpolation
of the requested order. Here is an example that interpolates a 2D array at (0.5, 0.5) and (1, 2):
>>> a = arange(12, shape=(4,3), type = numarray.Float64)
>>> print a
[[ 0.
1.
2.]
[ 3.
4.
5.]
[ 6.
7.
8.]
[ 9. 10. 11.]]
>>> print map_coordinates(a, [[0.5, 2], [0.5, 1]])
[ 1.3625 7.
]
The affine_transform function applies an affine transformation to the input array. The given transformation matrix and offset are used to find for each point in the output the corresponding coordinates
in the input. The value of the input at the calculated coordinates is determined by spline interpolation
of the requested order. The transformation matrix must be two-dimensional or can also be given as a
one-dimensional sequence or array. In the latter case, it is assumed that the matrix is diagonal. A more
efficient interpolation algorithm is then applied that exploits the separability of the problem. The output
shape and output type can optionally be provided. If not given they are equal to the input shape and type.
The shift function returns a shifted version of the input, using spline interpolation of the requested
order.
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The zoom function returns a rescaled version of the input, using spline interpolation of the requested
order.
The rotate function returns the input array rotated in the plane defined by the two axes given by the
parameter axes, using spline interpolation of the requested order. The angle must be given in degrees. If
reshape is true, then the size of the output array is adapted to contain the rotated input.
1.11.5 Morphology
Binary morphology
Binary morphology (need something to put here).
The generate_binary_structure functions generates a binary structuring element for use in binary morphology operations. The rank of the structure must be provided. The size of the structure that
is returned is equal to three in each direction. The value of each element is equal to one if the square of
the Euclidean distance from the element to the center is less or equal to connectivity. For instance, two
dimensional 4-connected and 8-connected structures are generated as follows:
>>>
[[0
[1
[0
>>>
[[1
[1
[1
print generate_binary_structure(2, 1)
1 0]
1 1]
1 0]]
print generate_binary_structure(2, 2)
1 1]
1 1]
1 1]]
Most binary morphology functions can be expressed in terms of the basic operations erosion and dilation:
The binary_erosion function implements binary erosion of arrays of arbitrary rank with the given
structuring element. The origin parameter controls the placement of the structuring element as described
in Filter functions. If no structuring element is provided, an element with connectivity equal to one is
generated using generate_binary_structure. The border_value parameter gives the value of
the array outside boundaries. The erosion is repeated iterations times. If iterations is less than one, the
erosion is repeated until the result does not change anymore. If a mask array is given, only those elements
with a true value at the corresponding mask element are modified at each iteration.
The binary_dilation function implements binary dilation of arrays of arbitrary rank with the given
structuring element. The origin parameter controls the placement of the structuring element as described
in Filter functions. If no structuring element is provided, an element with connectivity equal to one is
generated using generate_binary_structure. The border_value parameter gives the value of
the array outside boundaries. The dilation is repeated iterations times. If iterations is less than one, the
dilation is repeated until the result does not change anymore. If a mask array is given, only those elements
with a true value at the corresponding mask element are modified at each iteration.
Here is an example of using binary_dilation to find all elements that touch the border, by repeatedly
dilating an empty array from the border using the data array as the mask:
>>>
>>>
>>>
[[1
[1
[0
[0
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struct = array([[0, 1, 0], [1, 1, 1], [0, 1, 0]])
a = array([[1,0,0,0,0], [1,1,0,1,0], [0,0,1,1,0], [0,0,0,0,0]])
print a
0 0 0 0]
1 0 1 0]
0 1 1 0]
0 0 0 0]]
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>>>
[[1
[1
[0
[0
print
0 0 0
1 0 0
0 0 0
0 0 0
binary_dilation(zeros(a.shape), struct, -1, a, border_value=1)
0]
0]
0]
0]]
The binary_erosion and binary_dilation functions both have an iterations parameter which allows the
erosion or dilation to be repeated a number of times. Repeating an erosion or a dilation with a given structure n times
is equivalent to an erosion or a dilation with a structure that is n-1 times dilated with itself. A function is provided that
allows the calculation of a structure that is dilated a number of times with itself:
The iterate_structure function returns a structure by dilation of the input structure iteration - 1
times with itself. For instance:
>>>
>>>
[[0
[1
[0
>>>
[[0
[0
[1
[0
[0
struct = generate_binary_structure(2, 1)
print struct
1 0]
1 1]
1 0]]
print iterate_structure(struct, 2)
0 1 0 0]
1 1 1 0]
1 1 1 1]
1 1 1 0]
0 1 0 0]]
If the origin of the original structure is equal to 0, then it is also equal to 0 for the iterated structure. If not,
the origin must also be adapted if the equivalent of the iterations erosions or dilations must be achieved
with the iterated structure. The adapted origin is simply obtained by multiplying with the number of
iterations. For convenience the iterate_structure also returns the adapted origin if the origin
parameter is not None:
>>> print iterate_structure(struct, 2, -1)
(array([[0, 0, 1, 0, 0],
[0, 1, 1, 1, 0],
[1, 1, 1, 1, 1],
[0, 1, 1, 1, 0],
[0, 0, 1, 0, 0]], type=Bool), [-2, -2])
Other morphology operations can be defined in terms of erosion and d dilation. Following functions provide a few of
these operations for convenience:
The binary_opening function implements binary opening of arrays of arbitrary rank with the given
structuring element. Binary opening is equivalent to a binary erosion followed by a binary dilation with
the same structuring element. The origin parameter controls the placement of the structuring element as
described in Filter functions. If no structuring element is provided, an element with connectivity equal to
one is generated using generate_binary_structure. The iterations parameter gives the number
of erosions that is performed followed by the same number of dilations.
The binary_closing function implements binary closing of arrays of arbitrary rank with the given
structuring element. Binary closing is equivalent to a binary dilation followed by a binary erosion with
the same structuring element. The origin parameter controls the placement of the structuring element as
described in Filter functions. If no structuring element is provided, an element with connectivity equal to
one is generated using generate_binary_structure. The iterations parameter gives the number
of dilations that is performed followed by the same number of erosions.
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The binary_fill_holes function is used to close holes in objects in a binary image, where the
structure defines the connectivity of the holes. The origin parameter controls the placement of the structuring element as described in Filter functions. If no structuring element is provided, an element with
connectivity equal to one is generated using generate_binary_structure.
The binary_hit_or_miss function implements a binary hit-or-miss transform of arrays of arbitrary
rank with the given structuring elements. The hit-or-miss transform is calculated by erosion of the input
with the first structure, erosion of the logical not of the input with the second structure, followed by the
logical and of these two erosions. The origin parameters control the placement of the structuring elements
as described in Filter functions. If origin2 equals None it is set equal to the origin1 parameter. If the first
structuring element is not provided, a structuring element with connectivity equal to one is generated
using generate_binary_structure, if structure2 is not provided, it is set equal to the logical not
of structure1.
Grey-scale morphology
Grey-scale morphology operations are the equivalents of binary morphology operations that operate on arrays with
arbitrary values. Below we describe the grey-scale equivalents of erosion, dilation, opening and closing. These
operations are implemented in a similar fashion as the filters described in Filter functions, and we refer to this section
for the description of filter kernels and footprints, and the handling of array borders. The grey-scale morphology
operations optionally take a structure parameter that gives the values of the structuring element. If this parameter
is not given the structuring element is assumed to be flat with a value equal to zero. The shape of the structure
can optionally be defined by the footprint parameter. If this parameter is not given, the structure is assumed to be
rectangular, with sizes equal to the dimensions of the structure array, or by the size parameter if structure is not given.
The size parameter is only used if both structure and footprint are not given, in which case the structuring element
is assumed to be rectangular and flat with the dimensions given by size. The size parameter, if provided, must be a
sequence of sizes or a single number in which case the size of the filter is assumed to be equal along each axis. The
footprint parameter, if provided, must be an array that defines the shape of the kernel by its non-zero elements.
Similar to binary erosion and dilation there are operations for grey-scale erosion and dilation:
The grey_erosion function calculates a multi-dimensional grey- scale erosion.
The grey_dilation function calculates a multi-dimensional grey- scale dilation.
Grey-scale opening and closing operations can be defined similar to their binary counterparts:
The grey_opening function implements grey-scale opening of arrays of arbitrary rank. Grey-scale
opening is equivalent to a grey-scale erosion followed by a grey-scale dilation.
The grey_closing function implements grey-scale closing of arrays of arbitrary rank. Grey-scale
opening is equivalent to a grey-scale dilation followed by a grey-scale erosion.
The morphological_gradient function implements a grey-scale morphological gradient of arrays
of arbitrary rank. The grey-scale morphological gradient is equal to the difference of a grey-scale dilation
and a grey-scale erosion.
The morphological_laplace function implements a grey-scale morphological laplace of arrays of
arbitrary rank. The grey-scale morphological laplace is equal to the sum of a grey-scale dilation and a
grey-scale erosion minus twice the input.
The white_tophat function implements a white top-hat filter of arrays of arbitrary rank. The white
top-hat is equal to the difference of the input and a grey-scale opening.
The black_tophat function implements a black top-hat filter of arrays of arbitrary rank. The black
top-hat is equal to the difference of the a grey-scale closing and the input.
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1.11.6 Distance transforms
Distance transforms are used to calculate the minimum distance from each element of an object to the background.
The following functions implement distance transforms for three different distance metrics: Euclidean, City Block,
and Chessboard distances.
The function distance_transform_cdt uses a chamfer type algorithm to calculate the distance
transform of the input, by replacing each object element (defined by values larger than zero) with
the shortest distance to the background (all non-object elements). The structure determines the type
of chamfering that is done. If the structure is equal to ‘cityblock’ a structure is generated using
generate_binary_structure with a squared distance equal to 1. If the structure is equal to ‘chessboard’, a structure is generated using generate_binary_structure with a squared distance equal
to the rank of the array. These choices correspond to the common interpretations of the cityblock and the
chessboard distancemetrics in two dimensions.
In addition to the distance transform, the feature transform can be calculated. In this case the index of
the closest background element is returned along the first axis of the result. The return_distances, and
return_indices flags can be used to indicate if the distance transform, the feature transform, or both must
be returned.
The distances and indices arguments can be used to give optional output arrays that must be of the correct
size and type (both Int32).
The basics of the algorithm used to implement this function is described in: G. Borgefors, “Distance
transformations in arbitrary dimensions.”, Computer Vision, Graphics, and Image Processing, 27:321345, 1984.
The function distance_transform_edt calculates the exact euclidean distance transform of the
input, by replacing each object element (defined by values larger than zero) with the shortest euclidean
distance to the background (all non-object elements).
In addition to the distance transform, the feature transform can be calculated. In this case the index of
the closest background element is returned along the first axis of the result. The return_distances, and
return_indices flags can be used to indicate if the distance transform, the feature transform, or both must
be returned.
Optionally the sampling along each axis can be given by the sampling parameter which should be a
sequence of length equal to the input rank, or a single number in which the sampling is assumed to be
equal along all axes.
The distances and indices arguments can be used to give optional output arrays that must be of the correct
size and type (Float64 and Int32).
The algorithm used to implement this function is described in: C. R. Maurer, Jr., R. Qi, and V. Raghavan,
“A linear time algorithm for computing exact euclidean distance transforms of binary images in arbitrary
dimensions. IEEE Trans. PAMI 25, 265-270, 2003.
The function distance_transform_bf uses a brute-force algorithm to calculate the distance transform of the input, by replacing each object element (defined by values larger than zero) with the shortest
distance to the background (all non-object elements). The metric must be one of “euclidean”, “cityblock”,
or “chessboard”.
In addition to the distance transform, the feature transform can be calculated. In this case the index of
the closest background element is returned along the first axis of the result. The return_distances, and
return_indices flags can be used to indicate if the distance transform, the feature transform, or both must
be returned.
Optionally the sampling along each axis can be given by the sampling parameter which should be a
sequence of length equal to the input rank, or a single number in which the sampling is assumed to be
equal along all axes. This parameter is only used in the case of the euclidean distance transform.
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The distances and indices arguments can be used to give optional output arrays that must be of the correct
size and type (Float64 and Int32).
Note: This function uses a slow brute-force algorithm, the function distance_transform_cdt
can be used to more efficiently calculate cityblock and chessboard distance transforms. The function
distance_transform_edt can be used to more efficiently calculate the exact euclidean distance
transform.
1.11.7 Segmentation and labeling
Segmentation is the process of separating objects of interest from the background. The most simple approach is
probably intensity thresholding, which is easily done with numpy functions:
>>>
...
...
...
>>>
[[0
[0
[0
[0
a = array([[1,2,2,1,1,0],
[0,2,3,1,2,0],
[1,1,1,3,3,2],
[1,1,1,1,2,1]])
print where(a > 1, 1, 0)
1 1 0 0 0]
1 1 0 1 0]
0 0 1 1 1]
0 0 0 1 0]]
The result is a binary image, in which the individual objects still need to be identified and labeled. The function
label generates an array where each object is assigned a unique number:
The label function generates an array where the objects in the input are labeled with an integer index. It
returns a tuple consisting of the array of object labels and the number of objects found, unless the output
parameter is given, in which case only the number of objects is returned. The connectivity of the objects
is defined by a structuring element. For instance, in two dimensions using a four-connected structuring
element gives:
>>> a = array([[0,1,1,0,0,0],[0,1,1,0,1,0],[0,0,0,1,1,1],[0,0,0,0,1,0]])
>>> s = [[0, 1, 0], [1,1,1], [0,1,0]]
>>> print label(a, s)
(array([[0, 1, 1, 0, 0, 0],
[0, 1, 1, 0, 2, 0],
[0, 0, 0, 2, 2, 2],
[0, 0, 0, 0, 2, 0]]), 2)
These two objects are not connected because there is no way in which we can place the structuring
element such that it overlaps with both objects. However, an 8-connected structuring element results in
only a single object:
>>>
>>>
>>>
[[0
[0
[0
[0
a = array([[0,1,1,0,0,0],[0,1,1,0,1,0],[0,0,0,1,1,1],[0,0,0,0,1,0]])
s = [[1,1,1], [1,1,1], [1,1,1]]
print label(a, s)[0]
1 1 0 0 0]
1 1 0 1 0]
0 0 1 1 1]
0 0 0 1 0]]
If no structuring element is provided, one is generated by calling generate_binary_structure
(see Binary morphology) using a connectivity of one (which in 2D is the 4-connected structure of the first
example). The input can be of any type, any value not equal to zero is taken to be part of an object. This
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is useful if you need to ‘re-label’ an array of object indices, for instance after removing unwanted objects.
Just apply the label function again to the index array. For instance:
>>> l, n = label([1, 0, 1, 0, 1])
>>> print l
[1 0 2 0 3]
>>> l = where(l != 2, l, 0)
>>> print l
[1 0 0 0 3]
>>> print label(l)[0]
[1 0 0 0 2]
Note: The structuring element used by label is assumed to be symmetric.
There is a large number of other approaches for segmentation, for instance from an estimation of the borders of
the objects that can be obtained for instance by derivative filters. One such an approach is watershed segmentation.
The function watershed_ift generates an array where each object is assigned a unique label, from an array that
localizes the object borders, generated for instance by a gradient magnitude filter. It uses an array containing initial
markers for the objects:
The watershed_ift function applies a watershed from markers algorithm, using an Iterative Forest
Transform, as described in: P. Felkel, R. Wegenkittl, and M. Bruckschwaiger, “Implementation and Complexity of the Watershed-from-Markers Algorithm Computed as a Minimal Cost Forest.”, Eurographics
2001, pp. C:26-35.
The inputs of this function are the array to which the transform is applied, and an array of markers that
designate the objects by a unique label, where any non-zero value is a marker. For instance:
>>>
...
...
...
...
...
...
>>>
...
...
...
...
...
...
>>>
[[1
[1
[1
[1
[1
[1
[1
input = array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 1, 1, 1, 0],
[0, 1, 0, 0, 0, 1, 0],
[0, 1, 0, 0, 0, 1, 0],
[0, 1, 0, 0, 0, 1, 0],
[0, 1, 1, 1, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0]], numarray.UInt8)
markers = array([[1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 2, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]], numarray.Int8)
print watershed_ift(input, markers)
1 1 1 1 1 1]
1 2 2 2 1 1]
2 2 2 2 2 1]
2 2 2 2 2 1]
2 2 2 2 2 1]
1 2 2 2 1 1]
1 1 1 1 1 1]]
Here two markers were used to designate an object (marker = 2) and the background (marker = 1). The
order in which these are processed is arbitrary: moving the marker for the background to the lower right
corner of the array yields a different result:
>>> markers = array([[0, 0, 0, 0, 0, 0, 0],
...
[0, 0, 0, 0, 0, 0, 0],
...
[0, 0, 0, 0, 0, 0, 0],
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...
...
...
...
>>>
[[1
[1
[1
[1
[1
[1
[1
print
1 1 1
1 1 1
1 2 2
1 2 2
1 2 2
1 1 1
1 1 1
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
watershed_ift(input,
1 1 1]
1 1 1]
2 1 1]
2 1 1]
2 1 1]
1 1 1]
1 1 1]]
2, 0, 0,
0, 0, 0,
0, 0, 0,
0, 0, 0,
markers)
0],
0],
0],
1]], numarray.Int8)
The result is that the object (marker = 2) is smaller because the second marker was processed earlier. This
may not be the desired effect if the first marker was supposed to designate a background object. Therefore
watershed_ift treats markers with a negative value explicitly as background markers and processes
them after the normal markers. For instance, replacing the first marker by a negative marker gives a result
similar to the first example:
>>> markers = array([[0, 0, 0,
...
[0, 0, 0,
...
[0, 0, 0,
...
[0, 0, 0,
...
[0, 0, 0,
...
[0, 0, 0,
...
[0, 0, 0,
>>> print watershed_ift(input,
[[-1 -1 -1 -1 -1 -1 -1]
[-1 -1 2 2 2 -1 -1]
[-1 2 2 2 2 2 -1]
[-1 2 2 2 2 2 -1]
[-1 2 2 2 2 2 -1]
[-1 -1 2 2 2 -1 -1]
[-1 -1 -1 -1 -1 -1 -1]]
0, 0, 0,
0, 0, 0,
0, 0, 0,
2, 0, 0,
0, 0, 0,
0, 0, 0,
0, 0, 0,
markers)
0],
0],
0],
0],
0],
0],
-1]], numarray.Int8)
The connectivity of the objects is defined by a structuring element. If no structuring element is provided,
one is generated by calling generate_binary_structure (see Binary morphology) using a connectivity of one (which in 2D is a 4-connected structure.) For example, using an 8-connected structure
with the last example yields a different object:
>>> print watershed_ift(input, markers,
...
structure = [[1,1,1], [1,1,1], [1,1,1]])
[[-1 -1 -1 -1 -1 -1 -1]
[-1 2 2 2 2 2 -1]
[-1 2 2 2 2 2 -1]
[-1 2 2 2 2 2 -1]
[-1 2 2 2 2 2 -1]
[-1 2 2 2 2 2 -1]
[-1 -1 -1 -1 -1 -1 -1]]
Note: The implementation of watershed_ift limits the data types of the input to UInt8 and
UInt16.
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1.11.8 Object measurements
Given an array of labeled objects, the properties of the individual objects can be measured. The find_objects
function can be used to generate a list of slices that for each object, give the smallest sub-array that fully contains the
object:
The find_objects function finds all objects in a labeled array and returns a list of slices that correspond to the smallest regions in the array that contains the object. For instance:
>>>
>>>
>>>
>>>
[[1
[1
>>>
[[0
[1
[0
a = array([[0,1,1,0,0,0],[0,1,1,0,1,0],[0,0,0,1,1,1],[0,0,0,0,1,0]])
l, n = label(a)
f = find_objects(l)
print a[f[0]]
1]
1]]
print a[f[1]]
1 0]
1 1]
1 0]]
find_objects returns slices for all objects, unless the max_label parameter is larger then zero, in
which case only the first max_label objects are returned. If an index is missing in the label array, None is
return instead of a slice. For example:
>>> print find_objects([1, 0, 3, 4], max_label = 3)
[(slice(0, 1, None),), None, (slice(2, 3, None),)]
The list of slices generated by find_objects is useful to find the position and dimensions of the objects in the
array, but can also be used to perform measurements on the individual objects. Say we want to find the sum of the
intensities of an object in image:
>>>
>>>
>>>
>>>
image = arange(4*6,shape=(4,6))
mask = array([[0,1,1,0,0,0],[0,1,1,0,1,0],[0,0,0,1,1,1],[0,0,0,0,1,0]])
labels = label(mask)[0]
slices = find_objects(labels)
Then we can calculate the sum of the elements in the second object:
>>> print where(labels[slices[1]] == 2, image[slices[1]], 0).sum()
80
That is however not particularly efficient, and may also be more complicated for other types of measurements. Therefore a few measurements functions are defined that accept the array of object labels and the index of the object to be
measured. For instance calculating the sum of the intensities can be done by:
>>> print sum(image, labels, 2)
80.0
For large arrays and small objects it is more efficient to call the measurement functions after slicing the array:
>>> print sum(image[slices[1]], labels[slices[1]], 2)
80.0
Alternatively, we can do the measurements for a number of labels with a single function call, returning a list of results.
For instance, to measure the sum of the values of the background and the second object in our example we give a list
of labels:
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>>> print sum(image, labels, [0, 2])
[178.0, 80.0]
The measurement functions described below all support the index parameter to indicate which object(s) should be
measured. The default value of index is None. This indicates that all elements where the label is larger than zero
should be treated as a single object and measured. Thus, in this case the labels array is treated as a mask defined by
the elements that are larger than zero. If index is a number or a sequence of numbers it gives the labels of the objects
that are measured. If index is a sequence, a list of the results is returned. Functions that return more than one result,
return their result as a tuple if index is a single number, or as a tuple of lists, if index is a sequence.
The sum function calculates the sum of the elements of the object with label(s) given by index, using the
labels array for the object labels. If index is None, all elements with a non-zero label value are treated as
a single object. If label is None, all elements of input are used in the calculation.
The mean function calculates the mean of the elements of the object with label(s) given by index, using
the labels array for the object labels. If index is None, all elements with a non-zero label value are treated
as a single object. If label is None, all elements of input are used in the calculation.
The variance function calculates the variance of the elements of the object with label(s) given by index,
using the labels array for the object labels. If index is None, all elements with a non-zero label value are
treated as a single object. If label is None, all elements of input are used in the calculation.
The standard_deviation function calculates the standard deviation of the elements of the object
with label(s) given by index, using the labels array for the object labels. If index is None, all elements
with a non-zero label value are treated as a single object. If label is None, all elements of input are used
in the calculation.
The minimum function calculates the minimum of the elements of the object with label(s) given by index,
using the labels array for the object labels. If index is None, all elements with a non-zero label value are
treated as a single object. If label is None, all elements of input are used in the calculation.
The maximum function calculates the maximum of the elements of the object with label(s) given by index,
using the labels array for the object labels. If index is None, all elements with a non-zero label value are
treated as a single object. If label is None, all elements of input are used in the calculation.
The minimum_position function calculates the position of the minimum of the elements of the object
with label(s) given by index, using the labels array for the object labels. If index is None, all elements
with a non-zero label value are treated as a single object. If label is None, all elements of input are used
in the calculation.
The maximum_position function calculates the position of the maximum of the elements of the object
with label(s) given by index, using the labels array for the object labels. If index is None, all elements
with a non-zero label value are treated as a single object. If label is None, all elements of input are used
in the calculation.
The extrema function calculates the minimum, the maximum, and their positions, of the elements of
the object with label(s) given by index, using the labels array for the object labels. If index is None, all
elements with a non-zero label value are treated as a single object. If label is None, all elements of input
are used in the calculation. The result is a tuple giving the minimum, the maximum, the position of the
minimum and the postition of the maximum. The result is the same as a tuple formed by the results of the
functions minimum, maximum, minimum_position, and maximum_position that are described above.
The center_of_mass function calculates the center of mass of the of the object with label(s) given
by index, using the labels array for the object labels. If index is None, all elements with a non-zero label
value are treated as a single object. If label is None, all elements of input are used in the calculation.
The histogram function calculates a histogram of the of the object with label(s) given by index, using
the labels array for the object labels. If index is None, all elements with a non-zero label value are treated
as a single object. If label is None, all elements of input are used in the calculation. Histograms are
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defined by their minimum (min), maximum (max) and the number of bins (bins). They are returned as
one-dimensional arrays of type Int32.
1.11.9 Extending ndimage in C
A few functions in the scipy.ndimage take a call-back argument. This can be a python function, but also a
PyCObject containing a pointer to a C function. To use this feature, you must write your own C extension that
defines the function, and define a Python function that returns a PyCObject containing a pointer to this function.
An example of a function that supports this is geometric_transform (see Interpolation functions). You can pass
it a python callable object that defines a mapping from all output coordinates to corresponding coordinates in the input
array. This mapping function can also be a C function, which generally will be much more efficient, since the overhead
of calling a python function at each element is avoided.
For example to implement a simple shift function we define the following function:
static int
_shift_function(int *output_coordinates, double* input_coordinates,
int output_rank, int input_rank, void *callback_data)
{
int ii;
/* get the shift from the callback data pointer: */
double shift = *(double*)callback_data;
/* calculate the coordinates: */
for(ii = 0; ii < irank; ii++)
icoor[ii] = ocoor[ii] - shift;
/* return OK status: */
return 1;
}
This function is called at every element of the output array, passing the current coordinates in the output_coordinates
array. On return, the input_coordinates array must contain the coordinates at which the input is interpolated. The ranks
of the input and output array are passed through output_rank and input_rank. The value of the shift is passed through
the callback_data argument, which is a pointer to void. The function returns an error status, in this case always 1,
since no error can occur.
A pointer to this function and a pointer to the shift value must be passed to geometric_transform. Both are
passed by a single PyCObject which is created by the following python extension function:
static PyObject *
py_shift_function(PyObject *obj, PyObject *args)
{
double shift = 0.0;
if (!PyArg_ParseTuple(args, "d", &shift)) {
PyErr_SetString(PyExc_RuntimeError, "invalid parameters");
return NULL;
} else {
/* assign the shift to a dynamically allocated location: */
double *cdata = (double*)malloc(sizeof(double));
*cdata = shift;
/* wrap function and callback_data in a CObject: */
return PyCObject_FromVoidPtrAndDesc(_shift_function, cdata,
_destructor);
}
}
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The value of the shift is obtained and then assigned to a dynamically allocated memory location. Both this data pointer
and the function pointer are then wrapped in a PyCObject, which is returned. Additionally, a pointer to a destructor
function is given, that will free the memory we allocated for the shift value when the PyCObject is destroyed. This
destructor is very simple:
static void
_destructor(void* cobject, void *cdata)
{
if (cdata)
free(cdata);
}
To use these functions, an extension module is built:
static PyMethodDef methods[] = {
{"shift_function", (PyCFunction)py_shift_function, METH_VARARGS, ""},
{NULL, NULL, 0, NULL}
};
void
initexample(void)
{
Py_InitModule("example", methods);
}
This extension can then be used in Python, for example:
>>> import example
>>> array = arange(12, shape=(4,3), type = Float64)
>>> fnc = example.shift_function(0.5)
>>> print geometric_transform(array, fnc)
[[ 0.
0.
0.
]
[ 0.
1.3625 2.7375]
[ 0.
4.8125 6.1875]
[ 0.
8.2625 9.6375]]
C callback functions for use with ndimage functions must all be written according to this scheme. The next section
lists the ndimage functions that acccept a C callback function and gives the prototype of the callback function.
1.11.10 Functions that support C callback functions
The ndimage functions that support C callback functions are described here. Obviously, the prototype of the function that is provided to these functions must match exactly that what they expect. Therefore we give here the prototypes of the callback functions. All these callback functions accept a void callback_data pointer that must be
wrapped in a PyCObject using the Python PyCObject_FromVoidPtrAndDesc function, which can also accept a pointer to a destructor function to free any memory allocated for callback_data. If callback_data is not needed,
PyCObject_FromVoidPtr may be used instead. The callback functions must return an integer error status that is
equal to zero if something went wrong, or 1 otherwise. If an error occurs, you should normally set the python error
status with an informative message before returning, otherwise, a default error message is set by the calling function.
The function generic_filter (see Generic filter functions) accepts a callback function with the following prototype:
The calling function iterates over the elements of the input and output arrays, calling the callback function
at each element. The elements within the footprint of the filter at the current element are passed through
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the buffer parameter, and the number of elements within the footprint through filter_size. The calculated
valued should be returned in the return_value argument.
The function generic_filter1d (see Generic filter functions) accepts a callback function with the following
prototype:
The calling function iterates over the lines of the input and output arrays, calling the callback function
at each line. The current line is extended according to the border conditions set by the calling function,
and the result is copied into the array that is passed through the input_line array. The length of the input
line (after extension) is passed through input_length. The callback function should apply the 1D filter and
store the result in the array passed through output_line. The length of the output line is passed through
output_length.
The function geometric_transform (see Interpolation functions) expects a function with the following prototype:
The calling function iterates over the elements of the output array, calling the callback function at each
element. The coordinates of the current output element are passed through output_coordinates. The
callback function must return the coordinates at which the input must be interpolated in input_coordinates.
The rank of the input and output arrays are given by input_rank and output_rank respectively.
1.12 File IO (scipy.io)
See Also:
numpy-reference.routines.io (in numpy)
1.12.1 Matlab files
loadmat (file_name[, mdict, appendmat, **kwargs)
savemat (file_name, mdict[, appendmat, format, ...])
Load Matlab(tm) file
Save a dictionary of names and arrays into the MATLAB-style
.mat file.
Getting started:
>>> import scipy.io as sio
If you are using IPython, try tab completing on sio. You’ll find:
sio.loadmat
sio.savemat
These are the high-level functions you will most likely use. You’ll also find:
sio.matlab
This is the package from which loadmat and savemat are imported. Within sio.matlab, you will find the
mio module - containing the machinery that loadmat and savemat use. From time to time you may find yourself
re-using this machinery.
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How do I start?
You may have a .mat file that you want to read into Scipy. Or, you want to pass some variables from Scipy / Numpy
into Matlab.
To save us using a Matlab license, let’s start in Octave. Octave has Matlab-compatible save / load functions. Start
Octave (octave at the command line for me):
octave:1> a = 1:12
a =
1
2
3
4
5
6
7
8
9
10
11
12
octave:2> a = reshape(a, [1 3 4])
a =
ans(:,:,1) =
1
2
3
ans(:,:,2) =
4
5
6
ans(:,:,3) =
7
8
9
ans(:,:,4) =
10
11
12
octave:3> save -6 octave_a.mat a % Matlab 6 compatible
octave:4> ls octave_a.mat
octave_a.mat
Now, to Python:
>>> mat_contents = sio.loadmat(’octave_a.mat’)
>>> print mat_contents
{’a’: array([[[ 1.,
4.,
7., 10.],
[ 2.,
5.,
8., 11.],
[ 3.,
6.,
9., 12.]]]), ’__version__’: ’1.0’, ’__header__’: ’MATLAB 5.0 MAT-file, writte
>>> oct_a = mat_contents[’a’]
>>> print oct_a
[[[ 1.
4.
7. 10.]
[ 2.
5.
8. 11.]
[ 3.
6.
9. 12.]]]
>>> print oct_a.shape
(1, 3, 4)
Now let’s try the other way round:
>>> import numpy as np
>>> vect = np.arange(10)
>>> print vect.shape
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(10,)
>>> sio.savemat(’np_vector.mat’, {’vect’:vect})
/Users/mb312/usr/local/lib/python2.6/site-packages/scipy/io/matlab/mio.py:196: FutureWarning: Us
oned_as=oned_as)
Then back to Octave:
octave:5> load np_vector.mat
octave:6> vect
vect =
0
1
2
3
4
5
6
7
8
9
octave:7> size(vect)
ans =
10
1
Note the deprecation warning. The oned_as keyword determines the way in which one-dimensional vectors are
stored. In the future, this will default to row instead of column:
>>> sio.savemat(’np_vector.mat’, {’vect’:vect}, oned_as=’row’)
We can load this in Octave or Matlab:
octave:8> load np_vector.mat
octave:9> vect
vect =
0
1
2
3
4
5
6
7
8
9
octave:10> size(vect)
ans =
1
10
Matlab structs
Matlab structs are a little bit like Python dicts, except the field names must be strings. Any Matlab object can be a
value of a field. As for all objects in Matlab, structs are in fact arrays of structs, where a single struct is an array of
shape (1, 1).
octave:11> my_struct = struct(’field1’, 1, ’field2’, 2)
my_struct =
{
field1 = 1
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field2 =
2
}
octave:12> save -6 octave_struct.mat my_struct
We can load this in Python:
>>> mat_contents = sio.loadmat(’octave_struct.mat’)
>>> print mat_contents
{’my_struct’: array([[([[1.0]], [[2.0]])]],
dtype=[(’field1’, ’|O8’), (’field2’, ’|O8’)]), ’__version__’: ’1.0’, ’__header__’: ’MATLAB 5.0
>>> oct_struct = mat_contents[’my_struct’]
>>> print oct_struct.shape
(1, 1)
>>> val = oct_struct[0,0]
>>> print val
([[1.0]], [[2.0]])
>>> print val[’field1’]
[[ 1.]]
>>> print val[’field2’]
[[ 2.]]
>>> print val.dtype
[(’field1’, ’|O8’), (’field2’, ’|O8’)]
In this version of Scipy (0.8.0), Matlab structs come back as numpy structured arrays, with fields named for the struct
fields. You can see the field names in the dtype output above. Note also:
>>> val = oct_struct[0,0]
and:
octave:13> size(my_struct)
ans =
1
1
So, in Matlab, the struct array must be at least 2D, and we replicate that when we read into Scipy. If you want all
length 1 dimensions squeezed out, try this:
>>> mat_contents = sio.loadmat(’octave_struct.mat’, squeeze_me=True)
>>> oct_struct = mat_contents[’my_struct’]
>>> oct_struct.shape
()
Sometimes, it’s more convenient to load the matlab structs as python objects rather than numpy structured arrarys - it can make the access syntax in python a bit more similar to that in matlab. In order to do this, use the
struct_as_record=False parameter to loadmat.
>>> mat_contents = sio.loadmat(’octave_struct.mat’, struct_as_record=False)
>>> oct_struct = mat_contents[’my_struct’]
>>> oct_struct[0,0].field1
array([[ 1.]])
struct_as_record=False works nicely with squeeze_me:
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>>> mat_contents = sio.loadmat(’octave_struct.mat’, struct_as_record=False, squeeze_me=True)
>>> oct_struct = mat_contents[’my_struct’]
>>> oct_struct.shape # but no - it’s a scalar
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
AttributeError: ’mat_struct’ object has no attribute ’shape’
>>> print type(oct_struct)
<class ’scipy.io.matlab.mio5_params.mat_struct’>
>>> print oct_struct.field1
1.0
Saving struct arrays can be done in various ways. One simple method is to use dicts:
>>> a_dict = {’field1’: 0.5, ’field2’: ’a string’}
>>> sio.savemat(’saved_struct.mat’, {’a_dict’: a_dict})
loaded as:
octave:21> load saved_struct
octave:22> a_dict
a_dict =
{
field2 = a string
field1 = 0.50000
}
You can also save structs back again to Matlab (or Octave in our case) like this:
>>> dt = [(’f1’, ’f8’), (’f2’, ’S10’)]
>>> arr = np.zeros((2,), dtype=dt)
>>> print arr
[(0.0, ’’) (0.0, ’’)]
>>> arr[0][’f1’] = 0.5
>>> arr[0][’f2’] = ’python’
>>> arr[1][’f1’] = 99
>>> arr[1][’f2’] = ’not perl’
>>> sio.savemat(’np_struct_arr.mat’, {’arr’: arr})
Matlab cell arrays
Cell arrays in Matlab are rather like python lists, in the sense that the elements in the arrays can contain any type of
Matlab object. In fact they are most similar to numpy object arrays, and that is how we load them into numpy.
octave:14> my_cells = {1, [2, 3]}
my_cells =
{
[1,1] =
[1,2] =
2
1
3
}
octave:15> save -6 octave_cells.mat my_cells
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Back to Python:
>>> mat_contents = sio.loadmat(’octave_cells.mat’)
>>> oct_cells = mat_contents[’my_cells’]
>>> print oct_cells.dtype
object
>>> val = oct_cells[0,0]
>>> print val
[[ 1.]]
>>> print val.dtype
float64
Saving to a Matlab cell array just involves making a numpy object array:
>>> obj_arr = np.zeros((2,), dtype=np.object)
>>> obj_arr[0] = 1
>>> obj_arr[1] = ’a string’
>>> print obj_arr
[1 a string]
>>> sio.savemat(’np_cells.mat’, {’obj_arr’:obj_arr})
octave:16> load np_cells.mat
octave:17> obj_arr
obj_arr =
{
[1,1] = 1
[2,1] = a string
}
1.12.2 Matrix Market files
mminfo (source)
mmread (source)
mmwrite (target, a[, comment, field, ...])
Queries the contents of the Matrix Market file ‘filename’ to extract size and
storage information.
Reads the contents of a Matrix Market file ‘filename’ into a matrix.
Writes the sparse or dense matrix A to a Matrix Market formatted file.
1.12.3 Other
save_as_module ([file_name, data])
Save the dictionary “data” into a module and shelf named save.
1.12.4 Wav sound files (scipy.io.wavfile)
read (file)
write (filename, rate, data)
Return the sample rate (in samples/sec) and data from a WAV file
Write a numpy array as a WAV file
1.12.5 Arff files (scipy.io.arff)
Module to read arff files (weka format).
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arff is a simple file format which support numerical, string and data values. It supports sparse data too.
See http://weka.sourceforge.net/wekadoc/index.php/en:ARFF_(3.4.6) for more details about arff format and available
datasets.
loadarff (filename)
Read an arff file.
1.12.6 Netcdf (scipy.io.netcdf)
netcdf_file
A netcdf_file object has two standard attributes: dimensions and variables. The values
of both are dictionaries, mapping dimension names to their associated lengths and variable names to
variables, respectively. Application programs should never modify these dictionaries.
Allows reading of NetCDF files (version of pupynere package)
1.13 Weave
1.13.1 Outline
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Contents
• Weave
– Outline
– Introduction
– Requirements
– Installation
– Testing
* Testing Notes:
– Benchmarks
– Inline
* More with printf
* More examples
· Binary search
· Dictionary Sort
· NumPy – cast/copy/transpose
· wxPython
* Keyword Option
* Inline Arguments
* Distutils keywords
· Keyword Option Examples
· Returning Values
· The issue with locals()
· A quick look at the code
* Technical Details
* Passing Variables in/out of the C/C++ code
* Type Conversions
· NumPy Argument Conversion
· String, List, Tuple, and Dictionary Conversion
· File Conversion
· Callable, Instance, and Module Conversion
· Customizing Conversions
* The Catalog
· Function Storage
· Catalog search paths and the PYTHONCOMPILED variable
– Blitz
* Requirements
* Limitations
* NumPy efficiency issues: What compilation buys you
* The Tools
· Parser
· Blitz and NumPy
* Type definitions and coersion
* Cataloging Compiled Functions
* Checking Array Sizes
* Creating the Extension Module
– Extension Modules
* A Simple Example
* Fibonacci Example
– Customizing Type Conversions – Type Factories
– Things I wish weave did
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1.13.2 Introduction
The scipy.weave (below just weave) package provides tools for including C/C++ code within in Python code.
This offers both another level of optimization to those who need it, and an easy way to modify and extend any supported
extension libraries such as wxPython and hopefully VTK soon. Inlining C/C++ code within Python generally results
in speed ups of 1.5x to 30x speed-up over algorithms written in pure Python (However, it is also possible to slow things
down...). Generally algorithms that require a large number of calls to the Python API don’t benefit as much from the
conversion to C/C++ as algorithms that have inner loops completely convertable to C.
There are three basic ways to use weave. The weave.inline() function executes C code directly within Python,
and weave.blitz() translates Python NumPy expressions to C++ for fast execution. blitz() was the original
reason weave was built. For those interested in building extension libraries, the ext_tools module provides classes
for building extension modules within Python.
Most of weave’s functionality should work on Windows and Unix, although some of its functionality requires gcc
or a similarly modern C++ compiler that handles templates well. Up to now, most testing has been done on Windows
2000 with Microsoft’s C++ compiler (MSVC) and with gcc (mingw32 2.95.2 and 2.95.3-6). All tests also pass on
Linux (RH 7.1 with gcc 2.96), and I’ve had reports that it works on Debian also (thanks Pearu).
The inline and blitz provide new functionality to Python (although I’ve recently learned about the PyInline
project which may offer similar functionality to inline). On the other hand, tools for building Python extension
modules already exists (SWIG, SIP, pycpp, CXX, and others). As of yet, I’m not sure where weave fits in this
spectrum. It is closest in flavor to CXX in that it makes creating new C/C++ extension modules pretty easy. However,
if you’re wrapping a gaggle of legacy functions or classes, SWIG and friends are definitely the better choice. weave
is set up so that you can customize how Python types are converted to C types in weave. This is great for inline(),
but, for wrapping legacy code, it is more flexible to specify things the other way around – that is how C types map to
Python types. This weave does not do. I guess it would be possible to build such a tool on top of weave, but with
good tools like SWIG around, I’m not sure the effort produces any new capabilities. Things like function overloading
are probably easily implemented in weave and it might be easier to mix Python/C code in function calls, but nothing
beyond this comes to mind. So, if you’re developing new extension modules or optimizing Python functions in C,
weave.ext_tools() might be the tool for you. If you’re wrapping legacy code, stick with SWIG.
The next several sections give the basics of how to use weave. We’ll discuss what’s happening under the covers in
more detail later on. Serious users will need to at least look at the type conversion section to understand how Python
variables map to C/C++ types and how to customize this behavior. One other note. If you don’t know C or C++ then
these docs are probably of very little help to you. Further, it’d be helpful if you know something about writing Python
extensions. weave does quite a bit for you, but for anything complex, you’ll need to do some conversions, reference
counting, etc.
Note: weave is actually part of the SciPy package. However, it also works fine as a standalone package (you can
check out the sources using svn co http://svn.scipy.org/svn/scipy/trunk/Lib/weave weave
and install as python setup.py install). The examples here are given as if it is used as a stand alone package. If you are
using from within scipy, you can use ‘‘ from scipy import weave‘‘ and the examples will work identically.
1.13.3 Requirements
• Python
I use 2.1.1. Probably 2.0 or higher should work.
• C++ compiler
weave uses distutils to actually build extension modules, so it uses whatever compiler was originally
used to build Python. weave itself requires a C++ compiler. If you used a C++ compiler to build Python, your
probably fine.
On Unix gcc is the preferred choice because I’ve done a little testing with it. All testing has been done with gcc,
but I expect the majority of compilers should work for inline and ext_tools. The one issue I’m not sure
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about is that I’ve hard coded things so that compilations are linked with the stdc++ library. Is this standard
across Unix compilers, or is this a gcc-ism?
For blitz(), you’ll need a reasonably recent version of gcc. 2.95.2 works on windows and 2.96 looks fine on
Linux. Other versions are likely to work. Its likely that KAI’s C++ compiler and maybe some others will work,
but I haven’t tried. My advice is to use gcc for now unless your willing to tinker with the code some.
On Windows, either MSVC or gcc (mingw32) should work. Again, you’ll need gcc for blitz() as the MSVC
compiler doesn’t handle templates well.
I have not tried Cygwin, so please report success if it works for you.
• NumPy
The python NumPy module is required for blitz() to work and for numpy.distutils which is used by weave.
1.13.4 Installation
There are currently two ways to get weave. First, weave is part of SciPy and installed automatically (as a subpackage) whenever SciPy is installed. Second, since weave is useful outside of the scientific community, it has been
setup so that it can be used as a stand-alone module.
The stand-alone version can be downloaded from here. Instructions for installing should be found there as well.
setup.py file to simplify installation.
1.13.5 Testing
Once weave is installed, fire up python and run its unit tests.
>>> import weave
>>> weave.test()
runs long time... spews tons of output and a few warnings
.
.
.
..............................................................
................................................................
..................................................
---------------------------------------------------------------------Ran 184 tests in 158.418s
OK
>>>
This takes a while, usually several minutes. On Unix with remote file systems, I’ve had it take 15 or so minutes. In the
end, it should run about 180 tests and spew some speed results along the way. If you get errors, they’ll be reported at
the end of the output. Please report errors that you find. Some tests are known to fail at this point.
If you only want to test a single module of the package, you can do this by running test() for that specific module.
>>> import weave.scalar_spec
>>> weave.scalar_spec.test()
.......
---------------------------------------------------------------------Ran 7 tests in 23.284s
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Testing Notes:
• Windows 1
I’ve had some test fail on windows machines where I have msvc, gcc-2.95.2 (in c:gcc-2.95.2), and gcc-2.95.3-6
(in c:gcc) all installed. My environment has c:gcc in the path and does not have c:gcc-2.95.2 in the path. The test
process runs very smoothly until the end where several test using gcc fail with cpp0 not found by g++. If I check
os.system(‘gcc -v’) before running tests, I get gcc-2.95.3-6. If I check after running tests (and after failure), I
get gcc-2.95.2. ??huh??. The os.environ[’PATH’] still has c:gcc first in it and is not corrupted (msvc/distutils
messes with the environment variables, so we have to undo its work in some places). If anyone else sees this, let
me know - - it may just be an quirk on my machine (unlikely). Testing with the gcc- 2.95.2 installation always
works.
• Windows 2
If you run the tests from PythonWin or some other GUI tool, you’ll get a ton of DOS windows popping up
periodically as weave spawns the compiler multiple times. Very annoying. Anyone know how to fix this?
• wxPython
wxPython tests are not enabled by default because importing wxPython on a Unix machine without access to a
X-term will cause the program to exit. Anyone know of a safe way to detect whether wxPython can be imported
and whether a display exists on a machine?
1.13.6 Benchmarks
This section has not been updated from old scipy weave and Numeric....
This section has a few benchmarks – thats all people want to see anyway right? These are mostly taken from running
files in the weave/example directory and also from the test scripts. Without more information about what the test
actually do, their value is limited. Still, their here for the curious. Look at the example scripts for more specifics about
what problem was actually solved by each run. These examples are run under windows 2000 using Microsoft Visual
C++ and python2.1 on a 850 MHz PIII laptop with 320 MB of RAM. Speed up is the improvement (degredation)
factor of weave compared to conventional Python functions. The blitz() comparisons are shown compared to
NumPy.
Table 1.1: inline and ext_tools
Algorithm
binary search
fibonacci (recursive)
fibonacci (loop)
return None
map
dictionary sort
vector quantization
Speed up
1.50
82.10
9.17
0.14
1.20
2.54
37.40
Table 1.2: blitz – double precision
Algorithm
a = b + c 512x512
a = b + c + d 512x512
5 pt avg. filter, 2D Image 512x512
Electromagnetics (FDTD) 100x100x100
Speed up
3.05
4.59
9.01
8.61
The benchmarks shown blitz in the best possible light. NumPy (at least on my machine) is significantly worse for
double precision than it is for single precision calculations. If your interested in single precision results, you can pretty
much divide the double precision speed up by 3 and you’ll be close.
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1.13.7 Inline
inline() compiles and executes C/C++ code on the fly. Variables in the local and global Python scope are also
available in the C/C++ code. Values are passed to the C/C++ code by assignment much like variables are passed into
a standard Python function. Values are returned from the C/C++ code through a special argument called return_val.
Also, the contents of mutable objects can be changed within the C/C++ code and the changes remain after the C code
exits and returns to Python. (more on this later)
Here’s a trivial printf example using inline():
>>> import weave
>>> a = 1
>>> weave.inline(’printf("%d\\n",a);’,[’a’])
1
In this, its most basic form, inline(c_code, var_list) requires two arguments. c_code is a string of valid
C/C++ code. var_list is a list of variable names that are passed from Python into C/C++. Here we have a simple
printf statement that writes the Python variable a to the screen. The first time you run this, there will be a pause
while the code is written to a .cpp file, compiled into an extension module, loaded into Python, cataloged for future
use, and executed. On windows (850 MHz PIII), this takes about 1.5 seconds when using Microsoft’s C++ compiler
(MSVC) and 6-12 seconds using gcc (mingw32 2.95.2). All subsequent executions of the code will happen very
quickly because the code only needs to be compiled once. If you kill and restart the interpreter and then execute the
same code fragment again, there will be a much shorter delay in the fractions of seconds range. This is because weave
stores a catalog of all previously compiled functions in an on disk cache. When it sees a string that has been compiled,
it loads the already compiled module and executes the appropriate function.
Note: If you try the printf example in a GUI shell such as IDLE, PythonWin, PyShell, etc., you’re unlikely to
see the output. This is because the C code is writing to stdout, instead of to the GUI window. This doesn’t mean that
inline doesn’t work in these environments – it only means that standard out in C is not the same as the standard out for
Python in these cases. Non input/output functions will work as expected.
Although effort has been made to reduce the overhead associated with calling inline, it is still less efficient for simple
code snippets than using equivalent Python code. The simple printf example is actually slower by 30% or so
than using Python print statement. And, it is not difficult to create code fragments that are 8-10 times slower
using inline than equivalent Python. However, for more complicated algorithms, the speed up can be worth while
– anywhwere from 1.5- 30 times faster. Algorithms that have to manipulate Python objects (sorting a list) usually
only see a factor of 2 or so improvement. Algorithms that are highly computational or manipulate NumPy arrays can
see much larger improvements. The examples/vq.py file shows a factor of 30 or more improvement on the vector
quantization algorithm that is used heavily in information theory and classification problems.
More with printf
MSVC users will actually see a bit of compiler output that distutils does not supress the first time the code executes:
>>> weave.inline(r’printf("%d\n",a);’,[’a’])
sc_e013937dbc8c647ac62438874e5795131.cpp
Creating library C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp
\Release\sc_e013937dbc8c647ac62438874e5795131.lib and
object C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\sc_e013937dbc8c647ac62438874e
1
Nothing bad is happening, its just a bit annoying. * Anyone know how to turn this off?*
This example also demonstrates using ‘raw strings’. The r preceeding the code string in the last example denotes
that this is a ‘raw string’. In raw strings, the backslash character is not interpreted as an escape character, and so it
isn’t necessary to use a double backslash to indicate that the ‘n’ is meant to be interpreted in the C printf statement
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instead of by Python. If your C code contains a lot of strings and control characters, raw strings might make things
easier. Most of the time, however, standard strings work just as well.
The printf statement in these examples is formatted to print out integers. What happens if a is a string? inline
will happily, compile a new version of the code to accept strings as input, and execute the code. The result?
>>> a = ’string’
>>> weave.inline(r’printf("%d\n",a);’,[’a’])
32956972
In this case, the result is non-sensical, but also non-fatal. In other situations, it might produce a compile time error
because a is required to be an integer at some point in the code, or it could produce a segmentation fault. Its possible
to protect against passing inline arguments of the wrong data type by using asserts in Python.
>>> a = ’string’
>>> def protected_printf(a):
...
assert(type(a) == type(1))
...
weave.inline(r’printf("%d\n",a);’,[’a’])
>>> protected_printf(1)
1
>>> protected_printf(’string’)
AssertError...
For printing strings, the format statement needs to be changed. Also, weave doesn’t convert strings to char*. Instead
it uses CXX Py::String type, so you have to do a little more work. Here we convert it to a C++ std::string and then ask
cor the char* version.
>>> a = ’string’
>>> weave.inline(r’printf("%s\n",std::string(a).c_str());’,[’a’])
string
XXX
This is a little convoluted. Perhaps strings should convert to std::string objects instead of CXX objects. Or
maybe to char*.
As in this case, C/C++ code fragments often have to change to accept different types. For the given printing task,
however, C++ streams provide a way of a single statement that works for integers and strings. By default, the stream
objects live in the std (standard) namespace and thus require the use of std::.
>>> weave.inline(’std::cout << a << std::endl;’,[’a’])
1
>>> a = ’string’
>>> weave.inline(’std::cout << a << std::endl;’,[’a’])
string
Examples using printf and cout are included in examples/print_example.py.
More examples
This section shows several more advanced uses of inline. It includes a few algorithms from the Python Cookbook
that have been re-written in inline C to improve speed as well as a couple examples using NumPy and wxPython.
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Binary search
Lets look at the example of searching a sorted list of integers for a value. For inspiration, we’ll use Kalle Svensson’s
binary_search() algorithm from the Python Cookbook. His recipe follows:
def binary_search(seq, t):
min = 0; max = len(seq) - 1
while 1:
if max < min:
return -1
m = (min + max) / 2
if seq[m] < t:
min = m + 1
elif seq[m] > t:
max = m - 1
else:
return m
This Python version works for arbitrary Python data types. The C version below is specialized to handle integer values.
There is a little type checking done in Python to assure that we’re working with the correct data types before heading
into C. The variables seq and t don’t need to be declared beacuse weave handles converting and declaring them in
the C code. All other temporary variables such as min, max, etc. must be declared – it is C after all. Here’s the new
mixed Python/C function:
def c_int_binary_search(seq,t):
# do a little type checking in Python
assert(type(t) == type(1))
assert(type(seq) == type([]))
# now the C code
code = """
#line 29 "binary_search.py"
int val, m, min = 0;
int max = seq.length() - 1;
PyObject *py_val;
for(;;)
{
if (max < min )
{
return_val = Py::new_reference_to(Py::Int(-1));
break;
}
m = (min + max) /2;
val = py_to_int(PyList_GetItem(seq.ptr(),m),"val");
if (val < t)
min = m + 1;
else if (val > t)
max = m - 1;
else
{
return_val = Py::new_reference_to(Py::Int(m));
break;
}
}
"""
return inline(code,[’seq’,’t’])
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We have two variables seq and t passed in. t is guaranteed (by the assert) to be an integer. Python integers are
converted to C int types in the transition from Python to C. seq is a Python list. By default, it is translated to a CXX
list object. Full documentation for the CXX library can be found at its website. The basics are that the CXX provides
C++ class equivalents for Python objects that simplify, or at least object orientify, working with Python objects in
C/C++. For example, seq.length() returns the length of the list. A little more about CXX and its class methods,
etc. is in the ** type conversions ** section.
Note: CXX uses templates and therefore may be a little less portable than another alternative by Gordan McMillan
called SCXX which was inspired by CXX. It doesn’t use templates so it should compile faster and be more portable.
SCXX has a few less features, but it appears to me that it would mesh with the needs of weave quite well. Hopefully
xxx_spec files will be written for SCXX in the future, and we’ll be able to compare on a more empirical basis. Both
sets of spec files will probably stick around, it just a question of which becomes the default.
Most of the algorithm above looks similar in C to the original Python code. There are two main differences. The first is
the setting of return_val instead of directly returning from the C code with a return statement. return_val
is an automatically defined variable of type PyObject* that is returned from the C code back to Python. You’ll
have to handle reference counting issues when setting this variable. In this example, CXX classes and functions
handle the dirty work. All CXX functions and classes live in the namespace Py::. The following code converts the integer m to a CXX Int() object and then to a PyObject* with an incremented reference count using
Py::new_reference_to().
return_val = Py::new_reference_to(Py::Int(m));
The second big differences shows up in the retrieval of integer values from the Python list. The simple Python seq[i]
call balloons into a C Python API call to grab the value out of the list and then a separate call to py_to_int() that
converts the PyObject* to an integer. py_to_int() includes both a NULL cheack and a PyInt_Check() call as
well as the conversion call. If either of the checks fail, an exception is raised. The entire C++ code block is executed
with in a try/catch block that handles exceptions much like Python does. This removes the need for most error
checking code.
It is worth note that CXX lists do have indexing operators that result in code that looks much like Python. However,
the overhead in using them appears to be relatively high, so the standard Python API was used on the seq.ptr()
which is the underlying PyObject* of the List object.
The #line directive that is the first line of the C code block isn’t necessary, but it’s nice for debugging. If the
compilation fails because of the syntax error in the code, the error will be reported as an error in the Python file
“binary_search.py” with an offset from the given line number (29 here).
So what was all our effort worth in terms of efficiency? Well not a lot in this case. The examples/binary_search.py file
runs both Python and C versions of the functions As well as using the standard bisect module. If we run it on a 1
million element list and run the search 3000 times (for 0- 2999), here are the results we get:
C:\home\ej\wrk\scipy\weave\examples> python binary_search.py
Binary search for 3000 items in 1000000 length list of integers:
speed in python: 0.159999966621
speed of bisect: 0.121000051498
speed up: 1.32
speed in c: 0.110000014305
speed up: 1.45
speed in c(no asserts): 0.0900000333786
speed up: 1.78
So, we get roughly a 50-75% improvement depending on whether we use the Python asserts in our C version. If
we move down to searching a 10000 element list, the advantage evaporates. Even smaller lists might result in the
Python version being faster. I’d like to say that moving to NumPy lists (and getting rid of the GetItem() call) offers a
substantial speed up, but my preliminary efforts didn’t produce one. I think the log(N) algorithm is to blame. Because
the algorithm is nice, there just isn’t much time spent computing things, so moving to C isn’t that big of a win. If
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there are ways to reduce conversion overhead of values, this may improve the C/Python speed up. Anyone have other
explanations or faster code, please let me know.
Dictionary Sort
The demo in examples/dict_sort.py is another example from the Python CookBook. This submission, by Alex Martelli,
demonstrates how to return the values from a dictionary sorted by their keys:
def sortedDictValues3(adict):
keys = adict.keys()
keys.sort()
return map(adict.get, keys)
Alex provides 3 algorithms and this is the 3rd and fastest of the set. The C version of this same algorithm follows:
def c_sort(adict):
assert(type(adict) == type({}))
code = """
#line 21 "dict_sort.py"
Py::List keys = adict.keys();
Py::List items(keys.length()); keys.sort();
PyObject* item = NULL;
for(int i = 0; i < keys.length();i++)
{
item = PyList_GET_ITEM(keys.ptr(),i);
item = PyDict_GetItem(adict.ptr(),item);
Py_XINCREF(item);
PyList_SetItem(items.ptr(),i,item);
}
return_val = Py::new_reference_to(items);
"""
return inline_tools.inline(code,[’adict’],verbose=1)
Like the original Python function, the C++ version can handle any Python dictionary regardless of the key/value pair
types. It uses CXX objects for the most part to declare python types in C++, but uses Python API calls to manipulate
their contents. Again, this choice is made for speed. The C++ version, while more complicated, is about a factor of 2
faster than Python.
C:\home\ej\wrk\scipy\weave\examples> python dict_sort.py
Dict sort of 1000 items for 300 iterations:
speed in python: 0.319999933243
[0, 1, 2, 3, 4]
speed in c: 0.151000022888
speed up: 2.12
[0, 1, 2, 3, 4]
NumPy – cast/copy/transpose
CastCopyTranspose is a function called quite heavily by Linear Algebra routines in the NumPy library. Its needed
in part because of the row-major memory layout of multi-demensional Python (and C) arrays vs. the col-major order
of the underlying Fortran algorithms. For small matrices (say 100x100 or less), a significant portion of the common
routines such as LU decompisition or singular value decompostion are spent in this setup routine. This shouldn’t
happen. Here is the Python version of the function using standard NumPy operations.
def _castCopyAndTranspose(type, array):
if a.typecode() == type:
cast_array = copy.copy(NumPy.transpose(a))
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else:
cast_array = copy.copy(NumPy.transpose(a).astype(type))
return cast_array
And the following is a inline C version of the same function:
from weave.blitz_tools import blitz_type_factories
from weave import scalar_spec
from weave import inline
def _cast_copy_transpose(type,a_2d):
assert(len(shape(a_2d)) == 2)
new_array = zeros(shape(a_2d),type)
NumPy_type = scalar_spec.NumPy_to_blitz_type_mapping[type]
code = \
"""
for(int i = 0;i < _Na_2d[0]; i++)
for(int j = 0; j < _Na_2d[1]; j++)
new_array(i,j) = (%s) a_2d(j,i);
""" % NumPy_type
inline(code,[’new_array’,’a_2d’],
type_factories = blitz_type_factories,compiler=’gcc’)
return new_array
This example uses blitz++ arrays instead of the standard representation of NumPy arrays so that indexing is simplier
to write. This is accomplished by passing in the blitz++ “type factories” to override the standard Python to C++ type
conversions. Blitz++ arrays allow you to write clean, fast code, but they also are sloooow to compile (20 seconds
or more for this snippet). This is why they aren’t the default type used for Numeric arrays (and also because most
compilers can’t compile blitz arrays...). inline() is also forced to use ‘gcc’ as the compiler because the default
compiler on Windows (MSVC) will not compile blitz code. (‘gcc’ I think will use the standard compiler on Unix
machine instead of explicitly forcing gcc (check this)) Comparisons of the Python vs inline C++ code show a factor
of 3 speed up. Also shown are the results of an “inplace” transpose routine that can be used if the output of the
linear algebra routine can overwrite the original matrix (this is often appropriate). This provides another factor of 2
improvement.
#C:\home\ej\wrk\scipy\weave\examples> python cast_copy_transpose.py
# Cast/Copy/Transposing (150,150)array 1 times
# speed in python: 0.870999932289
# speed in c: 0.25
# speed up: 3.48
# inplace transpose c: 0.129999995232
# speed up: 6.70
wxPython
inline knows how to handle wxPython objects. Thats nice in and of itself, but it also demonstrates that the type
conversion mechanism is reasonably flexible. Chances are, it won’t take a ton of effort to support special types you
might have. The examples/wx_example.py borrows the scrolled window example from the wxPython demo, accept
that it mixes inline C code in the middle of the drawing function.
def DoDrawing(self, dc):
red = wxNamedColour("RED");
blue = wxNamedColour("BLUE");
grey_brush = wxLIGHT_GREY_BRUSH;
code = \
"""
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#line 108 "wx_example.py"
dc->BeginDrawing();
dc->SetPen(wxPen(*red,4,wxSOLID));
dc->DrawRectangle(5,5,50,50);
dc->SetBrush(*grey_brush);
dc->SetPen(wxPen(*blue,4,wxSOLID));
dc->DrawRectangle(15, 15, 50, 50);
"""
inline(code,[’dc’,’red’,’blue’,’grey_brush’])
dc.SetFont(wxFont(14, wxSWISS, wxNORMAL, wxNORMAL))
dc.SetTextForeground(wxColour(0xFF, 0x20, 0xFF))
te = dc.GetTextExtent("Hello World")
dc.DrawText("Hello World", 60, 65)
dc.SetPen(wxPen(wxNamedColour(’VIOLET’), 4))
dc.DrawLine(5, 65+te[1], 60+te[0], 65+te[1])
...
Here, some of the Python calls to wx objects were just converted to C++ calls. There isn’t any benefit, it just demonstrates the capabilities. You might want to use this if you have a computationally intensive loop in your drawing code
that you want to speed up. On windows, you’ll have to use the MSVC compiler if you use the standard wxPython
DLLs distributed by Robin Dunn. Thats because MSVC and gcc, while binary compatible in C, are not binary compatible for C++. In fact, its probably best, no matter what platform you’re on, to specify that inline use the same
compiler that was used to build wxPython to be on the safe side. There isn’t currently a way to learn this info from the
library – you just have to know. Also, at least on the windows platform, you’ll need to install the wxWindows libraries
and link to them. I think there is a way around this, but I haven’t found it yet – I get some linking errors dealing with
wxString. One final note. You’ll probably have to tweak weave/wx_spec.py or weave/wx_info.py for your machine’s
configuration to point at the correct directories etc. There. That should sufficiently scare people into not even looking
at this... :)
Keyword Option
The basic definition of the inline() function has a slew of optional variables. It also takes keyword arguments that
are passed to distutils as compiler options. The following is a formatted cut/paste of the argument section of
inline’s doc-string. It explains all of the variables. Some examples using various options will follow.
def inline(code,arg_names,local_dict = None, global_dict = None,
force = 0,
compiler=’’,
verbose = 0,
support_code = None,
customize=None,
type_factories = None,
auto_downcast=1,
**kw):
inline has quite a few options as listed below. Also, the keyword arguments for distutils extension modules are
accepted to specify extra information needed for compiling.
Inline Arguments
code string. A string of valid C++ code. It should not specify a return statement. Instead it should assign results that
need to be returned to Python in the return_val. arg_names list of strings. A list of Python variable names that should
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be transferred from Python into the C/C++ code. local_dict optional. dictionary. If specified, it is a dictionary of
values that should be used as the local scope for the C/C++ code. If local_dict is not specified the local dictionary of
the calling function is used. global_dict optional. dictionary. If specified, it is a dictionary of values that should be
used as the global scope for the C/C++ code. If global_dict is not specified the global dictionary of the calling function
is used. force optional. 0 or 1. default 0. If 1, the C++ code is compiled every time inline is called. This is really only
useful for debugging, and probably only useful if you’re editing support_code a lot. compiler optional. string. The
name of compiler to use when compiling. On windows, it understands ‘msvc’ and ‘gcc’ as well as all the compiler
names understood by distutils. On Unix, it’ll only understand the values understoof by distutils. (I should add ‘gcc’
though to this).
On windows, the compiler defaults to the Microsoft C++ compiler. If this isn’t available, it looks for mingw32 (the
gcc compiler).
On Unix, it’ll probably use the same compiler that was used when compiling Python. Cygwin’s behavior should be
similar.
verbose optional. 0,1, or 2. defualt 0. Speficies how much much information is printed during the compile phase
of inlining code. 0 is silent (except on windows with msvc where it still prints some garbage). 1 informs you when
compiling starts, finishes, and how long it took. 2 prints out the command lines for the compilation process and can
be useful if you’re having problems getting code to work. Its handy for finding the name of the .cpp file if you need
to examine it. verbose has no affect if the compilation isn’t necessary. support_code optional. string. A string of
valid C++ code declaring extra code that might be needed by your compiled function. This could be declarations of
functions, classes, or structures. customize optional. base_info.custom_info object. An alternative way to specifiy
support_code, headers, etc. needed by the function see the weave.base_info module for more details. (not sure this’ll
be used much). type_factories optional. list of type specification factories. These guys are what convert Python data
types to C/C++ data types. If you’d like to use a different set of type conversions than the default, specify them here.
Look in the type conversions section of the main documentation for examples. auto_downcast optional. 0 or 1. default
1. This only affects functions that have Numeric arrays as input variables. Setting this to 1 will cause all floating point
values to be cast as float instead of double if all the NumPy arrays are of type float. If even one of the arrays has type
double or double complex, all variables maintain there standard types.
Distutils keywords
inline() also accepts a number of distutils keywords for controlling how the code is compiled. The following
descriptions have been copied from Greg Ward’s distutils.extension.Extension class doc- strings for
convenience: sources [string] list of source filenames, relative to the distribution root (where the setup script lives), in
Unix form (slash- separated) for portability. Source files may be C, C++, SWIG (.i), platform- specific resource files,
or whatever else is recognized by the “build_ext” command as source for a Python extension. Note: The module_path
file is always appended to the front of this list include_dirs [string] list of directories to search for C/C++ header files
(in Unix form for portability) define_macros [(name : string, value : string|None)] list of macros to define; each macro
is defined using a 2-tuple, where ‘value’ is either the string to define it to or None to define it without a particular value
(equivalent of “#define FOO” in source or -DFOO on Unix C compiler command line) undef_macros [string] list of
macros to undefine explicitly library_dirs [string] list of directories to search for C/C++ libraries at link time libraries
[string] list of library names (not filenames or paths) to link against runtime_library_dirs [string] list of directories to
search for C/C++ libraries at run time (for shared extensions, this is when the extension is loaded) extra_objects [string]
list of extra files to link with (eg. object files not implied by ‘sources’, static library that must be explicitly specified,
binary resource files, etc.) extra_compile_args [string] any extra platform- and compiler-specific information to use
when compiling the source files in ‘sources’. For platforms and compilers where “command line” makes sense, this is
typically a list of command-line arguments, but for other platforms it could be anything. extra_link_args [string] any
extra platform- and compiler-specific information to use when linking object files together to create the extension (or
to create a new static Python interpreter). Similar interpretation as for ‘extra_compile_args’. export_symbols [string]
list of symbols to be exported from a shared extension. Not used on all platforms, and not generally necessary for
Python extensions, which typically export exactly one symbol: “init” + extension_name.
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Keyword Option Examples
We’ll walk through several examples here to demonstrate the behavior of inline and also how the various arguments
are used. In the simplest (most) cases, code and arg_names are the only arguments that need to be specified. Here’s
a simple example run on Windows machine that has Microsoft VC++ installed.
>>> from weave import inline
>>> a = ’string’
>>> code = """
...
int l = a.length();
...
return_val = Py::new_reference_to(Py::Int(l));
...
"""
>>> inline(code,[’a’])
sc_86e98826b65b047ffd2cd5f479c627f12.cpp
Creating
library C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\sc_86e98826b65b047ffd2cd5f47
and object C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\sc_86e98826b65b047ff
d2cd5f479c627f12.exp
6
>>> inline(code,[’a’])
6
When inline is first run, you’ll notice that pause and some trash printed to the screen. The “trash” is acutually part of
the compilers output that distutils does not supress. The name of the extension file, sc_bighonkingnumber.cpp,
is generated from the md5 check sum of the C/C++ code fragment. On Unix or windows machines with only gcc
installed, the trash will not appear. On the second call, the code fragment is not compiled since it already exists, and
only the answer is returned. Now kill the interpreter and restart, and run the same code with a different string.
>>>
>>>
>>>
...
...
...
>>>
15
from weave import inline
a = ’a longer string’
code = """
int l = a.length();
return_val = Py::new_reference_to(Py::Int(l));
"""
inline(code,[’a’])
Notice this time, inline() did not recompile the code because it found the compiled function in the persistent
catalog of functions. There is a short pause as it looks up and loads the function, but it is much shorter than compiling
would require.
You can specify the local and global dictionaries if you’d like (much like exec or eval() in Python), but if they
aren’t specified, the “expected” ones are used – i.e. the ones from the function that called inline(). This is
accomplished through a little call frame trickery. Here is an example where the local_dict is specified using the same
code example from above:
>>>
>>>
>>>
>>>
15
>>>
21
a = ’a longer string’
b = ’an even longer string’
my_dict = {’a’:b}
inline(code,[’a’])
inline(code,[’a’],my_dict)
Everytime, the code is changed, inline does a recompile. However, changing any of the other options in inline
does not force a recompile. The force option was added so that one could force a recompile when tinkering with
other variables. In practice, it is just as easy to change the code by a single character (like adding a space some place)
to force the recompile.
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Note: It also might be nice to add some methods for purging the cache and on disk catalogs.
I use verbose sometimes for debugging. When set to 2, it’ll output all the information (including the name of
the .cpp file) that you’d expect from running a make file. This is nice if you need to examine the generated code to
see where things are going haywire. Note that error messages from failed compiles are printed to the screen even if
verbose is set to 0.
The following example demonstrates using gcc instead of the standard msvc compiler on windows using same code
fragment as above. Because the example has already been compiled, the force=1 flag is needed to make inline()
ignore the previously compiled version and recompile using gcc. The verbose flag is added to show what is printed
out:
>>>inline(code,[’a’],compiler=’gcc’,verbose=2,force=1)
running build_ext
building ’sc_86e98826b65b047ffd2cd5f479c627f13’ extension
c:\gcc-2.95.2\bin\g++.exe -mno-cygwin -mdll -O2 -w -Wstrict-prototypes -IC:
\home\ej\wrk\scipy\weave -IC:\Python21\Include -c C:\DOCUME~1\eric\LOCAL
S~1\Temp\python21_compiled\sc_86e98826b65b047ffd2cd5f479c627f13.cpp
-o C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\sc_86e98826b65b04ffd2cd5f479c627f13.
skipping C:\home\ej\wrk\scipy\weave\CXX\cxxextensions.c
(C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\cxxextensions.o up-to-date)
skipping C:\home\ej\wrk\scipy\weave\CXX\cxxsupport.cxx
(C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\cxxsupport.o up-to-date)
skipping C:\home\ej\wrk\scipy\weave\CXX\IndirectPythonInterface.cxx
(C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\indirectpythoninterface.o up-to-date)
skipping C:\home\ej\wrk\scipy\weave\CXX\cxx_extensions.cxx
(C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\cxx_extensions.o
up-to-date)
writing C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\sc_86e98826b65b047ffd2cd5f479c6
c:\gcc-2.95.2\bin\dllwrap.exe --driver-name g++ -mno-cygwin
-mdll -static --output-lib
C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\libsc_86e98826b65b047ffd2cd5f479c627f13
C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\sc_86e98826b65b047ffd2cd5f479c627f13.de
-sC:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\sc_86e98826b65b047ffd2cd5f479c627f13.
C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\cxxextensions.o
C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\cxxsupport.o
C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\indirectpythoninterface.o
C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\temp\Release\cxx_extensions.o -LC:\Python21\libs
-lpython21 -o
C:\DOCUME~1\eric\LOCALS~1\Temp\python21_compiled\sc_86e98826b65b047ffd2cd5f479c627f13.pyd
15
That’s quite a bit of output. verbose=1 just prints the compile time.
>>>inline(code,[’a’],compiler=’gcc’,verbose=1,force=1)
Compiling code...
finished compiling (sec): 6.00800001621
15
Note: I’ve only used the compiler option for switching between ‘msvc’ and ‘gcc’ on windows. It may have use on
Unix also, but I don’t know yet.
The support_code argument is likely to be used a lot. It allows you to specify extra code fragments such as
function, structure or class definitions that you want to use in the code string. Note that changes to support_code
do not force a recompile. The catalog only relies on code (for performance reasons) to determine whether recompiling
is necessary. So, if you make a change to support_code, you’ll need to alter code in some way or use the force
argument to get the code to recompile. I usually just add some inocuous whitespace to the end of one of the lines in
code somewhere. Here’s an example of defining a separate method for calculating the string length:
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>>>
>>>
>>>
...
...
...
...
...
...
>>>
...
15
from weave import inline
a = ’a longer string’
support_code = """
PyObject* length(Py::String a)
{
int l = a.length();
return Py::new_reference_to(Py::Int(l));
}
"""
inline("return_val = length(a);",[’a’],
support_code = support_code)
customize is a left over from a previous way of specifying compiler options. It is a custom_info object that can
specify quite a bit of information about how a file is compiled. These info objects are the standard way of defining
compile information for type conversion classes. However, I don’t think they are as handy here, especially since we’ve
exposed all the keyword arguments that distutils can handle. Between these keywords, and the support_code
option, I think customize may be obsolete. We’ll see if anyone cares to use it. If not, it’ll get axed in the next
version.
The type_factories variable is important to people who want to customize the way arguments are converted
from Python to C. We’ll talk about this in the next chapter xx of this document when we discuss type conversions.
auto_downcast handles one of the big type conversion issues that is common when using NumPy arrays in conjunction with Python scalar values. If you have an array of single precision values and multiply that array by a Python
scalar, the result is upcast to a double precision array because the scalar value is double precision. This is not usually the desired behavior because it can double your memory usage. auto_downcast goes some distance towards
changing the casting precedence of arrays and scalars. If your only using single precision arrays, it will automatically
downcast all scalar values from double to single precision when they are passed into the C++ code. This is the default
behavior. If you want all values to keep there default type, set auto_downcast to 0.
Returning Values
Python variables in the local and global scope transfer seemlessly from Python into the C++ snippets. And, if inline
were to completely live up to its name, any modifications to variables in the C++ code would be reflected in the Python
variables when control was passed back to Python. For example, the desired behavior would be something like:
# THIS DOES NOT WORK
>>> a = 1
>>> weave.inline("a++;",[’a’])
>>> a
2
Instead you get:
>>> a = 1
>>> weave.inline("a++;",[’a’])
>>> a
1
Variables are passed into C++ as if you are calling a Python function. Python’s calling convention is sometimes called
“pass by assignment”. This means its as if a c_a = a assignment is made right before inline call is made and the
c_a variable is used within the C++ code. Thus, any changes made to c_a are not reflected in Python’s a variable.
Things do get a little more confusing, however, when looking at variables with mutable types. Changes made in C++
to the contents of mutable types are reflected in the Python variables.
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>>>
>>>
>>>
[3,
a= [1,2]
weave.inline("PyList_SetItem(a.ptr(),0,PyInt_FromLong(3));",[’a’])
print a
2]
So modifications to the contents of mutable types in C++ are seen when control is returned to Python. Modifications
to immutable types such as tuples, strings, and numbers do not alter the Python variables. If you need to make changes
to an immutable variable, you’ll need to assign the new value to the “magic” variable return_val in C++. This
value is returned by the inline() function:
>>> a = 1
>>> a = weave.inline("return_val = Py::new_reference_to(Py::Int(a+1));",[’a’])
>>> a
2
The return_val variable can also be used to return newly created values. This is possible by returning a tuple. The
following trivial example illustrates how this can be done:
# python version
def multi_return():
return 1, ’2nd’
# C version.
def c_multi_return():
code = """
py::tuple results(2);
results[0] = 1;
results[1] = "2nd";
return_val = results;
"""
return inline_tools.inline(code)
The example is available in examples/tuple_return.py. It also has the dubious honor of demonstrating how
much inline() can slow things down. The C version here is about 7-10 times slower than the Python version. Of
course, something so trivial has no reason to be written in C anyway.
The issue with locals() inline passes the locals() and globals() dictionaries from Python into the
C++ function from the calling function. It extracts the variables that are used in the C++ code from these dictionaries,
converts then to C++ variables, and then calculates using them. It seems like it would be trivial, then, after the
calculations were finished to then insert the new values back into the locals() and globals() dictionaries so
that the modified values were reflected in Python. Unfortunately, as pointed out by the Python manual, the locals()
dictionary is not writable.
I suspect locals() is not writable because there are some optimizations done to speed lookups of the local namespace. I’m guessing local lookups don’t always look at a dictionary to find values. Can someone “in the know” confirm
or correct this? Another thing I’d like to know is whether there is a way to write to the local namespace of another
stack frame from C/C++. If so, it would be possible to have some clean up code in compiled functions that wrote
final values of variables in C++ back to the correct Python stack frame. I think this goes a long way toward making
inline truely live up to its name. I don’t think we’ll get to the point of creating variables in Python for variables
created in C – although I suppose with a C/C++ parser you could do that also.
A quick look at the code
weave generates a C++ file holding an extension function for each inline code snippet. These file names are
generated using from the md5 signature of the code snippet and saved to a location specified by the PYTHONCOM-
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PILED environment variable (discussed later). The cpp files are generally about 200-400 lines long and include quite
a few functions to support type conversions, etc. However, the actual compiled function is pretty simple. Below is the
familiar printf example:
>>> import weave
>>> a = 1
>>> weave.inline(’printf("%d\\n",a);’,[’a’])
1
And here is the extension function generated by inline:
static PyObject* compiled_func(PyObject*self, PyObject* args)
{
py::object return_val;
int exception_occured = 0;
PyObject *py__locals = NULL;
PyObject *py__globals = NULL;
PyObject *py_a;
py_a = NULL;
if(!PyArg_ParseTuple(args,"OO:compiled_func",&py__locals,&py__globals))
return NULL;
try
{
PyObject* raw_locals = py_to_raw_dict(py__locals,"_locals");
PyObject* raw_globals = py_to_raw_dict(py__globals,"_globals");
/* argument conversion code */
py_a = get_variable("a",raw_locals,raw_globals);
int a = convert_to_int(py_a,"a");
/* inline code */
/* NDARRAY API VERSION 90907 */
printf("%d\n",a);
/*I would like to fill in changed locals and globals here...*/
}
catch(...)
{
return_val = py::object();
exception_occured = 1;
}
/* cleanup code */
if(!(PyObject*)return_val && !exception_occured)
{
return_val = Py_None;
}
return return_val.disown();
}
Every inline function takes exactly two arguments – the local and global dictionaries for the current scope. All variable
values are looked up out of these dictionaries. The lookups, along with all inline code execution, are done within
a C++ try block. If the variables aren’t found, or there is an error converting a Python variable to the appropriate
type in C++, an exception is raised. The C++ exception is automatically converted to a Python exception by SCXX
and returned to Python. The py_to_int() function illustrates how the conversions and exception handling works.
py_to_int first checks that the given PyObject* pointer is not NULL and is a Python integer. If all is well, it calls the
Python API to convert the value to an int. Otherwise, it calls handle_bad_type() which gathers information
about what went wrong and then raises a SCXX TypeError which returns to Python as a TypeError.
int py_to_int(PyObject* py_obj,char* name)
{
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if (!py_obj || !PyInt_Check(py_obj))
handle_bad_type(py_obj,"int", name);
return (int) PyInt_AsLong(py_obj);
}
void handle_bad_type(PyObject* py_obj, char* good_type, char* var_name)
{
char msg[500];
sprintf(msg,"received ’%s’ type instead of ’%s’ for variable ’%s’",
find_type(py_obj),good_type,var_name);
throw Py::TypeError(msg);
}
char* find_type(PyObject* py_obj)
{
if(py_obj == NULL) return "C NULL value";
if(PyCallable_Check(py_obj)) return "callable";
if(PyString_Check(py_obj)) return "string";
if(PyInt_Check(py_obj)) return "int";
if(PyFloat_Check(py_obj)) return "float";
if(PyDict_Check(py_obj)) return "dict";
if(PyList_Check(py_obj)) return "list";
if(PyTuple_Check(py_obj)) return "tuple";
if(PyFile_Check(py_obj)) return "file";
if(PyModule_Check(py_obj)) return "module";
//should probably do more interagation (and thinking) on these.
if(PyCallable_Check(py_obj) && PyInstance_Check(py_obj)) return "callable";
if(PyInstance_Check(py_obj)) return "instance";
if(PyCallable_Check(py_obj)) return "callable";
return "unkown type";
}
Since the inline is also executed within the try/catch block, you can use CXX exceptions within your code. It
is usually a bad idea to directly return from your code, even if an error occurs. This skips the clean up section of
the extension function. In this simple example, there isn’t any clean up code, but in more complicated examples, there
may be some reference counting that needs to be taken care of here on converted variables. To avoid this, either uses
exceptions or set return_val to NULL and use if/then’s to skip code after errors.
Technical Details
There are several main steps to using C/C++ code withing Python:
1. Type conversion
2. Generating C/C++ code
3. Compile the code to an extension module
4. Catalog (and cache) the function for future use
Items 1 and 2 above are related, but most easily discussed separately. Type conversions are customizable by the user if
needed. Understanding them is pretty important for anything beyond trivial uses of inline. Generating the C/C++
code is handled by ext_function and ext_module classes and . For the most part, compiling the code is handled
by distutils. Some customizations were needed, but they were relatively minor and do not require changes to distutils
itself. Cataloging is pretty simple in concept, but surprisingly required the most code to implement (and still likely
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needs some work). So, this section covers items 1 and 4 from the list. Item 2 is covered later in the chapter covering
the ext_tools module, and distutils is covered by a completely separate document xxx.
Passing Variables in/out of the C/C++ code
Note: Passing variables into the C code is pretty straight forward, but there are subtlties to how variable modifications
in C are returned to Python. see Returning Values for a more thorough discussion of this issue.
Type Conversions
Note: Maybe xxx_converter instead of xxx_specification is a more descriptive name. Might change in
future version?
By default, inline() makes the following type conversions between Python and C++ types.
Table 1.3: Default Data Type Conversions
Python
int
float
complex
string
list
dict
tuple
file
callable
instance
numpy.ndarray
wxXXX
C++
int
double
std::complex
py::string
py::list
py::dict
py::tuple
FILE*
py::object
py::object
PyArrayObject*
wxXXX*
The Py:: namespace is defined by the SCXX library which has C++ class equivalents for many Python types. std::
is the namespace of the standard library in C++.
Note:
• I haven’t figured out how to handle long int yet (I think they are currenlty converted to int - - check this).
• Hopefully VTK will be added to the list soon
Python to C++ conversions fill in code in several locations in the generated inline extension function. Below is the
basic template for the function. This is actually the exact code that is generated by calling weave.inline("").
The /* inline code */ section is filled with the code passed to the inline() function call. The
/*argument convserion code*/ and /* cleanup code */ sections are filled with code that handles
conversion from Python to C++ types and code that deallocates memory or manipulates reference counts before the
function returns. The following sections demostrate how these two areas are filled in by the default conversion methods. * Note: I’m not sure I have reference counting correct on a few of these. The only thing I increase/decrease the
ref count on is NumPy arrays. If you see an issue, please let me know.
NumPy Argument Conversion
Integer, floating point, and complex arguments are handled in a very similar fashion. Consider the following inline
function that has a single integer variable passed in:
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>>> a = 1
>>> inline("",[’a’])
The argument conversion code inserted for a is:
/* argument conversion code */
int a = py_to_int (get_variable("a",raw_locals,raw_globals),"a");
get_variable() reads the variable a from the local and global namespaces. py_to_int() has the following
form:
static int py_to_int(PyObject* py_obj,char* name)
{
if (!py_obj || !PyInt_Check(py_obj))
handle_bad_type(py_obj,"int", name);
return (int) PyInt_AsLong(py_obj);
}
Similarly, the float and complex conversion routines look like:
static double py_to_float(PyObject* py_obj,char* name)
{
if (!py_obj || !PyFloat_Check(py_obj))
handle_bad_type(py_obj,"float", name);
return PyFloat_AsDouble(py_obj);
}
static std::complex py_to_complex(PyObject* py_obj,char* name)
{
if (!py_obj || !PyComplex_Check(py_obj))
handle_bad_type(py_obj,"complex", name);
return std::complex(PyComplex_RealAsDouble(py_obj),
PyComplex_ImagAsDouble(py_obj));
}
NumPy conversions do not require any clean up code.
String, List, Tuple, and Dictionary Conversion
Strings, Lists, Tuples and Dictionary conversions are all converted to SCXX types by default. For the following code,
>>> a = [1]
>>> inline("",[’a’])
The argument conversion code inserted for a is:
/* argument conversion code */
Py::List a = py_to_list(get_variable("a",raw_locals,raw_globals),"a");
get_variable() reads the variable a from the local and global namespaces. py_to_list() and its friends has
the following form:
static Py::List py_to_list(PyObject* py_obj,char* name)
{
if (!py_obj || !PyList_Check(py_obj))
handle_bad_type(py_obj,"list", name);
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return Py::List(py_obj);
}
static Py::String py_to_string(PyObject* py_obj,char* name)
{
if (!PyString_Check(py_obj))
handle_bad_type(py_obj,"string", name);
return Py::String(py_obj);
}
static Py::Dict py_to_dict(PyObject* py_obj,char* name)
{
if (!py_obj || !PyDict_Check(py_obj))
handle_bad_type(py_obj,"dict", name);
return Py::Dict(py_obj);
}
static Py::Tuple py_to_tuple(PyObject* py_obj,char* name)
{
if (!py_obj || !PyTuple_Check(py_obj))
handle_bad_type(py_obj,"tuple", name);
return Py::Tuple(py_obj);
}
SCXX handles reference counts on for strings, lists, tuples, and dictionaries, so clean up code isn’t necessary.
File Conversion
For the following code,
>>> a = open("bob",’w’)
>>> inline("",[’a’])
The argument conversion code is:
/* argument conversion code */
PyObject* py_a = get_variable("a",raw_locals,raw_globals);
FILE* a = py_to_file(py_a,"a");
get_variable() reads the variable a from the local and global namespaces. py_to_file() converts PyObject*
to a FILE* and increments the reference count of the PyObject*:
FILE* py_to_file(PyObject* py_obj, char* name)
{
if (!py_obj || !PyFile_Check(py_obj))
handle_bad_type(py_obj,"file", name);
Py_INCREF(py_obj);
return PyFile_AsFile(py_obj);
}
Because the PyObject* was incremented, the clean up code needs to decrement the counter
/* cleanup code */
Py_XDECREF(py_a);
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Its important to understand that file conversion only works on actual files – i.e. ones created using the open()
command in Python. It does not support converting arbitrary objects that support the file interface into C FILE*
pointers. This can affect many things. For example, in initial printf() examples, one might be tempted to solve the
problem of C and Python IDE’s (PythonWin, PyCrust, etc.) writing to different stdout and stderr by using fprintf()
and passing in sys.stdout and sys.stderr. For example, instead of
>>> weave.inline(’printf("hello\\n");’)
You might try:
>>> buf = sys.stdout
>>> weave.inline(’fprintf(buf,"hello\\n");’,[’buf’])
This will work as expected from a standard python interpreter, but in PythonWin, the following occurs:
>>> buf = sys.stdout
>>> weave.inline(’fprintf(buf,"hello\\n");’,[’buf’])
The traceback tells us that inline() was unable to convert ‘buf’ to a C++ type (If instance conversion was implemented, the error would have occurred at runtime instead). Why is this? Let’s look at what the buf object really
is:
>>> buf
pywin.framework.interact.InteractiveView instance at 00EAD014
PythonWin has reassigned sys.stdout to a special object that implements the Python file interface. This works
great in Python, but since the special object doesn’t have a FILE* pointer underlying it, fprintf doesn’t know what to
do with it (well this will be the problem when instance conversion is implemented...).
Callable, Instance, and Module Conversion
Note: Need to look into how ref counts should be handled. Also, Instance and Module conversion are not currently
implemented.
>>> def a():
pass
>>> inline("",[’a’])
Callable and instance variables are converted to PyObject*. Nothing is done to there reference counts.
/* argument conversion code */
PyObject* a = py_to_callable(get_variable("a",raw_locals,raw_globals),"a");
get_variable() reads the variable a from the local and global namespaces. The py_to_callable() and
py_to_instance() don’t currently increment the ref count.
PyObject* py_to_callable(PyObject* py_obj, char* name)
{
if (!py_obj || !PyCallable_Check(py_obj))
handle_bad_type(py_obj,"callable", name);
return py_obj;
}
PyObject* py_to_instance(PyObject* py_obj, char* name)
{
if (!py_obj || !PyFile_Check(py_obj))
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handle_bad_type(py_obj,"instance", name);
return py_obj;
}
There is no cleanup code for callables, modules, or instances.
Customizing Conversions
Converting from Python to C++ types is handled by xxx_specification classes. A type specification class actually
serve in two related but different roles. The first is in determining whether a Python variable that needs to be converted
should be represented by the given class. The second is as a code generator that generate C++ code needed to convert
from Python to C++ types for a specific variable.
When
>>> a = 1
>>> weave.inline(’printf("%d",a);’,[’a’])
is called for the first time, the code snippet has to be compiled. In this process, the variable ‘a’ is tested against a
list of type specifications (the default list is stored in weave/ext_tools.py). The first specification in the list is used to
represent the variable.
Examples of xxx_specification are scattered throughout numerous “xxx_spec.py” files in the weave package. Closely related to the xxx_specification classes are yyy_info classes. These classes contain compiler,
header, and support code information necessary for including a certain set of capabilities (such as blitz++ or CXX
support) in a compiled module. xxx_specification classes have one or more yyy_info classes associated
with them. If you’d like to define your own set of type specifications, the current best route is to examine some of the
existing spec and info files. Maybe looking over sequence_spec.py and cxx_info.py are a good place to start. After
defining specification classes, you’ll need to pass them into inline using the type_factories argument. A
lot of times you may just want to change how a specific variable type is represented. Say you’d rather have Python
strings converted to std::string or maybe char* instead of using the CXX string object, but would like all other
type conversions to have default behavior. This requires that a new specification class that handles strings is written
and then prepended to a list of the default type specifications. Since it is closer to the front of the list, it effectively
overrides the default string specification. The following code demonstrates how this is done: ...
The Catalog
catalog.py has a class called catalog that helps keep track of previously compiled functions. This prevents
inline() and related functions from having to compile functions everytime they are called. Instead, catalog will
check an in memory cache to see if the function has already been loaded into python. If it hasn’t, then it starts searching
through persisent catalogs on disk to see if it finds an entry for the given function. By saving information about
compiled functions to disk, it isn’t necessary to re-compile functions everytime you stop and restart the interpreter.
Functions are compiled once and stored for future use.
When inline(cpp_code) is called the following things happen:
1. A fast local cache of functions is checked for the last function called for cpp_code. If an entry for cpp_code
doesn’t exist in the cache or the cached function call fails (perhaps because the function doesn’t have compatible
types) then the next step is to check the catalog.
2. The catalog class also keeps an in-memory cache with a list of all the functions compiled for cpp_code. If
cpp_code has ever been called, then this cache will be present (loaded from disk). If the cache isn’t present,
then it is loaded from disk.
If the cache is present, each function in the cache is called until one is found that was compiled for the correct
argument types. If none of the functions work, a new function is compiled with the given argument types. This
function is written to the on-disk catalog as well as into the in-memory cache.
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3. When a lookup for cpp_code fails, the catalog looks through the on-disk function catalogs for the entries. The PYTHONCOMPILED variable determines where to search for these catalogs and in what order.
If PYTHONCOMPILED is not present several platform dependent locations are searched. All functions found
for cpp_code in the path are loaded into the in-memory cache with functions found earlier in the search path
closer to the front of the call list.
If the function isn’t found in the on-disk catalog, then the function is compiled, written to the first writable
directory in the PYTHONCOMPILED path, and also loaded into the in-memory cache.
Function Storage
Function caches are stored as dictionaries where the key is the entire C++ code string and the value is either a single
function (as in the “level 1” cache) or a list of functions (as in the main catalog cache). On disk catalogs are stored in
the same manor using standard Python shelves.
Early on, there was a question as to whether md5 check sums of the C++ code strings should be used instead of the
actual code strings. I think this is the route inline Perl took. Some (admittedly quick) tests of the md5 vs. the entire
string showed that using the entire string was at least a factor of 3 or 4 faster for Python. I think this is because it is
more time consuming to compute the md5 value than it is to do look-ups of long strings in the dictionary. Look at the
examples/md5_speed.py file for the test run.
Catalog search paths and the PYTHONCOMPILED variable
The default location for catalog files on Unix is is ~/.pythonXX_compiled where XX is version of Python being used.
If this directory doesn’t exist, it is created the first time a catalog is used. The directory must be writable. If, for any
reason it isn’t, then the catalog attempts to create a directory based on your user id in the /tmp directory. The directory
permissions are set so that only you have access to the directory. If this fails, I think you’re out of luck. I don’t think
either of these should ever fail though. On Windows, a directory called pythonXX_compiled is created in the user’s
temporary directory.
The actual catalog file that lives in this directory is a Python shelve with a platform specific name such as
“nt21compiled_catalog” so that multiple OSes can share the same file systems without trampling on each other. Along
with the catalog file, the .cpp and .so or .pyd files created by inline will live in this directory. The catalog file simply
contains keys which are the C++ code strings with values that are lists of functions. The function lists point at functions within these compiled modules. Each function in the lists executes the same C++ code string, but compiled for
different input variables.
You can use the PYTHONCOMPILED environment variable to specify alternative locations for compiled functions.
On Unix this is a colon (‘:’) separated list of directories. On windows, it is a (‘;’) separated list of directories. These
directories will be searched prior to the default directory for a compiled function catalog. Also, the first writable
directory in the list is where all new compiled function catalogs, .cpp and .so or .pyd files are written. Relative
directory paths (‘.’ and ‘..’) should work fine in the PYTHONCOMPILED variable as should environement variables.
There is a “special” path variable called MODULE that can be placed in the PYTHONCOMPILED variable. It
specifies that the compiled catalog should reside in the same directory as the module that called it. This is useful if an
admin wants to build a lot of compiled functions during the build of a package and then install them in site-packages
along with the package. User’s who specify MODULE in their PYTHONCOMPILED variable will have access to
these compiled functions. Note, however, that if they call the function with a set of argument types that it hasn’t
previously been built for, the new function will be stored in their default directory (or some other writable directory in
the PYTHONCOMPILED path) because the user will not have write access to the site-packages directory.
An example of using the PYTHONCOMPILED path on bash follows:
PYTHONCOMPILED=MODULE:/some/path;export PYTHONCOMPILED;
If you are using python21 on linux, and the module bob.py in site-packages has a compiled function in it, then the
catalog search order when calling that function for the first time in a python session would be:
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/usr/lib/python21/site-packages/linuxpython_compiled
/some/path/linuxpython_compiled
~/.python21_compiled/linuxpython_compiled
The default location is always included in the search path.
Note: hmmm. see a possible problem here. I should probably make a sub- directory such as /usr/lib/python21/sitepackages/python21_compiled/linuxpython_compiled so that library files compiled with python21 are tried to link with
python22 files in some strange scenarios. Need to check this.
The in-module cache (in weave.inline_tools reduces the overhead of calling inline functions by about a factor
of 2. It can be reduced a little more for type loop calls where the same function is called over and over again if the
cache was a single value instead of a dictionary, but the benefit is very small (less than 5%) and the utility is quite a bit
less. So, we’ll stick with a dictionary as the cache.
1.13.8 Blitz
Note: most of this section is lifted from old documentation. It should be pretty accurate, but there may be a few
discrepancies.
weave.blitz() compiles NumPy Python expressions for fast execution. For most applications, compiled expressions should provide a factor of 2-10 speed-up over NumPy arrays. Using compiled expressions is meant to be as
unobtrusive as possible and works much like pythons exec statement. As an example, the following code fragment
takes a 5 point average of the 512x512 2d image, b, and stores it in array, a:
from scipy import * # or from NumPy import *
a = ones((512,512), Float64)
b = ones((512,512), Float64)
# ...do some stuff to fill in b...
# now average
a[1:-1,1:-1] = (b[1:-1,1:-1] + b[2:,1:-1] + b[:-2,1:-1] \
+ b[1:-1,2:] + b[1:-1,:-2]) / 5.
To compile the expression, convert the expression to a string by putting quotes around it and then use weave.blitz:
import weave
expr = "a[1:-1,1:-1] = (b[1:-1,1:-1] + b[2:,1:-1] + b[:-2,1:-1]" \
"+ b[1:-1,2:] + b[1:-1,:-2]) / 5."
weave.blitz(expr)
The first time weave.blitz is run for a given expression and set of arguements, C++ code that accomplishes the
exact same task as the Python expression is generated and compiled to an extension module. This can take up to a
couple of minutes depending on the complexity of the function. Subsequent calls to the function are very fast. Futher,
the generated module is saved between program executions so that the compilation is only done once for a given
expression and associated set of array types. If the given expression is executed with a new set of array types, the
code most be compiled again. This does not overwrite the previously compiled function – both of them are saved and
available for exectution.
The following table compares the run times for standard NumPy code and compiled code for the 5 point averaging.
Method Run Time (seconds) Standard NumPy 0.46349 blitz (1st time compiling) 78.95526 blitz (subsequent calls)
0.05843 (factor of 8 speedup)
These numbers are for a 512x512 double precision image run on a 400 MHz Celeron processor under RedHat Linux
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Because of the slow compile times, its probably most effective to develop algorithms as you usually do using the
capabilities of scipy or the NumPy module. Once the algorithm is perfected, put quotes around it and execute it using
weave.blitz. This provides the standard rapid prototyping strengths of Python and results in algorithms that run
close to that of hand coded C or Fortran.
Requirements
Currently, the weave.blitz has only been tested under Linux with gcc-2.95-3 and on Windows with Mingw32
(2.95.2). Its compiler requirements are pretty heavy duty (see the blitz++ home page), so it won’t work with just any
compiler. Particularly MSVC++ isn’t up to snuff. A number of other compilers such as KAI++ will also work, but my
suspicions are that gcc will get the most use.
Limitations
1. Currently, weave.blitz handles all standard mathematic operators except for the ** power operator. The
built-in trigonmetric, log, floor/ceil, and fabs functions might work (but haven’t been tested). It also handles all
types of array indexing supported by the NumPy module. numarray’s NumPy compatible array indexing modes
are likewise supported, but numarray’s enhanced (array based) indexing modes are not supported.
weave.blitz does not currently support operations that use array broadcasting, nor have any of the special
purpose functions in NumPy such as take, compress, etc. been implemented. Note that there are no obvious
reasons why most of this functionality cannot be added to scipy.weave, so it will likely trickle into future
versions. Using slice() objects directly instead of start:stop:step is also not supported.
2. Currently Python only works on expressions that include assignment such as
>>> result = b + c + d
This means that the result array must exist before calling weave.blitz. Future versions will allow the
following:
>>> result = weave.blitz_eval("b + c + d")
3. weave.blitz works best when algorithms can be expressed in a “vectorized” form. Algorithms that have a
large number of if/thens and other conditions are better hand written in C or Fortran. Further, the restrictions
imposed by requiring vectorized expressions sometimes preclude the use of more efficient data structures or
algorithms. For maximum speed in these cases, hand-coded C or Fortran code is the only way to go.
4. weave.blitz can produce different results than NumPy in certain situations. It can happen when the array
receiving the results of a calculation is also used during the calculation. The NumPy behavior is to carry out the
entire calculation on the right hand side of an equation and store it in a temporary array. This temprorary array is
assigned to the array on the left hand side of the equation. blitz, on the other hand, does a “running” calculation
of the array elements assigning values from the right hand side to the elements on the left hand side immediately
after they are calculated. Here is an example, provided by Prabhu Ramachandran, where this happens:
# 4 point average.
>>> expr = "u[1:-1, 1:-1] = (u[0:-2, 1:-1] + u[2:, 1:-1] + \
...
"u[1:-1,0:-2] + u[1:-1, 2:])*0.25"
>>> u = zeros((5, 5), ’d’); u[0,:] = 100
>>> exec (expr)
>>> u
array([[ 100., 100., 100., 100., 100.],
[
0.,
25.,
25.,
25.,
0.],
[
0.,
0.,
0.,
0.,
0.],
[
0.,
0.,
0.,
0.,
0.],
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[
0.,
0.,
0.,
0.,
0.]])
>>> u = zeros((5, 5), ’d’); u[0,:] = 100
>>> weave.blitz (expr)
>>> u
array([[ 100. , 100.
, 100.
[
0. ,
25.
,
31.25
[
0. ,
6.25
,
9.375
[
0. ,
1.5625
,
2.734375
[
0. ,
0.
,
0.
,
,
,
,
,
100.
,
32.8125
,
10.546875 ,
3.3203125,
0.
,
100. ],
0. ],
0. ],
0. ],
0. ]])
You can prevent this behavior by using a temporary array.
>>> u = zeros((5, 5), ’d’); u[0,:] = 100
>>> temp = zeros((4, 4), ’d’);
>>> expr = "temp = (u[0:-2, 1:-1] + u[2:, 1:-1] + "\
...
"u[1:-1,0:-2] + u[1:-1, 2:])*0.25;"\
...
"u[1:-1,1:-1] = temp"
>>> weave.blitz (expr)
>>> u
array([[ 100., 100., 100., 100., 100.],
[
0.,
25.,
25.,
25.,
0.],
[
0.,
0.,
0.,
0.,
0.],
[
0.,
0.,
0.,
0.,
0.],
[
0.,
0.,
0.,
0.,
0.]])
5. One other point deserves mention lest people be confused. weave.blitz is not a general purpose Python->C
compiler. It only works for expressions that contain NumPy arrays and/or Python scalar values. This focused
scope concentrates effort on the compuationally intensive regions of the program and sidesteps the difficult
issues associated with a general purpose Python->C compiler.
NumPy efficiency issues: What compilation buys you
Some might wonder why compiling NumPy expressions to C++ is beneficial since operations on NumPy array operations are already executed within C loops. The problem is that anything other than the simplest expression are
executed in less than optimal fashion. Consider the following NumPy expression:
a = 1.2 * b + c * d
When NumPy calculates the value for the 2d array, a, it does the following steps:
temp1 = 1.2 * b
temp2 = c * d
a = temp1 + temp2
Two things to note. Since c is an (perhaps large) array, a large temporary array must be created to store the results of
1.2 * b. The same is true for temp2. Allocation is slow. The second thing is that we have 3 loops executing, one
to calculate temp1, one for temp2 and one for adding them up. A C loop for the same problem might look like:
for(int i = 0; i < M; i++)
for(int j = 0; j < N; j++)
a[i,j] = 1.2 * b[i,j] + c[i,j] * d[i,j]
Here, the 3 loops have been fused into a single loop and there is no longer a need for a temporary array. This provides
a significant speed improvement over the above example (write me and tell me what you get).
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So, converting NumPy expressions into C/C++ loops that fuse the loops and eliminate temporary arrays can provide
big gains. The goal then,is to convert NumPy expression to C/C++ loops, compile them in an extension module, and
then call the compiled extension function. The good news is that there is an obvious correspondence between the
NumPy expression above and the C loop. The bad news is that NumPy is generally much more powerful than this
simple example illustrates and handling all possible indexing possibilities results in loops that are less than straight
forward to write. (take a peak in NumPy for confirmation). Luckily, there are several available tools that simplify the
process.
The Tools
weave.blitz relies heavily on several remarkable tools. On the Python side, the main facilitators are Jermey
Hylton’s parser module and Travis Oliphant’s NumPy module. On the compiled language side, Todd Veldhuizen’s
blitz++ array library, written in C++ (shhhh. don’t tell David Beazley), does the heavy lifting. Don’t assume that,
because it’s C++, it’s much slower than C or Fortran. Blitz++ uses a jaw dropping array of template techniques
(metaprogramming, template expression, etc) to convert innocent looking and readable C++ expressions into to code
that usually executes within a few percentage points of Fortran code for the same problem. This is good. Unfortunately
all the template raz-ma-taz is very expensive to compile, so the 200 line extension modules often take 2 or more
minutes to compile. This isn’t so good. weave.blitz works to minimize this issue by remembering where compiled
modules live and reusing them instead of re-compiling every time a program is re-run.
Parser
Tearing NumPy expressions apart, examining the pieces, and then rebuilding them as C++ (blitz) expressions requires
a parser of some sort. I can imagine someone attacking this problem with regular expressions, but it’d likely be ugly
and fragile. Amazingly, Python solves this problem for us. It actually exposes its parsing engine to the world through
the parser module. The following fragment creates an Abstract Syntax Tree (AST) object for the expression and
then converts to a (rather unpleasant looking) deeply nested list representation of the tree.
>>> import parser
>>> import scipy.weave.misc
>>> ast = parser.suite("a = b * c + d")
>>> ast_list = ast.tolist()
>>> sym_list = scipy.weave.misc.translate_symbols(ast_list)
>>> pprint.pprint(sym_list)
[’file_input’,
[’stmt’,
[’simple_stmt’,
[’small_stmt’,
[’expr_stmt’,
[’testlist’,
[’test’,
[’and_test’,
[’not_test’,
[’comparison’,
[’expr’,
[’xor_expr’,
[’and_expr’,
[’shift_expr’,
[’arith_expr’,
[’term’,
[’factor’, [’power’, [’atom’, [’NAME’, ’a’]]]]]]]]]]]]]]],
[’EQUAL’, ’=’],
[’testlist’,
[’test’,
[’and_test’,
[’not_test’,
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[’comparison’,
[’expr’,
[’xor_expr’,
[’and_expr’,
[’shift_expr’,
[’arith_expr’,
[’term’,
[’factor’, [’power’, [’atom’, [’NAME’, ’b’]]]],
[’STAR’, ’*’],
[’factor’, [’power’, [’atom’, [’NAME’, ’c’]]]]],
[’PLUS’, ’+’],
[’term’,
[’factor’, [’power’, [’atom’, [’NAME’, ’d’]]]]]]]]]]]]]]]]],
[’NEWLINE’, ’’]]],
[’ENDMARKER’, ’’]]
Despite its looks, with some tools developed by Jermey H., its possible to search these trees for specific patterns (subtrees), extract the sub-tree, manipulate them converting python specific code fragments to blitz code fragments, and
then re-insert it in the parse tree. The parser module documentation has some details on how to do this. Traversing the
new blitzified tree, writing out the terminal symbols as you go, creates our new blitz++ expression string.
Blitz and NumPy
The other nice discovery in the project is that the data structure used for NumPy arrays and blitz arrays is nearly
identical. NumPy stores “strides” as byte offsets and blitz stores them as element offsets, but other than that, they are
the same. Further, most of the concept and capabilities of the two libraries are remarkably similar. It is satisfying that
two completely different implementations solved the problem with similar basic architectures. It is also fortuitous.
The work involved in converting NumPy expressions to blitz expressions was greatly diminished. As an example,
consider the code for slicing an array in Python with a stride:
>>> a = b[0:4:2] + c
>>> a
[0,2,4]
In Blitz it is as follows:
Array<2,int> b(10);
Array<2,int> c(3);
// ...
Array<2,int> a = b(Range(0,3,2)) + c;
Here the range object works exactly like Python slice objects with the exception that the top index (3) is inclusive
where as Python’s (4) is exclusive. Other differences include the type declaraions in C++ and parentheses instead of
brackets for indexing arrays. Currently, weave.blitz handles the inclusive/exclusive issue by subtracting one from
upper indices during the translation. An alternative that is likely more robust/maintainable in the long run, is to write
a PyRange class that behaves like Python’s range. This is likely very easy.
The stock blitz also doesn’t handle negative indices in ranges. The current implementation of the blitz() has a
partial solution to this problem. It calculates and index that starts with a ‘-‘ sign by subtracting it from the maximum
index in the array so that:
upper index limit
/-----\
b[:-1] -> b(Range(0,Nb[0]-1-1))
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This approach fails, however, when the top index is calculated from other values. In the following scenario, if i+j
evaluates to a negative value, the compiled code will produce incorrect results and could even core- dump. Right now,
all calculated indices are assumed to be positive.
b[:i-j] -> b(Range(0,i+j))
A solution is to calculate all indices up front using if/then to handle the +/- cases. This is a little work and results in
more code, so it hasn’t been done. I’m holding out to see if blitz++ can be modified to handle negative indexing, but
haven’t looked into how much effort is involved yet. While it needs fixin’, I don’t think there is a ton of code where
this is an issue.
The actual translation of the Python expressions to blitz expressions is currently a two part process. First, all x:y:z
slicing expression are removed from the AST, converted to slice(x,y,z) and re-inserted into the tree. Any math needed
on these expressions (subtracting from the maximum index, etc.) are also preformed here. _beg and _end are used as
special variables that are defined as blitz::fromBegin and blitz::toEnd.
a[i+j:i+j+1,:] = b[2:3,:]
becomes a more verbose:
a[slice(i+j,i+j+1),slice(_beg,_end)] = b[slice(2,3),slice(_beg,_end)]
The second part does a simple string search/replace to convert to a blitz expression with the following translations:
slice(_beg,_end)
slice
[
]
_stp
->
->
->
->
->
_all # not strictly needed, but cuts down on code.
blitz::Range
(
)
1
_all is defined in the compiled function as blitz::Range.all(). These translations could of course happen
directly in the syntax tree. But the string replacement is slightly easier. Note that name spaces are maintained in the
C++ code to lessen the likelyhood of name clashes. Currently no effort is made to detect name clashes. A good rule
of thumb is don’t use values that start with ‘_’ or ‘py_’ in compiled expressions and you’ll be fine.
Type definitions and coersion
So far we’ve glossed over the dynamic vs. static typing issue between Python and C++. In Python, the type of value
that a variable holds can change through the course of program execution. C/C++, on the other hand, forces you to
declare the type of value a variables will hold prior at compile time. weave.blitz handles this issue by examining
the types of the variables in the expression being executed, and compiling a function for those explicit types. For
example:
a = ones((5,5),Float32)
b = ones((5,5),Float32)
weave.blitz("a = a + b")
When compiling this expression to C++, weave.blitz sees that the values for a and b in the local scope have type
Float32, or ‘float’ on a 32 bit architecture. As a result, it compiles the function using the float type (no attempt has
been made to deal with 64 bit issues).
What happens if you call a compiled function with array types that are different than the ones for which it was
originally compiled? No biggie, you’ll just have to wait on it to compile a new version for your new types. This
doesn’t overwrite the old functions, as they are still accessible. See the catalog section in the inline() documentation
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to see how this is handled. Suffice to say, the mechanism is transparent to the user and behaves like dynamic typing
with the occasional wait for compiling newly typed functions.
When working with combined scalar/array operations, the type of the array is always used. This is similar to the savespace flag that was recently added to NumPy. This prevents issues with the following expression perhaps unexpectedly
being calculated at a higher (more expensive) precision that can occur in Python:
>>> a = array((1,2,3),typecode = Float32)
>>> b = a * 2.1 # results in b being a Float64 array.
In this example,
>>> a = ones((5,5),Float32)
>>> b = ones((5,5),Float32)
>>> weave.blitz("b = a * 2.1")
the 2.1 is cast down to a float before carrying out the operation. If you really want to force the calculation to be a
double, define a and b as double arrays.
One other point of note. Currently, you must include both the right hand side and left hand side (assignment side)
of your equation in the compiled expression. Also, the array being assigned to must be created prior to calling
weave.blitz. I’m pretty sure this is easily changed so that a compiled_eval expression can be defined, but no
effort has been made to allocate new arrays (and decern their type) on the fly.
Cataloging Compiled Functions
See The Catalog section in the weave.inline() documentation.
Checking Array Sizes
Surprisingly, one of the big initial problems with compiled code was making sure all the arrays in an operation were
of compatible type. The following case is trivially easy:
a = b + c
It only requires that arrays a, b, and c have the same shape. However, expressions like:
a[i+j:i+j+1,:] = b[2:3,:] + c
are not so trivial. Since slicing is involved, the size of the slices, not the input arrays must be checked. Broadcasting
complicates things further because arrays and slices with different dimensions and shapes may be compatible for math
operations (broadcasting isn’t yet supported by weave.blitz). Reductions have a similar effect as their results are
different shapes than their input operand. The binary operators in NumPy compare the shapes of their two operands just
before they operate on them. This is possible because NumPy treats each operation independently. The intermediate
(temporary) arrays created during sub-operations in an expression are tested for the correct shape before they are
combined by another operation. Because weave.blitz fuses all operations into a single loop, this isn’t possible.
The shape comparisons must be done and guaranteed compatible before evaluating the expression.
The solution chosen converts input arrays to “dummy arrays” that only represent the dimensions of the arrays, not the
data. Binary operations on dummy arrays check that input array sizes are comptible and return a dummy array with
the size correct size. Evaluating an expression of dummy arrays traces the changing array sizes through all operations
and fails if incompatible array sizes are ever found.
The machinery for this is housed in weave.size_check. It basically involves writing a new class (dummy array)
and overloading it math operators to calculate the new sizes correctly. All the code is in Python and there is a fair
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amount of logic (mainly to handle indexing and slicing) so the operation does impose some overhead. For large arrays
(ie. 50x50x50), the overhead is negligible compared to evaluating the actual expression. For small arrays (ie. 16x16),
the overhead imposed for checking the shapes with this method can cause the weave.blitz to be slower than
evaluating the expression in Python.
What can be done to reduce the overhead? (1) The size checking code could be moved into C. This would likely
remove most of the overhead penalty compared to NumPy (although there is also some calling overhead), but no effort
has been made to do this. (2) You can also call weave.blitz with check_size=0 and the size checking isn’t
done. However, if the sizes aren’t compatible, it can cause a core-dump. So, foregoing size_checking isn’t advisable
until your code is well debugged.
Creating the Extension Module
weave.blitz uses the same machinery as weave.inline to build the extension module. The only difference is
the code included in the function is automatically generated from the NumPy array expression instead of supplied by
the user.
1.13.9 Extension Modules
weave.inline and weave.blitz are high level tools that generate extension modules automatically. Under
the covers, they use several classes from weave.ext_tools to help generate the extension module. The main two
classes are ext_module and ext_function (I’d like to add ext_class and ext_method also). These classes
simplify the process of generating extension modules by handling most of the “boiler plate” code automatically.
Note: inline actually sub-classes weave.ext_tools.ext_function to generate slightly different code
than the standard ext_function. The main difference is that the standard class converts function arguments to C
types, while inline always has two arguments, the local and global dicts, and the grabs the variables that need to be
convereted to C from these.
A Simple Example
The following simple example demonstrates how to build an extension module within a Python function:
# examples/increment_example.py
from weave import ext_tools
def build_increment_ext():
""" Build a simple extension with functions that increment numbers.
The extension will be built in the local directory.
"""
mod = ext_tools.ext_module(’increment_ext’)
a = 1 # effectively a type declaration for ’a’ in the
# following functions.
ext_code = "return_val = Py::new_reference_to(Py::Int(a+1));"
func = ext_tools.ext_function(’increment’,ext_code,[’a’])
mod.add_function(func)
ext_code = "return_val = Py::new_reference_to(Py::Int(a+2));"
func = ext_tools.ext_function(’increment_by_2’,ext_code,[’a’])
mod.add_function(func)
mod.compile()
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The function build_increment_ext() creates an extension module named increment_ext and compiles
it to a shared library (.so or .pyd) that can be loaded into Python.. increment_ext contains two functions,
increment and increment_by_2. The first line of build_increment_ext(),
mod = ext_tools.ext_module(‘increment_ext’)
creates an ext_module instance that is ready to have ext_function instances added to it. ext_function
instances are created much with a calling convention similar to weave.inline(). The most common call includes
a C/C++ code snippet and a list of the arguments for the function. The following
ext_code
=
“return_val
=
Py::new_reference_to(Py::Int(a+1));”
ext_tools.ext_function(‘increment’,ext_code,[’a’])
func
=
creates a C/C++ extension function that is equivalent to the following Python function:
def increment(a):
return a + 1
A second method is also added to the module and then,
mod.compile()
is called to build the extension module. By default, the module is created in the current working directory. This example is available in the examples/increment_example.py file found in the weave directory. At the bottom of
the file in the module’s “main” program, an attempt to import increment_ext without building it is made. If this
fails (the module doesn’t exist in the PYTHONPATH), the module is built by calling build_increment_ext().
This approach only takes the time consuming ( a few seconds for this example) process of building the module if it
hasn’t been built before.
if __name__ == "__main__":
try:
import increment_ext
except ImportError:
build_increment_ext()
import increment_ext
a = 1
print ’a, a+1:’, a, increment_ext.increment(a)
print ’a, a+2:’, a, increment_ext.increment_by_2(a)
Note: If we were willing to always pay the penalty of building the C++ code for a module, we could store
the md5 checksum of the C++ code along with some information about the compiler, platform, etc. Then,
ext_module.compile() could try importing the module before it actually compiles it, check the md5 checksum and other meta-data in the imported module with the meta-data of the code it just produced and only compile the
code if the module didn’t exist or the meta-data didn’t match. This would reduce the above code to:
if __name__ == "__main__":
build_increment_ext()
a = 1
print ’a, a+1:’, a, increment_ext.increment(a)
print ’a, a+2:’, a, increment_ext.increment_by_2(a)
Note: There would always be the overhead of building the C++ code, but it would only actually compile the code
once. You pay a little in overhead and get cleaner “import” code. Needs some thought.
If you run increment_example.py from the command line, you get the following:
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[[email protected]]$ python increment_example.py
a, a+1: 1 2
a, a+2: 1 3
If the module didn’t exist before it was run, the module is created. If it did exist, it is just imported and used.
Fibonacci Example
examples/fibonacci.py provides a little more complex example of how to use ext_tools. Fibonacci numbers are a series of numbers where each number in the series is the sum of the previous two: 1, 1, 2, 3, 5, 8, etc. Here,
the first two numbers in the series are taken to be 1. One approach to calculating Fibonacci numbers uses recursive
function calls. In Python, it might be written as:
def fib(a):
if a <= 2:
return 1
else:
return fib(a-2) + fib(a-1)
In C, the same function would look something like this:
int fib(int a)
{
if(a <= 2)
return 1;
else
return fib(a-2) + fib(a-1);
}
Recursion is much faster in C than in Python, so it would be beneficial to use the C version for fibonacci number
calculations instead of the Python version. We need an extension function that calls this C function to do this. This
is possible by including the above code snippet as “support code” and then calling it from the extension function.
Support code snippets (usually structure definitions, helper functions and the like) are inserted into the extension
module C/C++ file before the extension function code. Here is how to build the C version of the fibonacci number
generator:
def build_fibonacci():
""" Builds an extension module with fibonacci calculators.
"""
mod = ext_tools.ext_module(’fibonacci_ext’)
a = 1 # this is effectively a type declaration
# recursive fibonacci in C
fib_code = """
int fib1(int a)
{
if(a <= 2)
return 1;
else
return fib1(a-2) + fib1(a-1);
}
"""
ext_code = """
int val = fib1(a);
return_val = Py::new_reference_to(Py::Int(val));
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"""
fib = ext_tools.ext_function(’fib’,ext_code,[’a’])
fib.customize.add_support_code(fib_code)
mod.add_function(fib)
mod.compile()
XXX More about custom_info, and what xxx_info instances are good for.
Note: recursion is not the fastest way to calculate fibonacci numbers, but this approach serves nicely for this example.
1.13.10 Customizing Type Conversions – Type Factories
not written
1.13.11 Things I wish weave did
It is possible to get name clashes if you uses a variable name that is already defined in a header automatically included
(such as stdio.h) For instance, if you try to pass in a variable named stdout, you’ll get a cryptic error report due
to the fact that stdio.h also defines the name. weave should probably try and handle this in some way. Other
things...
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CHAPTER
TWO
RELEASE NOTES
2.1 SciPy 0.8.0 Release Notes
Contents
• Release Notes
– SciPy 0.8.0 Release Notes
* Python 3
* Major documentation improvements
* Deprecated features
· Swapping inputs for correlation functions (scipy.signal)
· Obsolete code deprecated (scipy.misc)
· Additional deprecations
* New features
· DCT support (scipy.fftpack)
· Single precision support for fft functions (scipy.fftpack)
· Correlation functions now implement the usual definition (scipy.signal)
· Additions and modification to LTI functions (scipy.signal)
· Improved waveform generators (scipy.signal)
· New functions and other changes in scipy.linalg
· New function and changes in scipy.optimize
· New sparse least squares solver
· ARPACK-based sparse SVD
· Alternative behavior available for scipy.constants.find
· Incomplete sparse LU decompositions
· Faster matlab file reader and default behavior change
· Faster evaluation of orthogonal polynomials
· Lambert W function
· Improved hypergeometric 2F1 function
· More flexible interface for Radial basis function interpolation
* Removed features
· scipy.io
SciPy 0.8.0 is the culmination of 17 months of hard work. It contains many new features, numerous bug-fixes,
improved test coverage and better documentation. There have been a number of deprecations and API changes in
this release, which are documented below. All users are encouraged to upgrade to this release, as there are a large
number of bug-fixes and optimizations. Moreover, our development attention will now shift to bug-fix releases on the
0.8.x branch, and on adding new features on the development trunk. This release requires Python 2.4 - 2.6 and NumPy
1.4.1 or greater.
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Please note that SciPy is still considered to have “Beta” status, as we work toward a SciPy 1.0.0 release. The 1.0.0
release will mark a major milestone in the development of SciPy, after which changing the package structure or API
will be much more difficult. Whilst these pre-1.0 releases are considered to have “Beta” status, we are committed to
making them as bug-free as possible.
However, until the 1.0 release, we are aggressively reviewing and refining the functionality, organization, and interface.
This is being done in an effort to make the package as coherent, intuitive, and useful as possible. To achieve this, we
need help from the community of users. Specifically, we need feedback regarding all aspects of the project - everything
- from which algorithms we implement, to details about our function’s call signatures.
2.1.1 Python 3
Python 3 compatibility is planned and is currently technically feasible, since Numpy has been ported. However, since
the Python 3 compatible Numpy 1.5 has not been released yet, support for Python 3 in Scipy is not yet included in
Scipy 0.8. SciPy 0.9, planned for fall 2010, will very likely include experimental support for Python 3.
2.1.2 Major documentation improvements
SciPy documentation is greatly improved.
2.1.3 Deprecated features
Swapping inputs for correlation functions (scipy.signal)
Concern correlate, correlate2d, convolve and convolve2d. If the second input is larger than the first input, the inputs
are swapped before calling the underlying computation routine. This behavior is deprecated, and will be removed in
scipy 0.9.0.
Obsolete code deprecated (scipy.misc)
The modules helpmod, ppimport and pexec from scipy.misc are deprecated. They will be removed from SciPy in
version 0.9.
Additional deprecations
• linalg: The function solveh_banded currently returns a tuple containing the Cholesky factorization and the
solution to the linear system. In SciPy 0.9, the return value will be just the solution.
• The function constants.codata.find will generate a DeprecationWarning. In Scipy version 0.8.0, the keyword
argument ‘disp’ was added to the function, with the default value ‘True’. In 0.9.0, the default will be ‘False’.
• The qshape keyword argument of signal.chirp is deprecated. Use the argument vertex_zero instead.
• Passing the coefficients of a polynomial as the argument f0 to signal.chirp is deprecated. Use the function
signal.sweep_poly instead.
• The io.recaster module has been deprecated and will be removed in 0.9.0.
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2.1.4 New features
DCT support (scipy.fftpack)
New realtransforms have been added, namely dct and idct for Discrete Cosine Transform; type I, II and III are available.
Single precision support for fft functions (scipy.fftpack)
fft functions can now handle single precision inputs as well: fft(x) will return a single precision array if x is single
precision.
At the moment, for FFT sizes that are not composites of 2, 3, and 5, the transform is computed internally in double
precision to avoid rounding error in FFTPACK.
Correlation functions now implement the usual definition (scipy.signal)
The outputs should now correspond to their matlab and R counterparts, and do what most people expect if the
old_behavior=False argument is passed:
• correlate, convolve and their 2d counterparts do not swap their inputs depending on their relative shape anymore;
• correlation functions now conjugate their second argument while computing the slided sum-products, which
correspond to the usual definition of correlation.
Additions and modification to LTI functions (scipy.signal)
• The functions impulse2 and step2 were added to scipy.signal.
They use the function
scipy.signal.lsim2 to compute the impulse and step response of a system, respectively.
• The function scipy.signal.lsim2 was changed to pass any additional keyword arguments to the ODE
solver.
Improved waveform generators (scipy.signal)
Several improvements to the chirp function in scipy.signal were made:
• The waveform generated when method=”logarithmic” was corrected; it now generates a waveform that is also
known as an “exponential” or “geometric” chirp. (See http://en.wikipedia.org/wiki/Chirp.)
• A new chirp method, “hyperbolic”, was added.
• Instead of the keyword qshape, chirp now uses the keyword vertex_zero, a boolean.
• chirp no longer handles an arbitrary polynomial. This functionality has been moved to a new function,
sweep_poly.
A new function, sweep_poly, was added.
New functions and other changes in scipy.linalg
The functions cho_solve_banded, circulant, companion, hadamard and leslie were added to scipy.linalg.
The function block_diag was enhanced to accept scalar and 1D arguments, along with the usual 2D arguments.
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New function and changes in scipy.optimize
The curve_fit function has been added; it takes a function and uses non-linear least squares to fit that to the provided
data.
The leastsq and fsolve functions now return an array of size one instead of a scalar when solving for a single parameter.
New sparse least squares solver
The lsqr function was added to scipy.sparse. This routine finds a least-squares solution to a large, sparse, linear
system of equations.
ARPACK-based sparse SVD
A naive implementation of SVD for sparse matrices is available in scipy.sparse.linalg.eigen.arpack. It is based on
using an symmetric solver on <A, A>, and as such may not be very precise.
Alternative behavior available for scipy.constants.find
The keyword argument disp was added to the function scipy.constants.find, with the default value True.
When disp is True, the behavior is the same as in Scipy version 0.7. When False, the function returns the list of keys
instead of printing them. (In SciPy version 0.9, the default will be reversed.)
Incomplete sparse LU decompositions
Scipy now wraps SuperLU version 4.0, which supports incomplete sparse LU decompositions. These can be accessed
via scipy.sparse.linalg.spilu. Upgrade to SuperLU 4.0 also fixes some known bugs.
Faster matlab file reader and default behavior change
We’ve rewritten the matlab file reader in Cython and it should now read matlab files at around the same speed that
Matlab does.
The reader reads matlab named and anonymous functions, but it can’t write them.
Until scipy 0.8.0 we have returned arrays of matlab structs as numpy object arrays, where the objects have attributes
named for the struct fields. As of 0.8.0, we return matlab structs as numpy structured arrays. You can get the older
behavior by using the optional struct_as_record=False keyword argument to scipy.io.loadmat and
friends.
There is an inconsistency in the matlab file writer, in that it writes numpy 1D arrays as column vectors in matlab 5
files, and row vectors in matlab 4 files. We will change this in the next version, so both write row vectors. There is
a FutureWarning when calling the writer to warn of this change; for now we suggest using the oned_as=’row’
keyword argument to scipy.io.savemat and friends.
Faster evaluation of orthogonal polynomials
Values of orthogonal polynomials can be evaluated with new vectorized functions in scipy.special:
eval_legendre, eval_chebyt, eval_chebyu, eval_chebyc, eval_chebys, eval_jacobi, eval_laguerre, eval_genlaguerre,
eval_hermite, eval_hermitenorm, eval_gegenbauer, eval_sh_legendre, eval_sh_chebyt, eval_sh_chebyu,
eval_sh_jacobi. This is faster than constructing the full coefficient representation of the polynomials, which
was previously the only available way.
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Note that the previous orthogonal polynomial routines will now also invoke this feature, when possible.
Lambert W function
scipy.special.lambertw can now be used for evaluating the Lambert W function.
Improved hypergeometric 2F1 function
Implementation of scipy.special.hyp2f1 for real parameters was revised. The new version should produce
accurate values for all real parameters.
More flexible interface for Radial basis function interpolation
The scipy.interpolate.Rbf class now accepts a callable as input for the “function” argument, in addition to
the built-in radial basis functions which can be selected with a string argument.
2.1.5 Removed features
scipy.stsci: the package was removed
The module scipy.misc.limits was removed.
The IO code in both NumPy and SciPy is being extensively reworked. NumPy will be where basic code for reading
and writing NumPy arrays is located, while SciPy will house file readers and writers for various data formats (data,
audio, video, images, matlab, etc.).
Several functions in scipy.io are removed in the 0.8.0 release including: npfile, save, load, create_module, create_shelf, objload, objsave, fopen, read_array, write_array, fread, fwrite, bswap, packbits, unpackbits, and convert_objectarray. Some of these functions have been replaced by NumPy’s raw reading and writing capabilities,
memory-mapping capabilities, or array methods. Others have been moved from SciPy to NumPy, since basic array
reading and writing capability is now handled by NumPy.
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CHAPTER
THREE
REFERENCE
3.1 Clustering package (scipy.cluster)
3.1.1 Hierarchical clustering (scipy.cluster.hierarchy)
Warning: This documentation is work-in-progress and unorganized.
Function Reference
These functions cut hierarchical clusterings into flat clusterings or find the roots of the forest formed by a cut by
providing the flat cluster ids of each observation.
Function
fcluster
fclusterdata
leaders
Description
forms flat clusters from hierarchical clusters.
forms flat clusters directly from data.
singleton root nodes for flat cluster.
These are routines for agglomerative clustering.
Function
linkage
single
complete
average
weighted
centroid
median
ward
Description
agglomeratively clusters original observations.
the single/min/nearest algorithm. (alias)
the complete/max/farthest algorithm. (alias)
the average/UPGMA algorithm. (alias)
the weighted/WPGMA algorithm. (alias)
the centroid/UPGMC algorithm. (alias)
the median/WPGMC algorithm. (alias)
the Ward/incremental algorithm. (alias)
These routines compute statistics on hierarchies.
Function
cophenet
from_mlab_linkage
inconsistent
maxinconsts
maxdists
maxRstat
to_mlab_linkage
Description
computes the cophenetic distance between leaves.
converts a linkage produced by MATLAB(TM).
the inconsistency coefficients for cluster.
the maximum inconsistency coefficient for each cluster.
the maximum distance for each cluster.
the maximum specific statistic for each cluster.
converts a linkage to one MATLAB(TM) can understand.
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Routines for visualizing flat clusters.
Function
dendrogram
Description
visualizes linkages (requires matplotlib).
These are data structures and routines for representing hierarchies as tree objects.
Function
ClusterNode
leaves_list
to_tree
Description
represents cluster nodes in a cluster hierarchy.
a left-to-right traversal of the leaves.
represents a linkage matrix as a tree object.
These are predicates for checking the validity of linkage and inconsistency matrices as well as for checking isomorphism of two flat cluster assignments.
Function
is_valid_im
is_valid_linkage
is_isomorphic
is_monotonic
correspond
num_obs_linkage
Description
checks for a valid inconsistency matrix.
checks for a valid hierarchical clustering.
checks if two flat clusterings are isomorphic.
checks if a linkage is monotonic.
checks whether a condensed distance matrix corresponds with a linkage
the number of observations corresponding to a linkage matrix.
• MATLAB and MathWorks are registered trademarks of The MathWorks, Inc.
• Mathematica is a registered trademark of The Wolfram Research, Inc.
References
Copyright Notice
Copyright (C) Damian Eads, 2007-2008. New BSD License.
class ClusterNode(id, left=None, right=None, dist=0, count=1)
A tree node class for representing a cluster. Leaf nodes correspond to original observations, while non-leaf
nodes correspond to non-singleton clusters.
The to_tree function converts a matrix returned by the linkage function into an easy-to-use tree representation.
Seealso
• to_tree: for converting a linkage matrix Z into a tree object.
Methods
get_count()
The number of leaf nodes (original observations) belonging to the cluster node nd. If the target node is a
leaf, 1 is returned.
Returns
c
[int] The number of leaf nodes below the target node.
get_id()
The identifier of the target node. For 0 ≤ i < n, i corresponds to original observation i. For n ≤ i <
2n − 1, i corresponds to non-singleton cluster formed at iteration i − n.
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Returns
id
[int] The identifier of the target node.
get_left()
Returns a reference to the left child tree object. If the node is a leaf, None is returned.
Returns
left
[ClusterNode] The left child of the target node.
get_right()
Returns a reference to the right child tree object. If the node is a leaf, None is returned.
Returns
right
[ClusterNode] The left child of the target node.
is_leaf()
Returns True iff the target node is a leaf.
Returns
leafness
[bool] True if the target node is a leaf node.
pre_order(func=<function <lambda> at 0x3090870>)
Performs preorder traversal without recursive function calls. When a leaf node is first encountered, func
is called with the leaf node as its argument, and its result is appended to the list.
For example, the statement:
ids = root.pre_order(lambda x: x.id)
returns a list of the node ids corresponding to the leaf nodes of the tree as they appear from left to right.
Parameters
• func : function Applied to each leaf ClusterNode object in the pre-order traversal. Given
the i’th leaf node in the pre-order traversal n[i], the result of func(n[i]) is stored in L[i].
If not provided, the index of the original observation to which the node corresponds is
used.
Returns
• L : list The pre-order traversal.
average(y)
Performs average/UPGMA linkage on the condensed distance matrix y. See linkage for more information
on the return structure and algorithm.
Parameters
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y
[ndarray] The upper triangular of the distance matrix. The result of pdist is returned in
this form.
Returns
Z
[ndarray] A linkage matrix containing the hierarchical clustering. See the linkage function documentation for more information on its structure.
Seealso
• linkage: for advanced creation of hierarchical clusterings.
centroid(y)
Performs centroid/UPGMC linkage. See linkage for more information on the return structure and algorithm.
The following are common calling conventions:
1.Z = centroid(y)
Performs centroid/UPGMC linkage on the condensed distance matrix y. See linkage for more information on the return structure and algorithm.
2.Z = centroid(X)
Performs centroid/UPGMC linkage on the observation matrix X using Euclidean distance as the distance
metric. See linkage for more information on the return structure and algorithm.
Parameters
Q
[ndarray] A condensed or redundant distance matrix. A condensed distance matrix is a flat
array containing the upper triangular of the distance matrix. This is the form that pdist
returns. Alternatively, a collection of m observation vectors in n dimensions may be passed
as a m by n array.
Returns
Z
[ndarray] A linkage matrix containing the hierarchical clustering. See the linkage function documentation for more information on its structure.
Seealso
• linkage: for advanced creation of hierarchical clusterings.
complete(y)
Performs complete complete/max/farthest point linkage on the condensed distance matrix y. See linkage for
more information on the return structure and algorithm.
Parameters
y
[ndarray] The upper triangular of the distance matrix. The result of pdist is returned in
this form.
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Returns
Z
[ndarray] A linkage matrix containing the hierarchical clustering. See the linkage function documentation for more information on its structure.
cophenet(Z, Y=None)
Calculates the cophenetic distances between each observation in the hierarchical clustering defined by the
linkage Z.
Suppose p and q are original observations in disjoint clusters s and t, respectively and s and t are joined by
a direct parent cluster u. The cophenetic distance between observations i and j is simply the distance between
clusters s and t.
Parameters
• Z : ndarray The hierarchical clustering encoded as an array (see linkage function).
• Y : ndarray (optional) Calculates the cophenetic correlation coefficient c of a hierarchical
clustering defined by the linkage matrix Z of a set of n observations in m dimensions. Y is
the condensed distance matrix from which Z was generated.
Returns
(c, {d}) - c : ndarray
The cophentic correlation distance (if y is passed).
• d : ndarray The cophenetic distance matrix in condensed form. The ij th entry is the cophenetic distance between original observations i and j.
correspond(Z, Y)
Checks if a linkage matrix Z and condensed distance matrix Y could possibly correspond to one another.
They must have the same number of original observations for the check to succeed.
This function is useful as a sanity check in algorithms that make extensive use of linkage and distance matrices
that must correspond to the same set of original observations.
Arguments
• Z
[ndarray] The linkage matrix to check for correspondance.
• Y
[ndarray] The condensed distance matrix to check for correspondance.
Returns
• b
[bool] A boolean indicating whether the linkage matrix and distance matrix could possibly
correspond to one another.
dendrogram(Z, p=30,
truncate_mode=None,
color_threshold=None,
get_leaves=True,
orientation=’top’, labels=None, count_sort=False, distance_sort=False, show_leaf_counts=True,
no_plot=False, no_labels=False, color_list=None, leaf_font_size=None, leaf_rotation=None,
leaf_label_func=None, no_leaves=False, show_contracted=False, link_color_func=None)
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Plots the hiearchical clustering defined by the linkage Z as a dendrogram. The dendrogram illustrates
how each cluster is composed by drawing a U-shaped link between a non-singleton cluster and its
children. The height of the top of the U-link is the distance between its children clusters. It is also
the cophenetic distance between original observations in the two children clusters. It is expected that
the distances in Z[:,2] be monotonic, otherwise crossings appear in the dendrogram.
Arguments
• Z : ndarray The linkage matrix encoding the hierarchical clustering to render as a
dendrogram. See the linkage function for more information on the format of Z.
• truncate_mode : string The dendrogram can be hard to read when the original observation matrix from which the linkage is derived is large. Truncation is used to
condense the dendrogram. There are several modes:
– None/’none’: no truncation is performed (Default)
– ‘lastp’: the last p non-singleton formed in the linkage
are the only non-leaf nodes in the linkage; they correspond to to rows
Z[n-p-2:end] in Z. All other non-singleton clusters are contracted into leaf
nodes.
– ‘mlab’: This corresponds to MATLAB(TM) behavior. (not
implemented yet)
– ‘level’/’mtica’: no more than p levels of the
dendrogram tree are displayed. This corresponds to Mathematica(TM) behavior.
• p : int The p parameter for truncate_mode.
‘
• color_threshold : double For brevity, let t be the color_threshold. Colors all the descendent
links below a cluster node k the same color if k is the first node below the cut threshold t. All links
connecting nodes with distances greater than or equal to the threshold are colored blue. If t is less
than or equal to zero, all nodes are colored blue. If color_threshold is None or ‘default’,
corresponding with MATLAB(TM) behavior, the threshold is set to 0.7*max(Z[:,2]).
• get_leaves : bool Includes a list R[’leaves’]=H in the result dictionary. For each i, H[i]
== j, cluster node j appears in the i th position in the left-to-right traversal of the leaves, where
j < 2n − 1 and i < n.
• orientation : string The direction to plot the dendrogram, which can be any of the following strings
– ‘top’: plots the root at the top, and plot descendent
links going downwards. (default).
– ‘bottom’: plots the root at the bottom, and plot descendent
links going upwards.
– ‘left’: plots the root at the left, and plot descendent
links going right.
– ‘right’: plots the root at the right, and plot descendent
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links going left.
• labels : ndarray By default labels is None so the index of the original observation is used to label
the leaf nodes.
Otherwise, this is an n -sized list (or tuple). The labels[i] value is the text to put under the i th
leaf node only if it corresponds to an original observation and not a non-singleton cluster.
• count_sort : string/bool For each node n, the order (visually, from left-to-right) n’s two descendent
links are plotted is determined by this parameter, which can be any of the following values:
– False: nothing is done.
– ‘ascending’/True: the child with the minimum number of
original objects in its cluster is plotted first.
– ‘descendent’: the child with the maximum number of
original objects in its cluster is plotted first.
Note distance_sort and count_sort cannot both be True.
• distance_sort : string/bool For each node n, the order (visually, from left-to-right) n’s two descendent
links are plotted is determined by this parameter, which can be any of the following values:
– False: nothing is done.
– ‘ascending’/True: the child with the minimum distance
between its direct descendents is plotted first.
– ‘descending’: the child with the maximum distance
between its direct descendents is plotted first.
Note distance_sort and count_sort cannot both be True.
• show_leaf_counts : bool
When True, leaf nodes representing k > 1 original observation are labeled with the number of
observations they contain in parentheses.
• no_plot : bool When True, the final rendering is not performed. This is useful if only the data
structures computed for the rendering are needed or if matplotlib is not available.
• no_labels : bool When True, no labels appear next to the leaf nodes in the rendering of the dendrogram.
• leaf_label_rotation : double
Specifies the angle (in degrees) to rotate the leaf labels. When unspecified, the rotation based on the
number of nodes in the dendrogram. (Default=0)
• leaf_font_size : int Specifies the font size (in points) of the leaf labels. When unspecified, the size
based on the number of nodes in the dendrogram.
• leaf_label_func : lambda or function
When leaf_label_func is a callable function, for each leaf with cluster index k < 2n − 1. The
function is expected to return a string with the label for the leaf.
Indices k < n correspond to original observations while indices k ≥ n correspond to non-singleton
clusters.
For example, to label singletons with their node id and non-singletons with their id, count, and
inconsistency coefficient, simply do:
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# First define the leaf label function.
def llf(id):
if id < n:
return str(id)
else:
return ’[%d %d %1.2f]’ % (id, count, R[n-id,3])
# The text for the leaf nodes is going to be big so force
# a rotation of 90 degrees.
dendrogram(Z, leaf_label_func=llf, leaf_rotation=90)
• show_contracted : bool When True the heights of non-singleton nodes contracted into a leaf node
are plotted as crosses along the link connecting that leaf node. This really is only useful when
truncation is used (see truncate_mode parameter).
• link_color_func : lambda/function When a callable function, link_color_function is called with each
non-singleton id corresponding to each U-shaped link it will paint. The function is expected to return
the color to paint the link, encoded as a matplotlib color string code.
For example:
dendrogram(Z, link_color_func=lambda k: colors[k])
colors the direct links below each untruncated non-singleton node k using colors[k].
Returns
• R : dict A dictionary of data structures computed to render the dendrogram. Its has
the following keys:
– ‘icoords’: a list of lists [I1, I2, ..., Ip] where
Ik is a list of 4 independent variable coordinates corresponding to the line that
represents the k’th link painted.
– ‘dcoords’: a list of lists [I2, I2, ..., Ip] where
Ik is a list of 4 independent variable coordinates corresponding to the line that
represents the k’th link painted.
– ‘ivl’: a list of labels corresponding to the leaf nodes.
– ‘leaves’: for each i, H[i] == j, cluster node
j appears in the i th position in the left-to-right traversal of the leaves, where
j < 2n − 1 and i < n. If j is less than n, the i th leaf node corresponds to an
original observation. Otherwise, it corresponds to a non-singleton cluster.
fcluster(Z, t, criterion=’inconsistent’, depth=2, R=None, monocrit=None)
Forms flat clusters from the hierarchical clustering defined by the linkage matrix Z. The threshold t is a required
parameter.
Arguments
• Z : ndarray The hierarchical clustering encoded with the matrix returned by the linkage
function.
• t : double The threshold to apply when forming flat clusters.
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• criterion : string (optional) The criterion to use in forming flat clusters. This can be any
of the following values:
– ‘inconsistent’: If a cluster node and all its
decendents have an inconsistent value less than or equal to t then all its leaf
descendents belong to the same flat cluster. When no non-singleton cluster meets
this criterion, every node is assigned to its own cluster. (Default)
– ‘distance’: Forms flat clusters so that the original
observations in each flat cluster have no greater a cophenetic distance than t.
– ‘maxclust’: Finds a minimum threshold r so that
the cophenetic distance between any two original observations in the same flat
cluster is no more than r and no more than t flat clusters are formed.
– ‘monocrit’: Forms a flat cluster from a cluster node c
with index i when monocrit[j] <= t.
For example, to threshold on the maximum mean distance as computed in the
inconsistency matrix R with a threshold of 0.8 do:
MR = maxRstat(Z, R, 3)
cluster(Z, t=0.8, criterion=’monocrit’, monocrit=MR)
– ‘maxclust_monocrit’: Forms a flat cluster from a
non-singleton cluster node c when monocrit[i] <= r for all cluster indices
i below and including c. r is minimized such that no more than t flat clusters
are formed. monocrit must be monotonic. For example, to minimize the threshold t on maximum inconsistency values so that no more than 3 flat clusters are
formed, do:
MI = maxinconsts(Z, R) cluster(Z, t=3, criterion=’maxclust_monocrit’,
monocrit=MI)
• depth : int (optional) The maximum depth to perform the inconsistency calculation. It
has no meaning for the other criteria. (default=2)
• R : ndarray (optional) The inconsistency matrix to use for the ‘inconsistent’ criterion.
This matrix is computed if not provided.
• monocrit : ndarray (optional) A (n-1) numpy vector of doubles. monocrit[i] is
the statistics upon which non-singleton i is thresholded. The monocrit vector must be
monotonic, i.e. given a node c with index i, for all node indices j corresponding to nodes
below c, monocrit[i] >= monocrit[j].
Returns
• T
[ndarray] A vector of length n. T[i] is the flat cluster number to which original
observation i belongs.
fclusterdata(X, t, criterion=’inconsistent’, metric=’euclidean’, depth=2, method=’single’, R=None)
T = fclusterdata(X, t)
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Clusters the original observations in the n by m data matrix X (n observations in m dimensions), using the
euclidean distance metric to calculate distances between original observations, performs hierarchical clustering
using the single linkage algorithm, and forms flat clusters using the inconsistency method with t as the cut-off
threshold.
A one-dimensional numpy array T of length n is returned. T[i] is the index of the flat cluster to which the
original observation i belongs.
Arguments
• X : ndarray n by m data matrix with n observations in m dimensions.
• t : double The threshold to apply when forming flat clusters.
• criterion : string Specifies the criterion for forming flat clusters. Valid values are ‘inconsistent’, ‘distance’, or ‘maxclust’ cluster formation algorithms. See fcluster for
descriptions.
• method : string The linkage method to use (single, complete, average, weighted, median
centroid, ward). See linkage for more information.
• metric : string The distance metric for calculating pairwise distances. See distance.pdist
for descriptions and linkage to verify compatibility with the linkage method.
• t : double The cut-off threshold for the cluster function or the maximum number of
clusters (criterion=’maxclust’).
• depth : int The maximum depth for the inconsistency calculation. See inconsistent
for more information.
• R : ndarray The inconsistency matrix. It will be computed if necessary if it is not passed.
Returns
• T : ndarray A vector of length n. T[i] is the flat cluster number to which original
observation i belongs.
Notes
This function is similar to MATLAB(TM) clusterdata function.
from_mlab_linkage(Z)
Converts a linkage matrix generated by MATLAB(TM) to a new linkage matrix compatible with this module.
The conversion does two things:
•the indices are converted from 1..N to 0..(N-1) form, and
•a fourth column Z[:,3] is added where Z[i,3] is represents the number of original observations (leaves) in
the non-singleton cluster i.
This function is useful when loading in linkages from legacy data files generated by MATLAB.
Arguments
• Z
[ndarray] A linkage matrix generated by MATLAB(TM)
Returns
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• ZS
[ndarray] A linkage matrix compatible with this library.
inconsistent(Z, d=2)
Calculates inconsistency statistics on a linkage.
Note: This function behaves similarly to the MATLAB(TM) inconsistent function.
Parameters
• d
[int] The number of links up to d levels below each non-singleton cluster
• Z
[ndarray] The (n − 1) by 4 matrix encoding the linkage (hierarchical clustering). See
linkage documentation for more information on its form.
Returns
• R
[ndarray] A (n − 1) by 5 matrix where the i‘th row contains the link statistics for
the non-singleton cluster i. The link statistics are computed over the link heights for
links d levels below the cluster i. R[i,0] and R[i,1] are the mean and standard
deviation of the link heights, respectively; R[i,2] is the number of links included
in the calculation; and R[i,3] is the inconsistency coefficient,
Z[i, 2] − R[i, 0]
.
R[i, 1]
is_isomorphic(T1, T2)
Determines if two different cluster assignments T1 and T2 are equivalent.
Arguments
• T1 : ndarray An assignment of singleton cluster ids to flat cluster ids.
• T2 : ndarray An assignment of singleton cluster ids to flat cluster ids.
Returns
• b : boolean Whether the flat cluster assignments T1 and T2 are equivalent.
is_monotonic(Z)
Returns True if the linkage passed is monotonic. The linkage is monotonic if for every cluster s and t joined,
the distance between them is no less than the distance between any previously joined clusters.
Arguments
• Z : ndarray The linkage matrix to check for monotonicity.
Returns
• b : bool A boolean indicating whether the linkage is monotonic.
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is_valid_im(R, warning=False, throw=False, name=None)
Returns True if the inconsistency matrix passed is valid. It must be a n by 4 numpy array of doubles. The
standard deviations R[:,1] must be nonnegative. The link counts R[:,2] must be positive and no greater
than n − 1.
Arguments
• R : ndarray The inconsistency matrix to check for validity.
• warning : bool When True, issues a Python warning if the linkage matrix passed is
invalid.
• throw : bool When True, throws a Python exception if the linkage matrix passed is
invalid.
• name : string This string refers to the variable name of the invalid linkage matrix.
Returns
• b : bool True iff the inconsistency matrix is valid.
is_valid_linkage(Z, warning=False, throw=False, name=None)
Checks the validity of a linkage matrix. A linkage matrix is valid if it is a two dimensional nd-array (type
double) with n rows and 4 columns. The first two columns must contain indices between 0 and 2n − 1. For a
given row i, 0 ≤ Z[i, 0] ≤ i + n − 1 and 0 ≤ Z[i, 1] ≤ i + n − 1 (i.e. a cluster cannot join another cluster
unless the cluster being joined has been generated.)
Arguments
• warning : bool When True, issues a Python warning if the linkage matrix passed is
invalid.
• throw : bool When True, throws a Python exception if the linkage matrix passed is
invalid.
• name : string This string refers to the variable name of the invalid linkage matrix.
Returns
• b
[bool] True iff the inconsistency matrix is valid.
leaders(Z, T)
(L, M) = leaders(Z, T):
Returns the root nodes in a hierarchical clustering corresponding to a cut defined by a flat cluster assignment
vector T. See the fcluster function for more information on the format of T.
For each flat cluster j of the k flat clusters represented in the n-sized flat cluster assignment vector T, this
function finds the lowest cluster node i in the linkage tree Z such that:
•leaf descendents belong only to flat cluster j (i.e. T[p]==j for all p in S(i) where S(i) is the set of leaf
ids of leaf nodes descendent with cluster node i)
•there does not exist a leaf that is not descendent with i that also belongs to cluster j (i.e. T[q]!=j for
all q not in S(i)). If this condition is violated, T is not a valid cluster assignment vector, and an exception
will be thrown.
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Arguments
• Z
[ndarray] The hierarchical clustering encoded as a matrix. See linkage for more
information.
• T
[ndarray] The flat cluster assignment vector.
Returns
(L, M)
• L
[ndarray] The leader linkage node id’s stored as a k-element 1D array where k is the
number of flat clusters found in T.
L[j]=i is the linkage cluster node id that is the leader of flat cluster with id M[j].
If i < n, i corresponds to an original observation, otherwise it corresponds to a
non-singleton cluster.
For example: if L[3]=2 and M[3]=8, the flat cluster with id 8’s leader is linkage
node 2.
• M
[ndarray] The leader linkage node id’s stored as a k-element 1D array where k is the
number of flat clusters found in T. This allows the set of flat cluster ids to be any
arbitrary set of k integers.
leaves_list(Z)
Returns a list of leaf node ids (corresponding to observation vector index) as they appear in the tree from left to
right. Z is a linkage matrix.
Arguments
• Z
[ndarray] The hierarchical clustering encoded as a matrix. See linkage for more
information.
Returns
• L
[ndarray] The list of leaf node ids.
linkage(y, method=’single’, metric=’euclidean’)
Performs hierarchical/agglomerative clusteringon the
condensed distance matrix y. y must be a n2 sized vector where n is the number of original observations
paired in the distance matrix. The behavior of this function is very similar to the MATLAB(TM) linkage
function.
A 4 by (n − 1) matrix Z is returned. At the i-th iteration, clusters with indices Z[i, 0] and Z[i, 1]
are combined to form cluster n+i. A cluster with an index less than n corresponds to one of the n original
observations. The distance between clusters Z[i, 0] and Z[i, 1] is given by Z[i, 2]. The fourth
value Z[i, 3] represents the number of original observations in the newly formed cluster.
The following linkage methods are used to compute the distance d(s, t) between two clusters s and t. The
algorithm begins with a forest of clusters that have yet to be used in the hierarchy being formed. When
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two clusters s and t from this forest are combined into a single cluster u, s and t are removed from the
forest, and u is added to the forest. When only one cluster remains in the forest, the algorithm stops, and
this cluster becomes the root.
A distance matrix is maintained at each iteration. The d[i,j] entry corresponds to the distance between
cluster i and j in the original forest.
At each iteration, the algorithm must update the distance matrix to reflect the distance of the newly formed
cluster u with the remaining clusters in the forest.
Suppose there are |u| original observations u[0], . . . , u[|u| − 1] in cluster u and |v| original objects
v[0], . . . , v[|v| − 1] in cluster v. Recall s and t are combined to form cluster u. Let v be any remaining cluster in the forest that is not u.
The following are methods for calculating the distance between the newly formed cluster u and each v.
• method=’single’ assigns
d(u, v) = min(dist(u[i], v[j]))
for all points i in cluster u and j in cluster v. This is also known as the Nearest Point Algorithm.
• method=’complete’ assigns
d(u, v) = max(dist(u[i], v[j]))
for all points i in cluster u and j in cluster v. This is also known by the Farthest Point Algorithm or
Voor Hees Algorithm.
• method=’average’ assigns
d(u, v) =
X d(u[i], v[j])
ij
(|u| ∗ |v|)
for all points i and j where |u| and |v| are the cardinalities of clusters u and v, respectively. This is
also called the UPGMA algorithm. This is called UPGMA.
• method=’weighted’ assigns
d(u, v) = (dist(s, v) + dist(t, v))/2
where cluster u was formed with cluster s and t and v is a remaining cluster in the forest. (also called
WPGMA)
• method=’centroid’ assigns
dist(s, t) = ||cs − ct ||2
where cs and ct are the centroids of clusters s and t, respectively. When two clusters s and t are
combined into a new cluster u, the new centroid is computed over all the original objects in clusters s
and t. The distance then becomes the Euclidean distance between the centroid of u and the centroid
of a remaining cluster v in the forest. This is also known as the UPGMC algorithm.
• method=’median’ assigns math:d(s,t) like the centroid method. When two clusters s and t are
combined into a new cluster u, the average of centroids s and t give the new centroid u. This is also
known as the WPGMC algorithm.
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• method=’ward’ uses the Ward variance minimization algorithm. The new entry d(u, v) is computed
as follows,
r
|v| + |s|
|v| + |t|
|v|
d(u, v) =
d(v, s)2 +
d(v, t)2 +
d(s, t)2
T
T
T
where u is the newly joined cluster consisting of clusters s and t, v is an unused cluster in the forest,
T = |v| + |s| + |t|, and | ∗ | is the cardinality of its argument. This is also known as the incremental
algorithm.
Warning: When the minimum distance pair in the forest is chosen, there may be two or more pairs with the
same minimum distance. This implementation may chose a different minimum than the MATLAB(TM)
version.
Parameters
• y
[ndarray] A condensed or redundant distance matrix. A condensed distance matrix
is a flat array containing the upper triangular of the distance matrix. This is the
form that pdist returns. Alternatively, a collection of m observation vectors in n
dimensions may be passed as an m by n array.
• method
[string] The linkage algorithm to use. See the Linkage Methods section below
for full descriptions.
• metric
[string] The distance metric to use. See the distance.pdist function for a list
of valid distance metrics.
Returns
• Z
[ndarray] The hierarchical clustering encoded as a linkage matrix.
maxRstat(Z, R, i)
Returns the maximum statistic for each non-singleton cluster and its descendents.
Arguments
• Z
[ndarray] The hierarchical clustering encoded as a matrix. See linkage for more
information.
• R
[ndarray] The inconsistency matrix.
• i
[int] The column of R to use as the statistic.
Returns
• MR : ndarray Calculates the maximum statistic for the i’th column of the inconsistency
matrix R for each non-singleton cluster node. MR[j] is the maximum over R[Q(j)-n,
i] where Q(j) the set of all node ids corresponding to nodes below and including j.
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maxdists(Z)
Returns the maximum distance between any cluster for each non-singleton cluster.
Arguments
• Z
[ndarray] The hierarchical clustering encoded as a matrix. See linkage for more
information.
Returns
• MD : ndarray A (n-1) sized numpy array of doubles; MD[i] represents the maximum
distance between any cluster (including singletons) below and including the node with
index i. More specifically, MD[i] = Z[Q(i)-n, 2].max() where Q(i) is the set
of all node indices below and including node i.
maxinconsts(Z, R)
Returns the maximum inconsistency coefficient for each non-singleton cluster and its descendents.
Arguments
• Z
[ndarray] The hierarchical clustering encoded as a matrix. See linkage for more
information.
• R
[ndarray] The inconsistency matrix.
Returns
• MI
[ndarray] A monotonic (n-1)-sized numpy array of doubles.
median(y)
Performs median/WPGMC linkage. See linkage for more information on the return structure and algorithm.
The following are common calling conventions:
1.Z = median(y)
Performs median/WPGMC linkage on the condensed distance matrix y. See linkage for more information on the return structure and algorithm.
2.Z = median(X)
Performs median/WPGMC linkage on the observation matrix X using Euclidean distance as the distance
metric. See linkage for more information on the return structure and algorithm.
Parameters
Q
[ndarray] A condensed or redundant distance matrix. A condensed distance matrix is a
flat array containing the upper triangular of the distance matrix. This is the form that
pdist returns. Alternatively, a collection of m observation vectors in n dimensions may
be passed as a m by n array.
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Returns
• Z
[ndarray] The hierarchical clustering encoded as a linkage matrix.
Seealso
• linkage: for advanced creation of hierarchical clusterings.
num_obs_linkage(Z)
Returns the number of original observations of the linkage matrix passed.
Arguments
• Z
[ndarray] The linkage matrix on which to perform the operation.
Returns
• n
[int] The number of original observations in the linkage.
set_link_color_palette(palette)
Changes the list of matplotlib color codes to use when coloring links with the dendrogram color_threshold
feature.
Arguments
• palette : A list of matplotlib color codes. The order of
the color codes is the order in which the colors are cycled through when color thresholding in
the dendrogram.
single(y)
Performs single/min/nearest linkage on the condensed distance matrix y. See linkage for more information
on the return structure and algorithm.
Parameters
y
[ndarray] The upper triangular of the distance matrix. The result of pdist is returned in
this form.
Returns
Z
[ndarray] The linkage matrix.
Seealso
• linkage: for advanced creation of hierarchical clusterings.
to_mlab_linkage(Z)
Converts a linkage matrix Z generated by the linkage function of this module to a MATLAB(TM) compatible
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one. The return linkage matrix has the last column removed and the cluster indices are converted to 1..N
indexing.
Arguments
• Z
[ndarray] A linkage matrix generated by this library.
Returns
• ZM
[ndarray] A linkage matrix compatible with MATLAB(TM)’s hierarchical clustering
functions.
to_tree(Z, rd=False)
Converts a hierarchical clustering encoded in the matrix Z (by linkage) into an easy-to-use tree object. The
reference r to the root ClusterNode object is returned.
Each ClusterNode object has a left, right, dist, id, and count attribute. The left and right attributes point to
ClusterNode objects that were combined to generate the cluster. If both are None then the ClusterNode object
is a leaf node, its count must be 1, and its distance is meaningless but set to 0.
Note: This function is provided for the convenience of the library user. ClusterNodes are not used as input to
any of the functions in this library.
Parameters
• Z : ndarray The linkage matrix in proper form (see the linkage function documentation).
• r : bool When False, a reference to the root ClusterNode object is returned. Otherwise,
a tuple (r,d) is returned. r is a reference to the root node while d is a dictionary mapping
cluster ids to ClusterNode references. If a cluster id is less than n, then it corresponds to
a singleton cluster (leaf node). See linkage for more information on the assignment of
cluster ids to clusters.
Returns
• L : list The pre-order traversal.
ward(y)
Performs Ward’s linkage on a condensed or redundant distance matrix. See linkage for more information on the
return structure and algorithm.
The following are common calling conventions:
1.Z = ward(y) Performs Ward’s linkage on the condensed distance matrix Z. See linkage for more
information on the return structure and algorithm.
2.Z = ward(X) Performs Ward’s linkage on the observation matrix X using Euclidean distance as the
distance metric. See linkage for more information on the return structure and algorithm.
Parameters
Q
[ndarray] A condensed or redundant distance matrix. A condensed distance matrix is a
flat array containing the upper triangular of the distance matrix. This is the form that
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pdist returns. Alternatively, a collection of m observation vectors in n dimensions may
be passed as a m by n array.
Returns
• Z
[ndarray] The hierarchical clustering encoded as a linkage matrix.
Seealso
• linkage: for advanced creation of hierarchical clusterings.
weighted(y)
Performs weighted/WPGMA linkage on the condensed distance matrix y. See linkage for more information
on the return structure and algorithm.
Parameters
y
[ndarray] The upper triangular of the distance matrix. The result of pdist is returned in
this form.
Returns
Z
[ndarray] A linkage matrix containing the hierarchical clustering. See the linkage
function documentation for more information on its structure.
Seealso
• linkage: for advanced creation of hierarchical clusterings.
3.1.2 K-means clustering and vector quantization (scipy.cluster.vq)
K-means Clustering and Vector Quantization Module
Provides routines for k-means clustering, generating code books from k-means models, and quantizing vectors by
comparing them with centroids in a code book.
The k-means algorithm takes as input the number of clusters to generate, k, and a set of observation vectors to cluster.
It returns a set of centroids, one for each of the k clusters. An observation vector is classified with the cluster number
or centroid index of the centroid closest to it.
A vector v belongs to cluster i if it is closer to centroid i than any other centroids. If v belongs to i, we say centroid i is
the dominating centroid of v. Common variants of k-means try to minimize distortion, which is defined as the sum of
the distances between each observation vector and its dominating centroid. Each step of the k-means algorithm refines
the choices of centroids to reduce distortion. The change in distortion is often used as a stopping criterion: when the
change is lower than a threshold, the k-means algorithm is not making sufficient progress and terminates.
Since vector quantization is a natural application for k-means, information theory terminology is often used. The
centroid index or cluster index is also referred to as a “code” and the table mapping codes to centroids and vice
versa is often referred as a “code book”. The result of k-means, a set of centroids, can be used to quantize vectors.
Quantization aims to find an encoding of vectors that reduces the expected distortion.
For example, suppose we wish to compress a 24-bit color image (each pixel is represented by one byte for red, one
for blue, and one for green) before sending it over the web. By using a smaller 8-bit encoding, we can reduce the
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amount of data by two thirds. Ideally, the colors for each of the 256 possible 8-bit encoding values should be chosen
to minimize distortion of the color. Running k-means with k=256 generates a code book of 256 codes, which fills up
all possible 8-bit sequences. Instead of sending a 3-byte value for each pixel, the 8-bit centroid index (or code word)
of the dominating centroid is transmitted. The code book is also sent over the wire so each 8-bit code can be translated
back to a 24-bit pixel value representation. If the image of interest was of an ocean, we would expect many 24-bit
blues to be represented by 8-bit codes. If it was an image of a human face, more flesh tone colors would be represented
in the code book.
All routines expect obs to be a M by N array where the rows are the observation vectors. The codebook is a k by N
array where the i’th row is the centroid of code word i. The observation vectors and centroids have the same feature
dimension.
whiten(obs) –
Normalize a group of observations so each feature has unit variance.
vq(obs,code_book) –
Calculate code book membership of a set of observation vectors.
kmeans(obs,k_or_guess,iter=20,thresh=1e-5) –
Clusters a set of observation vectors. Learns centroids with the k-means algorithm, trying to minimize distortion.
A code book is generated that can be used to quantize vectors.
kmeans2 –
A different implementation of k-means with more methods for initializing centroids. Uses maximum number of
iterations as opposed to a distortion threshold as its stopping criterion.
whiten(obs)
Normalize a group of observations on a per feature basis.
Before running k-means, it is beneficial to rescale each feature dimension of the observation set with whitening.
Each feature is divided by its standard deviation across all observations to give it unit variance.
Parameters
obs
[ndarray] Each row of the array is an observation. The columns are the features seen
during each observation.
#
obs = [[
[
[
[
f0
1.,
2.,
3.,
4.,
f1
1.,
2.,
3.,
4.,
f2
1.],
2.],
3.],
4.]])
#o0
#o1
#o2
#o3
XXX perhaps should have an axis variable here.
Returns
result
[ndarray] Contains the values in obs scaled by the standard devation of each column.
Examples
>>> from numpy import array
>>> from scipy.cluster.vq import whiten
>>> features = array([[ 1.9,2.3,1.7],
...
[ 1.5,2.5,2.2],
...
[ 0.8,0.6,1.7,]])
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>>> whiten(features)
array([[ 3.41250074,
[ 2.69407953,
[ 1.43684242,
2.20300046,
2.39456571,
0.57469577,
5.88897275],
7.62102355],
5.88897275]])
vq(obs, code_book)
Vector Quantization: assign codes from a code book to observations.
Assigns a code from a code book to each observation. Each observation vector in the M by N obs array is
compared with the centroids in the code book and assigned the code of the closest centroid.
The features in obs should have unit variance, which can be acheived by passing them through the whiten
function. The code book can be created with the k-means algorithm or a different encoding algorithm.
Parameters
obs
[ndarray] Each row of the NxM array is an observation. The columns are the “features”
seen during each observation. The features must be whitened first using the whiten function or something equivalent.
code_book
[ndarray.] The code book is usually generated using the k-means algorithm. Each row of
the array holds a different code, and the columns are the features of the code.
#
code_book = [[
[
[
f0
1.,
1.,
1.,
f1
2.,
2.,
2.,
f2
3.,
3.,
3.,
f3
4.], #c0
4.], #c1
4.]]) #c2
Returns
code
[ndarray] A length N array holding the code book index for each observation.
dist
[ndarray] The distortion (distance) between the observation and its nearest code.
Notes
This currently forces 32-bit math precision for speed. Anyone know of a situation where this undermines the
accuracy of the algorithm?
Examples
>>> from numpy import array
>>> from scipy.cluster.vq import vq
>>> code_book = array([[1.,1.,1.],
...
[2.,2.,2.]])
>>> features = array([[ 1.9,2.3,1.7],
...
[ 1.5,2.5,2.2],
...
[ 0.8,0.6,1.7]])
>>> vq(features,code_book)
(array([1, 1, 0],’i’), array([ 0.43588989,
0.73484692,
0.83066239]))
kmeans(obs, k_or_guess, iter=20, thresh=1.0000000000000001e-05)
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Performs k-means on a set of observation vectors forming k
clusters. This yields a code book mapping centroids to codes and vice versa. The k-means algorithm
adjusts the centroids until sufficient progress cannot be made, i.e. the change in distortion since the last
iteration is less than some threshold.
Parameters
obs
[ndarray] Each row of the M by N array is an observation vector. The columns are
the features seen during each observation. The features must be whitened first with the
whiten function.
k_or_guess
[int or ndarray] The number of centroids to generate. A code is assigned to each centroid,
which is also the row index of the centroid in the code_book matrix generated.
The initial k centroids are chosen by randomly selecting observations from the observation matrix. Alternatively, passing a k by N array specifies the initial k centroids.
iter
[int] The number of times to run k-means, returning the codebook with the lowest distortion. This argument is ignored if initial centroids are specified with an array for the
k_or_guess paramter. This parameter does not represent the number of iterations of the
k-means algorithm.
thresh
[float] Terminates the k-means algorithm if the change in distortion since the last k-means
iteration is less than thresh.
Returns
codebook
[ndarray] A k by N array of k centroids. The i’th centroid codebook[i] is represented with
the code i. The centroids and codes generated represent the lowest distortion seen, not
necessarily the globally minimal distortion.
distortion
[float] The distortion between the observations passed and the centroids generated.
Seealso
• kmeans2: a different implementation of k-means clustering with more methods for generating initial centroids but without using a distortion change threshold as a stopping
criterion.
• whiten: must be called prior to passing an observation matrix to kmeans.
Examples
>>> from numpy import array
>>> from scipy.cluster.vq import vq, kmeans, whiten
>>> features = array([[ 1.9,2.3],
...
[ 1.5,2.5],
...
[ 0.8,0.6],
...
[ 0.4,1.8],
...
[ 0.1,0.1],
...
[ 0.2,1.8],
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...
[ 2.0,0.5],
...
[ 0.3,1.5],
...
[ 1.0,1.0]])
>>> whitened = whiten(features)
>>> book = array((whitened[0],whitened[2]))
>>> kmeans(whitened,book)
(array([[ 2.3110306 , 2.86287398],
[ 0.93218041, 1.24398691]]), 0.85684700941625547)
>>> from numpy import random
>>> random.seed((1000,2000))
>>> codes = 3
>>> kmeans(whitened,codes)
(array([[ 2.3110306 , 2.86287398],
[ 1.32544402, 0.65607529],
[ 0.40782893, 2.02786907]]), 0.5196582527686241)
kmeans2(data, k, iter=10, thresh=1.0000000000000001e-05, minit=’random’, missing=’warn’)
Classify a set of observations into k clusters using the k-means
algorithm.
The algorithm attempts to minimize the Euclidian distance between observations and centroids. Several initialization methods are included.
Parameters
data
[ndarray] A M by N array of M observations in N dimensions or a length M array of M
one-dimensional observations.
k
[int or ndarray] The number of clusters to form as well as the number of centroids to
generate. If minit initialization string is ‘matrix’, or if a ndarray is given instead, it is
interpreted as initial cluster to use instead.
iter
[int] Number of iterations of the k-means algrithm to run. Note that this differs in meaning
from the iters parameter to the kmeans function.
thresh
[float] (not used yet).
minit
[string] Method for initialization. Available methods are ‘random’, ‘points’, ‘uniform’,
and ‘matrix’:
‘random’: generate k centroids from a Gaussian with mean and variance estimated from
the data.
‘points’: choose k observations (rows) at random from data for the initial centroids.
‘uniform’: generate k observations from the data from a uniform distribution defined by
the data set (unsupported).
‘matrix’: interpret the k parameter as a k by M (or length k array for one-dimensional
data) array of initial centroids.
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Returns
centroid
[ndarray] A k by N array of centroids found at the last iteration of k-means.
label
[ndarray] label[i] is the code or index of the centroid the i’th observation is closest to.
3.1.3 Vector Quantization / Kmeans
Clustering algorithms are useful in information theory, target detection, communications, compression,
and other areas. The vq module only supports vector quantization and the k-means algorithms. Development of self-organizing maps (SOM) and other approaches is underway.
3.1.4 Hierarchical Clustering
The hierarchy module provides functions for hierarchical and agglomerative clustering. Its features include generating hierarchical clusters from distance matrices, computing distance matrices from observation vectors, calculating statistics on clusters, cutting linkages to generate flat clusters, and visualizing
clusters with dendrograms.
3.1.5 Distance Computation
The distance module provides functions for computing distances between pairs of vectors from a set of
observation vectors.
3.2 Constants (scipy.constants)
Physical and mathematical constants and units.
3.2.1 Mathematical constants
pi
golden
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Golden ratio
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3.2.2 Physical constants
c
mu_0
epsilon_0
h
hbar
G
g
e
R
alpha
N_A
k
sigma
Wien
Rydberg
m_e
m_p
m_n
speed of light in vacuum
the magnetic constant µ0
the electric constant (vacuum permittivity), 0
the Planck constant h
h̄ = h/(2π)
Newtonian constant of gravitation
standard acceleration of gravity
elementary charge
molar gas constant
fine-structure constant
Avogadro constant
Boltzmann constant
Stefan-Boltzmann constant σ
Wien displacement law constant
Rydberg constant
electron mass
proton mass
neutron mass
3.2.3 Constants database
In addition to the above variables containing physical constants, scipy.constants also contains a database of
additional physical constants.
value (key)
unit (key)
precision (key)
find (sub[, disp])
Value in physical_constants indexed by key
Unit in physical_constants indexed by key
Relative precision in physical_constants indexed by key
Find the codata.physical_constant keys containing a given string.
value(key)
Value in physical_constants indexed by key
Parameters
key : Python string or unicode
Key in dictionary physical_constants
Returns
value : float
Value in physical_constants corresponding to key
See Also:
codata
Contains the description of physical_constants, which, as a dictionary literal object, does not itself possess
a docstring.
Examples
>>> from scipy.constants import codata
>>> codata.value(’elementary charge’)
1.60217653e-019
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unit(key)
Unit in physical_constants indexed by key
Parameters
key : Python string or unicode
Key in dictionary physical_constants
Returns
unit : Python string
Unit in physical_constants corresponding to key
See Also:
codata
Contains the description of physical_constants, which, as a dictionary literal object, does not itself possess
a docstring.
Examples
>>> from scipy.constants import codata
>>> codata.unit(u’proton mass’)
’kg’
precision(key)
Relative precision in physical_constants indexed by key
Parameters
key : Python string or unicode
Key in dictionary physical_constants
Returns
prec : float
Relative precision in physical_constants corresponding to key
See Also:
codata
Contains the description of physical_constants, which, as a dictionary literal object, does not itself possess
a docstring.
Examples
>>> from scipy.constants import codata
>>> codata.precision(u’proton mass’)
1.7338050694080732e-007
find(sub, disp=True)
Find the codata.physical_constant keys containing a given string.
Parameters
sub : str or unicode
Sub-string to search keys for
disp : bool
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If True, print the keys that are found, and return None. Otherwise, return the list of
keys without printing anything.
Returns
keys : None or list
If disp is False, the list of keys is returned. Otherwise, None is returned.
See Also:
codata
Contains the description of physical_constants, which, as a dictionary literal object, does not itself possess
a docstring.
physical_constants
Dictionary of physical constants, of the format physical_constants[name] = (value, unit,
uncertainty).
Available constants:
alpha particle mass
alpha particle mass energy equivalent
alpha particle mass energy equivalent in MeV
alpha particle mass in u
alpha particle molar mass
alpha particle-electron mass ratio
alpha particle-proton mass ratio
Angstrom star
atomic mass constant
atomic mass constant energy equivalent
atomic mass constant energy equivalent in MeV
atomic mass unit-electron volt relationship
atomic mass unit-hartree relationship
atomic mass unit-hertz relationship
atomic mass unit-inverse meter relationship
atomic mass unit-joule relationship
atomic mass unit-kelvin relationship
atomic mass unit-kilogram relationship
atomic unit of 1st hyperpolarizablity
atomic unit of 2nd hyperpolarizablity
atomic unit of action
atomic unit of charge
atomic unit of charge density
atomic unit of current
atomic unit of electric dipole moment
atomic unit of electric field
atomic unit of electric field gradient
atomic unit of electric polarizablity
atomic unit of electric potential
atomic unit of electric quadrupole moment
atomic unit of energy
atomic unit of force
atomic unit of length
atomic unit of magnetic dipole moment
atomic unit of magnetic flux density
Continued on next page
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Table 3.1 – continued from previous page
atomic unit of magnetizability
atomic unit of mass
atomic unit of momentum
atomic unit of permittivity
atomic unit of time
atomic unit of velocity
Avogadro constant
Bohr magneton
Bohr magneton in eV/T
Bohr magneton in Hz/T
Bohr magneton in inverse meters per tesla
Bohr magneton in K/T
Bohr radius
Boltzmann constant
Boltzmann constant in eV/K
Boltzmann constant in Hz/K
Boltzmann constant in inverse meters per kelvin
characteristic impedance of vacuum
classical electron radius
Compton wavelength
Compton wavelength over 2 pi
conductance quantum
conventional value of Josephson constant
conventional value of von Klitzing constant
Cu x unit
deuteron magnetic moment
deuteron magnetic moment to Bohr magneton ratio
deuteron magnetic moment to nuclear magneton ratio
deuteron mass
deuteron mass energy equivalent
deuteron mass energy equivalent in MeV
deuteron mass in u
deuteron molar mass
deuteron rms charge radius
deuteron-electron magnetic moment ratio
deuteron-electron mass ratio
deuteron-neutron magnetic moment ratio
deuteron-proton magnetic moment ratio
deuteron-proton mass ratio
electric constant
electron charge to mass quotient
electron g factor
electron gyromagnetic ratio
electron gyromagnetic ratio over 2 pi
electron magnetic moment
electron magnetic moment anomaly
electron magnetic moment to Bohr magneton ratio
electron magnetic moment to nuclear magneton ratio
electron mass
electron mass energy equivalent
electron mass energy equivalent in MeV
electron mass in u
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Table 3.1 – continued from previous page
electron molar mass
electron to alpha particle mass ratio
electron to shielded helion magnetic moment ratio
electron to shielded proton magnetic moment ratio
electron volt
electron volt-atomic mass unit relationship
electron volt-hartree relationship
electron volt-hertz relationship
electron volt-inverse meter relationship
electron volt-joule relationship
electron volt-kelvin relationship
electron volt-kilogram relationship
electron-deuteron magnetic moment ratio
electron-deuteron mass ratio
electron-muon magnetic moment ratio
electron-muon mass ratio
electron-neutron magnetic moment ratio
electron-neutron mass ratio
electron-proton magnetic moment ratio
electron-proton mass ratio
electron-tau mass ratio
elementary charge
elementary charge over h
Faraday constant
Faraday constant for conventional electric current
Fermi coupling constant
fine-structure constant
first radiation constant
first radiation constant for spectral radiance
Hartree energy
Hartree energy in eV
hartree-atomic mass unit relationship
hartree-electron volt relationship
hartree-hertz relationship
hartree-inverse meter relationship
hartree-joule relationship
hartree-kelvin relationship
hartree-kilogram relationship
helion mass
helion mass energy equivalent
helion mass energy equivalent in MeV
helion mass in u
helion molar mass
helion-electron mass ratio
helion-proton mass ratio
hertz-atomic mass unit relationship
hertz-electron volt relationship
hertz-hartree relationship
hertz-inverse meter relationship
hertz-joule relationship
hertz-kelvin relationship
hertz-kilogram relationship
Continued on next page
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Table 3.1 – continued from previous page
inverse fine-structure constant
inverse meter-atomic mass unit relationship
inverse meter-electron volt relationship
inverse meter-hartree relationship
inverse meter-hertz relationship
inverse meter-joule relationship
inverse meter-kelvin relationship
inverse meter-kilogram relationship
inverse of conductance quantum
Josephson constant
joule-atomic mass unit relationship
joule-electron volt relationship
joule-hartree relationship
joule-hertz relationship
joule-inverse meter relationship
joule-kelvin relationship
joule-kilogram relationship
kelvin-atomic mass unit relationship
kelvin-electron volt relationship
kelvin-hartree relationship
kelvin-hertz relationship
kelvin-inverse meter relationship
kelvin-joule relationship
kelvin-kilogram relationship
kilogram-atomic mass unit relationship
kilogram-electron volt relationship
kilogram-hartree relationship
kilogram-hertz relationship
kilogram-inverse meter relationship
kilogram-joule relationship
kilogram-kelvin relationship
lattice parameter of silicon
Loschmidt constant (273.15 K, 101.325 kPa)
magnetic constant
magnetic flux quantum
Mo x unit
molar gas constant
molar mass constant
molar mass of carbon-12
molar Planck constant
molar Planck constant times c
molar volume of ideal gas (273.15 K, 100 kPa)
molar volume of ideal gas (273.15 K, 101.325 kPa)
molar volume of silicon
muon Compton wavelength
muon Compton wavelength over 2 pi
muon g factor
muon magnetic moment
muon magnetic moment anomaly
muon magnetic moment to Bohr magneton ratio
muon magnetic moment to nuclear magneton ratio
muon mass
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Table 3.1 – continued from previous page
muon mass energy equivalent
muon mass energy equivalent in MeV
muon mass in u
muon molar mass
muon-electron mass ratio
muon-neutron mass ratio
muon-proton magnetic moment ratio
muon-proton mass ratio
muon-tau mass ratio
natural unit of action
natural unit of action in eV s
natural unit of energy
natural unit of energy in MeV
natural unit of length
natural unit of mass
natural unit of momentum
natural unit of momentum in MeV/c
natural unit of time
natural unit of velocity
neutron Compton wavelength
neutron Compton wavelength over 2 pi
neutron g factor
neutron gyromagnetic ratio
neutron gyromagnetic ratio over 2 pi
neutron magnetic moment
neutron magnetic moment to Bohr magneton ratio
neutron magnetic moment to nuclear magneton ratio
neutron mass
neutron mass energy equivalent
neutron mass energy equivalent in MeV
neutron mass in u
neutron molar mass
neutron to shielded proton magnetic moment ratio
neutron-electron magnetic moment ratio
neutron-electron mass ratio
neutron-muon mass ratio
neutron-proton magnetic moment ratio
neutron-proton mass ratio
neutron-tau mass ratio
Newtonian constant of gravitation
Newtonian constant of gravitation over h-bar c
nuclear magneton
nuclear magneton in eV/T
nuclear magneton in inverse meters per tesla
nuclear magneton in K/T
nuclear magneton in MHz/T
Planck constant
Planck constant in eV s
Planck constant over 2 pi
Planck constant over 2 pi in eV s
Planck constant over 2 pi times c in MeV fm
Planck length
Continued on next page
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Table 3.1 – continued from previous page
Planck mass
Planck temperature
Planck time
proton charge to mass quotient
proton Compton wavelength
proton Compton wavelength over 2 pi
proton g factor
proton gyromagnetic ratio
proton gyromagnetic ratio over 2 pi
proton magnetic moment
proton magnetic moment to Bohr magneton ratio
proton magnetic moment to nuclear magneton ratio
proton magnetic shielding correction
proton mass
proton mass energy equivalent
proton mass energy equivalent in MeV
proton mass in u
proton molar mass
proton rms charge radius
proton-electron mass ratio
proton-muon mass ratio
proton-neutron magnetic moment ratio
proton-neutron mass ratio
proton-tau mass ratio
quantum of circulation
quantum of circulation times 2
Rydberg constant
Rydberg constant times c in Hz
Rydberg constant times hc in eV
Rydberg constant times hc in J
Sackur-Tetrode constant (1 K, 100 kPa)
Sackur-Tetrode constant (1 K, 101.325 kPa)
second radiation constant
shielded helion gyromagnetic ratio
shielded helion gyromagnetic ratio over 2 pi
shielded helion magnetic moment
shielded helion magnetic moment to Bohr magneton ratio
shielded helion magnetic moment to nuclear magneton ratio
shielded helion to proton magnetic moment ratio
shielded helion to shielded proton magnetic moment ratio
shielded proton gyromagnetic ratio
shielded proton gyromagnetic ratio over 2 pi
shielded proton magnetic moment
shielded proton magnetic moment to Bohr magneton ratio
shielded proton magnetic moment to nuclear magneton ratio
speed of light in vacuum
standard acceleration of gravity
standard atmosphere
Stefan-Boltzmann constant
tau Compton wavelength
tau Compton wavelength over 2 pi
tau mass
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Table 3.1 – continued from previous page
tau mass energy equivalent
tau mass energy equivalent in MeV
tau mass in u
tau molar mass
tau-electron mass ratio
tau-muon mass ratio
tau-neutron mass ratio
tau-proton mass ratio
Thomson cross section
unified atomic mass unit
von Klitzing constant
weak mixing angle
Wien displacement law constant
{220} lattice spacing of silicon
3.2.4 Unit prefixes
SI
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deka
deci
centi
milli
micro
nano
pico
femto
atto
zepto
1024
1021
1018
1015
1012
109
106
103
102
101
10−1
10−2
10−3
10−6
10−9
10−12
10−15
10−18
10−21
Binary
kibi
mebi
gibi
tebi
pebi
exbi
zebi
yobi
210
220
230
240
250
260
270
280
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3.2.5 Units
Weight
gram
metric_ton
grain
lb
oz
stone
grain
long_ton
short_ton
troy_ounce
troy_pound
carat
m_u
10−3 kg
103 kg
one grain in kg
one pound (avoirdupous) in kg
one ounce in kg
one stone in kg
one grain in kg
one long ton in kg
one short ton in kg
one Troy ounce in kg
one Troy pound in kg
one carat in kg
atomic mass constant (in kg)
Angle
degree
arcmin
arcsec
degree in radians
arc minute in radians
arc second in radians
Time
minute
hour
day
week
year
Julian_year
one minute in seconds
one hour in seconds
one day in seconds
one week in seconds
one year (365 days) in seconds
one Julian year (365.25 days) in seconds
Length
inch
foot
yard
mile
mil
pt
survey_foot
survey_mile
nautical_mile
fermi
angstrom
micron
au
light_year
parsec
166
one inch in meters
one foot in meters
one yard in meters
one mile in meters
one mil in meters
one point in meters
one survey foot in meters
one survey mile in meters
one nautical mile in meters
one Fermi in meters
one Ångström in meters
one micron in meters
one astronomical unit in meters
one light year in meters
one parsec in meters
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Pressure
atm
bar
torr
psi
standard atmosphere in pascals
one bar in pascals
one torr (mmHg) in pascals
one psi in pascals
Area
one hectare in square meters
one acre in square meters
hectare
acre
Volume
liter
gallon
gallon_imp
fluid_ounce
fluid_ounce_imp
bbl
one liter in cubic meters
one gallon (US) in cubic meters
one gallon (UK) in cubic meters
one fluid ounce (US) in cubic meters
one fluid ounce (UK) in cubic meters
one barrel in cubic meters
Speed
kmh
mph
mach
knot
kilometers per hour in meters per second
miles per hour in meters per second
one Mach (approx., at 15 °C, 1 atm) in meters per second
one knot in meters per second
Temperature
zero_Celsius
degree_Fahrenheit
C2K (C)
K2C (K)
F2C (F)
C2F (C)
F2K (F)
K2F (K)
zero of Celsius scale in Kelvin
one Fahrenheit (only differences) in Kelvins
Convert Celsius to Kelvin
Convert Kelvin to Celsius
Convert Fahrenheit to Celsius
Convert Celsius to Fahrenheit
Convert Fahrenheit to Kelvin
Convert Kelvin to Fahrenheit
C2K(C)
Convert Celsius to Kelvin
Parameters
C : float-like scalar or array-like
Celsius temperature(s) to be converted
Returns
K : float or a numpy array of floats, corresponding to type of Parameters
Equivalent Kelvin temperature(s)
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Notes
Computes K = C + zero_Celsius where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute
zero” as measured in Celsius.
Examples
>>> from scipy.constants.constants import C2K
>>> C2K(np.array([-40, 40.0]))
array([ 233.15, 313.15])
K2C(K)
Convert Kelvin to Celsius
Parameters
K : float-like scalar or array-like
Kelvin temperature(s) to be converted
Returns
C : float or a numpy array of floats, corresponding to type of Parameters
Equivalent Celsius temperature(s)
Notes
Computes C = K - zero_Celsius where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute
zero” as measured in Celsius.
Examples
>>> from scipy.constants.constants import K2C
>>> K2C(np.array([233.15, 313.15]))
array([-40., 40.])
F2C(F)
Convert Fahrenheit to Celsius
Parameters
F : float-like scalar or array-like
Fahrenheit temperature(s) to be converted
Returns
C : float or a numpy array of floats, corresponding to type of Parameters
Equivalent Celsius temperature(s)
Notes
Computes C = (F - 32) / 1.8
Examples
>>> from scipy.constants.constants import F2C
>>> F2C(np.array([-40, 40.0]))
array([-40.
,
4.44444444])
C2F(C)
Convert Celsius to Fahrenheit
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Parameters
C : float-like scalar or array-like
Celsius temperature(s) to be converted
Returns
F : float or a numpy array of floats, corresponding to type of Parameters
Equivalent Fahrenheit temperature(s)
Notes
Computes F = 1.8 * C + 32
Examples
>>> from scipy.constants.constants import C2F
>>> C2F(np.array([-40, 40.0]))
array([ -40., 104.])
F2K(F)
Convert Fahrenheit to Kelvin
Parameters
F : float-like scalar or array-like
Fahrenheit temperature(s) to be converted
Returns
K : float or a numpy array of floats, corresponding to type of Parameters
Equivalent Kelvin temperature(s)
Notes
Computes K = (F - 32)/1.8 + zero_Celsius where zero_Celsius = 273.15, i.e., (the absolute value of) temperature
“absolute zero” as measured in Celsius.
Examples
>>> from scipy.constants.constants import F2K
>>> F2K(np.array([-40, 104]))
array([ 233.15, 313.15])
K2F(K)
Convert Kelvin to Fahrenheit
Parameters
K : float-like scalar or array-like
Kelvin temperature(s) to be converted
Returns
F : float or a numpy array of floats, corresponding to type of Parameters
Equivalent Fahrenheit temperature(s)
Notes
Computes F = 1.8 * (K - zero_Celsius ) + 32 where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute zero” as measured in Celsius.
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Examples
>>> from scipy.constants.constants import K2F
>>> K2F(np.array([233.15, 313.15]))
array([ -40., 104.])
Energy
eV
calorie
calorie_IT
erg
Btu
Btu_th
ton_TNT
one electron volt in Joules
one calorie (thermochemical) in Joules
one calorie (International Steam Table calorie, 1956) in Joules
one erg in Joules
one British thermal unit (International Steam Table) in Joules
one British thermal unit (thermochemical) in Joules
one ton of TNT in Joules
Power
one horsepower in watts
hp
Force
one dyne in newtons
one pound force in newtons
one kilogram force in newtons
dyn
lbf
kgf
Optics
lambda2nu (lambda_)
nu2lambda (nu)
Convert wavelength to optical frequency
Convert optical frequency to wavelength.
lambda2nu(lambda_)
Convert wavelength to optical frequency
Parameters
lambda : float-like scalar or array-like
Wavelength(s) to be converted
Returns
nu : float or a numpy array of floats, corresponding to type of Parameters
Equivalent optical frequency(ies)
Notes
Computes ν = c/λ where c = 299792458.0, i.e., the (vacuum) speed of light in meters/second.
Examples
>>> from scipy.constants.constants import lambda2nu
>>> lambda2nu(np.array((1, speed_of_light)))
array([ 2.99792458e+08,
1.00000000e+00])
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nu2lambda(nu)
Convert optical frequency to wavelength.
Parameters
nu : float-like scalar or array-like
Optical frequency(ies) to be converted
Returns
lambda : float or a numpy array of floats, corresp. to type of Parameters
Equivalent wavelength(s)
Notes
Computes λ = c/ν where c = 299792458.0, i.e., the (vacuum) speed of light in meters/second.
Examples
>>> from scipy.constants.constants import nu2lambda
>>> nu2lambda(np.array((1, speed_of_light)))
array([ 2.99792458e+08,
1.00000000e+00])
3.3 Fourier transforms (scipy.fftpack)
3.3.1 Fast Fourier transforms
fft (x[, n, axis, overwrite_x])
ifft (x[, n, axis, overwrite_x])
fftn (x[, shape, axes, overwrite_x])
ifftn (x[, shape, axes, overwrite_x])
fft2 (x[, shape, axes, -1), ...])
ifft2 (x[, shape, axes, -1), ...])
rfft (x[, n, axis, overwrite_x])
irfft (x[, n, axis, overwrite_x])
Return discrete Fourier transform of arbitrary type sequence x.
ifft(x, n=None, axis=-1, overwrite_x=0) -> y
fftn(x, shape=None, axes=None, overwrite_x=0) -> y
Return inverse multi-dimensional discrete Fourier transform of arbitrary type
sequence x.
2-D discrete Fourier transform.
ifft2(x, shape=None, axes=(-2,-1), overwrite_x=0) -> y
rfft(x, n=None, axis=-1, overwrite_x=0) -> y
irfft(x, n=None, axis=-1, overwrite_x=0) -> y
fft(x, n=None, axis=-1, overwrite_x=0)
Return discrete Fourier transform of arbitrary type sequence x.
Parameters
x : array-like
array to fourier transform.
n : int, optional
Length of the Fourier transform.
If n<x.shape[axis], x is truncated.
n>x.shape[axis], x is zero-padded. (Default n=x.shape[axis]).
If
axis : int, optional
Axis along which the fft’s are computed. (default=-1)
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)
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Returns
z : complex ndarray
with the elements:
[y(0),y(1),..,y(n/2-1),y(-n/2),...,y(-1)] if n is even [y(0),y(1),..,y((n-1)/2),y(-(n1)/2),...,y(-1)] if n is odd
where
y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1
Note that y(-j) = y(n-j).conjugate().
See Also:
ifft
Inverse FFT
rfft
FFT of a real sequence
Notes
The packing of the result is “standard”: If A = fft(a, n), then A[0] contains the zero-frequency term, A[1:n/2+1]
contains the positive-frequency terms, and A[n/2+1:] contains the negative-frequency terms, in order of decreasingly negative frequency. So for an 8-point transform, the frequencies of the result are [ 0, 1, 2, 3, 4, -3, -2,
-1].
This is most efficient for n a power of two.
Note: In scipy 0.8.0 fft in single precision is available, but only for input array sizes which can be factorized
into (combinations of) 2, 3 and 5. For other sizes the computation will be done in double precision.
Examples
>>> x = np.arange(5)
>>> np.all(np.abs(x-fft(ifft(x))<1.e-15) #within numerical accuracy.
True
ifft(x, n=None, axis=-1, overwrite_x=0)
ifft(x, n=None, axis=-1, overwrite_x=0) -> y
Return inverse discrete Fourier transform of arbitrary type sequence x.
The returned complex array contains
[y(0),y(1),...,y(n-1)]
where
y(j) = 1/n sum[k=0..n-1] x[k] * exp(sqrt(-1)*j*k* 2*pi/n)
Optional input: see fft.__doc__
fftn(x, shape=None, axes=None, overwrite_x=0)
fftn(x, shape=None, axes=None, overwrite_x=0) -> y
Return multi-dimensional discrete Fourier transform of arbitrary type sequence x.
The returned array contains
y[j_1,..,j_d] = sum[k_1=0..n_1-1, ..., k_d=0..n_d-1]
x[k_1,..,k_d] * prod[i=1..d] exp(-sqrt(-1)*2*pi/n_i * j_i * k_i)
where d = len(x.shape) and n = x.shape. Note that y[..., -j_i, ...] = y[..., n_i-j_i, ...].conjugate().
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Optional input:
shape
Defines the shape of the Fourier transform.
If shape is not specified then
shape=take(x.shape,axes,axis=0). If shape[i]>x.shape[i] then the i-th dimension is padded
with zeros. If shape[i]<x.shape[i], then the i-th dimension is truncated to desired length shape[i].
axes
The transform is applied along the given axes of the input array (or the newly constructed array if
shape argument was used).
overwrite_x
If set to true, the contents of x can be destroyed.
Notes:
y == fftn(ifftn(y)) within numerical accuracy.
ifftn(x, shape=None, axes=None, overwrite_x=0)
Return inverse multi-dimensional discrete Fourier transform of arbitrary type sequence x.
The returned array contains:
y[j_1,..,j_d] = 1/p * sum[k_1=0..n_1-1, ..., k_d=0..n_d-1]
x[k_1,..,k_d] * prod[i=1..d] exp(sqrt(-1)*2*pi/n_i * j_i * k_i)
where d = len(x.shape), n = x.shape, and p = prod[i=1..d] n_i.
For description of parameters see fftn.
See Also:
fftn
for detailed information.
fft2(x, shape=None, axes=(-2, -1), overwrite_x=0)
2-D discrete Fourier transform.
Return the two-dimensional discrete Fourier transform of the 2-D argument x.
See Also:
fftn
for detailed information.
ifft2(x, shape=None, axes=(-2, -1), overwrite_x=0)
ifft2(x, shape=None, axes=(-2,-1), overwrite_x=0) -> y
Return inverse two-dimensional discrete Fourier transform of arbitrary type sequence x.
See ifftn.__doc__ for more information.
rfft(x, n=None, axis=-1, overwrite_x=0)
rfft(x, n=None, axis=-1, overwrite_x=0) -> y
Return discrete Fourier transform of real sequence x.
The returned real arrays contains
[y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2))] if n is even [y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2)),Im(y(n/2))] if n is
odd
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where
y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n) j = 0..n-1
Note that y(-j) = y(n-j).conjugate().
Optional input:
n
Defines the length of the Fourier transform. If n is not specified then n=x.shape[axis] is set. If
n<x.shape[axis], x is truncated. If n>x.shape[axis], x is zero-padded.
axis
The transform is applied along the given axis of the input array (or the newly constructed array if n
argument was used).
overwrite_x
If set to true, the contents of x can be destroyed.
Notes:
y == rfft(irfft(y)) within numerical accuracy.
irfft(x, n=None, axis=-1, overwrite_x=0)
irfft(x, n=None, axis=-1, overwrite_x=0) -> y
Return inverse discrete Fourier transform of real sequence x. The contents of x is interpreted as the output of
rfft(..) function.
The returned real array contains
[y(0),y(1),...,y(n-1)]
where for n is even
y(j) = 1/n (sum[k=1..n/2-1] (x[2*k-1]+sqrt(-1)*x[2*k])
• exp(sqrt(-1)*j*k* 2*pi/n)
• c.c. + x[0] + (-1)**(j) x[n-1])
and for n is odd
y(j) = 1/n (sum[k=1..(n-1)/2] (x[2*k-1]+sqrt(-1)*x[2*k])
• exp(sqrt(-1)*j*k* 2*pi/n)
• c.c. + x[0])
c.c. denotes complex conjugate of preceeding expression.
Optional input: see rfft.__doc__
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3.3.2 Differential and pseudo-differential operators
diff (x[, order, period, _cache])
tilbert (x, h[, period, _cache])
itilbert (x, h[, period, _cache])
hilbert (x[, _cache])
ihilbert (x)
cs_diff (x, a, b[, period, _cache])
sc_diff (x, a, b[, period, _cache])
ss_diff (x, a, b[, period, _cache])
cc_diff (x, a, b[, period, _cache])
shift (x, a[, period, _cache])
diff(x, order=1, period=2*pi) -> y
tilbert(x, h, period=2*pi) -> y
itilbert(x, h, period=2*pi) -> y
hilbert(x) -> y
ihilbert(x) -> y
cs_diff(x, a, b, period=2*pi) -> y
Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x.
ss_diff(x, a, b, period=2*pi) -> y
cc_diff(x, a, b, period=2*pi) -> y
shift(x, a, period=2*pi) -> y
diff(x, order=1, period=None, _cache={})
diff(x, order=1, period=2*pi) -> y
Return k-th derivative (or integral) of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then
y_j = pow(sqrt(-1)*j*2*pi/period, order) * x_j y_0 = 0 if order is not 0.
Optional input:
order
The order of differentiation. Default order is 1. If order is negative, then integration is carried out
under the assumption that x_0==0.
period
The assumed period of the sequence. Default is 2*pi.
Notes:
If sum(x,axis=0)=0 then
diff(diff(x,k),-k)==x (within numerical accuracy)
For odd order and even len(x), the Nyquist mode is taken zero.
tilbert(x, h, period=None, _cache={})
tilbert(x, h, period=2*pi) -> y
Return h-Tilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then
y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j y_0 = 0
Input:
h
Defines the parameter of the Tilbert transform.
period
The assumed period of the sequence. Default period is 2*pi.
Notes:
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If sum(x,axis=0)==0 and n=len(x) is odd then
tilbert(itilbert(x)) == x
If 2*pi*h/period is approximately 10 or larger then numerically
tilbert == hilbert
(theoretically oo-Tilbert == Hilbert). For even len(x), the Nyquist mode of x is taken zero.
itilbert(x, h, period=None, _cache={})
itilbert(x, h, period=2*pi) -> y
Return inverse h-Tilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then
y_j = -sqrt(-1)*tanh(j*h*2*pi/period) * x_j y_0 = 0
Optional input: see tilbert.__doc__
hilbert(x, _cache={})
hilbert(x) -> y
Return Hilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then
y_j = sqrt(-1)*sign(j) * x_j y_0 = 0
Notes:
If sum(x,axis=0)==0 then
hilbert(ihilbert(x)) == x
For even len(x), the Nyquist mode of x is taken zero.
ihilbert(x)
ihilbert(x) -> y
Return inverse Hilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then
y_j = -sqrt(-1)*sign(j) * x_j y_0 = 0
cs_diff(x, a, b, period=None, _cache={})
cs_diff(x, a, b, period=2*pi) -> y
Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then
y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j y_0 = 0
Input:
a,b
Defines the parameters of the cosh/sinh pseudo-differential operator.
period
The period of the sequence. Default period is 2*pi.
Notes:
For even len(x), the Nyquist mode of x is taken zero.
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sc_diff(x, a, b, period=None, _cache={})
Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:
y_j = sqrt(-1)*sinh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
y_0 = 0
Parameters
x : array_like
Input array.
a,b : float
Defines the parameters of the sinh/cosh pseudo-differential operator.
period : float, optional
The period of the sequence x. Default is 2*pi.
Notes
sc_diff(cs_diff(x,a,b),b,a) == x For even len(x), the Nyquist mode of x is taken as zero.
ss_diff(x, a, b, period=None, _cache={})
ss_diff(x, a, b, period=2*pi) -> y
Return (a,b)-sinh/sinh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then
y_j = sinh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j y_0 = a/b * x_0
Input:
a,b
Defines the parameters of the sinh/sinh pseudo-differential operator.
period
The period of the sequence x. Default is 2*pi.
Notes:
ss_diff(ss_diff(x,a,b),b,a) == x
cc_diff(x, a, b, period=None, _cache={})
cc_diff(x, a, b, period=2*pi) -> y
Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then
y_j = cosh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
Input:
a,b
Defines the parameters of the sinh/sinh pseudo-differential operator.
Optional input:
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period
The period of the sequence x. Default is 2*pi.
Notes:
cc_diff(cc_diff(x,a,b),b,a) == x
shift(x, a, period=None, _cache={})
shift(x, a, period=2*pi) -> y
Shift periodic sequence x by a: y(u) = x(u+a).
If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then
y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f
Optional input:
period
The period of the sequences x and y. Default period is 2*pi.
3.3.3 Helper functions
fftshift (x[, axes])
ifftshift (x[, axes])
dftfreq
rfftfreq (n[, d])
Shift the zero-frequency component to the center of the spectrum.
The inverse of fftshift.
rfftfreq(n, d=1.0) -> f
fftshift(x, axes=None)
Shift the zero-frequency component to the center of the spectrum.
This function swaps half-spaces for all axes listed (defaults to all). Note that y[0] is the Nyquist component
only if len(x) is even.
Parameters
x : array_like
Input array.
axes : int or shape tuple, optional
Axes over which to shift. Default is None, which shifts all axes.
Returns
y : ndarray
The shifted array.
See Also:
ifftshift
The inverse of fftshift.
Examples
>>> freqs = np.fft.fftfreq(10, 0.1)
>>> freqs
array([ 0., 1., 2., 3., 4., -5., -4., -3., -2., -1.])
>>> np.fft.fftshift(freqs)
array([-5., -4., -3., -2., -1., 0., 1., 2., 3., 4.])
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Shift the zero-frequency component only along the second axis:
>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
>>> freqs
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
>>> np.fft.fftshift(freqs, axes=(1,))
array([[ 2., 0., 1.],
[-4., 3., 4.],
[-1., -3., -2.]])
ifftshift(x, axes=None)
The inverse of fftshift.
Parameters
x : array_like
Input array.
axes : int or shape tuple, optional
Axes over which to calculate. Defaults to None, which shifts all axes.
Returns
y : ndarray
The shifted array.
See Also:
fftshift
Shift zero-frequency component to the center of the spectrum.
Examples
>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
>>> freqs
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
>>> np.fft.ifftshift(np.fft.fftshift(freqs))
array([[ 0., 1., 2.],
[ 3., 4., -4.],
[-3., -2., -1.]])
rfftfreq(n, d=1.0)
rfftfreq(n, d=1.0) -> f
DFT sample frequencies (for usage with rfft,irfft).
The returned float array contains the frequency bins in cycles/unit (with zero at the start) given a window length
n and a sample spacing d:
f = [0,1,1,2,2,...,n/2-1,n/2-1,n/2]/(d*n) if n is even f = [0,1,1,2,2,...,n/2-1,n/2-1,n/2,n/2]/(d*n) if n is
odd
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3.3.4 Convolutions (scipy.fftpack.convolve)
convolve ()
convolve - Function signature: y = convolve(x,omega,[swap_real_imag,overwrite_x])
Required arguments: x : input rank-1 array(‘d’) with bounds (n) omega : input rank-1
array(‘d’) with bounds (n) Optional arguments: overwrite_x := 0 input int swap_real_imag
:= 0 input int Return objects: y : rank-1 array(‘d’) with bounds (n) and x storage
convolve_z ()
convolve_z - Function signature: y =
convolve_z(x,omega_real,omega_imag,[overwrite_x]) Required arguments: x : input
rank-1 array(‘d’) with bounds (n) omega_real : input rank-1 array(‘d’) with bounds (n)
omega_imag : input rank-1 array(‘d’) with bounds (n) Optional arguments: overwrite_x :=
0 input int Return objects: y : rank-1 array(‘d’) with bounds (n) and x storage
init_convolution_kernel
init_convolution_kernel
()
- Function signature: omega =
init_convolution_kernel(n,kernel_func,[d,zero_nyquist,kernel_func_extra_args]) Required
arguments: n : input int kernel_func : call-back function Optional arguments: d := 0 input
int kernel_func_extra_args := () input tuple zero_nyquist := d%2 input int Return objects:
omega : rank-1 array(‘d’) with bounds (n) Call-back functions: def kernel_func(k): return
kernel_func Required arguments: k : input int Return objects: kernel_func : float
destroy_convolve_cache
destroy_convolve_cache
()
- Function signature: destroy_convolve_cache()
convolve()
convolve - Function signature:
y = convolve(x,omega,[swap_real_imag,overwrite_x])
Required arguments:
x : input rank-1 array(‘d’) with bounds (n) omega : input rank-1 array(‘d’) with bounds (n)
Optional arguments:
overwrite_x := 0 input int swap_real_imag := 0 input int
Return objects:
y : rank-1 array(‘d’) with bounds (n) and x storage
convolve_z()
convolve_z - Function signature:
y = convolve_z(x,omega_real,omega_imag,[overwrite_x])
Required arguments:
x : input rank-1 array(‘d’) with bounds (n) omega_real : input rank-1 array(‘d’) with bounds (n)
omega_imag : input rank-1 array(‘d’) with bounds (n)
Optional arguments:
overwrite_x := 0 input int
Return objects:
y : rank-1 array(‘d’) with bounds (n) and x storage
init_convolution_kernel()
init_convolution_kernel - Function signature:
omega = init_convolution_kernel(n,kernel_func,[d,zero_nyquist,kernel_func_extra_args])
Required arguments:
n : input int kernel_func : call-back function
Optional arguments:
d := 0 input int kernel_func_extra_args := () input tuple zero_nyquist := d%2 input int
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Return objects:
omega : rank-1 array(‘d’) with bounds (n)
Call-back functions:
def kernel_func(k): return kernel_func Required arguments:
k : input int
Return objects:
kernel_func : float
destroy_convolve_cache()
destroy_convolve_cache - Function signature: destroy_convolve_cache()
3.3.5 Other (scipy.fftpack._fftpack)
drfft ()
drfft - Function signature: y = drfft(x,[n,direction,normalize,overwrite_x]) Required
arguments: x : input rank-1 array(‘d’) with bounds (*) Optional arguments: overwrite_x := 0
input int n := size(x) input int direction := 1 input int normalize := (direction<0) input int
Return objects: y : rank-1 array(‘d’) with bounds (*) and x storage
zfft ()
zfft - Function signature: y = zfft(x,[n,direction,normalize,overwrite_x]) Required
arguments: x : input rank-1 array(‘D’) with bounds (*) Optional arguments: overwrite_x := 0
input int n := size(x) input int direction := 1 input int normalize := (direction<0) input int
Return objects: y : rank-1 array(‘D’) with bounds (*) and x storage
zrfft ()
zrfft - Function signature: y = zrfft(x,[n,direction,normalize,overwrite_x]) Required
arguments: x : input rank-1 array(‘D’) with bounds (*) Optional arguments: overwrite_x := 1
input int n := size(x) input int direction := 1 input int normalize := (direction<0) input int
Return objects: y : rank-1 array(‘D’) with bounds (*) and x storage
zfftnd ()
zfftnd - Function signature: y = zfftnd(x,[s,direction,normalize,overwrite_x]) Required
arguments: x : input rank-1 array(‘D’) with bounds (*) Optional arguments: overwrite_x := 0
input int s := old_shape(x,j++) input rank-1 array(‘i’) with bounds (r) direction := 1 input int
normalize := (direction<0) input int Return objects: y : rank-1 array(‘D’) with bounds (*)
and x storage
destroy_drfft_cache
destroy_drfft_cache
()
- Function signature: destroy_drfft_cache()
destroy_zfft_cache
destroy_zfft_cache
()
- Function signature: destroy_zfft_cache()
destroy_zfftnd_cache
destroy_zfftnd_cache
()
- Function signature: destroy_zfftnd_cache()
drfft()
drfft - Function signature:
y = drfft(x,[n,direction,normalize,overwrite_x])
Required arguments:
x : input rank-1 array(‘d’) with bounds (*)
Optional arguments:
overwrite_x := 0 input int n := size(x) input int direction := 1 input int normalize := (direction<0) input int
Return objects:
y : rank-1 array(‘d’) with bounds (*) and x storage
zfft()
zfft - Function signature:
y = zfft(x,[n,direction,normalize,overwrite_x])
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Required arguments:
x : input rank-1 array(‘D’) with bounds (*)
Optional arguments:
overwrite_x := 0 input int n := size(x) input int direction := 1 input int normalize := (direction<0) input int
Return objects:
y : rank-1 array(‘D’) with bounds (*) and x storage
zrfft()
zrfft - Function signature:
y = zrfft(x,[n,direction,normalize,overwrite_x])
Required arguments:
x : input rank-1 array(‘D’) with bounds (*)
Optional arguments:
overwrite_x := 1 input int n := size(x) input int direction := 1 input int normalize := (direction<0) input int
Return objects:
y : rank-1 array(‘D’) with bounds (*) and x storage
zfftnd()
zfftnd - Function signature:
y = zfftnd(x,[s,direction,normalize,overwrite_x])
Required arguments:
x : input rank-1 array(‘D’) with bounds (*)
Optional arguments:
overwrite_x := 0 input int s := old_shape(x,j++) input rank-1 array(‘i’) with bounds (r) direction := 1 input
int normalize := (direction<0) input int
Return objects:
y : rank-1 array(‘D’) with bounds (*) and x storage
destroy_drfft_cache()
destroy_drfft_cache - Function signature: destroy_drfft_cache()
destroy_zfft_cache()
destroy_zfft_cache - Function signature: destroy_zfft_cache()
destroy_zfftnd_cache()
destroy_zfftnd_cache - Function signature: destroy_zfftnd_cache()
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3.4 Integration and ODEs (scipy.integrate)
3.4.1 Integrating functions, given function object
quad (func, a, b[, args=(), full_output, ...])
Compute a definite integral.
dblquad (func, a, b, gfun, hfun[, args=(), epsabs, ...])Compute the double integral of func2d(y,x) from x=a..b and
y=gfun(x)..hfun(x).
tplquad (func, a, b, gfun, hfun, qfun, rfun[, args=(), epsabs,
Compute
...])
a triple (definite) integral.
fixed_quad (func, a, b[, args=(), n])
Compute a definite integral using fixed-order Gaussian
quadrature.
quadrature (func, a, b[, args=(), tol, maxCompute a definite integral using fixed-tolerance Gaussian
iter, ...])
quadrature.
romberg (function, a, b[, args=(), tol, show, ...])
Romberg integration of a callable function or method.
quad(func, a, b, args=(), full_output=0, epsabs=1.4899999999999999e-08, epsrel=1.4899999999999999e-08,
limit=50, points=None, weight=None, wvar=None, wopts=None, maxp1=50, limlst=50)
Compute a definite integral.
Integrate func from a to b (possibly infinite interval) using a technique from the Fortran library QUADPACK.
If func takes many arguments, it is integrated along the axis corresponding to the first argument. Use the keyword
argument args to pass the other arguments.
Run scipy.integrate.quad_explain() for more information on the more esoteric inputs and outputs.
Parameters
func : function
A Python function or method to integrate.
a : float
Lower limit of integration (use -scipy.integrate.Inf for -infinity).
b : float
Upper limit of integration (use scipy.integrate.Inf for +infinity).
args : tuple, optional
extra arguments to pass to func
full_output : int
Non-zero to return a dictionary of integration information. If non-zero, warning
messages are also suppressed and the message is appended to the output tuple.
Returns
y : float
The integral of func from a to b.
abserr : float
an estimate of the absolute error in the result.
infodict : dict
a dictionary containing additional information. Run scipy.integrate.quad_explain()
for more information.
message : :
a convergence message.
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explain : :
appended only with ‘cos’ or ‘sin’ weighting and infinite integration limits, it contains
an explanation of the codes in infodict[’ierlst’]
Examples
R4
Calculate 0 x2 dx and compare with an analytic result
>>> from scipy import integrate
>>> x2 = lambda x: x**2
>>> integrate.quad(x,0.,4.)
(21.333333333333332, 2.3684757858670003e-13)
>> print 4.**3/3
21.3333333333
Calculate
R∞
0
e−x dx
>>> invexp = lambda x: exp(-x)
>>> integrate.quad(invexp,0,inf)
(0.99999999999999989, 5.8426061711142159e-11)
>>>
>>>
>>>
0.5
>>>
>>>
1.5
f = lambda x,a : a*x
y, err = integrate.quad(f, 0, 1, args=(1,))
y
y, err = integrate.quad(f, 0, 1, args=(3,))
y
dblquad(func, a, b, gfun, hfun, args=(), epsabs=1.4899999999999999e-08, epsrel=1.4899999999999999e-08)
Compute the double integral of func2d(y,x) from x=a..b and y=gfun(x)..hfun(x).
Parameters
func2d : function
a Python function or method of at least two variables: y must be the first
argument and x the second argument.
(a,b)
[tuple] the limits of integration in x: a < b
gfun
[function] the lower boundary curve in y which is a function taking a single floating point argument (x) and returning a floating point result: a lambda function
can be useful here.
hfun
[function] the upper boundary curve in y (same requirements as gfun).
args :
extra arguments to pass to func2d.
epsabs
[float] absolute tolerance passed directly to the inner 1-D quadrature integration.
epsrel
[float] relative tolerance of the inner 1-D integrals.
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Returns
y : float
the resultant integral.
abserr
[float] an estimate of the error.
See also: :
quad - single integral tplquad - triple integral fixed_quad - fixed-order Gaussian
quadrature quadrature - adaptive Gaussian quadrature odeint, ode - ODE integrators simps, trapz, romb - integrators for sampled data scipy.special - for coefficients
and roots of orthogonal polynomials
tplquad(func,
a,
b,
gfun,
hfun,
epsrel=1.4899999999999999e-08)
Compute a triple (definite) integral.
qfun,
rfun,
args=(),
epsabs=1.4899999999999999e-08,
Return the triple integral of func3d(z, y,x) from x=a..b, y=gfun(x)..hfun(x), and z=qfun(x,y)..rfun(x,y)
Parameters
func3d : function
A Python function or method of at least three variables in the order (z, y, x).
(a,b) : tuple
The limits of integration in x: a < b
gfun : function
The lower boundary curve in y which is a function taking a single floating point
argument (x) and returning a floating point result: a lambda function can be useful
here.
hfun : function
The upper boundary curve in y (same requirements as gfun).
qfun : function
The lower boundary surface in z. It must be a function that takes two floats in the
order (x, y) and returns a float.
rfun : function
The upper boundary surface in z. (Same requirements as qfun.)
args : Arguments
Extra arguments to pass to func3d.
epsabs : float
Absolute tolerance passed directly to the innermost 1-D quadrature integration.
epsrel : float
Relative tolerance of the innermost 1-D integrals.
Returns
y : float
The resultant integral.
abserr : float
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An estimate of the error.
See Also:
quad
Adaptive quadrature using QUADPACK
quadrature
Adaptive Gaussian quadrature
fixed_quad
Fixed-order Gaussian quadrature
dblquad
Double integrals
romb
Integrators for sampled data
trapz
Integrators for sampled data
simps
Integrators for sampled data
ode
ODE integrators
odeint
ODE integrators
scipy.special
For coefficients and roots of orthogonal polynomials
fixed_quad(func, a, b, args=(), n=5)
Compute a definite integral using fixed-order Gaussian quadrature.
Integrate func from a to b using Gaussian quadrature of order n.
Parameters
func : callable
A Python function or method to integrate (must accept vector inputs).
a : float
Lower limit of integration.
b : float
Upper limit of integration.
args : tuple, optional
Extra arguments to pass to function, if any.
n : int, optional
Order of quadrature integration. Default is 5.
Returns
val : float
Gaussian quadrature approximation to the integral
See Also:
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quad
adaptive quadrature using QUADPACK
dblquad, tplquad
romberg
adaptive Romberg quadrature
quadrature
adaptive Gaussian quadrature
romb, simps, trapz
cumtrapz
cumulative integration for sampled data
ode, odeint
quadrature(func, a, b, args=(), tol=1.4899999999999999e-08, maxiter=50, vec_func=True)
Compute a definite integral using fixed-tolerance Gaussian quadrature.
Integrate func from a to b using Gaussian quadrature with absolute tolerance tol.
Parameters
func : function
A Python function or method to integrate.
a : float
Lower limit of integration.
b : float
Upper limit of integration.
args : tuple, optional
Extra arguments to pass to function.
tol : float, optional
Iteration stops when error between last two iterates is less than tolerance.
maxiter : int, optional
Maximum number of iterations.
vec_func : bool, optional
True or False if func handles arrays as arguments (is a “vector” function). Default is
True.
Returns
val : float
Gaussian quadrature approximation (within tolerance) to integral.
err : float
Difference between last two estimates of the integral.
See Also:
romberg
adaptive Romberg quadrature
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fixed_quad
fixed-order Gaussian quadrature
quad
adaptive quadrature using QUADPACK
dblquad
double integrals
tplquad
triple integrals
romb
integrator for sampled data
simps
integrator for sampled data
trapz
integrator for sampled data
cumtrapz
cumulative integration for sampled data
ode
ODE integrator
odeint
ODE integrator
romberg(function, a, b, args=(), tol=1.48e-08, show=False, divmax=10, vec_func=False)
Romberg integration of a callable function or method.
Returns the integral of function (a function of one variable) over the interval (a, b).
If show is 1, the triangular array of the intermediate results will be printed. If vec_func is True (default is False),
then function is assumed to support vector arguments.
Parameters
function : callable
Function to be integrated.
a : float
Lower limit of integration.
b : float
Upper limit of integration.
Returns
results : float
Result of the integration.
See Also:
fixed_quad
Fixed-order Gaussian quadrature.
quad
Adaptive quadrature using QUADPACK.
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dblquad, tplquad, romb, simps, trapz
cumtrapz
Cumulative integration for sampled data.
ode, odeint
References
[R1]
Examples
Integrate a gaussian from 0,1 and compare to the error function.
>>> from scipy.special import erf
>>> gaussian = lambda x: 1/np.sqrt(np.pi) * np.exp(-x**2)
>>> result = romberg(gaussian, 0, 1, show=True)
Romberg integration of <function vfunc at 0x101eceaa0> from [0, 1]
Steps
1
2
4
8
16
32
StepSize
1.000000
0.500000
0.250000
0.125000
0.062500
0.031250
Results
0.385872
0.412631
0.419184
0.420810
0.421215
0.421317
0.421551
0.421368
0.421352
0.421350
0.421350
0.421356
0.421350
0.421350
0.421350
0.421350
0.421350
0.421350
0.421350
0.421350
0.421350
The final result is 0.421350396475 after 33 function evaluations.
>>> print 2*result,erf(1)
0.84270079295 0.84270079295
3.4.2 Integrating functions, given fixed samples
trapz (y[, x, dx, axis])
Integrate along the given axis using the composite trapezoidal rule.
cumtrapz (y[, x, dx, Cumulatively
axis])
integrate y(x) using samples along the given axis and the composite
trapezoidal rule. If x is None, spacing given by dx is assumed.
simps (y[, x, dx, axis,Integrate
even]) y(x) using samples along the given axis and the composite Simpson’s rule. If x is
None, spacing of dx is assumed.
romb (y[, dx, axis, show])
Romberg integration using samples of a function
trapz(y, x=None, dx=1.0, axis=-1)
Integrate along the given axis using the composite trapezoidal rule.
Integrate y (x) along given axis.
Parameters
y : array_like
Input array to integrate.
x : array_like, optional
If x is None, then spacing between all y elements is dx.
dx : scalar, optional
If x is None, spacing given by dx is assumed. Default is 1.
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axis : int, optional
Specify the axis.
Returns
out : float
Definite integral as approximated by trapezoidal rule.
See Also:
sum, cumsum
Notes
Image [R3] illustrates trapezoidal rule – y-axis locations of points will be taken from y array, by default x-axis
distances between points will be 1.0, alternatively they can be provided with x array or with dx scalar. Return
value will be equal to combined area under the red lines.
References
[R2], [R3]
Examples
>>> np.trapz([1,2,3])
4.0
>>> np.trapz([1,2,3], x=[4,6,8])
8.0
>>> np.trapz([1,2,3], dx=2)
8.0
>>> a = np.arange(6).reshape(2, 3)
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> np.trapz(a, axis=0)
array([ 1.5, 2.5, 3.5])
>>> np.trapz(a, axis=1)
array([ 2., 8.])
cumtrapz(y, x=None, dx=1.0, axis=-1)
Cumulatively integrate y(x) using samples along the given axis and the composite trapezoidal rule. If x is None,
spacing given by dx is assumed.
Parameters
y : array
x : array, optional
dx : int, optional
axis : int, optional
Specifies the axis to cumulate:
• -1 –> X axis
• 0 –> Z axis
• 1 –> Y axis
See Also:
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quad
adaptive quadrature using QUADPACK
romberg
adaptive Romberg quadrature
quadrature
adaptive Gaussian quadrature
fixed_quad
fixed-order Gaussian quadrature
dblquad
double integrals
tplquad
triple integrals
romb
integrators for sampled data
trapz
integrators for sampled data
cumtrapz
cumulative integration for sampled data
ode
ODE integrators
odeint
ODE integrators
simps(y, x=None, dx=1, axis=-1, even=’avg’)
Integrate y(x) using samples along the given axis and the composite Simpson’s rule. If x is None, spacing of dx
is assumed.
If there are an even number of samples, N, then there are an odd number of intervals (N-1), but Simpson’s rule
requires an even number of intervals. The parameter ‘even’ controls how this is handled.
Parameters
y : array_like
Array to be integrated.
x : array_like, optional
If given, the points at which y is sampled.
dx : int, optional
Spacing of integration points along axis of y. Only used when x is None. Default is
1.
axis : int, optional
Axis along which to integrate. Default is the last axis.
even : {‘avg’, ‘first’, ‘str’}, optional
‘avg’
[Average two results:1) use the first N-2 intervals with] a trapezoidal rule on the
last interval and 2) use the last N-2 intervals with a trapezoidal rule on the first
interval.
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‘first’
[Use Simpson’s rule for the first N-2 intervals with] a trapezoidal rule on the last
interval.
‘last’
[Use Simpson’s rule for the last N-2 intervals with a] trapezoidal rule on the first
interval.
See Also:
quad
adaptive quadrature using QUADPACK
romberg
adaptive Romberg quadrature
quadrature
adaptive Gaussian quadrature
fixed_quad
fixed-order Gaussian quadrature
dblquad
double integrals
tplquad
triple integrals
romb
integrators for sampled data
trapz
integrators for sampled data
cumtrapz
cumulative integration for sampled data
ode
ODE integrators
odeint
ODE integrators
Notes
For an odd number of samples that are equally spaced the result is exact if the function is a polynomial of order
3 or less. If the samples are not equally spaced, then the result is exact only if the function is a polynomial of
order 2 or less.
romb(y, dx=1.0, axis=-1, show=False)
Romberg integration using samples of a function
Parameters
y : array like
a vector of 2**k + 1 equally-spaced samples of a function
dx
[array like] the sample spacing.
axis
[array like?] the axis along which to integrate
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show
[Boolean] When y is a single 1-d array, then if this argument is True print the
table showing Richardson extrapolation from the samples.
Returns
ret : array_like?
The integrated result for each axis.
See also: :
quad - adaptive quadrature using QUADPACK romberg - adaptive Romberg quadrature quadrature - adaptive Gaussian quadrature fixed_quad - fixed-order Gaussian
quadrature dblquad, tplquad - double and triple integrals simps, trapz - integrators
for sampled data cumtrapz - cumulative integration for sampled data ode, odeint ODE integrators
See Also:
scipy.special for orthogonal polynomials (special) for Gaussian quadrature roots and weights for other weighting factors and regions.
3.4.3 Integrators of ODE systems
odeint (func, y0, t[, args=(), Dfun, col_deriv, ...])
ode
Integrate a system of ordinary differential equations.
A generic interface class to numeric integrators.
odeint(func, y0, t, args=(), Dfun=None, col_deriv=0, full_output=0, ml=None, mu=None, rtol=None, atol=None,
tcrit=None, h0=0.0, hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12, mxords=5, printmessg=0)
Integrate a system of ordinary differential equations.
Solve a system of ordinary differential equations using lsoda from the FORTRAN library odepack.
Solves the initial value problem for stiff or non-stiff systems of first order ode-s:
dy/dt = func(y,t0,...)
where y can be a vector.
Parameters
func : callable(y, t0, ...)
Computes the derivative of y at t0.
y0 : array
Initial condition on y (can be a vector).
t : array
A sequence of time points for which to solve for y. The initial value point should be
the first element of this sequence.
args : tuple
Extra arguments to pass to function.
Dfun : callable(y, t0, ...)
Gradient (Jacobian) of func.
col_deriv : boolean
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True if Dfun defines derivatives down columns (faster), otherwise Dfun should define
derivatives across rows.
full_output : boolean
True if to return a dictionary of optional outputs as the second output
printmessg : boolean
Whether to print the convergence message
Returns
y : array, shape (len(y0), len(t))
Array containing the value of y for each desired time in t, with the initial value y0 in
the first row.
infodict : dict, only returned if full_output == True
Dictionary containing additional output information
key
‘hu’
‘tcur’
‘tolsf’
meaning
vector of step sizes successfully used for each time step.
vector with the value of t reached for each time step. (will always be at least as large as the input time
vector of tolerance scale factors, greater than 1.0, computed when a request for too much accuracy wa
detected.
‘tsw’
value of t at the time of the last method switch (given for each time step)
‘nst’
cumulative number of time steps
‘nfe’
cumulative number of function evaluations for each time step
‘nje’
cumulative number of jacobian evaluations for each time step
‘nqu’
a vector of method orders for each successful step.
‘imxer’ index of the component of largest magnitude in the weighted local error vector (e / ewt) on an error
return, -1 otherwise.
‘lenrw’ the length of the double work array required.
‘leniw’ the length of integer work array required.
‘mused’ a vector of method indicators for each successful time step: 1: adams (nonstiff), 2: bdf (stiff)
See Also:
ode
a more object-oriented integrator based on VODE
quad
for finding the area under a curve
class ode(f, jac=None)
A generic interface class to numeric integrators.
See Also:
odeint
an integrator with a simpler interface based on lsoda from ODEPACK
quad
for finding the area under a curve
Examples
A problem to integrate and the corresponding jacobian:
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>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
from scipy import eye
from scipy.integrate import ode
y0, t0 = [1.0j, 2.0], 0
def f(t, y, arg1):
return [1j*arg1*y[0] + y[1], -arg1*y[1]**2]
def jac(t, y, arg1):
return [[1j*arg1, 1], [0, -arg1*2*y[1]]]
The integration:
>>>
>>>
>>>
>>>
>>>
>>>
>>>
r = ode(f, jac).set_integrator(’zvode’, method=’bdf’, with_jacobian=True)
r.set_initial_value(y0, t0).set_f_params(2.0).set_jac_params(2.0)
t1 = 10
dt = 1
while r.successful() and r.t < t1:
r.integrate(r.t+dt)
print r.t, r.y
Methods
3.5 Interpolation (scipy.interpolate)
3.5.1 Univariate interpolation
Interpolate a 1D function.
The interpolating polynomial for a set of points
The interpolating polynomial for a set of points
Piecewise polynomial curve specified by points and
derivatives
barycentric_interpolate (xi, yi, x)
Convenience function for polynomial interpolation
krogh_interpolate (xi, yi, x[, der])
Convenience function for polynomial interpolation.
piecewise_polynomial_interpolate (xi, yi, x[, or- Convenience function for piecewise polynomial
ders, der])
interpolation
interp1d
BarycentricInterpolator
KroghInterpolator
PiecewisePolynomial
class interp1d(x, y, kind=’linear’, axis=-1, copy=True, bounds_error=True, fill_value=nan)
Interpolate a 1D function.
See Also:
splrep, splev, UnivariateSpline
class BarycentricInterpolator(xi, yi=None)
The interpolating polynomial for a set of points
Constructs a polynomial that passes through a given set of points. Allows evaluation of the polynomial, efficient
changing of the y values to be interpolated, and updating by adding more x values. For reasons of numerical
stability, this function does not compute the coefficients of the polynomial.
This class uses a “barycentric interpolation” method that treats the problem as a special case of rational function
interpolation. This algorithm is quite stable, numerically, but even in a world of exact computation, unless the
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x coordinates are chosen very carefully - Chebyshev zeros (e.g. cos(i*pi/n)) are a good choice - polynomial
interpolation itself is a very ill-conditioned process due to the Runge phenomenon.
Based on Berrut and Trefethen 2004, “Barycentric Lagrange Interpolation”.
Methods
class KroghInterpolator(xi, yi)
The interpolating polynomial for a set of points
Constructs a polynomial that passes through a given set of points, optionally with specified derivatives at those
points. Allows evaluation of the polynomial and all its derivatives. For reasons of numerical stability, this
function does not compute the coefficients of the polynomial, although they can be obtained by evaluating all
the derivatives.
Be aware that the algorithms implemented here are not necessarily the most numerically stable known. Moreover, even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev
zeros (e.g. cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due
to the Runge phenomenon. In general, even with well-chosen x values, degrees higher than about thirty cause
problems with numerical instability in this code.
Based on [R4].
Parameters
xi : array-like, length N
Known x-coordinates
yi : array-like, N by R
Known y-coordinates, interpreted as vectors of length R, or scalars if R=1. When
an xi occurs two or more times in a row, the corresponding yi’s represent derivative
values.
References
[R4]
Methods
class PiecewisePolynomial(xi, yi, orders=None, direction=None)
Piecewise polynomial curve specified by points and derivatives
This class represents a curve that is a piecewise polynomial. It passes through a list of points and has specified
derivatives at each point. The degree of the polynomial may very from segment to segment, as may the number
of derivatives available. The degree should not exceed about thirty.
Appending points to the end of the curve is efficient.
Methods
barycentric_interpolate(xi, yi, x)
Convenience function for polynomial interpolation
Constructs a polynomial that passes through a given set of points, then evaluates the polynomial. For reasons of
numerical stability, this function does not compute the coefficients of the polynomial.
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This function uses a “barycentric interpolation” method that treats the problem as a special case of rational
function interpolation. This algorithm is quite stable, numerically, but even in a world of exact computation,
unless the x coordinates are chosen very carefully - Chebyshev zeros (e.g. cos(i*pi/n)) are a good choice polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon.
Based on Berrut and Trefethen 2004, “Barycentric Lagrange Interpolation”.
Parameters
xi : array-like of length N
The x coordinates of the points the polynomial should pass through
yi : array-like N by R
The y coordinates of the points the polynomial should pass through; if R>1 the
polynomial is vector-valued.
x : scalar or array-like of length M
Returns
y : scalar or array-like of length R or length M or M by R
The shape of y depends on the shape of x and whether the interpolator is vectorvalued or scalar-valued.
Notes
Construction of the interpolation weights is a relatively slow process. If you want to call this many times with
the same xi (but possibly varying yi or x) you should use the class BarycentricInterpolator. This is what this
function uses internally.
krogh_interpolate(xi, yi, x, der=0)
Convenience function for polynomial interpolation.
Constructs a polynomial that passes through a given set of points, optionally with specified derivatives at those
points. Evaluates the polynomial or some of its derivatives. For reasons of numerical stability, this function does
not compute the coefficients of the polynomial, although they can be obtained by evaluating all the derivatives.
Be aware that the algorithms implemented here are not necessarily the most numerically stable known. Moreover, even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev
zeros (e.g. cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due
to the Runge phenomenon. In general, even with well-chosen x values, degrees higher than about thirty cause
problems with numerical instability in this code.
Based on Krogh 1970, “Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation”
The polynomial passes through all the pairs (xi,yi). One may additionally specify a number of derivatives at
each point xi; this is done by repeating the value xi and specifying the derivatives as successive yi values.
Parameters
xi : array-like, length N
known x-coordinates
yi : array-like, N by R
known y-coordinates, interpreted as vectors of length R, or scalars if R=1
x : scalar or array-like of length N
Point or points at which to evaluate the derivatives
der : integer or list
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How many derivatives to extract; None for all potentially nonzero derivatives (that
is a number equal to the number of points), or a list of derivatives to extract. This
number includes the function value as 0th derivative.
Returns :
——- :
d : array
If the interpolator’s values are R-dimensional then the returned array will be the
number of derivatives by N by R. If x is a scalar, the middle dimension will be
dropped; if the yi are scalars then the last dimension will be dropped.
Notes
Construction of the interpolating polynomial is a relatively expensive process. If you want to evaluate it repeatedly consider using the class KroghInterpolator (which is what this function uses).
piecewise_polynomial_interpolate(xi, yi, x, orders=None, der=0)
Convenience function for piecewise polynomial interpolation
Parameters
xi : array-like of length N
a sorted list of x-coordinates
yi : list of lists of length N
yi[i] is the list of derivatives known at xi[i]
x : scalar or array-like of length M
orders : list of integers, or integer
a list of polynomial orders, or a single universal order
der : integer
which single derivative to extract
Returns
y : scalar or array-like of length R or length M or M by R
Notes
If orders is None, or orders[i] is None, then the degree of the polynomial segment is exactly the degree required
to match all i available derivatives at both endpoints. If orders[i] is not None, then some derivatives will be
ignored. The code will try to use an equal number of derivatives from each end; if the total number of derivatives
needed is odd, it will prefer the rightmost endpoint. If not enough derivatives are available, an exception is raised.
Construction of these piecewise polynomials can be an expensive process; if you repeatedly evaluate the same
polynomial, consider using the class PiecewisePolynomial (which is what this function does).
3.5.2 Multivariate interpolation
interp2d (x, y, z[, kind, copy, bounds_error,
Interpolate
...])
over a 2D grid.
Rbf (*args)
A class for radial basis function approximation/interpolation of
n-dimensional scattered data.
class interp2d(x, y, z, kind=’linear’, copy=True, bounds_error=False, fill_value=nan)
Interpolate over a 2D grid.
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Parameters
x, y : 1D arrays
Arrays defining the coordinates of a 2D grid. If the points lie on a regular grid, x can
specify the column coordinates and y the row coordinates, e.g.:
x = [0,1,2];
y = [0,3,7]
otherwise x and y must specify the full coordinates, i.e.:
x = [0,1,2,0,1,2,0,1,2];
y = [0,0,0,3,3,3,7,7,7]
If x and y are multi-dimensional, they are flattened before use.
z : 1D array
The values of the interpolated function on the grid points. If z is a multi-dimensional
array, it is flattened before use.
kind : {‘linear’, ‘cubic’, ‘quintic’}
The kind of interpolation to use.
copy : bool
If True, then data is copied, otherwise only a reference is held.
bounds_error : bool
If True, when interpolated values are requested outside of the domain of the input
data, an error is raised. If False, then fill_value is used.
fill_value : number
If provided, the value to use for points outside of the interpolation domain. Defaults
to NaN.
Raises
ValueError when inputs are invalid. :
See Also:
bisplrep, bisplev
BivariateSpline
a more recent wrapper of the FITPACK routines
class Rbf(*args)
A class for radial basis function approximation/interpolation of n-dimensional scattered data.
Parameters
*args : arrays
x, y, z, ..., d, where x, y, z, ... are the coordinates of the nodes and d is the array of
values at the nodes
function : str or callable, optional
The radial basis function, based on the radius, r, given by the norm (defult is Euclidean distance); the default is ‘multiquadric’:
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’multiquadric’: sqrt((r/self.epsilon)**2 + 1)
’inverse’: 1.0/sqrt((r/self.epsilon)**2 + 1)
’gaussian’: exp(-(r/self.epsilon)**2)
’linear’: r
’cubic’: r**3
’quintic’: r**5
’thin_plate’: r**2 * log(r)
If callable, then it must take 2 arguments (self, r). The epsilon parameter will be
available as self.epsilon. Other keyword arguments passed in will be available as
well.
epsilon : float, optional
Adjustable constant for gaussian or multiquadrics functions - defaults to approximate
average distance between nodes (which is a good start).
smooth : float, optional
Values greater than zero increase the smoothness of the approximation. 0 is for
interpolation (default), the function will always go through the nodal points in this
case.
norm : callable, optional
A function that returns the ‘distance’ between two points, with inputs as arrays of
positions (x, y, z, ...), and an output as an array of distance. E.g, the default:
def euclidean_norm(x1, x2):
return sqrt( ((x1 - x2)**2).sum(axis=0) )
which is called with x1=x1[ndims,newaxis,:] and x2=x2[ndims,:,newaxis] such that
the result is a matrix of the distances from each point in x1 to each point in x2.
Examples
>>> rbfi = Rbf(x, y, z, d)
>>> di = rbfi(xi, yi, zi)
# radial basis function interpolator instance
# interpolated values
3.5.3 1-D Splines
UnivariateSpline
InterpolatedUnivariateSpline
LSQUnivariateSpline
One-dimensional smoothing spline fit to a given set of data points.
One-dimensional interpolating spline for a given set of data points.
One-dimensional spline with explicit internal knots.
class UnivariateSpline(x, y, w=None, bbox=, [None, None], k=3, s=None)
One-dimensional smoothing spline fit to a given set of data points.
Fits a spline y=s(x) of degree k to the provided x,‘y‘ data. s specifies the number of knots by specifying a
smoothing condition.
Parameters
x : sequence
input dimension of data points – must be increasing
y : sequence
input dimension of data points
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w : sequence or None, optional
weights for spline fitting. Must be positive. If None (default), weights are all equal.
bbox : sequence or None, optional
2-sequence specifying the boundary of the approximation interval. If None (default),
bbox=[x[0],x[-1]].
k : int, optional
Degree of the smoothing spline. Must be <= 5.
s : float or None, optional
Positive smoothing factor used to choose the number of knots. Number of knots will
be increased until the smoothing condition is satisfied:
sum((w[i]*(y[i]-s(x[i])))**2,axis=0) <= s
If None (default), s=len(w) which should be a good value if 1/w[i] is an estimate of
the standard deviation of y[i]. If 0, spline will interpolate through all data points.
See Also:
InterpolatedUnivariateSpline
Subclass with smoothing forced to 0
LSQUnivariateSpline
Subclass in which knots are user-selected instead of being set by smoothing condition
splrep
An older, non object-oriented wrapping of FITPACK
splev, sproot, splint, spalde
BivariateSpline
A similar class for two-dimensional spline interpolation
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
from numpy import linspace,exp
from numpy.random import randn
from scipy.interpolate import UnivariateSpline
x = linspace(-3,3,100)
y = exp(-x**2) + randn(100)/10
s = UnivariateSpline(x,y,s=1)
xs = linspace(-3,3,1000)
ys = s(xs)
xs,ys is now a smoothed, super-sampled version of the noisy gaussian x,y
Methods
class InterpolatedUnivariateSpline(x, y, w=None, bbox=, [None, None], k=3)
One-dimensional interpolating spline for a given set of data points.
Fits a spline y=s(x) of degree k to the provided x,‘y‘ data. Spline function passes through all provided points.
Equivalent to UnivariateSpline with s=0.
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Parameters
x : sequence
input dimension of data points – must be increasing
y : sequence
input dimension of data points
w : sequence or None, optional
weights for spline fitting. Must be positive. If None (default), weights are all equal.
bbox : sequence or None, optional
2-sequence specifying the boundary of the approximation interval. If None (default),
bbox=[x[0],x[-1]].
k : int, optional
Degree of the smoothing spline. Must be <= 5.
See Also:
UnivariateSpline
Superclass – allows knots to be selected by a smoothing condition
LSQUnivariateSpline
spline for which knots are user-selected
splrep
An older, non object-oriented wrapping of FITPACK
splev, sproot, splint, spalde
BivariateSpline
A similar class for two-dimensional spline interpolation
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
from numpy import linspace,exp
from numpy.random import randn
from scipy.interpolate import UnivariateSpline
x = linspace(-3,3,100)
y = exp(-x**2) + randn(100)/10
s = UnivariateSpline(x,y,s=1)
xs = linspace(-3,3,1000)
ys = s(xs)
xs,ys is now a smoothed, super-sampled version of the noisy gaussian x,y
Methods
class LSQUnivariateSpline(x, y, t, w=None, bbox=, [None, None], k=3)
One-dimensional spline with explicit internal knots.
Fits a spline y=s(x) of degree k to the provided x,‘y‘ data. t specifies the internal knots of the spline
Parameters
x : sequence
input dimension of data points – must be increasing
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y : sequence
input dimension of data points
t: sequence :
interior knots of the spline. Must be in ascending order and bbox[0]<t[0]<...<t[1]<bbox[-1]
w : sequence or None, optional
weights for spline fitting. Must be positive. If None (default), weights are all equal.
bbox : sequence or None, optional
2-sequence specifying the boundary of the approximation interval. If None (default),
bbox=[x[0],x[-1]].
k : int, optional
Degree of the smoothing spline. Must be <= 5.
Raises
ValueError :
If the interior knots do not satisfy the Schoenberg-Whitney conditions
See Also:
UnivariateSpline
Superclass – knots are specified by setting a smoothing condition
InterpolatedUnivariateSpline
spline passing through all points
splrep
An older, non object-oriented wrapping of FITPACK
splev, sproot, splint, spalde
BivariateSpline
A similar class for two-dimensional spline interpolation
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
from numpy import linspace,exp
from numpy.random import randn
from scipy.interpolate import LSQUnivariateSpline
x = linspace(-3,3,100)
y = exp(-x**2) + randn(100)/10
t = [-1,0,1]
s = LSQUnivariateSpline(x,y,t)
xs = linspace(-3,3,1000)
ys = s(xs)
xs,ys is now a smoothed, super-sampled version of the noisy gaussian x,y with knots [-3,-1,0,1,3]
Methods
The above univariate spline classes have the following methods:
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UnivariateSpline.__call__ Evaluate
(self, x[, nu])
spline (or its nu-th derivative) at positions x. Note: x can be
unordered but the evaluation is more efficient if x is (partially) ordered.
UnivariateSpline.derivatives
Return
(self,
all x)
derivatives of the spline at the point x.
UnivariateSpline.integral Return
(self, a,definite
b)
integral of the spline between two given points.
UnivariateSpline.roots (self)Return the zeros of the spline.
UnivariateSpline.get_coeffs
Return
(self)spline coefficients.
UnivariateSpline.get_knots
Return
(self) the positions of (boundary and interior) knots of the spline.
UnivariateSpline.get_residual
Return(self)
weighted sum of squared residuals of the spline approximation: sum
((w[i]*(y[i]-s(x[i])))**2,axis=0)
UnivariateSpline.set_smoothing_factor
Continue spline computation
(self, s)
with the given smoothing factor s and with the
knots found at the last call.
__call__(x, nu=None)
Evaluate spline (or its nu-th derivative) at positions x. Note: x can be unordered but the evaluation is more
efficient if x is (partially) ordered.
derivatives(x)
Return all derivatives of the spline at the point x.
integral(a, b)
Return definite integral of the spline between two given points.
roots()
Return the zeros of the spline.
Restriction: only cubic splines are supported by fitpack.
get_coeffs()
Return spline coefficients.
get_knots()
Return the positions of (boundary and interior) knots of the spline.
get_residual()
Return weighted sum of squared residuals of the spline approximation: sum ((w[i]*(y[i]-s(x[i])))**2,axis=0)
set_smoothing_factor(s)
Continue spline computation with the given smoothing factor s and with the knots found at the last call.
Low-level interface to FITPACK functions:
splrep (x, y[, w, xb, xe, k, task, ...])
splprep (x[, w, u, ub, ue, k, ...])
splev (x, tck[, der])
splint (a, b, tck[, full_output])
sproot (tck[, mest])
spalde (x, tck)
bisplrep (x, y, z[, w, xb, xe, yb, ye, ...])
bisplev (x, y, tck[, dx, dy])
Find the B-spline representation of 1-D curve.
Find the B-spline representation of an N-dimensional curve.
Evaluate a B-spline and its derivatives.
Evaluate the definite integral of a B-spline.
Find the roots of a cubic B-spline.
Evaluate all derivatives of a B-spline.
Find a bivariate B-spline representation of a surface.
Evaluate a bivariate B-spline and its derivatives.
splrep(x, y, w=None, xb=None, xe=None, k=3, task=0, s=None, t=None, full_output=0, per=0, quiet=1)
Find the B-spline representation of 1-D curve.
Description:
Given the set of data points (x[i], y[i]) determine a smooth spline approximation of degree k on the
interval xb <= x <= xe. The coefficients, c, and the knot points, t, are returned. Uses the FORTRAN
routine curfit from FITPACK.
Inputs:
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x, y – The data points defining a curve y = f(x). w – Strictly positive rank-1 array of weights the
same length as x and y.
The weights are used in computing the weighted least-squares spline fit. If the errors in
the y values have standard-deviation given by the vector d, then w should be 1/d. Default
is ones(len(x)).
xb, xe – The interval to fit. If None, these default to x[0] and x[-1]
respectively.
k – The order of the spline fit. It is recommended to use cubic splines.
Even order splines should be avoided especially with small s values. 1 <= k <= 5
task – If task==0 find t and c for a given smoothing factor, s.
If task==1 find t and c for another value of the
smoothing factor, s. There must have been a previous call with task=0 or task=1 for the
same set of data (t will be stored an used internally)
If task=-1 find the weighted least square spline for
a given set of knots, t. These should be interior knots as knots on the ends will be added
automatically.
s – A smoothing condition. The amount of smoothness is determined by
satisfying the conditions: sum((w * (y - g))**2,axis=0) <= s where g(x) is the smoothed
interpolation of (x,y). The user can use s to control the tradeoff between closeness and
smoothness of fit. Larger s means more smoothing while smaller values of s indicate less
smoothing. Recommended values of s depend on the weights, w. If the weights represent
the inverse of the standard-deviation of y, then a good s value should be found in the range
(m-sqrt(2*m),m+sqrt(2*m)) where m is the number of datapoints in x, y, and w. default :
s=m-sqrt(2*m) if weights are supplied.
s = 0.0 (interpolating) if no weights are supplied.
t – The knots needed for task=-1. If given then task is automatically
set to -1.
full_output – If non-zero, then return optional outputs. per – If non-zero, data points are considered
periodic with period
x[m-1] - x[0] and a smooth periodic spline approximation is returned. Values of y[m-1]
and w[m-1] are not used.
quiet – Non-zero to suppress messages.
Outputs: (tck, {fp, ier, msg})
tck – (t,c,k) a tuple containing the vector of knots, the B-spline
coefficients, and the degree of the spline.
fp – The weighted sum of squared residuals of the spline approximation. ier – An integer flag about
splrep success. Success is indicated if
ier<=0. If ier in [1,2,3] an error occurred but was not raised. Otherwise an error is raised.
msg – A message corresponding to the integer flag, ier.
Remarks:
See splev for evaluation of the spline and its derivatives.
Example:
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x = linspace(0, 10, 10) y = sin(x) tck = splrep(x, y) x2 = linspace(0, 10, 200) y2 = splev(x2, tck)
plot(x, y, ‘o’, x2, y2)
See also:
splprep, splev, sproot, spalde, splint - evaluation, roots, integral bisplrep, bisplev - bivariate splines UnivariateSpline, BivariateSpline - an alternative wrapping
of the FITPACK functions
Notes:
Based on algorithms described in:
Dierckx P.
[An algorithm for smoothing, differentiation and integ-] ration of experimental data using spline
functions, J.Comp.Appl.Maths 1 (1975) 165-184.
Dierckx P.
[A fast algorithm for smoothing data on a rectangular] grid while using spline functions, SIAM
J.Numer.Anal. 19 (1982) 1286-1304.
Dierckx P.
[An improved algorithm for curve fitting with spline] functions, report tw54, Dept. Computer Science,K.U. Leuven, 1981.
Dierckx P.
[Curve and surface fitting with splines, Monographs on] Numerical Analysis, Oxford University
Press, 1993.
splprep(x, w=None, u=None, ub=None, ue=None, k=3, task=0, s=None, t=None, full_output=0, nest=None,
per=0, quiet=1)
Find the B-spline representation of an N-dimensional curve.
Description:
Given a list of N rank-1 arrays, x, which represent a curve in N-dimensional space parametrized
by u, find a smooth approximating spline curve g(u). Uses the FORTRAN routine parcur from
FITPACK
Inputs:
x – A list of sample vector arrays representing the curve. u – An array of parameter values. If not
given, these values are
calculated automatically as (M = len(x[0])): v[0] = 0 v[i] = v[i-1] + distance(x[i],x[i-1])
u[i] = v[i] / v[M-1]
ub, ue – The end-points of the parameters interval. Defaults to
u[0] and u[-1].
k – Degree of the spline. Cubic splines are recommended. Even values of
k should be avoided especially with a small s-value. 1 <= k <= 5.
task – If task==0 find t and c for a given smoothing factor, s.
If task==1 find t and c for another value of the smoothing factor,
s. There must have been a previous call with task=0 or task=1 for the same set of data.
If task=-1 find the weighted least square spline for a given set of
knots, t.
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s – A smoothing condition. The amount of smoothness is determined by
satisfying the conditions: sum((w * (y - g))**2,axis=0) <= s where g(x) is the smoothed
interpolation of (x,y). The user can use s to control the tradeoff between closeness and
smoothness of fit. Larger s means more smoothing while smaller values of s indicate less
smoothing. Recommended values of s depend on the weights, w. If the weights represent
the inverse of the standard-deviation of y, then a good s value should be found in the range
(m-sqrt(2*m),m+sqrt(2*m)) where m is the number of datapoints in x, y, and w.
t – The knots needed for task=-1. full_output – If non-zero, then return optional outputs. nest – An
over-estimate of the total number of knots of the spline to
help in determining the storage space. By default nest=m/2. Always large enough is
nest=m+k+1.
per – If non-zero, data points are considered periodic with period
x[m-1] - x[0] and a smooth periodic spline approximation is returned. Values of y[m-1] and
w[m-1] are not used.
quiet – Non-zero to suppress messages.
Outputs: (tck, u, {fp, ier, msg})
tck – (t,c,k) a tuple containing the vector of knots, the B-spline
coefficients, and the degree of the spline.
u – An array of the values of the parameter.
fp – The weighted sum of squared residuals of the spline approximation. ier – An integer flag about
splrep success. Success is indicated
if ier<=0. If ier in [1,2,3] an error occurred but was not raised. Otherwise an error is
raised.
msg – A message corresponding to the integer flag, ier.
Remarks:
SEE splev for evaluation of the spline and its derivatives.
See also:
splrep, splev, sproot, spalde, splint - evaluation, roots, integral bisplrep, bisplev - bivariate splines UnivariateSpline, BivariateSpline - an alternative wrapping
of the FITPACK functions
Notes:
Dierckx P.
[Algorithms for smoothing data with periodic and] parametric splines, Computer Graphics and Image Processing 20 (1982) 171-184.
Dierckx P.
[Algorithms for smoothing data with periodic and param-] etric splines, report tw55, Dept. Computer Science, K.U.Leuven, 1981.
Dierckx P.
[Curve and surface fitting with splines, Monographs on] Numerical Analysis, Oxford University
Press, 1993.
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splev(x, tck, der=0)
Evaluate a B-spline and its derivatives.
Given the knots and coefficients of a B-spline representation, evaluate the value of the smoothing polynomial
and it’s derivatives. This is a wrapper around the FORTRAN routines splev and splder of FITPACK.
Parameters
x (u) – a 1-D array of points at which to return the value of the :
smoothed spline or its derivatives. If tck was returned from splprep, then the parameter values, u should be given.
tck – A sequence of length 3 returned by splrep or splprep containg the :
knots, coefficients, and degree of the spline.
der – The order of derivative of the spline to compute (must be less than :
or equal to k).
Returns
y – an array of values representing the spline function or curve. :
If tck was returned from splrep, then this is a list of arrays representing the curve in
N-dimensional space.
See Also:
splprep, splrep, sproot, spalde, splint, roots, integral, bisplrep, bisplev,
UnivariateSpline, BivariateSpline
References
[R5], [R6], [R7]
splint(a, b, tck, full_output=0)
Evaluate the definite integral of a B-spline.
Given the knots and coefficients of a B-spline, evaluate the definite integral of the smoothing polynomial between two given points.
Parameters
a, b – The end-points of the integration interval. :
tck – A length 3 sequence describing the given spline (See splev). :
full_output – Non-zero to return optional output. :
Returns
integral – The resulting integral. :
wrk – An array containing the integrals of the :
normalized B-splines defined on the set of knots.
See Also:
splprep, splrep, sproot, spalde, splev, roots, integral, bisplrep, bisplev,
UnivariateSpline, BivariateSpline
References
[R8], [R9]
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sproot(tck, mest=10)
Find the roots of a cubic B-spline.
Given the knots (>=8) and coefficients of a cubic B-spline return the roots of the spline.
Parameters
tck – A length 3 sequence describing the given spline (See splev). :
The number of knots must be >= 8. The knots must be a montonically increasing
sequence.
mest – An estimate of the number of zeros (Default is 10) :
Returns
zeros – An array giving the roots of the spline. :
See Also:
splprep, splrep, splint, spalde, splev, bisplrep, bisplev, UnivariateSpline,
BivariateSpline
References
[R10], [R11], [R12]
spalde(x, tck)
Evaluate all derivatives of a B-spline.
Description:
Given the knots and coefficients of a cubic B-spline compute all derivatives up to order k at a point
(or set of points).
Inputs:
tck – A length 3 sequence describing the given spline (See splev). x – A point or a set of points at
which to evaluate the derivatives.
Note that t(k) <= x <= t(n-k+1) must hold for each x.
Outputs: (results, )
results – An array (or a list of arrays) containing all derivatives
up to order k inclusive for each point x.
See also:
splprep, splrep, splint, sproot, splev - evaluation, roots, integral bisplrep, bisplev - bivariate splines UnivariateSpline, BivariateSpline - an alternative wrapping
of the FITPACK functions
Notes: Based on algorithms from:
de Boor C
[On calculating with b-splines, J. Approximation Theory] 6 (1972) 50-62.
Cox M.G.
[The numerical evaluation of b-splines, J. Inst. Maths] applics 10 (1972) 134-149.
Dierckx P.
[Curve and surface fitting with splines, Monographs on] Numerical Analysis, Oxford University Press, 1993.
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bisplrep(x, y, z, w=None, xb=None, xe=None, yb=None, ye=None, kx=3, ky=3, task=0, s=None,
eps=9.9999999999999998e-17, tx=None, ty=None, full_output=0, nxest=None, nyest=None, quiet=1)
Find a bivariate B-spline representation of a surface.
Description:
Given a set of data points (x[i], y[i], z[i]) representing a surface z=f(x,y), compute a B-spline
representation of the surface. Based on the routine SURFIT from FITPACK.
Inputs:
x, y, z – Rank-1 arrays of data points. w – Rank-1 array of weights. By default w=ones(len(x)). xb,
xe – End points of approximation interval in x. yb, ye – End points of approximation interval in y.
By default xb, xe, yb, ye = x.min(), x.max(), y.min(), y.max()
kx, ky – The degrees of the spline (1 <= kx, ky <= 5). Third order
(kx=ky=3) is recommended.
task – If task=0, find knots in x and y and coefficients for a given
smoothing factor, s.
If task=1, find knots and coefficients for another value of the
smoothing factor, s. bisplrep must have been previously called with task=0 or task=1.
If task=-1, find coefficients for a given set of knots tx, ty.
s – A non-negative smoothing factor. If weights correspond
to the inverse of the standard-deviation of the errors in z, then a good s-value should be found
in the range (m-sqrt(2*m),m+sqrt(2*m)) where m=len(x)
eps – A threshold for determining the effective rank of an
over-determined linear system of equations (0 < eps < 1) — not likely to need changing.
tx, ty – Rank-1 arrays of the knots of the spline for task=-1 full_output – Non-zero to return optional
outputs. nxest, nyest – Over-estimates of the total number of knots.
If None then nxest = max(kx+sqrt(m/2),2*kx+3),
nyest = max(ky+sqrt(m/2),2*ky+3)
quiet – Non-zero to suppress printing of messages.
Outputs: (tck, {fp, ier, msg})
tck – A list [tx, ty, c, kx, ky] containing the knots (tx, ty) and
coefficients (c) of the bivariate B-spline representation of the surface along with the degree of
the spline.
fp – The weighted sum of squared residuals of the spline approximation. ier – An integer flag about
splrep success. Success is indicated if
ier<=0. If ier in [1,2,3] an error occurred but was not raised. Otherwise an error is raised.
msg – A message corresponding to the integer flag, ier.
Remarks:
SEE bisplev to evaluate the value of the B-spline given its tck representation.
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See also:
splprep, splrep, splint, sproot, splev - evaluation, roots, integral UnivariateSpline, BivariateSpline - an
alternative wrapping
of the FITPACK functions
Notes: Based on algorithms from:
Dierckx P.
[An algorithm for surface fitting with spline functions] Ima J. Numer. Anal. 1 (1981) 267-283.
Dierckx P.
[An algorithm for surface fitting with spline functions] report tw50, Dept. Computer Science,K.U.Leuven, 1980.
Dierckx P.
[Curve and surface fitting with splines, Monographs on] Numerical Analysis, Oxford University Press, 1993.
bisplev(x, y, tck, dx=0, dy=0)
Evaluate a bivariate B-spline and its derivatives.
Description:
Return a rank-2 array of spline function values (or spline derivative values) at points given by the
cross-product of the rank-1 arrays x and y. In special cases, return an array or just a float if either x
or y or both are floats. Based on BISPEV from FITPACK.
Inputs:
x, y – Rank-1 arrays specifying the domain over which to evaluate the
spline or its derivative.
tck – A sequence of length 5 returned by bisplrep containing the knot
locations, the coefficients, and the degree of the spline: [tx, ty, c, kx, ky].
dx, dy – The orders of the partial derivatives in x and y respectively.
Outputs: (vals, )
vals – The B-pline or its derivative evaluated over the set formed by
the cross-product of x and y.
Remarks:
SEE bisprep to generate the tck representation.
See also:
splprep, splrep, splint, sproot, splev - evaluation, roots, integral UnivariateSpline, BivariateSpline - an
alternative wrapping
of the FITPACK functions
Notes: Based on algorithms from:
Dierckx P.
[An algorithm for surface fitting with spline functions] Ima J. Numer. Anal. 1 (1981) 267-283.
Dierckx P.
[An algorithm for surface fitting with spline functions] report tw50, Dept. Computer Science,K.U.Leuven, 1980.
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Dierckx P.
[Curve and surface fitting with splines, Monographs on] Numerical Analysis, Oxford University Press, 1993.
3.5.4 2-D Splines
See Also:
scipy.ndimage.map_coordinates
Bivariate spline s(x,y) of degrees kx and ky on the rectangle [xb,xe] x [yb, ye] calculated
from a given set of data points (x,y,z).
SmoothBivariateSpline
Smooth bivariate spline approximation.
LSQBivariateSpline
Weighted least-squares spline approximation. See also:
BivariateSpline
class BivariateSpline()
Bivariate spline s(x,y) of degrees kx and ky on the rectangle [xb,xe] x [yb, ye] calculated from a given set of
data points (x,y,z).
See also:
bisplrep, bisplev - an older wrapping of FITPACK UnivariateSpline - a similar class for univariate spline interpolation SmoothUnivariateSpline - to create a BivariateSpline through the
given points
LSQUnivariateSpline - to create a BivariateSpline using weighted
least-squares fitting
Methods
class SmoothBivariateSpline(x, y, z, w=None, bbox=, [None, None, None, None], kx=3, ky=3, s=None,
eps=None)
Smooth bivariate spline approximation.
See also:
bisplrep, bisplev - an older wrapping of FITPACK UnivariateSpline - a similar class for univariate spline interpolation LSQUnivariateSpline - to create a BivariateSpline using weighted
least-squares fitting
Methods
class LSQBivariateSpline(x, y, z, tx, ty, w=None, bbox=, [None, None, None, None], kx=3, ky=3, eps=None)
Weighted least-squares spline approximation. See also:
bisplrep, bisplev - an older wrapping of FITPACK UnivariateSpline - a similar class for univariate spline interpolation SmoothUnivariateSpline - to create a BivariateSpline through the
given points
Methods
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Low-level interface to FITPACK functions:
bisplrep (x, y, z[, w, xb, xe, yb, ye, ...])
bisplev (x, y, tck[, dx, dy])
Find a bivariate B-spline representation of a surface.
Evaluate a bivariate B-spline and its derivatives.
bisplrep(x, y, z, w=None, xb=None, xe=None, yb=None, ye=None, kx=3, ky=3, task=0, s=None,
eps=9.9999999999999998e-17, tx=None, ty=None, full_output=0, nxest=None, nyest=None, quiet=1)
Find a bivariate B-spline representation of a surface.
Description:
Given a set of data points (x[i], y[i], z[i]) representing a surface z=f(x,y), compute a B-spline
representation of the surface. Based on the routine SURFIT from FITPACK.
Inputs:
x, y, z – Rank-1 arrays of data points. w – Rank-1 array of weights. By default w=ones(len(x)). xb,
xe – End points of approximation interval in x. yb, ye – End points of approximation interval in y.
By default xb, xe, yb, ye = x.min(), x.max(), y.min(), y.max()
kx, ky – The degrees of the spline (1 <= kx, ky <= 5). Third order
(kx=ky=3) is recommended.
task – If task=0, find knots in x and y and coefficients for a given
smoothing factor, s.
If task=1, find knots and coefficients for another value of the
smoothing factor, s. bisplrep must have been previously called with task=0 or task=1.
If task=-1, find coefficients for a given set of knots tx, ty.
s – A non-negative smoothing factor. If weights correspond
to the inverse of the standard-deviation of the errors in z, then a good s-value should be found
in the range (m-sqrt(2*m),m+sqrt(2*m)) where m=len(x)
eps – A threshold for determining the effective rank of an
over-determined linear system of equations (0 < eps < 1) — not likely to need changing.
tx, ty – Rank-1 arrays of the knots of the spline for task=-1 full_output – Non-zero to return optional
outputs. nxest, nyest – Over-estimates of the total number of knots.
If None then nxest = max(kx+sqrt(m/2),2*kx+3),
nyest = max(ky+sqrt(m/2),2*ky+3)
quiet – Non-zero to suppress printing of messages.
Outputs: (tck, {fp, ier, msg})
tck – A list [tx, ty, c, kx, ky] containing the knots (tx, ty) and
coefficients (c) of the bivariate B-spline representation of the surface along with the degree of
the spline.
fp – The weighted sum of squared residuals of the spline approximation. ier – An integer flag about
splrep success. Success is indicated if
ier<=0. If ier in [1,2,3] an error occurred but was not raised. Otherwise an error is raised.
msg – A message corresponding to the integer flag, ier.
Remarks:
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SEE bisplev to evaluate the value of the B-spline given its tck representation.
See also:
splprep, splrep, splint, sproot, splev - evaluation, roots, integral UnivariateSpline, BivariateSpline - an
alternative wrapping
of the FITPACK functions
Notes: Based on algorithms from:
Dierckx P.
[An algorithm for surface fitting with spline functions] Ima J. Numer. Anal. 1 (1981) 267-283.
Dierckx P.
[An algorithm for surface fitting with spline functions] report tw50, Dept. Computer Science,K.U.Leuven, 1980.
Dierckx P.
[Curve and surface fitting with splines, Monographs on] Numerical Analysis, Oxford University Press, 1993.
bisplev(x, y, tck, dx=0, dy=0)
Evaluate a bivariate B-spline and its derivatives.
Description:
Return a rank-2 array of spline function values (or spline derivative values) at points given by the
cross-product of the rank-1 arrays x and y. In special cases, return an array or just a float if either x
or y or both are floats. Based on BISPEV from FITPACK.
Inputs:
x, y – Rank-1 arrays specifying the domain over which to evaluate the
spline or its derivative.
tck – A sequence of length 5 returned by bisplrep containing the knot
locations, the coefficients, and the degree of the spline: [tx, ty, c, kx, ky].
dx, dy – The orders of the partial derivatives in x and y respectively.
Outputs: (vals, )
vals – The B-pline or its derivative evaluated over the set formed by
the cross-product of x and y.
Remarks:
SEE bisprep to generate the tck representation.
See also:
splprep, splrep, splint, sproot, splev - evaluation, roots, integral UnivariateSpline, BivariateSpline - an
alternative wrapping
of the FITPACK functions
Notes: Based on algorithms from:
Dierckx P.
[An algorithm for surface fitting with spline functions] Ima J. Numer. Anal. 1 (1981) 267-283.
Dierckx P.
[An algorithm for surface fitting with spline functions] report tw50, Dept. Computer Science,K.U.Leuven, 1980.
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[Curve and surface fitting with splines, Monographs on] Numerical Analysis, Oxford University Press, 1993.
3.5.5 Additional tools
lagrange (x, w)
approximate_taylor_polynomial (f, x, degree, scale[, order])
Return the Lagrange interpolating polynomial of the
data-points (x,w)
Estimate the Taylor polynomial of f at x by polynomial
fitting.
lagrange(x, w)
Return the Lagrange interpolating polynomial of the data-points (x,w)
Warning: This implementation is numerically unstable; do not expect to be able to use more than about 20 points
even if they are chosen optimally.
approximate_taylor_polynomial(f, x, degree, scale, order=None)
Estimate the Taylor polynomial of f at x by polynomial fitting.
Parameters
f : callable
The function whose Taylor polynomial is sought. Should accept a vector of x values.
x : scalar
The point at which the polynomial is to be evaluated.
degree : int
The degree of the Taylor polynomial
scale : scalar
The width of the interval to use to evaluate the Taylor polynomial. Function values spread over a range this wide are used to fit the polynomial. Must be chosen
carefully.
order : int or None
The order of the polynomial to be used in the fitting; f will be evaluated order+1
times. If None, use degree.
Returns
p : poly1d instance
The Taylor polynomial (translated to the origin, so that for example p(0)=f(x)).
Notes
The appropriate choice of “scale” is a trade-off; too large and the function differs from its Taylor polynomial too
much to get a good answer, too small and round-off errors overwhelm the higher-order terms. The algorithm
used becomes numerically unstable around order 30 even under ideal circumstances.
Choosing order somewhat larger than degree may improve the higher-order terms.
3.6 Input and output (scipy.io)
See Also:
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3.6.1 MATLAB® files
loadmat (file_name[, mdict, appendmat, **kwargs)
savemat (file_name, mdict[, appendmat, format, ...])
Load Matlab(tm) file
Save a dictionary of names and arrays into the MATLAB-style
.mat file.
loadmat(file_name, mdict=None, appendmat=True, **kwargs)
Load Matlab(tm) file
Parameters
file_name : string
Name of the mat file (do not need .mat extension if appendmat==True) Can also pass
open file-like object
m_dict : dict, optional
dictionary in which to insert matfile variables
appendmat : {True, False} optional
True to append the .mat extension to the end of the given filename, if not already
present
byte_order : {None, string}, optional
None by default, implying byte order guessed from mat file. Otherwise can be one
of (‘native’, ‘=’, ‘little’, ‘<’, ‘BIG’, ‘>’)
mat_dtype : {False, True} optional
If True, return arrays in same dtype as would be loaded into matlab (instead of the
dtype with which they are saved)
squeeze_me : {False, True} optional
whether to squeeze unit matrix dimensions or not
chars_as_strings : {True, False} optional
whether to convert char arrays to string arrays
matlab_compatible : {False, True}
returns matrices as would be loaded by matlab (implies squeeze_me=False,
chars_as_strings=False, mat_dtype=True, struct_as_record=True)
struct_as_record : {True, False} optional
Whether to load matlab structs as numpy record arrays, or as old-style numpy arrays
with dtype=object. Setting this flag to False replicates the behaviour of scipy version
0.7.x (returning numpy object arrays). The default setting is True, because it allows
easier round-trip load and save of matlab files.
Returns
mat_dict : dict
dictionary with variable names as keys, and loaded matrices as values
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Notes
v4 (Level 1.0), v6 and v7 to 7.2 matfiles are supported.
You will need an HDF5 python library to read matlab 7.3 format mat files. Because scipy does not supply one,
we do not implement the HDF5 / 7.3 interface here.
savemat(file_name, mdict, appendmat=True, format=’5’, long_field_names=False, do_compression=False,
oned_as=None)
Save a dictionary of names and arrays into the MATLAB-style .mat file.
This saves the arrayobjects in the given dictionary to a matlab style .mat file.
Parameters
file_name : {string, file-like object}
Name of the mat file (do not need .mat extension if appendmat==True) Can also pass
open file-like object
m_dict : dict
dictionary from which to save matfile variables
appendmat : {True, False} optional
True to append the .mat extension to the end of the given filename, if not already
present
format : {‘5’, ‘4’} string, optional
‘5’ for matlab 5 (up to matlab 7.2) ‘4’ for matlab 4 mat files
long_field_names : boolean, optional, default=False
• False - maximum field name length in a structure is 31 characters which is the documented maximum length
• True - maximum field name length in a structure is 63 characters which works for Matlab
7.6
do_compression : {False, True} bool, optional
Whether to compress matrices on write. Default is False
oned_as : {‘column’, ‘row’} string, optional
If ‘column’, write 1D numpy arrays as column vectors If ‘row’, write 1D numpy
arrays as row vectors
3.6.2 Matrix Market files
mminfo (source)
mmread (source)
mmwrite (target, a[, comment, field, ...])
Queries the contents of the Matrix Market file ‘filename’ to extract size and
storage information.
Reads the contents of a Matrix Market file ‘filename’ into a matrix.
Writes the sparse or dense matrix A to a Matrix Market formatted file.
mminfo(source)
Queries the contents of the Matrix Market file ‘filename’ to extract size and storage information.
Parameters
source : file
Matrix Market filename (extension .mtx) or open file object
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Returns
rows,cols : int
Number of matrix rows and columns
entries : int
Number of non-zero entries of a sparse matrix or rows*cols for a dense matrix
format : {‘coordinate’, ‘array’}
field : {‘real’, ‘complex’, ‘pattern’, ‘integer’}
symm : {‘general’, ‘symmetric’, ‘skew-symmetric’, ‘hermitian’}
mmread(source)
Reads the contents of a Matrix Market file ‘filename’ into a matrix.
Parameters
source : file
Matrix Market filename (extensions .mtx, .mtz.gz) or open file object.
Returns
a: :
Sparse or full matrix
mmwrite(target, a, comment=”, field=None, precision=None)
Writes the sparse or dense matrix A to a Matrix Market formatted file.
Parameters
target : file
Matrix Market filename (extension .mtx) or open file object
a : array like
Sparse or full matrix
comment : str
comments to be prepended to the Matrix Market file
field : {‘real’, ‘complex’, ‘pattern’, ‘integer’}, optional
precision : :
Number of digits to display for real or complex values.
3.6.3 Other
save_as_module ([file_name, data])
npfile
Save the dictionary “data” into a module and shelf named save.
save_as_module(file_name=None, data=None)
Save the dictionary “data” into a module and shelf named save.
Parameters
file_name : str, optional
File name of the module to save.
data : dict, optional
The dictionary to store in the module.
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3.6.4 Wav sound files (scipy.io.wavfile)
read (file)
write (filename, rate, data)
Return the sample rate (in samples/sec) and data from a WAV file
Write a numpy array as a WAV file
read(file)
Return the sample rate (in samples/sec) and data from a WAV file
Parameters
file : file
Input wav file.
Returns
rate : int
Sample rate of wav file
data : numpy array
Data read from wav file
Notes
•The file can be an open file or a filename.
•The returned sample rate is a Python integer
•The data is returned as a numpy array with a data-type determined from the file.
write(filename, rate, data)
Write a numpy array as a WAV file
Parameters
filename : file
The name of the file to write (will be over-written).
rate : int
The sample rate (in samples/sec).
data : ndarray
A 1-D or 2-D numpy array of integer data-type.
Notes
•Writes a simple uncompressed WAV file.
•The bits-per-sample will be determined by the data-type.
•To write multiple-channels, use a 2-D array of shape (Nsamples, Nchannels).
3.6.5 Arff files (scipy.io.arff)
Module to read arff files (weka format).
arff is a simple file format which support numerical, string and data values. It supports sparse data too.
See http://weka.sourceforge.net/wekadoc/index.php/en:ARFF_(3.4.6) for more details about arff format and available
datasets.
loadarff (filename)
Read an arff file.
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loadarff(filename)
Read an arff file.
Parameters
filename : str
the name of the file
Returns
data : record array
the data of the arff file. Each record corresponds to one attribute.
meta : MetaData
this contains information about the arff file, like type and names of attributes, the
relation (name of the dataset), etc...
Notes
This function should be able to read most arff files. Not implemented functionalities include:
•date type attributes
•string type attributes
It can read files with numeric and nominal attributes. It can read files with sparse data (? in the file).
3.6.6 Netcdf (scipy.io.netcdf)
netcdf_fileA netcdf_file object has two standard attributes: dimensions and variables. The
values of both are dictionaries, mapping dimension names to their associated lengths and variable
names to variables, respectively. Application programs should never modify these dictionaries.
netcdf_variable
netcdf_variable objects are constructed by calling the method createVariable on the
netcdf_file object.
class netcdf_file(filename, mode=’r’, mmap=None, version=1)
A netcdf_file object has two standard attributes: dimensions and variables. The values of both are
dictionaries, mapping dimension names to their associated lengths and variable names to variables, respectively.
Application programs should never modify these dictionaries.
All other attributes correspond to global attributes defined in the NetCDF file. Global file attributes are created
by assigning to an attribute of the netcdf_file object.
Methods
class netcdf_variable(data, typecode, shape, dimensions, attributes=None)
netcdf_variable objects are constructed by calling the method createVariable on the netcdf_file
object.
netcdf_variable objects behave much like array objects defined in Numpy, except that their data resides
in a file. Data is read by indexing and written by assigning to an indexed subset; the entire array can be accessed
by the index [:] or using the methods getValue and assignValue. netcdf_variable objects also
have attribute shape with the same meaning as for arrays, but the shape cannot be modified. There is another
read-only attribute dimensions, whose value is the tuple of dimension names.
All other attributes correspond to variable attributes defined in the NetCDF file. Variable attributes are created
by assigning to an attribute of the netcdf_variable object.
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Methods
3.7 Linear algebra (scipy.linalg)
3.7.1 Basics
inv (a[, overwrite_a])
solve (a, b[, sym_pos, lower, ...])
solve_banded ((l, u), ab, b[, overwrite_ab, overwrite_b, ...])
solveh_banded (ab, b[, overwrite_ab, overwrite_b, ...])
det (a[, overwrite_a])
norm (a[, ord])
lstsq (a, b[, cond, overwrite_a, ...])
pinv (a[, cond, rcond])
pinv2 (a[, cond, rcond])
Compute the inverse of a matrix.
Solve the equation a x = b for x
Solve the equation a x = b for x, assuming a is banded
matrix.
Solve equation a x = b. a is Hermitian positive-definite
banded matrix.
Compute the determinant of a matrix
Matrix or vector norm.
Compute least-squares solution to equation Ax = b.
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
inv(a, overwrite_a=False)
Compute the inverse of a matrix.
Parameters
a : array-like, shape (M, M)
Matrix to be inverted
Returns
ainv : array-like, shape (M, M)
Inverse of the matrix a
Raises LinAlgError if a is singular :
Examples
>>> a = array([[1., 2.], [3., 4.]])
>>> inv(a)
array([[-2. , 1. ],
[ 1.5, -0.5]])
>>> dot(a, inv(a))
array([[ 1., 0.],
[ 0., 1.]])
solve(a, b, sym_pos=False, lower=False, overwrite_a=False, overwrite_b=False, debug=False)
Solve the equation a x = b for x
Parameters
a : array, shape (M, M)
b : array, shape (M,) or (M, N)
sym_pos : boolean
Assume a is symmetric and positive definite
lower : boolean
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Use only data contained in the lower triangle of a, if sym_pos is true. Default is to
use upper triangle.
overwrite_a : boolean
Allow overwriting data in a (may enhance performance)
overwrite_b : boolean
Allow overwriting data in b (may enhance performance)
Returns
x : array, shape (M,) or (M, N) depending on b
Solution to the system a x = b
Raises LinAlgError if a is singular :
solve_banded((l, u), ab, b, overwrite_ab=False, overwrite_b=False, debug=False)
Solve the equation a x = b for x, assuming a is banded matrix.
The matrix a is stored in ab using the matrix diagonal orded form:
ab[u + i - j, j] == a[i,j]
Example of ab (shape of a is (6,6), u=1, l=2):
*
a00
a10
a20
a01
a11
a21
a31
a12
a22
a32
a42
a23
a33
a43
a53
a34
a44
a54
*
a45
a55
*
*
Parameters
(l, u) : (integer, integer)
Number of non-zero lower and upper diagonals
ab : array, shape (l+u+1, M)
Banded matrix
b : array, shape (M,) or (M, K)
Right-hand side
overwrite_ab : boolean
Discard data in ab (may enhance performance)
overwrite_b : boolean
Discard data in b (may enhance performance)
Returns
x : array, shape (M,) or (M, K)
The solution to the system a x = b
solveh_banded(ab, b, overwrite_ab=False, overwrite_b=False, lower=False)
Solve equation a x = b. a is Hermitian positive-definite banded matrix.
The matrix a is stored in ab either in lower diagonal or upper diagonal ordered form:
ab[u + i - j, j] == a[i,j] (if upper form; i <= j) ab[ i - j, j] == a[i,j] (if lower form; i >= j)
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Example of ab (shape of a is (6,6), u=2):
upper form:
a02 a13 a24 a35
*
*
a01 a12 a23 a34 a45
*
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 *
*
Cells marked with * are not used.
Parameters
ab : array, shape (u + 1, M)
Banded matrix
b : array, shape (M,) or (M, K)
Right-hand side
overwrite_ab : boolean
Discard data in ab (may enhance performance)
overwrite_b : boolean
Discard data in b (may enhance performance)
lower : boolean
Is the matrix in the lower form. (Default is upper form)
Returns
c : array, shape (u+1, M)
Cholesky factorization of a, in the same banded format as ab
x : array, shape (M,) or (M, K)
The solution to the system a x = b
Notes
The inclusion of c in the return value is deprecated. In SciPy version 0.9, the return value will be the solution x
only.
det(a, overwrite_a=False)
Compute the determinant of a matrix
Parameters
a : array, shape (M, M)
Returns
det : float or complex
Determinant of a
Notes
The determinant is computed via LU factorization, LAPACK routine z/dgetrf.
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norm(a, ord=None)
Matrix or vector norm.
This function is able to return one of seven different matrix norms, or one of an infinite number of vector norms
(described below), depending on the value of the ord parameter.
Parameters
x : array_like, shape (M,) or (M, N)
Input array.
ord : {non-zero int, inf, -inf, ‘fro’}, optional
Order of the norm (see table under Notes). inf means numpy’s inf object.
Returns
n : float
Norm of the matrix or vector.
Notes
For values of ord <= 0, the result is, strictly speaking, not a mathematical ‘norm’, but it may still be useful
for various numerical purposes.
The following norms can be calculated:
ord
None
‘fro’
inf
-inf
0
1
-1
2
-2
other
norm for matrices
Frobenius norm
Frobenius norm
max(sum(abs(x), axis=1))
min(sum(abs(x), axis=1))
–
max(sum(abs(x), axis=0))
min(sum(abs(x), axis=0))
2-norm (largest sing. value)
smallest singular value
–
norm for vectors
2-norm
–
max(abs(x))
min(abs(x))
sum(x != 0)
as below
as below
as below
as below
sum(abs(x)**ord)**(1./ord)
The Frobenius norm is given by [R14]:
P
||A||F = [ i,j abs(ai,j )2 ]1/2
References
[R14]
Examples
>>> from numpy import linalg as LA
>>> a = np.arange(9) - 4
>>> a
array([-4, -3, -2, -1, 0, 1, 2,
>>> b = a.reshape((3, 3))
>>> b
array([[-4, -3, -2],
[-1, 0, 1],
[ 2, 3, 4]])
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>>> LA.norm(a)
7.745966692414834
>>> LA.norm(b)
7.745966692414834
>>> LA.norm(b, ’fro’)
7.745966692414834
>>> LA.norm(a, np.inf)
4
>>> LA.norm(b, np.inf)
9
>>> LA.norm(a, -np.inf)
0
>>> LA.norm(b, -np.inf)
2
>>> LA.norm(a, 1)
20
>>> LA.norm(b, 1)
7
>>> LA.norm(a, -1)
-4.6566128774142013e-010
>>> LA.norm(b, -1)
6
>>> LA.norm(a, 2)
7.745966692414834
>>> LA.norm(b, 2)
7.3484692283495345
>>> LA.norm(a, -2)
nan
>>> LA.norm(b, -2)
1.8570331885190563e-016
>>> LA.norm(a, 3)
5.8480354764257312
>>> LA.norm(a, -3)
nan
lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False)
Compute least-squares solution to equation Ax = b.
Compute a vector x such that the 2-norm |b - A x| is minimized.
Parameters
a : array, shape (M, N)
Left hand side matrix (2-D array).
b : array, shape (M,) or (M, K)
Right hand side matrix or vector (1-D or 2-D array).
cond : float, optional
Cutoff for ‘small’ singular values; used to determine effective rank of a. Singular values smaller than rcond * largest_singular_value are considered
zero.
overwrite_a : bool, optional
Discard data in a (may enhance performance). Default is False.
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overwrite_b : bool, optional
Discard data in b (may enhance performance). Default is False.
Returns
x : array, shape (N,) or (N, K) depending on shape of b
Least-squares solution.
residues : ndarray, shape () or (1,) or (K,)
Sums of residues, squared 2-norm for each column in b - a x. If rank of matrix
a is < N or > M this is an empty array. If b was 1-D, this is an (1,) shape array,
otherwise the shape is (K,).
rank : int
Effective rank of matrix a.
s : array, shape (min(M,N),)
Singular values of a. The condition number of a is abs(s[0]/s[-1]).
Raises
LinAlgError : :
If computation does not converge.
See Also:
optimize.nnls
linear least squares with non-negativity constraint
pinv(a, cond=None, rcond=None)
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using a least-squares solver.
Parameters
a : array, shape (M, N)
Matrix to be pseudo-inverted
cond, rcond : float
Cutoff for ‘small’ singular values in the least-squares solver. Singular values smaller
than rcond*largest_singular_value are considered zero.
Returns
B : array, shape (N, M)
Raises LinAlgError if computation does not converge :
Examples
>>> from numpy import *
>>> a = random.randn(9, 6)
>>> B = linalg.pinv(a)
>>> allclose(a, dot(a, dot(B, a)))
True
>>> allclose(B, dot(B, dot(a, B)))
True
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pinv2(a, cond=None, rcond=None)
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its singular-value decomposition and including all ‘large’
singular values.
Parameters
a : array, shape (M, N)
Matrix to be pseudo-inverted
cond, rcond : float or None
Cutoff for ‘small’ singular values.
Singular
rcond*largest_singular_value are considered zero.
values
smaller
than
If None or -1, suitable machine precision is used.
Returns
B : array, shape (N, M)
Raises LinAlgError if SVD computation does not converge :
Examples
>>> from numpy import *
>>> a = random.randn(9, 6)
>>> B = linalg.pinv2(a)
>>> allclose(a, dot(a, dot(B, a)))
True
>>> allclose(B, dot(B, dot(a, B)))
True
3.7.2 Eigenvalue Problem
eig (a[, b, left, right, ...])
Solve an ordinary or generalized eigenvalue problem of a square
matrix.
eigvals (a[, b, overwrite_a])
Compute eigenvalues from an ordinary or generalized eigenvalue
problem.
eigh (a[, b, lower, eigvals_only, ...])
Solve an ordinary or generalized eigenvalue problem for a complex
Hermitian or real symmetric matrix.
eigvalsh (a[, b, lower, overSolve an ordinary or generalized eigenvalue problem for a complex
write_a, ...])
Hermitian or real symmetric matrix.
eig_banded (a_band[, lower, eigvals_only,Solve
...]) real symmetric or complex hermitian band matrix eigenvalue
problem.
eigvals_banded (a_band[, lower, over- Solve real symmetric or complex hermitian band matrix eigenvalue
write_a_band, ...])
problem.
eig(a, b=None, left=False, right=True, overwrite_a=False, overwrite_b=False)
Solve an ordinary or generalized eigenvalue problem of a square matrix.
Find eigenvalues w and right or left eigenvectors of a general matrix:
a
vr[:,i] = w[i]
b
vr[:,i]
a.H vl[:,i] = w[i].conj() b.H vl[:,i]
where .H is the Hermitean conjugation.
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Parameters
a : array, shape (M, M)
A complex or real matrix whose eigenvalues and eigenvectors will be computed.
b : array, shape (M, M)
Right-hand side matrix in a generalized eigenvalue problem. If omitted, identity
matrix is assumed.
left : boolean
Whether to calculate and return left eigenvectors
right : boolean
Whether to calculate and return right eigenvectors
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
overwrite_b : boolean
Whether to overwrite data in b (may improve performance)
Returns
w : double or complex array, shape (M,)
The eigenvalues, each repeated according to its multiplicity.
(if left == True) :
vl : double or complex array, shape (M, M)
The normalized left eigenvector corresponding to the eigenvalue w[i] is the column
v[:,i].
(if right == True) :
vr : double or complex array, shape (M, M)
The normalized right eigenvector corresponding to the eigenvalue w[i] is the column
vr[:,i].
Raises LinAlgError if eigenvalue computation does not converge :
See Also:
eigh
eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eigvals(a, b=None, overwrite_a=False)
Compute eigenvalues from an ordinary or generalized eigenvalue problem.
Find eigenvalues of a general matrix:
a
vr[:,i] = w[i]
b
vr[:,i]
Parameters
a : array, shape (M, M)
A complex or real matrix whose eigenvalues and eigenvectors will be computed.
b : array, shape (M, M)
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Right-hand side matrix in a generalized eigenvalue problem. If omitted, identity
matrix is assumed.
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
Returns
w : double or complex array, shape (M,)
The eigenvalues, each repeated according to its multiplicity, but not in any specific
order.
Raises LinAlgError if eigenvalue computation does not converge :
See Also:
eigvalsh
eigenvalues of symmetric or Hemitiean arrays
eig
eigenvalues and right eigenvectors of general arrays
eigh
eigenvalues and eigenvectors of symmetric/Hermitean arrays.
eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False, overwrite_b=False, turbo=True,
eigvals=None, type=1)
Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.
Find eigenvalues w and optionally eigenvectors v of matrix a, where b is positive definite:
a v[:,i] = w[i] b v[:,i]
v[i,:].conj() a v[:,i] = w[i]
v[i,:].conj() b v[:,i] = 1
Parameters
a : array, shape (M, M)
A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors
will be computed.
b : array, shape (M, M)
A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed.
lower : boolean
Whether the pertinent array data is taken from the lower or upper triangle of a. (Default: lower)
eigvals_only : boolean
Whether to calculate only eigenvalues and no eigenvectors. (Default: both are calculated)
turbo : boolean
Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if eigvals=None)
eigvals : tuple (lo, hi)
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Indexes of the smallest and largest (in ascending order) eigenvalues and corresponding eigenvectors to be returned: 0 <= lo < hi <= M-1. If omitted, all eigenvalues and
eigenvectors are returned.
type: integer :
Specifies the problem type to be solved:
type = 1: a v[:,i] = w[i] b v[:,i] type = 2: a b v[:,i] = w[i] v[:,i] type = 3: b a v[:,i]
= w[i] v[:,i]
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
overwrite_b : boolean
Whether to overwrite data in b (may improve performance)
Returns
w : real array, shape (N,)
The N (1<=N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity.
(if eigvals_only == False) :
v : complex array, shape (M, N)
The normalized selected eigenvector corresponding to the eigenvalue w[i] is the column v[:,i]. Normalization: type 1 and 3: v.conj() a v = w type 2: inv(v).conj() a
inv(v) = w type = 1 or 2: v.conj() b v = I type = 3 : v.conj() inv(b) v = I
Raises LinAlgError if eigenvalue computation does not converge, :
an error occurred, or b matrix is not definite positive. Note that :
if input matrices are not symmetric or hermitian, no error is reported :
but results will be wrong. :
See Also:
eig
eigenvalues and right eigenvectors for non-symmetric arrays
eigvalsh(a, b=None, lower=True, overwrite_a=False, overwrite_b=False, turbo=True, eigvals=None, type=1)
Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.
Find eigenvalues w of matrix a, where b is positive definite:
a v[:,i] = w[i] b v[:,i]
v[i,:].conj() a v[:,i] = w[i]
v[i,:].conj() b v[:,i] = 1
Parameters
a : array, shape (M, M)
A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors
will be computed.
b : array, shape (M, M)
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A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed.
lower : boolean
Whether the pertinent array data is taken from the lower or upper triangle of a. (Default: lower)
turbo : boolean
Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if eigvals=None)
eigvals : tuple (lo, hi)
Indexes of the smallest and largest (in ascending order) eigenvalues and corresponding eigenvectors to be returned: 0 <= lo < hi <= M-1. If omitted, all eigenvalues and
eigenvectors are returned.
type: integer :
Specifies the problem type to be solved:
type = 1: a v[:,i] = w[i] b v[:,i] type = 2: a b v[:,i] = w[i] v[:,i] type = 3: b a v[:,i]
= w[i] v[:,i]
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
overwrite_b : boolean
Whether to overwrite data in b (may improve performance)
Returns
w : real array, shape (N,)
The N (1<=N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity.
Raises LinAlgError if eigenvalue computation does not converge, :
an error occurred, or b matrix is not definite positive. Note that :
if input matrices are not symmetric or hermitian, no error is reported :
but results will be wrong. :
See Also:
eigvals
eigenvalues of general arrays
eigh
eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eig
eigenvalues and right eigenvectors for non-symmetric arrays
eig_banded(a_band,
lower=False,
eigvals_only=False,
overwrite_a_band=False,
lect_range=None, max_ev=0)
Solve real symmetric or complex hermitian band matrix eigenvalue problem.
select=’a’,
se-
Find eigenvalues w and optionally right eigenvectors v of a:
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a v[:,i] = w[i] v[:,i]
v.H v
= identity
The matrix a is stored in ab either in lower diagonal or upper diagonal ordered form:
ab[u + i - j, j] == a[i,j] (if upper form; i <= j) ab[ i - j, j] == a[i,j] (if lower form; i >= j)
Example of ab (shape of a is (6,6), u=2):
upper form:
a02 a13 a24 a35
*
*
a01 a12 a23 a34 a45
*
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 *
*
Cells marked with * are not used.
Parameters
a_band : array, shape (M, u+1)
Banded matrix whose eigenvalues to calculate
lower : boolean
Is the matrix in the lower form. (Default is upper form)
eigvals_only : boolean
Compute only the eigenvalues and no eigenvectors. (Default: calculate also eigenvectors)
overwrite_a_band: :
Discard data in a_band (may enhance performance)
select: {‘a’, ‘v’, ‘i’} :
Which eigenvalues to calculate
select
‘a’
‘v’
‘i’
calculated
All eigenvalues
Eigenvalues in the interval (min, max]
Eigenvalues with indices min <= i <= max
select_range : (min, max)
Range of selected eigenvalues
max_ev : integer
For select==’v’, maximum number of eigenvalues expected. For other values of
select, has no meaning.
In doubt, leave this parameter untouched.
Returns
w : array, shape (M,)
The eigenvalues, in ascending order, each repeated according to its multiplicity.
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v : double or complex double array, shape (M, M)
The normalized eigenvector corresponding to the eigenvalue w[i] is the column v[:,i].
Raises LinAlgError if eigenvalue computation does not converge :
eigvals_banded(a_band, lower=False, overwrite_a_band=False, select=’a’, select_range=None)
Solve real symmetric or complex hermitian band matrix eigenvalue problem.
Find eigenvalues w of a:
a v[:,i] = w[i] v[:,i]
v.H v
= identity
The matrix a is stored in ab either in lower diagonal or upper diagonal ordered form:
ab[u + i - j, j] == a[i,j] (if upper form; i <= j) ab[ i - j, j] == a[i,j] (if lower form; i >= j)
Example of ab (shape of a is (6,6), u=2):
upper form:
a02 a13 a24 a35
*
*
a01 a12 a23 a34 a45
*
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 *
*
Cells marked with * are not used.
Parameters
a_band : array, shape (M, u+1)
Banded matrix whose eigenvalues to calculate
lower : boolean
Is the matrix in the lower form. (Default is upper form)
overwrite_a_band: :
Discard data in a_band (may enhance performance)
select: {‘a’, ‘v’, ‘i’} :
Which eigenvalues to calculate
select
‘a’
‘v’
‘i’
calculated
All eigenvalues
Eigenvalues in the interval (min, max]
Eigenvalues with indices min <= i <= max
select_range : (min, max)
Range of selected eigenvalues
Returns
w : array, shape (M,)
The eigenvalues, in ascending order, each repeated according to its multiplicity.
Raises LinAlgError if eigenvalue computation does not converge :
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See Also:
eig_banded
eigenvalues and right eigenvectors for symmetric/Hermitian band matrices
eigvals
eigenvalues of general arrays
eigh
eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eig
eigenvalues and right eigenvectors for non-symmetric arrays
3.7.3 Decompositions
lu (a[, permute_l, overwrite_a])
lu_factor (a[, overwrite_a])
lu_solve ((lu, piv), b[, trans, overwrite_b])
svd (a[, full_matrices, compute_uv, ...])
svdvals (a[, overwrite_a])
diagsvd (s, M, N)
orth (A)
cholesky (a[, lower, overwrite_a])
cholesky_banded (ab[, overwrite_ab, lower])
cho_factor (a[, lower, overwrite_a])
cho_solve ((c, lower), b[, overwrite_b])
cho_solve_banded ((cb, lower), b[, overwrite_b])
qr (a[, overwrite_a, lwork, ...])
schur (a[, output, lwork, overwrite_a])
rsf2csf (T, Z)
hessenberg (a[, calc_q, overwrite_a])
Compute pivoted LU decompostion of a matrix.
Compute pivoted LU decomposition of a matrix.
Solve an equation system, a x = b, given the LU factorization of
a
Singular Value Decomposition.
Compute singular values of a matrix.
Construct the sigma matrix in SVD from singular values and
size M,N.
Construct an orthonormal basis for the range of A using SVD
Compute the Cholesky decomposition of a matrix.
Cholesky decompose a banded Hermitian positive-definite
matrix
Compute the Cholesky decomposition of a matrix, to use in
cho_solve
Solve the linear equations A x = b, given the Cholesky
factorization of A.
Solve the linear equations A x = b, given the Cholesky
factorization of A.
Compute QR decomposition of a matrix.
Compute Schur decomposition of a matrix.
Convert real Schur form to complex Schur form.
Compute Hessenberg form of a matrix.
lu(a, permute_l=False, overwrite_a=False)
Compute pivoted LU decompostion of a matrix.
The decomposition is:
A = P L U
where P is a permutation matrix, L lower triangular with unit diagonal elements, and U upper triangular.
Parameters
a : array, shape (M, N)
Array to decompose
permute_l : boolean
Perform the multiplication P*L (Default: do not permute)
overwrite_a : boolean
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Whether to overwrite data in a (may improve performance)
Returns
(If permute_l == False) :
p : array, shape (M, M)
Permutation matrix
l : array, shape (M, K)
Lower triangular or trapezoidal matrix with unit diagonal. K = min(M, N)
u : array, shape (K, N)
Upper triangular or trapezoidal matrix
(If permute_l == True) :
pl : array, shape (M, K)
Permuted L matrix. K = min(M, N)
u : array, shape (K, N)
Upper triangular or trapezoidal matrix
Notes
This is a LU factorization routine written for Scipy.
lu_factor(a, overwrite_a=False)
Compute pivoted LU decomposition of a matrix.
The decomposition is:
A = P L U
where P is a permutation matrix, L lower triangular with unit diagonal elements, and U upper triangular.
Parameters
a : array, shape (M, M)
Matrix to decompose
overwrite_a : boolean
Whether to overwrite data in A (may increase performance)
Returns
lu : array, shape (N, N)
Matrix containing U in its upper triangle, and L in its lower triangle. The unit diagonal elements of L are not stored.
piv : array, shape (N,)
Pivot indices representing the permutation matrix P: row i of matrix was interchanged with row piv[i].
See Also:
lu_solve
solve an equation system using the LU factorization of a matrix
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Notes
This is a wrapper to the *GETRF routines from LAPACK.
lu_solve((lu, piv), b, trans=0, overwrite_b=False)
Solve an equation system, a x = b, given the LU factorization of a
Parameters
(lu, piv) :
Factorization of the coefficient matrix a, as given by lu_factor
b : array
Right-hand side
trans : {0, 1, 2}
Type of system to solve:
trans
0
1
2
system
ax=b
a^T x = b
a^H x = b
Returns
x : array
Solution to the system
See Also:
lu_factor
LU factorize a matrix
svd(a, full_matrices=True, compute_uv=True, overwrite_a=False)
Singular Value Decomposition.
Factorizes the matrix a into two unitary matrices U and Vh and an 1d-array s of singular values (real, nonnegative) such that a == U S Vh if S is an suitably shaped matrix of zeros whose main diagonal is s.
Parameters
a : array, shape (M, N)
Matrix to decompose
full_matrices : boolean
If true, U, Vh are shaped (M,M), (N,N) If false, the shapes are (M,K), (K,N) where
K = min(M,N)
compute_uv : boolean
Whether to compute also U, Vh in addition to s (Default: true)
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
Returns
U: array, shape (M,M) or (M,K) depending on full_matrices :
s: array, shape (K,) :
The singular values, sorted so that s[i] >= s[i+1]. K = min(M, N)
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Vh: array, shape (N,N) or (K,N) depending on full_matrices :
For compute_uv = False, only s is returned. :
Raises LinAlgError if SVD computation does not converge :
See Also:
svdvals
return singular values of a matrix
diagsvd
return the Sigma matrix, given the vector s
Examples
>>> from scipy import random, linalg, allclose, dot
>>> a = random.randn(9, 6) + 1j*random.randn(9, 6)
>>> U, s, Vh = linalg.svd(a)
>>> U.shape, Vh.shape, s.shape
((9, 9), (6, 6), (6,))
>>> U, s, Vh = linalg.svd(a, full_matrices=False)
>>> U.shape, Vh.shape, s.shape
((9, 6), (6, 6), (6,))
>>> S = linalg.diagsvd(s, 6, 6)
>>> allclose(a, dot(U, dot(S, Vh)))
True
>>> s2 = linalg.svd(a, compute_uv=False)
>>> allclose(s, s2)
True
svdvals(a, overwrite_a=False)
Compute singular values of a matrix.
Parameters
a : array, shape (M, N)
Matrix to decompose
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
Returns
s: array, shape (K,) :
The singular values, sorted so that s[i] >= s[i+1]. K = min(M, N)
Raises LinAlgError if SVD computation does not converge :
See Also:
svd
return the full singular value decomposition of a matrix
diagsvd
return the Sigma matrix, given the vector s
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diagsvd(s, M, N)
Construct the sigma matrix in SVD from singular values and size M,N.
Parameters
s : array, shape (M,) or (N,)
Singular values
M : integer
N : integer
Size of the matrix whose singular values are s
Returns
S : array, shape (M, N)
The S-matrix in the singular value decomposition
orth(A)
Construct an orthonormal basis for the range of A using SVD
Parameters
A : array, shape (M, N)
Returns
Q : array, shape (M, K)
Orthonormal basis for the range of A. K = effective rank of A, as determined by
automatic cutoff
See Also:
svd
Singular value decomposition of a matrix
cholesky(a, lower=False, overwrite_a=False)
Compute the Cholesky decomposition of a matrix.
Returns the Cholesky decomposition, :lm:‘A = L L^*‘ or :lm:‘A = U^* U‘ of a Hermitian positive-definite
matrix :lm:‘A‘.
Parameters
a : array, shape (M, M)
Matrix to be decomposed
lower : boolean
Whether to compute the upper or lower triangular Cholesky factorization (Default:
upper-triangular)
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
Returns
c : array, shape (M, M)
Upper- or lower-triangular Cholesky factor of A
Raises LinAlgError if decomposition fails :
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Examples
>>> from scipy import array, linalg, dot
>>> a = array([[1,-2j],[2j,5]])
>>> L = linalg.cholesky(a, lower=True)
>>> L
array([[ 1.+0.j, 0.+0.j],
[ 0.+2.j, 1.+0.j]])
>>> dot(L, L.T.conj())
array([[ 1.+0.j, 0.-2.j],
[ 0.+2.j, 5.+0.j]])
cholesky_banded(ab, overwrite_ab=False, lower=False)
Cholesky decompose a banded Hermitian positive-definite matrix
The matrix a is stored in ab either in lower diagonal or upper diagonal ordered form:
ab[u + i - j, j] == a[i,j] (if upper form; i <= j) ab[ i - j, j] == a[i,j] (if lower form; i >= j)
Example of ab (shape of a is (6,6), u=2):
upper form:
a02 a13 a24 a35
*
*
a01 a12 a23 a34 a45
*
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 *
*
Parameters
ab : array, shape (u + 1, M)
Banded matrix
overwrite_ab : boolean
Discard data in ab (may enhance performance)
lower : boolean
Is the matrix in the lower form. (Default is upper form)
Returns
c : array, shape (u+1, M)
Cholesky factorization of a, in the same banded format as ab
cho_factor(a, lower=False, overwrite_a=False)
Compute the Cholesky decomposition of a matrix, to use in cho_solve
Returns a matrix containing the Cholesky decomposition, A = L L* or A = U* U of a Hermitian positivedefinite matrix a. The return value can be directly used as the first parameter to cho_solve.
Warning: The returned matrix also contains random data in the entries not used by the Cholesky decomposition. If you need to zero these entries, use the function cholesky instead.
Parameters
a : array, shape (M, M)
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Matrix to be decomposed
lower : boolean
Whether to compute the upper or lower triangular Cholesky factorization (Default:
upper-triangular)
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
Returns
c : array, shape (M, M)
Matrix whose upper or lower triangle contains the Cholesky factor of a. Other parts
of the matrix contain random data.
lower : boolean
Flag indicating whether the factor is in the lower or upper triangle
Raises
LinAlgError :
Raised if decomposition fails.
See Also:
cho_solve
Solve a linear set equations using the Cholesky factorization of a matrix.
cho_solve((c, lower), b, overwrite_b=False)
Solve the linear equations A x = b, given the Cholesky factorization of A.
Parameters
(c, lower) : tuple, (array, bool)
Cholesky factorization of a, as given by cho_factor
b : array
Right-hand side
Returns
x : array
The solution to the system A x = b
See Also:
cho_factor
Cholesky factorization of a matrix
cho_solve_banded((cb, lower), b, overwrite_b=False)
Solve the linear equations A x = b, given the Cholesky factorization of A.
Parameters
(cb, lower) : tuple, (array, bool)
cb is the Cholesky factorization of A, as given by cholesky_banded. lower must be
the same value that was given to cholesky_banded.
b : array
Right-hand side
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overwrite_b : bool
If True, the function will overwrite the values in b.
Returns
x : array
The solution to the system A x = b
See Also:
cholesky_banded
Cholesky factorization of a banded matrix
Notes
New in version 0.8.0.
qr(a, overwrite_a=False, lwork=None, econ=None, mode=’qr’)
Compute QR decomposition of a matrix.
Calculate the decomposition :lm:‘A = Q R‘ where Q is unitary/orthogonal and R upper triangular.
Parameters
a : array, shape (M, N)
Matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size is computed.
econ : boolean
Whether to compute the economy-size QR decomposition, making shapes of Q and
R (M, K) and (K, N) instead of (M,M) and (M,N). K=min(M,N). Default is False.
mode : {‘qr’, ‘r’}
Determines what information is to be returned: either both Q and R or only R.
Returns
(if mode == ‘qr’) :
Q : double or complex array, shape (M, M) or (M, K) for econ==True
(for any mode) :
R : double or complex array, shape (M, N) or (K, N) for econ==True
Size K = min(M, N)
Raises LinAlgError if decomposition fails :
Notes
This is an interface to the LAPACK routines dgeqrf, zgeqrf, dorgqr, and zungqr.
Examples
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>>> from scipy import random, linalg, dot
>>> a = random.randn(9, 6)
>>> q, r = linalg.qr(a)
>>> allclose(a, dot(q, r))
True
>>> q.shape, r.shape
((9, 9), (9, 6))
>>> r2 = linalg.qr(a, mode=’r’)
>>> allclose(r, r2)
>>> q3, r3 = linalg.qr(a, econ=True)
>>> q3.shape, r3.shape
((9, 6), (6, 6))
schur(a, output=’real’, lwork=None, overwrite_a=False)
Compute Schur decomposition of a matrix.
The Schur decomposition is
A = Z T Z^H
where Z is unitary and T is either upper-triangular, or for real Schur decomposition (output=’real’), quasi-upper
triangular. In the quasi-triangular form, 2x2 blocks describing complex-valued eigenvalue pairs may extrude
from the diagonal.
Parameters
a : array, shape (M, M)
Matrix to decompose
output : {‘real’, ‘complex’}
Construct the real or complex Schur decomposition (for real matrices).
lwork : integer
Work array size. If None or -1, it is automatically computed.
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
Returns
T : array, shape (M, M)
Schur form of A. It is real-valued for the real Schur decomposition.
Z : array, shape (M, M)
An unitary Schur transformation matrix for A. It is real-valued for the real Schur
decomposition.
See Also:
rsf2csf
Convert real Schur form to complex Schur form
rsf2csf(T, Z)
Convert real Schur form to complex Schur form.
Convert a quasi-diagonal real-valued Schur form to the upper triangular complex-valued Schur form.
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Parameters
T : array, shape (M, M)
Real Schur form of the original matrix
Z : array, shape (M, M)
Schur transformation matrix
Returns
T : array, shape (M, M)
Complex Schur form of the original matrix
Z : array, shape (M, M)
Schur transformation matrix corresponding to the complex form
See Also:
schur
Schur decompose a matrix
hessenberg(a, calc_q=False, overwrite_a=False)
Compute Hessenberg form of a matrix.
The Hessenberg decomposition is
A = Q H Q^H
where Q is unitary/orthogonal and H has only zero elements below the first subdiagonal.
Parameters
a : array, shape (M,M)
Matrix to bring into Hessenberg form
calc_q : boolean
Whether to compute the transformation matrix
overwrite_a : boolean
Whether to ovewrite data in a (may improve performance)
Returns
H : array, shape (M,M)
Hessenberg form of A
(If calc_q == True) :
Q : array, shape (M,M)
Unitary/orthogonal similarity transformation matrix s.t. A = Q H Q^H
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3.7.4 Matrix Functions
expm (A[, q])
expm2 (A)
expm3 (A[, q])
logm (A[, disp])
cosm (A)
sinm (A)
tanm (A)
coshm (A)
sinhm (A)
tanhm (A)
signm (a[, disp])
sqrtm (A[, disp])
funm (A, func[, disp])
Compute the matrix exponential using Pade approximation.
Compute the matrix exponential using eigenvalue decomposition.
Compute the matrix exponential using Taylor series.
Compute matrix logarithm.
Compute the matrix cosine.
Compute the matrix sine.
Compute the matrix tangent.
Compute the hyperbolic matrix cosine.
Compute the hyperbolic matrix sine.
Compute the hyperbolic matrix tangent.
Matrix sign function.
Matrix square root.
Evaluate a matrix function specified by a callable.
expm(A, q=7)
Compute the matrix exponential using Pade approximation.
Parameters
A : array, shape(M,M)
Matrix to be exponentiated
q : integer
Order of the Pade approximation
Returns
expA : array, shape(M,M)
Matrix exponential of A
expm2(A)
Compute the matrix exponential using eigenvalue decomposition.
Parameters
A : array, shape(M,M)
Matrix to be exponentiated
Returns
expA : array, shape(M,M)
Matrix exponential of A
expm3(A, q=20)
Compute the matrix exponential using Taylor series.
Parameters
A : array, shape(M,M)
Matrix to be exponentiated
q : integer
Order of the Taylor series
Returns
expA : array, shape(M,M)
Matrix exponential of A
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logm(A, disp=True)
Compute matrix logarithm.
The matrix logarithm is the inverse of expm: expm(logm(A)) == A
Parameters
A : array, shape(M,M)
Matrix whose logarithm to evaluate
disp : boolean
Print warning if error in the result is estimated large instead of returning estimated
error. (Default: True)
Returns
logA : array, shape(M,M)
Matrix logarithm of A
(if disp == False) :
errest : float
1-norm of the estimated error, ||err||_1 / ||A||_1
cosm(A)
Compute the matrix cosine.
This routine uses expm to compute the matrix exponentials.
Parameters
A : array, shape(M,M)
Returns
cosA : array, shape(M,M)
Matrix cosine of A
sinm(A)
Compute the matrix sine.
This routine uses expm to compute the matrix exponentials.
Parameters
A : array, shape(M,M)
Returns
sinA : array, shape(M,M)
Matrix cosine of A
tanm(A)
Compute the matrix tangent.
This routine uses expm to compute the matrix exponentials.
Parameters
A : array, shape(M,M)
Returns
tanA : array, shape(M,M)
Matrix tangent of A
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coshm(A)
Compute the hyperbolic matrix cosine.
This routine uses expm to compute the matrix exponentials.
Parameters
A : array, shape(M,M)
Returns
coshA : array, shape(M,M)
Hyperbolic matrix cosine of A
sinhm(A)
Compute the hyperbolic matrix sine.
This routine uses expm to compute the matrix exponentials.
Parameters
A : array, shape(M,M)
Returns
sinhA : array, shape(M,M)
Hyperbolic matrix sine of A
tanhm(A)
Compute the hyperbolic matrix tangent.
This routine uses expm to compute the matrix exponentials.
Parameters
A : array, shape(M,M)
Returns
tanhA : array, shape(M,M)
Hyperbolic matrix tangent of A
signm(a, disp=True)
Matrix sign function.
Extension of the scalar sign(x) to matrices.
Parameters
A : array, shape(M,M)
Matrix at which to evaluate the sign function
disp : boolean
Print warning if error in the result is estimated large instead of returning estimated
error. (Default: True)
Returns
sgnA : array, shape(M,M)
Value of the sign function at A
(if disp == False) :
errest : float
1-norm of the estimated error, ||err||_1 / ||A||_1
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Examples
>>> from scipy.linalg import signm, eigvals
>>> a = [[1,2,3], [1,2,1], [1,1,1]]
>>> eigvals(a)
array([ 4.12488542+0.j, -0.76155718+0.j, 0.63667176+0.j])
>>> eigvals(signm(a))
array([-1.+0.j, 1.+0.j, 1.+0.j])
sqrtm(A, disp=True)
Matrix square root.
Parameters
A : array, shape(M,M)
Matrix whose square root to evaluate
disp : boolean
Print warning if error in the result is estimated large instead of returning estimated
error. (Default: True)
Returns
sgnA : array, shape(M,M)
Value of the sign function at A
(if disp == False) :
errest : float
Frobenius norm of the estimated error, ||err||_F / ||A||_F
Notes
Uses algorithm by Nicholas J. Higham
funm(A, func, disp=True)
Evaluate a matrix function specified by a callable.
Returns the value of matrix-valued function f at A. The function f is an extension of the scalar-valued function
func to matrices.
Parameters
A : array, shape(M,M)
Matrix at which to evaluate the function
func : callable
Callable object that evaluates a scalar function f. Must be vectorized (eg. using
vectorize).
disp : boolean
Print warning if error in the result is estimated large instead of returning estimated
error. (Default: True)
Returns
fA : array, shape(M,M)
Value of the matrix function specified by func evaluated at A
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(if disp == False) :
errest : float
1-norm of the estimated error, ||err||_1 / ||A||_1
3.7.5 Special Matrices
block_diag (*arrs)
circulant (c)
companion (a)
hadamard (n[, dtype])
hankel (c[, r])
kron (a, b)
leslie (f, s)
toeplitz (c[, r])
tri (N[, M, k, dtype])
tril (m[, k])
triu (m[, k])
Create a block diagonal matrix from the provided arrays.
Construct a circulant matrix.
Create a companion matrix.
Construct a Hadamard matrix.
Construct a Hankel matrix.
Kronecker product of a and b.
Create a Leslie matrix.
Construct a Toeplitz matrix.
Construct (N, M) matrix filled with ones at and below the k-th diagonal.
Construct a copy of a matrix with elements above the k-th diagonal zeroed.
Construct a copy of a matrix with elements below the k-th diagonal zeroed.
block_diag(*arrs)
Create a block diagonal matrix from the provided arrays.
Given the inputs A, B and C, the output will have these arrays arranged on the diagonal:
[[A, 0, 0],
[0, B, 0],
[0, 0, C]]
If all the input arrays are square, the output is known as a block diagonal matrix.
Parameters
A, B, C, ... : array-like, up to 2D
Input arrays. A 1D array or array-like sequence with length n is treated as a 2D array
with shape (1,n).
Returns
D : ndarray
Array with A, B, C, ... on the diagonal. D has the same dtype as A.
References
[R13]
Examples
>>>
...
>>>
...
>>>
>>>
[[1
[0
[0
[0
248
A = [[1, 0],
[0, 1]]
B = [[3, 4, 5],
[6, 7, 8]]
C = [[7]]
print(block_diag(A, B, C))
0 0 0 0 0]
1 0 0 0 0]
0 3 4 5 0]
0 6 7 8 0]
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[0 0 0 0 0 7]]
>>> block_diag(1.0, [2,
array([[ 1., 0., 0.,
[ 0., 2., 3.,
[ 0., 0., 0.,
[ 0., 0., 0.,
3], [[4, 5], [6, 7]])
0., 0.],
0., 0.],
4., 5.],
6., 7.]])
circulant(c)
Construct a circulant matrix.
Parameters
c : array-like, 1D
First column of the matrix.
Returns
A : array, shape (len(c), len(c))
A circulant matrix whose first column is c.
See Also:
toeplitz
Toeplitz matrix
hankel
Hankel matrix
Notes
New in version 0.8.0.
Examples
>>> from scipy.linalg import circulant
>>> circulant([1, 2, 3])
array([[1, 3, 2],
[2, 1, 3],
[3, 2, 1]])
companion(a)
Create a companion matrix.
Create the companion matrix associated with the polynomial whose coefficients are given in a.
Parameters
a : array-like, 1D
Polynomial coefficients. The length of a must be at least two, and a[0] must not be
zero.
Returns
c : ndarray
A square ndarray with shape (n-1, n-1), where n is the length of a. The first row of c
is -a[1:]/a[0], and the first subdiagonal is all ones. The data type of the array is the
same as the data type of 1.0*a[0].
Notes
New in version 0.8.0.
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Examples
>>> companion([1, -10, 31, -30])
array([[ 10., -31., 30.],
[ 1.,
0.,
0.],
[ 0.,
1.,
0.]])
hadamard(n, dtype=<type ’int’>)
Construct a Hadamard matrix.
hadamard(n) constructs an n-by-n Hadamard matrix, using Sylvester’s construction. n must be a power of 2.
Parameters
n : int
The order of the matrix. n must be a power of 2.
dtype : numpy dtype
The data type of the array to be constructed.
Returns
H : ndarray with shape (n, n)
The Hadamard matrix.
Notes
New in version 0.8.0.
Examples
>>> hadamard(2, dtype=complex)
array([[ 1.+0.j, 1.+0.j],
[ 1.+0.j, -1.-0.j]])
>>> hadamard(4)
array([[ 1, 1, 1, 1],
[ 1, -1, 1, -1],
[ 1, 1, -1, -1],
[ 1, -1, -1, 1]])
hankel(c, r=None)
Construct a Hankel matrix.
The Hankel matrix has constant anti-diagonals, with c as its first column and r as its last row. If r is not given,
then r = zeros_like(c) is assumed.
Parameters
c : array-like, 1D
First column of the matrix. Whatever the actual shape of c, it will be converted to a
1D array.
r : array-like, 1D
Last row of the matrix. If None, r = zeros_like(c) is assumed. r[0] is ignored; the
last row of the returned matrix is [c[-1], r[1:]]. Whatever the actual shape of r, it
will be converted to a 1D array.
Returns
A : array, shape (len(c), len(r))
The Hankel matrix. dtype is the same as (c[0] + r[0]).dtype.
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See Also:
toeplitz
Toeplitz matrix
circulant
circulant matrix
Examples
>>> from scipy.linalg import hankel
>>> hankel([1, 17, 99])
array([[ 1, 17, 99],
[17, 99, 0],
[99, 0, 0]])
>>> hankel([1,2,3,4], [4,7,7,8,9])
array([[1, 2, 3, 4, 7],
[2, 3, 4, 7, 7],
[3, 4, 7, 7, 8],
[4, 7, 7, 8, 9]])
kron(a, b)
Kronecker product of a and b.
The result is the block matrix:
a[0,0]*b
a[1,0]*b
...
a[-1,0]*b
a[0,1]*b
a[1,1]*b
... a[0,-1]*b
... a[1,-1]*b
a[-1,1]*b ... a[-1,-1]*b
Parameters
a : array, shape (M, N)
b : array, shape (P, Q)
Returns
A : array, shape (M*P, N*Q)
Kronecker product of a and b
Examples
>>> from scipy import kron, array
>>> kron(array([[1,2],[3,4]]), array([[1,1,1]]))
array([[1, 1, 1, 2, 2, 2],
[3, 3, 3, 4, 4, 4]])
leslie(f, s)
Create a Leslie matrix.
Parameters
f : array-like, 1D
The “fecundity” coefficients.
s : array-like, 1D
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The “survival” coefficients. The length of s must be one less than the length of f, and
it must be at least 1.
Returns
L : ndarray, 2D
Returns a 2D numpy ndarray with shape (n,n), where n is the length of f. The array
is zero except for the first row, which is f, and the first subdiagonal, which is s. The
data type of the array will be the data type of f[0]+s[0].
Notes
New in version 0.8.0.
Examples
>>> leslie([0.1, 2.0, 1.0,
array([[ 0.1, 2. , 1. ,
[ 0.2, 0. , 0. ,
[ 0. , 0.8, 0. ,
[ 0. , 0. , 0.7,
0.1], [0.2, 0.8, 0.7])
0.1],
0. ],
0. ],
0. ]])
toeplitz(c, r=None)
Construct a Toeplitz matrix.
The Toepliz matrix has constant diagonals, with c as its first column and r as its first row. If r is not given, r ==
conjugate(c) is assumed.
Parameters
c : array-like, 1D
First column of the matrix. Whatever the actual shape of c, it will be converted to a
1D array.
r : array-like, 1D
First row of the matrix. If None, r = conjugate(c) is assumed; in this case, if c[0]
is real, the result is a Hermitian matrix. r[0] is ignored; the first row of the returned
matrix is [c[0], r[1:]]. Whatever the actual shape of r, it will be converted to a 1D
array.
Returns
A : array, shape (len(c), len(r))
The Toeplitz matrix. dtype is the same as (c[0] + r[0]).dtype.
See Also:
circulant
circulant matrix
hankel
Hankel matrix
Notes
The behavior when c or r is a scalar, or when c is complex and r is None, was changed in version 0.8.0. The
behavior in previous versions was undocumented and is no longer supported.
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Examples
>>> from scipy.linalg import toeplitz
>>> toeplitz([1,2,3], [1,4,5,6])
array([[1, 4, 5, 6],
[2, 1, 4, 5],
[3, 2, 1, 4]])
>>> toeplitz([1.0, 2+3j, 4-1j])
array([[ 1.+0.j, 2.-3.j, 4.+1.j],
[ 2.+3.j, 1.+0.j, 2.-3.j],
[ 4.-1.j, 2.+3.j, 1.+0.j]])
tri(N, M=None, k=0, dtype=None)
Construct (N, M) matrix filled with ones at and below the k-th diagonal.
The matrix has A[i,j] == 1 for i <= j + k
Parameters
N : integer
The size of the first dimension of the matrix.
M : integer or None
The size of the second dimension of the matrix. If M is None, M = N is assumed.
k : integer
Number of subdiagonal below which matrix is filled with ones. k = 0 is the main
diagonal, k < 0 subdiagonal and k > 0 superdiagonal.
dtype : dtype
Data type of the matrix.
Returns
A : array, shape (N, M)
Examples
>>> from scipy.linalg import tri
>>> tri(3, 5, 2, dtype=int)
array([[1, 1, 1, 0, 0],
[1, 1, 1, 1, 0],
[1, 1, 1, 1, 1]])
>>> tri(3, 5, -1, dtype=int)
array([[0, 0, 0, 0, 0],
[1, 0, 0, 0, 0],
[1, 1, 0, 0, 0]])
tril(m, k=0)
Construct a copy of a matrix with elements above the k-th diagonal zeroed.
Parameters
m : array
Matrix whose elements to return
k : integer
Diagonal above which to zero elements. k == 0 is the main diagonal, k < 0 subdiagonal and k > 0 superdiagonal.
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Returns
A : array, shape m.shape, dtype m.dtype
Examples
>>> from scipy.linalg import tril
>>> tril([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 0, 0, 0],
[ 4, 0, 0],
[ 7, 8, 0],
[10, 11, 12]])
triu(m, k=0)
Construct a copy of a matrix with elements below the k-th diagonal zeroed.
Parameters
m : array
Matrix whose elements to return
k : integer
Diagonal below which to zero elements. k == 0 is the main diagonal, k < 0 subdiagonal and k > 0 superdiagonal.
Returns
A : array, shape m.shape, dtype m.dtype
Examples
>>> from scipy.linalg import tril
>>> triu([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 1, 2, 3],
[ 4, 5, 6],
[ 0, 8, 9],
[ 0, 0, 12]])
3.8 Maximum entropy models (scipy.maxentropy)
3.8.1 Routines for fitting maximum entropy models
Contains two classes for fitting maximum entropy models (also known as “exponential family” models) subject to
linear constraints on the expectations of arbitrary feature statistics. One class, “model”, is for small discrete sample
spaces, using explicit summation. The other, “bigmodel”, is for sample spaces that are either continuous (and perhaps
high-dimensional) or discrete but too large to sum over, and uses importance sampling. conditional Monte Carlo
methods.
The maximum entropy model has exponential form
p(x) = exp
θT f~(x)
Z(θ)
!
with a real parameter vector theta of the same length as the feature statistic f(x), For more background, see, for
example, Cover and Thomas (1991), Elements of Information Theory.
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See the file bergerexample.py for a walk-through of how to use these routines when the sample space is small enough
to be enumerated.
See bergerexamplesimulated.py for a a similar walk-through using simulation.
Copyright: Ed Schofield, 2003-2006 License: BSD-style (see LICENSE.txt in main source directory)
3.8.2 Models
class basemodel()
A base class providing generic functionality for both small and large maximum entropy models. Cannot be
instantiated.
Methods
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basemodel.beginlogging (self,
Enable
file-logging params for each fn evaluation to files named
name[, freq])
‘filename.freq.pickle’, ‘filename.(2*freq).pickle’, ... each ‘freq’ iterations.
basemodel.endlogging (self)Stop logging param values whenever setparams() is called.
basemodel.clearcache (self)Clears the interim results of computations depending on the parameters and the
sample.
basemodel.crossentropy (self,
Returns
fx[, the
log_prior_x,
cross entropy
base])
H(q, p) of the empirical distribution q of the data
(with the given feature matrix fx) with respect to the model p. For discrete
distributions this is defined as:
basemodel.dual (self[, params,Computes
igthe Lagrangian dual L(theta) of the entropy of the model, for the
norepenalty, ...])
given vector theta=params. Minimizing this function (without constraints)
should fit the maximum entropy model subject to the given constraints. These
constraints are specified as the desired (target) values self.K for the
expectations of the feature statistic.
basemodel.fit (self, K[, al- Fit the maxent model p whose feature expectations are given by the vector K.
gorithm])
basemodel.grad (self[, params,Computes
igor estimates the gradient of the entropy dual.
norepenalty])
basemodel.log (self, params) This method is called every iteration during the optimization process. It calls
the user-supplied callback function (if any), logs the evolution of the entropy
dual and gradient norm, and checks whether the process appears to be
diverging, which would indicate inconsistent constraints (or, for bigmodel
instances, too large a variance in the estimates).
basemodel.logparams (self) Saves the model parameters if logging has been enabled and the # of iterations
since the last save has reached self.paramslogfreq.
basemodel.normconst (self) Returns the normalization constant, or partition function, for the current model.
Warning – this may be too large to represent; if so, this will result in numerical
overflow. In this case use lognormconst() instead.
basemodel.reset (self[, num-Resets the parameters self.params to zero, clearing the cache variables
features])
dependent on them. Also resets the number of function and gradient
evaluations to zero.
basemodel.setcallback (self[,
Setscallcallback functions to be called every iteration, every function evaluation,
back, callback_dual, ...])
or every gradient evaluation. All callback functions are passed one argument,
the current model object.
basemodel.setparams (self, params)
Set the parameter vector to params, replacing the existing parameters. params
must be a list or numpy array of the same length as the model’s feature vector f.
basemodel.setsmooth (sigma)
Speficies that the entropy dual and gradient should be computed with a
quadratic penalty term on magnitude of the parameters. This ‘smooths’ the
model to account for noise in the target expectation values or to improve
robustness when using simulation to fit models and when the sampling
distribution has high variance. The smoothing mechanism is described in Chen
and Rosenfeld, ‘A Gaussian prior for smoothing maximum entropy models’
(1999).
beginlogging(filename, freq=10)
Enable logging params for each fn evaluation to files named ‘filename.freq.pickle’, ‘filename.(2*freq).pickle’,
... each ‘freq’ iterations.
endlogging()
Stop logging param values whenever setparams() is called.
clearcache()
Clears the interim results of computations depending on the parameters and the sample.
crossentropy(fx, log_prior_x=None, base=2.7182818284590451)
Returns the cross entropy H(q, p) of the empirical distribution q of the data (with the given feature matrix fx)
with respect to the model p. For discrete distributions this is defined as:
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H(q, p) = - n^{-1} sum_{j=1}^n log p(x_j)
where x_j are the data elements assumed drawn from q whose features are given by the matrix fx = {f(x_j)},
j=1,...,n.
The ‘base’ argument specifies the base of the logarithm, which defaults to e.
For continuous distributions this makes no sense!
dual(params=None, ignorepenalty=False, ignoretest=False)
Computes the Lagrangian dual L(theta) of the entropy of the model, for the given vector theta=params. Minimizing this function (without constraints) should fit the maximum entropy model subject to the given
constraints. These constraints are specified as the desired (target) values self.K for the expectations of the
feature statistic.
This function is computed as:
L(theta) = log(Z) - theta^T . K
For ‘bigmodel’ objects, it estimates the entropy dual without actually computing p_theta. This is important if the
sample space is continuous or innumerable in practice. We approximate the norm constant Z using importance
sampling as in [Rosenfeld01whole]. This estimator is deterministic for any given sample. Note that the gradient
of this estimator is equal to the importance sampling ratio estimator of the gradient of the entropy dual [see my
thesis], justifying the use of this estimator in conjunction with grad() in optimization methods that use both the
function and gradient. Note, however, that convergence guarantees break down for most optimization algorithms
in the presence of stochastic error.
Note that, for ‘bigmodel’ objects, the dual estimate is deterministic for any given sample. It is given as:
L_est = log Z_est - sum_i{theta_i K_i}
where
Z_est = 1/m sum_{x in sample S_0} p_dot(x) / aux_dist(x),
and m = # observations in sample S_0, and K_i = the empirical expectation E_p_tilde f_i (X) = sum_x {p(x)
f_i(x)}.
fit(K, algorithm=’CG’)
Fit the maxent model p whose feature expectations are given by the vector K.
Model expectations are computed either exactly or using Monte Carlo simulation, depending on the ‘func’ and
‘grad’ parameters passed to this function.
For ‘model’ instances, expectations are computed exactly, by summing over the given sample space. If the
sample space is continuous or too large to iterate over, use the ‘bigmodel’ class instead.
For ‘bigmodel’ instances, the model expectations are not computed exactly (by summing or integrating over a
sample space) but approximately (by Monte Carlo simulation). Simulation is necessary when the sample space
is too large to sum or integrate over in practice, like a continuous sample space in more than about 4 dimensions
or a large discrete space like all possible sentences in a natural language.
Approximating the expectations by sampling requires an instrumental distribution that should be close to the
model for fast convergence. The tails should be fatter than the model. This instrumental distribution is specified
by calling setsampleFgen() with a user-supplied generator function that yields a matrix of features of a random
sample and its log pdf values.
The algorithm can be ‘CG’, ‘BFGS’, ‘LBFGSB’, ‘Powell’, or ‘Nelder-Mead’.
The CG (conjugate gradients) method is the default; it is quite fast and requires only linear space in the number
of parameters, (not quadratic, like Newton-based methods).
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The BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm is a variable metric Newton method. It is perhaps
faster than the CG method but requires O(N^2) instead of O(N) memory, so it is infeasible for more than about
10^3 parameters.
The Powell algorithm doesn’t require gradients. For small models it is slow but robust. For big models (where
func and grad are simulated) with large variance in the function estimates, this may be less robust than the
gradient-based algorithms.
grad(params=None, ignorepenalty=False)
Computes or estimates the gradient of the entropy dual.
log(params)
This method is called every iteration during the optimization process. It calls the user-supplied callback function
(if any), logs the evolution of the entropy dual and gradient norm, and checks whether the process appears to be
diverging, which would indicate inconsistent constraints (or, for bigmodel instances, too large a variance in the
estimates).
logparams()
Saves the model parameters if logging has been enabled and the # of iterations since the last save has reached
self.paramslogfreq.
normconst()
Returns the normalization constant, or partition function, for the current model. Warning – this may be too
large to represent; if so, this will result in numerical overflow. In this case use lognormconst() instead.
For ‘bigmodel’ instances, estimates the normalization term as Z = E_aux_dist [{exp (params.f(X))} /
aux_dist(X)] using a sample from aux_dist.
reset(numfeatures=None)
Resets the parameters self.params to zero, clearing the cache variables dependent on them. Also resets the
number of function and gradient evaluations to zero.
setcallback(callback=None, callback_dual=None, callback_grad=None)
Sets callback functions to be called every iteration, every function evaluation, or every gradient evaluation. All
callback functions are passed one argument, the current model object.
Note that line search algorithms in e.g. CG make potentially several function and gradient evaluations per
iteration, some of which we expect to be poor.
setparams(params)
Set the parameter vector to params, replacing the existing parameters. params must be a list or numpy array of
the same length as the model’s feature vector f.
setsmooth(sigma)
Speficies that the entropy dual and gradient should be computed with a quadratic penalty term on magnitude of
the parameters. This ‘smooths’ the model to account for noise in the target expectation values or to improve
robustness when using simulation to fit models and when the sampling distribution has high variance. The
smoothing mechanism is described in Chen and Rosenfeld, ‘A Gaussian prior for smoothing maximum entropy
models’ (1999).
The parameter ‘sigma’ will be squared and stored as self.sigma2.
class model(f=None, samplespace=None)
A maximum-entropy (exponential-form) model on a discrete sample space.
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model.expectations (self) The vector E_p[f(X)] under the model p_params of the vector of feature
functions f_i over the sample space.
model.lognormconst (self) Compute the log of the normalization constant (partition function) Z=sum_{x
in samplespace} p_0(x) exp(params . f(x)). The sample space must be discrete
and finite.
model.logpmf (self)
Returns an array indexed by integers representing the logarithms of the
probability mass function (pmf) at each point in the sample space under the
current model (with the current parameter vector self.params).
model.pmf_function (self[, f])
Returns the pmf p_theta(x) as a function taking values on the model’s sample
space. The returned pmf is defined as:
model.setfeaturesandsamplespace
Creates a new
(self,
matrix
f, samself.F of features f of all points in the sample space. f is a
plespace)
list of feature functions f_i mapping the sample space to real values. The
parameter vector self.params is initialized to zero.
expectations()
The vector E_p[f(X)] under the model p_params of the vector of feature functions f_i over the sample space.
lognormconst()
Compute the log of the normalization constant (partition function) Z=sum_{x in samplespace} p_0(x)
exp(params . f(x)). The sample space must be discrete and finite.
logpmf()
Returns an array indexed by integers representing the logarithms of the probability mass function (pmf) at each
point in the sample space under the current model (with the current parameter vector self.params).
pmf_function(f=None)
Returns the pmf p_theta(x) as a function taking values on the model’s sample space. The returned pmf is
defined as:
p_theta(x) = exp(theta.f(x) - log Z)
where theta is the current parameter vector self.params. The returned function p_theta also satisfies
all([p(x) for x in self.samplespace] == pmf()).
The feature statistic f should be a list of functions [f1(),...,fn(x)]. This must be passed unless the model already
contains an equivalent attribute ‘model.f’.
Requires that the sample space be discrete and finite, and stored as self.samplespace as a list or array.
setfeaturesandsamplespace(f, samplespace)
Creates a new matrix self.F of features f of all points in the sample space. f is a list of feature functions f_i
mapping the sample space to real values. The parameter vector self.params is initialized to zero.
We also compute f(x) for each x in the sample space and store them as self.F. This uses lots of memory but is
much faster.
This is only appropriate when the sample space is finite.
class bigmodel()
A maximum-entropy (exponential-form) model on a large sample space.
The model expectations are not computed exactly (by summing or integrating over a sample space) but approximately (by Monte Carlo estimation). Approximation is necessary when the sample space is too large to sum or
integrate over in practice, like a continuous sample space in more than about 4 dimensions or a large discrete
space like all possible sentences in a natural language.
Approximating the expectations by sampling requires an instrumental distribution that should be close to the
model for fast convergence. The tails should be fatter than the model.
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Methods
bigmodel.estimate (self)
This function approximates both the feature expectation vector E_p f(X)
and the log of the normalization term Z with importance sampling.
bigmodel.logpdf (self, fx[, log_prior_x])
Returns the log of the estimated density p(x) = p_theta(x) at the point x.
If log_prior_x is None, this is defined as: log p(x) = theta.f(x) - log Z
where f(x) is given by the (m x 1) array fx.
bigmodel.pdf (self, fx)
Returns the estimated density p_theta(x) at the point x with feature
statistic fx = f(x). This is defined as p_theta(x) = exp(theta.f(x)) /
Z(theta), where Z is the estimated value self.normconst() of the partition
function.
bigmodel.pdf_function (self)
Returns the estimated density p_theta(x) as a function p(f) taking a
vector f = f(x) of feature statistics at any point x. This is defined as:
p_theta(x) = exp(theta.f(x)) / Z
bigmodel.resample (self)
(Re)samples the matrix F of sample features.
bigmodel.setsampleFgen (self, samInitializes the Monte Carlo sampler to use the supplied generator of
pler[, staticsample])
samples’ features and log probabilities. This is an alternative to defining
a sampler in terms of a (fixed size) feature matrix sampleF and
accompanying vector samplelogprobs of log probabilities.
bigmodel.settestsamples (self, F_list,
Requests
log- that the model be tested every ‘testevery’ iterations during
prob_list[, testevery, priorlogfitting using the provided list F_list of feature matrices, each
prob_list])
representing a sample {x_j} from an auxiliary distribution q, together
with the corresponding log probabiltiy mass or density values log
{q(x_j)} in logprob_list. This is useful as an external check on the
fitting process with sample path optimization, which could otherwise
reflect the vagaries of the single sample being used for optimization,
rather than the population as a whole.
bigmodel.stochapprox (self, K) Tries to fit the model to the feature expectations K using stochastic
approximation, with the Robbins-Monro stochastic approximation
algorithm: theta_{k+1} = theta_k + a_k g_k - a_k e_k where g_k is the
gradient vector (= feature expectations E - K) evaluated at the point
theta_k, a_k is the sequence a_k = a_0 / k, where a_0 is some step size
parameter defined as self.a_0 in the model, and e_k is an unknown error
term representing the uncertainty of the estimate of g_k. We assume e_k
has nice enough properties for the algorithm to converge.
bigmodel.test (self)
Estimate the dual and gradient on the external samples, keeping track of
the parameters that yield the minimum such dual. The vector of desired
(target) feature expectations is stored as self.K.
estimate()
This function approximates both the feature expectation vector E_p f(X) and the log of the normalization term
Z with importance sampling.
It also computes the sample variance of the component estimates of the feature expectations as: varE = var(E_1,
..., E_T) where T is self.matrixtrials and E_t is the estimate of E_p f(X) approximated using the ‘t’th auxiliary
feature matrix.
It doesn’t return anything, but stores the member variables logZapprox, mu and varE. (This is done because
some optimization algorithms retrieve the dual fn and gradient fn in separate function calls, but we can compute
them more efficiently together.)
It uses a supplied generator sampleFgen whose .next() method returns features of random observations s_j
generated according to an auxiliary distribution aux_dist. It uses these either in a matrix (with multiple runs)
or with a sequential procedure, with more updating overhead but potentially stopping earlier (needing fewer
samples). In the matrix case, the features F={f_i(s_j)} and vector [log_aux_dist(s_j)] of log probabilities are
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generated by calling resample().
We use [Rosenfeld01Wholesentence]’s estimate of E_p[f_i] as:
{sum_j p(s_j)/aux_dist(s_j) f_i(s_j) }
/ {sum_j p(s_j) / aux_dist(s_j)}.
Note that this is consistent but biased.
This equals:
{sum_j p_dot(s_j)/aux_dist(s_j) f_i(s_j) }
/ {sum_j p_dot(s_j) / aux_dist(s_j)}
Compute the estimator E_p f_i(X) in log space as:
num_i / denom,
where
num_i = exp(logsumexp(theta.f(s_j) - log aux_dist(s_j)
• log f_i(s_j)))
and
denom = [n * Zapprox]
where Zapprox = exp(self.lognormconst()).
We can compute the denominator n*Zapprox directly as:
exp(logsumexp(log p_dot(s_j) - log aux_dist(s_j)))
= exp(logsumexp(theta.f(s_j) - log aux_dist(s_j)))
logpdf(fx, log_prior_x=None)
Returns the log of the estimated density p(x) = p_theta(x) at the point x. If log_prior_x is None, this is defined
as:
log p(x) = theta.f(x) - log Z
where f(x) is given by the (m x 1) array fx.
If, instead, fx is a 2-d (m x n) array, this function interprets each of its rows j=0,...,n-1 as a feature vector f(x_j),
and returns an array containing the log pdf value of each point x_j under the current model.
log Z is estimated using the sample provided with setsampleFgen().
The optional argument log_prior_x is the log of the prior density p_0 at the point x (or at each point x_j if fx is
2-dimensional). The log pdf of the model is then defined as
log p(x) = log p0(x) + theta.f(x) - log Z
and p then represents the model of minimum KL divergence D(p||p0) instead of maximum entropy.
pdf(fx)
Returns the estimated density p_theta(x) at the point x with feature statistic fx = f(x). This is defined as
p_theta(x) = exp(theta.f(x)) / Z(theta),
where Z is the estimated value self.normconst() of the partition function.
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pdf_function()
Returns the estimated density p_theta(x) as a function p(f) taking a vector f = f(x) of feature statistics at any
point x. This is defined as:
p_theta(x) = exp(theta.f(x)) / Z
resample()
(Re)samples the matrix F of sample features.
setsampleFgen(sampler, staticsample=True)
Initializes the Monte Carlo sampler to use the supplied generator of samples’ features and log probabilities.
This is an alternative to defining a sampler in terms of a (fixed size) feature matrix sampleF and accompanying
vector samplelogprobs of log probabilities.
Calling sampler.next() should generate tuples (F, lp), where F is an (m x n) matrix of features of the n sample
points x_1,...,x_n, and lp is an array of length n containing the (natural) log probability density (pdf or pmf) of
each point under the auxiliary sampling distribution.
The output of sampler.next() can optionally be a 3-tuple (F, lp, sample) instead of a 2-tuple (F, lp). In this case
the value ‘sample’ is then stored as a class variable self.sample. This is useful for inspecting the output and
understanding the model characteristics.
If matrixtrials > 1 and staticsample = True, (which is useful for estimating variance between the different feature
estimates), sampler.next() will be called once for each trial (0,...,matrixtrials) for each iteration. This allows
using a set of feature matrices, each of which stays constant over all iterations.
We now insist that sampleFgen.next() return the entire sample feature matrix to be used each iteration to avoid
overhead in extra function calls and memory copying (and extra code).
An alternative was to supply a list of samplers, sampler=[sampler0, sampler1, ..., sampler_{m-1}, samplerZ],
one for each feature and one for estimating the normalization constant Z. But this code was unmaintained, and
has now been removed (but it’s in Ed’s CVS repository :).
Example use: >>> import spmatrix >>> model = bigmodel() >>> def sampler(): ... n = 0 ... while True:
... f = spmatrix.ll_mat(1,3) ... f[0,0] = n+1; f[0,1] = n+1; f[0,2] = n+1 ... yield f, 1.0 ... n += 1 ... >>>
model.setsampleFgen(sampler()) >>> type(model.sampleFgen) <type ‘generator’> >>> [model.sampleF[0,i]
for i in range(3)] [1.0, 1.0, 1.0]
We now set matrixtrials as a class property instead, rather than passing it as an argument to this function, where
it can be written over (perhaps with the default function argument by accident) when we re-call this func (e.g.
to change the matrix size.)
settestsamples(F_list, logprob_list, testevery=1, priorlogprob_list=None)
Requests that the model be tested every ‘testevery’ iterations during fitting using the provided list F_list of
feature matrices, each representing a sample {x_j} from an auxiliary distribution q, together with the corresponding log probabiltiy mass or density values log {q(x_j)} in logprob_list. This is useful as an external check
on the fitting process with sample path optimization, which could otherwise reflect the vagaries of the single
sample being used for optimization, rather than the population as a whole.
If self.testevery > 1, only perform the test every self.testevery calls.
If priorlogprob_list is not None, it should be a list of arrays of log(p0(x_j)) values, j = 0,. ..., n - 1, specifying
the prior distribution p0 for the sample points x_j for each of the test samples.
stochapprox(K)
Tries to fit the model to the feature expectations K using stochastic approximation, with the Robbins-Monro
stochastic approximation algorithm: theta_{k+1} = theta_k + a_k g_k - a_k e_k where g_k is the gradient
vector (= feature expectations E - K) evaluated at the point theta_k, a_k is the sequence a_k = a_0 / k, where
a_0 is some step size parameter defined as self.a_0 in the model, and e_k is an unknown error term representing
the uncertainty of the estimate of g_k. We assume e_k has nice enough properties for the algorithm to converge.
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test()
Estimate the dual and gradient on the external samples, keeping track of the parameters that yield the minimum
such dual. The vector of desired (target) feature expectations is stored as self.K.
class conditionalmodel(F, counts, numcontexts)
A conditional maximum-entropy (exponential-form) model p(x|w) on a discrete sample space. This is useful
for classification problems: given the context w, what is the probability of each class x?
The form of such a model is:
p(x | w) = exp(theta . f(w, x)) / Z(w; theta)
where Z(w; theta) is a normalization term equal to:
Z(w; theta) = sum_x exp(theta . f(w, x)).
The sum is over all classes x in the set Y, which must be supplied to the constructor as the parameter ‘samplespace’.
Such a model form arises from maximizing the entropy of a conditional model p(x | w) subject to the constraints:
K_i = E f_i(W, X)
where the expectation is with respect to the distribution:
q(w) p(x | w)
where q(w) is the empirical probability mass function derived from observations of the context w in a training
set. Normally the vector K = {K_i} of expectations is set equal to the expectation of f_i(w, x) with respect to
the empirical distribution.
This method minimizes the Lagrangian dual L of the entropy, which is defined for conditional models as:
L(theta) = sum_w q(w) log Z(w; theta)
- sum_{w,x} q(w,x) [theta . f(w,x)]
Note that both sums are only over the training set {w,x}, not the entire sample space, since q(w,x) = 0 for all w,x
not in the training set.
The partial derivatives of L are:
dL / dtheta_i = K_i - E f_i(X, Y)
where the expectation is as defined above.
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conditionalmodel.dual (self[,
The entropy
params,dual
ig- function is defined for conditional models as
norepenalty])
conditionalmodel.expectations
The vector (self)
of expectations of the features with respect to the distribution
p_tilde(w) p(x | w), where p_tilde(w) is the empirical probability mass function
value stored as self.p_tilde_context[w].
conditionalmodel.fit (self[,
Fitsalthe conditional maximum entropy model subject to the constraints
gorithm])
conditionalmodel.lognormconst
Compute the
(self)
elementwise log of the normalization constant (partition function)
Z(w)=sum_{y in Y(w)} exp(theta . f(w, y)). The sample space must be discrete
and finite. This is a vector with one element for each context w.
conditionalmodel.logpmf
Returns
(self) a (sparse) row vector of logarithms of the conditional probability mass
function (pmf) values p(x | c) for all pairs (c, x), where c are contexts and x are
points in the sample space. The order of these is log p(x | c) = logpmf()[c *
numsamplepoints + x].
dual(params=None, ignorepenalty=False)
The entropy dual function is defined for conditional models as
L(theta) = sum_w q(w) log Z(w; theta)
• sum_{w,x} q(w,x) [theta . f(w,x)]
or equivalently as
L(theta) = sum_w q(w) log Z(w; theta) - (theta . k)
where K_i = sum_{w, x} q(w, x) f_i(w, x), and where q(w) is the empirical probability mass function derived
from observations of the context w in a training set. Normally q(w, x) will be 1, unless the same class label is
assigned to the same context more than once.
Note that both sums are only over the training set {w,x}, not the entire sample space, since q(w,x) = 0 for all w,x
not in the training set.
The entropy dual function is proportional to the negative log likelihood.
Compare to the entropy dual of an unconditional model:
L(theta) = log(Z) - theta^T . K
expectations()
The vector of expectations of the features with respect to the distribution p_tilde(w) p(x | w), where p_tilde(w)
is the empirical probability mass function value stored as self.p_tilde_context[w].
fit(algorithm=’CG’)
Fits the conditional maximum entropy model subject to the constraints
sum_{w, x} p_tilde(w) p(x | w) f_i(w, x) = k_i
for i=1,...,m, where k_i is the empirical expectation
k_i = sum_{w, x} p_tilde(w, x) f_i(w, x).
lognormconst()
Compute the elementwise log of the normalization constant (partition function) Z(w)=sum_{y in Y(w)}
exp(theta . f(w, y)). The sample space must be discrete and finite. This is a vector with one element for each
context w.
logpmf()
Returns a (sparse) row vector of logarithms of the conditional probability mass function (pmf) values p(x | c)
for all pairs (c, x), where c are contexts and x are points in the sample space. The order of these is log p(x | c) =
logpmf()[c * numsamplepoints + x].
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3.8.3 Utilities
arrayexp (x)
Returns the elementwise antilog of the real array x. We try to exponentiate with
numpy.exp() and, if that fails, with python’s math.exp(). numpy.exp() is about 10
times faster but throws an OverflowError exception for numerical underflow (e.g.
exp(-800), whereas python’s math.exp() just returns zero, which is much more
helpful.
arrayexpcomplex (x) Returns the elementwise antilog of the vector x. We try to exponentiate with
numpy.exp() and, if that fails, with python’s math.exp(). numpy.exp() is about 10
times faster but throws an OverflowError exception for numerical underflow (e.g.
exp(-800), whereas python’s math.exp() just returns zero, which is much more
helpful.
columnmeans (A)
This is a wrapper for general dense or sparse dot products. It is only necessary as a
common interface for supporting ndarray, scipy spmatrix, and PySparse arrays.
columnvariances (A) This is a wrapper for general dense or sparse dot products. It is not necessary except
as a common interface for supporting ndarray, scipy spmatrix, and PySparse arrays.
densefeaturematrix (f,
Returns
sam- an (m x n) dense array of non-zero evaluations of the scalar functions fi in the
ple)
list f at the points x_1,...,x_n in the list sample.
densefeatures (f, x) Returns a dense array of non-zero evaluations of the functions fi in the list f at the
point x.
dotprod (u, v)
This is a wrapper around general dense or sparse dot products. It is not necessary
except as a common interface for supporting ndarray, scipy spmatrix, and PySparse
arrays.
flatten (a)
Flattens the sparse matrix or dense array/matrix ‘a’ into a 1-dimensional array
innerprod (A, v)
This is a wrapper around general dense or sparse dot products. It is not necessary
except as a common interface for supporting ndarray, scipy spmatrix, and PySparse
arrays.
innerprodtranspose (A,
This
v)is a wrapper around general dense or sparse dot products. It is not necessary
except as a common interface for supporting ndarray, scipy spmatrix, and PySparse
arrays.
logsumexp (a)
Compute the log of the sum of exponentials log(e^{a_1}+...e^{a_n}) of the
components of the array a, avoiding numerical overflow.
logsumexp_naive (val- For testing logsumexp(). Subject to numerical overflow for large values (e.g. 720).
ues)
robustlog (x)
Returns log(x) if x > 0, the complex log cmath.log(x) if x < 0, or float(‘-inf’) if x == 0.
rowmeans (A)
This is a wrapper for general dense or sparse dot products. It is only necessary as a
common interface for supporting ndarray, scipy spmatrix, and PySparse arrays.
sample_wr (populaChooses k random elements (with replacement) from a population. (From the Python
tion, k)
Cookbook).
sparsefeaturematrixReturns
(f, sam- an (m x n) sparse matrix of non-zero evaluations of the scalar or vector
ple[, format])
functions f_1,...,f_m in the list f at the points x_1,...,x_n in the sequence ‘sample’.
sparsefeatures (f, x[, forReturns an Mx1 sparse matrix of non-zero evaluations of the scalar functions
mat])
f_1,...,f_m in the list f at the point x.
arrayexp(x)
Returns the elementwise antilog of the real array x. We try to exponentiate with numpy.exp() and, if that
fails, with python’s math.exp(). numpy.exp() is about 10 times faster but throws an OverflowError exception
for numerical underflow (e.g. exp(-800), whereas python’s math.exp() just returns zero, which is much more
helpful.
arrayexpcomplex(x)
Returns the elementwise antilog of the vector x. We try to exponentiate with numpy.exp() and, if that fails, with
python’s math.exp(). numpy.exp() is about 10 times faster but throws an OverflowError exception for numerical
underflow (e.g. exp(-800), whereas python’s math.exp() just returns zero, which is much more helpful.
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columnmeans(A)
This is a wrapper for general dense or sparse dot products. It is only necessary as a common interface for
supporting ndarray, scipy spmatrix, and PySparse arrays.
Returns a dense (1 x n) vector with the column averages of A, which can be an (m x n) sparse or dense matrix.
>>> a = numpy.array([[1,2],[3,4]],’d’)
>>> columnmeans(a)
array([ 2., 3.])
columnvariances(A)
This is a wrapper for general dense or sparse dot products. It is not necessary except as a common interface for
supporting ndarray, scipy spmatrix, and PySparse arrays.
Returns a dense (1 x n) vector with unbiased estimators for the column variances for each column of the (m x
n) sparse or dense matrix A. (The normalization is by (m - 1).)
>>> a = numpy.array([[1,2], [3,4]], ’d’)
>>> columnvariances(a)
array([ 2., 2.])
densefeaturematrix(f, sample)
Returns an (m x n) dense array of non-zero evaluations of the scalar functions fi in the list f at the points
x_1,...,x_n in the list sample.
densefeatures(f, x)
Returns a dense array of non-zero evaluations of the functions fi in the list f at the point x.
dotprod(u, v)
This is a wrapper around general dense or sparse dot products. It is not necessary except as a common interface
for supporting ndarray, scipy spmatrix, and PySparse arrays.
Returns the dot product of the (1 x m) sparse array u with the (m x 1) (dense) numpy array v.
flatten(a)
Flattens the sparse matrix or dense array/matrix ‘a’ into a 1-dimensional array
innerprod(A, v)
This is a wrapper around general dense or sparse dot products. It is not necessary except as a common interface
for supporting ndarray, scipy spmatrix, and PySparse arrays.
Returns the inner product of the (m x n) dense or sparse matrix A with the n-element dense array v. This is a
wrapper for A.dot(v) for dense arrays and spmatrix objects, and for A.matvec(v, result) for PySparse matrices.
innerprodtranspose(A, v)
This is a wrapper around general dense or sparse dot products. It is not necessary except as a common interface
for supporting ndarray, scipy spmatrix, and PySparse arrays.
Computes A^T V, where A is a dense or sparse matrix and V is a numpy array. If A is sparse, V must be a
rank-1 array, not a matrix. This function is efficient for large matrices A. This is a wrapper for u.T.dot(v) for
dense arrays and spmatrix objects, and for u.matvec_transp(v, result) for pysparse matrices.
logsumexp(a)
Compute the log of the sum of exponentials log(e^{a_1}+...e^{a_n}) of the components of the array a, avoiding
numerical overflow.
logsumexp_naive(values)
For testing logsumexp(). Subject to numerical overflow for large values (e.g. 720).
robustlog(x)
Returns log(x) if x > 0, the complex log cmath.log(x) if x < 0, or float(‘-inf’) if x == 0.
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rowmeans(A)
This is a wrapper for general dense or sparse dot products. It is only necessary as a common interface for
supporting ndarray, scipy spmatrix, and PySparse arrays.
Returns a dense (m x 1) vector representing the mean of the rows of A, which be an (m x n) sparse or dense
matrix.
>>> a = numpy.array([[1,2],[3,4]], float)
>>> rowmeans(a)
array([ 1.5, 3.5])
sample_wr(population, k)
Chooses k random elements (with replacement) from a population. (From the Python Cookbook).
sparsefeaturematrix(f, sample, format=’csc_matrix’)
Returns an (m x n) sparse matrix of non-zero evaluations of the scalar or vector functions f_1,...,f_m in the list f
at the points x_1,...,x_n in the sequence ‘sample’.
If format=’ll_mat’, the PySparse module (or a symlink to it) must be available in the Python site-packages/ directory. A trimmed-down version, patched for NumPy compatibility, is available in the SciPy sandbox/pysparse
directory.
sparsefeatures(f, x, format=’csc_matrix’)
Returns an Mx1 sparse matrix of non-zero evaluations of the scalar functions f_1,...,f_m in the list f at the point
x.
If format=’ll_mat’, the PySparse module (or a symlink to it) must be available in the Python site-packages/ directory. A trimmed-down version, patched for NumPy compatibility, is available in the SciPy sandbox/pysparse
directory.
3.9 Miscellaneous routines (scipy.misc)
Warning: This documentation is work-in-progress and unorganized.
Various utilities that don’t have another home.
who(vardict=None)
Print the Numpy arrays in the given dictionary.
If there is no dictionary passed in or vardict is None then returns Numpy arrays in the globals() dictionary (all
Numpy arrays in the namespace).
Parameters
vardict : dict, optional
A dictionary possibly containing ndarrays. Default is globals().
Returns
out : None
Returns ‘None’.
Notes
Prints out the name, shape, bytes and type of all of the ndarrays present in vardict.
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Examples
>>> a = np.arange(10)
>>> b = np.ones(20)
>>> np.who()
Name
Shape
Bytes
Type
===========================================================
a
10
40
int32
b
20
160
float64
Upper bound on total bytes =
200
>>> d = {’x’: np.arange(2.0), ’y’: np.arange(3.0), ’txt’: ’Some str’,
... ’idx’:5}
>>> np.whos(d)
Name
Shape
Bytes
Type
===========================================================
y
3
24
float64
x
2
16
float64
Upper bound on total bytes =
40
source(object, output=<open file ’<stdout>’, mode ’w’ at 0x323070>)
Print or write to a file the source code for a Numpy object.
The source code is only returned for objects written in Python. Many functions and classes are defined in C and
will therefore not return useful information.
Parameters
object : numpy object
Input object. This can be any object (function, class, module, ...).
output : file object, optional
If output not supplied then source code is printed to screen (sys.stdout). File object
must be created with either write ‘w’ or append ‘a’ modes.
See Also:
lookfor, info
Examples
>>> np.source(np.interp)
In file: /usr/lib/python2.6/dist-packages/numpy/lib/function_base.py
def interp(x, xp, fp, left=None, right=None):
""".... (full docstring printed)"""
if isinstance(x, (float, int, number)):
return compiled_interp([x], xp, fp, left, right).item()
else:
return compiled_interp(x, xp, fp, left, right)
The source code is only returned for objects written in Python.
>>> np.source(np.array)
Not available for this object.
info(object=None, maxwidth=76, output=<open file ’<stdout>’, mode ’w’ at 0x323070>, toplevel=’scipy’)
Get help information for a function, class, or module.
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Parameters
object : object or str, optional
Input object or name to get information about. If object is a numpy object, its docstring is given. If it is a string, available modules are searched for matching objects.
If None, information about info itself is returned.
maxwidth : int, optional
Printing width.
output : file like object, optional
File like object that the output is written to, default is stdout. The object has to be
opened in ‘w’ or ‘a’ mode.
toplevel : str, optional
Start search at this level.
See Also:
source, lookfor
Notes
When used interactively with an object, np.info(obj) is equivalent to help(obj) on the Python prompt
or obj? on the IPython prompt.
Examples
>>> np.info(np.polyval) # doctest: +SKIP
polyval(p, x)
Evaluate the polynomial p at x.
...
When using a string for object it is possible to get multiple results.
>>> np.info(’fft’) # doctest: +SKIP
*** Found in numpy ***
Core FFT routines
...
*** Found in numpy.fft ***
fft(a, n=None, axis=-1)
...
*** Repeat reference found in numpy.fft.fftpack ***
*** Total of 3 references found. ***
factorial(n, exact=0)
n! = special.gamma(n+1)
If exact==0, then floating point precision is used, otherwise exact long integer is computed.
Notes:
• Array argument accepted only for exact=0 case.
• If n<0, the return value is 0.
factorial2(n, exact=False)
Double factorial.
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This is the factorial with every second value is skipped, i.e., 7!!
mated numerically as:
n!! = special.gamma(n/2+1)*2**((m+1)/2)/sqrt(pi)
= 2**(n/2) * (n/2)!
= 7 * 5 * 3 * 1. It can be approxi-
n odd
n even
Parameters
n : int, array-like
Calculate n!!. Arrays are only supported with exact set to False. If n < 0, the
return value is 0.
exact : bool, optional
The result can be approximated rapidly using the gamma-formula above (default). If
exact is set to True, calculate the answer exactly using integer arithmetic.
Returns
nff : float or int
Double factorial of n, as an int or a float depending on exact.
References
[R67]
factorialk(n, k, exact=1)
n(!!...!) = multifactorial of order k k times
Parameters
n : int, array-like
Calculate multifactorial. Arrays are only supported with exact set to False. If n < 0,
the return value is 0.
exact : bool, optional
If exact is set to True, calculate the answer exactly using integer arithmetic.
Returns
val : int
Multi factorial of n.
Raises
NotImplementedError :
Raises when exact is False
comb(N, k, exact=0)
Combinations of N things taken k at a time.
Parameters
N : int, array
Nunmber of things.
k : int, array
Numner of elements taken.
exact : int, optional
If exact is 0, then floating point precision is used, otherwise exact long integer is
computed.
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Returns
val : int, array
The total number of combinations.
Notes
•Array arguments accepted only for exact=0 case.
•If k > N, N < 0, or k < 0, then a 0 is returned.
central_diff_weights(Np, ndiv=1)
Return weights for an Np-point central derivative of order ndiv assuming equally-spaced function points.
If weights are in the vector w, then derivative is w[0] * f(x-ho*dx) + ... + w[-1] * f(x+h0*dx)
Notes
Can be inaccurate for large number of points.
derivative(func, x0, dx=1.0, n=1, args=(), order=3)
Find the n-th derivative of a function at point x0.
Given a function, use a central difference formula with spacing dx to compute the n-th derivative at x0.
Parameters
func : function
Input function.
x0 : float
The point at which nth derivative is found.
dx : int, optional
Spacing.
n : int, optional
Order of the derivative. Default is 1.
args : tuple, optional
Arguments
order : int, optional
Number of points to use, must be odd.
Notes
Decreasing the step size too small can result in round-off error.
pade(an, m)
Given Taylor series coefficients in an, return a Pade approximation to the function as the ratio of two polynomials
p / q where the order of q is m.
3.10 Multi-dimensional image processing (scipy.ndimage)
Functions for multi-dimensional image processing.
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3.10.1 Filters scipy.ndimage.filters
convolve (input, weights[, output, mode, cval, ...])
convolve1d (input, weights[, axis, output, mode, ...])
correlate (input, weights[, output, mode, cval, ...])
correlate1d (input, weights[, axis, output, mode, ...])
gaussian_filter (input, sigma[, order, output, mode, ...])
gaussian_filter1d (input, sigma[, axis, order, output, ...])
gaussian_gradient_magnitude (input, sigma[, output, mode, cval])
gaussian_laplace (input, sigma[, output, mode, cval])
generic_filter (input, function[, size, footprint, ...])
generic_filter1d (input, function, filter_size[, axis, output, mode, ...])
generic_gradient_magnitude (input, derivative[, output, mode, cval, ...])
generic_laplace (input, derivative2[, output, mode, cval, ...])
laplace (input[, output, mode, cval])
maximum_filter (input[, size, footprint, ...])
maximum_filter1d (input, size[, axis, output, mode, ...])
median_filter (input[, size, footprint, ...])
minimum_filter (input[, size, footprint, ...])
minimum_filter1d (input, size[, axis, output, mode, ...])
percentile_filter (input, percentile[, size, footprint, ...])
prewitt (input[, axis, output, mode, ...])
rank_filter (input, rank[, size, footprint, ...])
sobel (input[, axis, output, mode, ...])
uniform_filter (input[, size, output, mode, ...])
uniform_filter1d (input, size[, axis, output, mode, ...])
Multi-dimensional convolution.
Calculate a one-dimensional convolution along the given axis.
Multi-dimensional correlation.
Calculate a one-dimensional correlation along the given axis.
Multi-dimensional Gaussian filter.
One-dimensional Gaussian filter.
Calculate a multidimensional gradient magnitude using
gaussian derivatives.
Calculate a multidimensional laplace filter using gaussian
second derivatives.
Calculates a multi-dimensional filter using the given function.
Calculate a one-dimensional filter along the given axis.
Calculate a gradient magnitude using the provided function for
the gradient.
Calculate a multidimensional laplace filter using the provided
second derivative function.
Calculate a multidimensional laplace filter using an estimation
for the second derivative based on differences.
Calculates a multi-dimensional maximum filter.
Calculate a one-dimensional maximum filter along the given
axis.
Calculates a multi-dimensional median filter.
Calculates a multi-dimensional minimum filter.
Calculate a one-dimensional minimum filter along the given
axis.
Calculates a multi-dimensional percentile filter.
Calculate a Prewitt filter.
Calculates a multi-dimensional rank filter.
Calculate a Sobel filter.
Multi-dimensional uniform filter.
Calculate a one-dimensional uniform filter along the given axis.
convolve(input, weights, output=None, mode=’reflect’, cval=0.0, origin=0)
Multi-dimensional convolution.
The array is convolved with the given kernel.
Parameters
input : array-like
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input array to filter
weights : ndarray
array of weights, same number of dimensions as input
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The origin parameter controls the placement of the filter. Default 0
See Also:
correlate
Correlate an image with a kernel.
convolve1d(input, weights, axis=-1, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculate a one-dimensional convolution along the given axis.
The lines of the array along the given axis are convolved with the given weights.
Parameters
input : array-like
input array to filter
weights : ndarray
one-dimensional sequence of numbers
axis : integer, optional
axis of input along which to calculate. Default is -1
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
correlate(input, weights, output=None, mode=’reflect’, cval=0.0, origin=0)
Multi-dimensional correlation.
The array is correlated with the given kernel.
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Parameters
input : array-like
input array to filter
weights : ndarray
array of weights, same number of dimensions as input
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The origin parameter controls the placement of the filter. Default 0
See Also:
convolve
Convolve an image with a kernel.
correlate1d(input, weights, axis=-1, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculate a one-dimensional correlation along the given axis.
The lines of the array along the given axis are correlated with the given weights.
Parameters
input : array-like
input array to filter
weights : array
one-dimensional sequence of numbers
axis : integer, optional
axis of input along which to calculate. Default is -1
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
gaussian_filter(input, sigma, order=0, output=None, mode=’reflect’, cval=0.0)
Multi-dimensional Gaussian filter.
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Parameters
input : array-like
input array to filter
sigma : scalar or sequence of scalars
standard deviation for Gaussian kernel. The standard deviations of the Gaussian
filter are given for each axis as a sequence, or as a single number, in which case it is
equal for all axes.
order : {0, 1, 2, 3} or sequence from same set, optional
The order of the filter along each axis is given as a sequence of integers, or as a single
number. An order of 0 corresponds to convolution with a Gaussian kernel. An order
of 1, 2, or 3 corresponds to convolution with the first, second or third derivatives of
a Gaussian. Higher order derivatives are not implemented
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
Notes
The multi-dimensional filter is implemented as a sequence of one-dimensional convolution filters. The intermediate arrays are stored in the same data type as the output. Therefore, for output types with a limited precision,
the results may be imprecise because intermediate results may be stored with insufficient precision.
gaussian_filter1d(input, sigma, axis=-1, order=0, output=None, mode=’reflect’, cval=0.0)
One-dimensional Gaussian filter.
Parameters
input : array-like
input array to filter
sigma : scalar
standard deviation for Gaussian kernel
axis : integer, optional
axis of input along which to calculate. Default is -1
order : {0, 1, 2, 3}, optional
An order of 0 corresponds to convolution with a Gaussian kernel. An order of 1, 2, or
3 corresponds to convolution with the first, second or third derivatives of a Gaussian.
Higher order derivatives are not implemented
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
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The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
gaussian_gradient_magnitude(input, sigma, output=None, mode=’reflect’, cval=0.0)
Calculate a multidimensional gradient magnitude using gaussian derivatives.
Parameters
input : array-like
input array to filter
sigma : scalar or sequence of scalars
The standard deviations of the Gaussian filter are given for each axis as a sequence,
or as a single number, in which case it is equal for all axes..
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
gaussian_laplace(input, sigma, output=None, mode=’reflect’, cval=0.0)
Calculate a multidimensional laplace filter using gaussian second derivatives.
Parameters
input : array-like
input array to filter
sigma : scalar or sequence of scalars
The standard deviations of the Gaussian filter are given for each axis as a sequence,
or as a single number, in which case it is equal for all axes..
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
generic_filter(input, function, size=None, footprint=None, output=None, mode=’reflect’, cval=0.0, origin=0,
extra_arguments=(), extra_keywords=None)
Calculates a multi-dimensional filter using the given function.
At each element the provided function is called. The input values within the filter footprint at that element are
passed to the function as a 1D array of double values.
Parameters
input : array-like
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input array to filter
function : callable
function to apply at each element
size : scalar or tuple, optional
See footprint, below
footprint : array, optional
Either size or footprint must be defined. size gives the shape that is taken
from the input array, at every element position, to define the input to the filter
function. footprint is a boolean array that specifies (implicitly) a shape, but
also which of the elements within this shape will get passed to the filter function.
Thus size=(n,m) is equivalent to footprint=np.ones((n,m)). We adjust
size to the number of dimensions of the input array, so that, if the input array is
shape (10,10,10), and size is 2, then the actual size used is (2,2,2).
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
extra_arguments : sequence, optional
Sequence of extra positional arguments to pass to passed function
extra_keywords : dict, optional
dict of extra keyword arguments to pass to passed function
generic_filter1d(input, function, filter_size, axis=-1, output=None, mode=’reflect’, cval=0.0, origin=0, extra_arguments=(), extra_keywords=None)
Calculate a one-dimensional filter along the given axis.
generic_filter1d iterates over the lines of the array, calling the given function at each line. The arguments of the
line are the input line, and the output line. The input and output lines are 1D double arrays. The input line is
extended appropriately according to the filter size and origin. The output line must be modified in-place with
the result.
Parameters
input : array-like
input array to filter
function : callable
function to apply along given axis
filter_size : scalar
length of the filter
axis : integer, optional
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axis of input along which to calculate. Default is -1
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
extra_arguments : sequence, optional
Sequence of extra positional arguments to pass to passed function
extra_keywords : dict, optional
dict of extra keyword arguments to pass to passed function
generic_gradient_magnitude(input, derivative, output=None, mode=’reflect’,
tra_arguments=(), extra_keywords=None)
Calculate a gradient magnitude using the provided function for the gradient.
cval=0.0,
ex-
Parameters
input : array-like
input array to filter
derivative : callable
Callable with the following signature::
derivative(input, axis, output, mode, cval,
*extra_arguments, **extra_keywords)
See extra_arguments, extra_keywords below derivative can assume
that input and output are ndarrays. Note that the output from derivative is
modified inplace; be careful to copy important inputs before returning them.
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
extra_keywords : dict, optional
dict of extra keyword arguments to pass to passed function
extra_arguments : sequence, optional
Sequence of extra positional arguments to pass to passed function
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generic_laplace(input, derivative2, output=None, mode=’reflect’, cval=0.0, extra_arguments=(), extra_keywords=None)
Calculate a multidimensional laplace filter using the provided second derivative function.
Parameters
input : array-like
input array to filter
derivative2 : callable
Callable with the following signature::
derivative2(input, axis, output, mode, cval,
*extra_arguments, **extra_keywords)
See extra_arguments, extra_keywords below
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
extra_keywords : dict, optional
dict of extra keyword arguments to pass to passed function
extra_arguments : sequence, optional
Sequence of extra positional arguments to pass to passed function
laplace(input, output=None, mode=’reflect’, cval=0.0)
Calculate a multidimensional laplace filter using an estimation for the second derivative based on differences.
Parameters
input : array-like
input array to filter
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
maximum_filter(input, size=None, footprint=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculates a multi-dimensional maximum filter.
Parameters
input : array-like
input array to filter
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size : scalar or tuple, optional
See footprint, below
footprint : array, optional
Either size or footprint must be defined. size gives the shape that is taken
from the input array, at every element position, to define the input to the filter
function. footprint is a boolean array that specifies (implicitly) a shape, but
also which of the elements within this shape will get passed to the filter function.
Thus size=(n,m) is equivalent to footprint=np.ones((n,m)). We adjust
size to the number of dimensions of the input array, so that, if the input array is
shape (10,10,10), and size is 2, then the actual size used is (2,2,2).
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
maximum_filter1d(input, size, axis=-1, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculate a one-dimensional maximum filter along the given axis.
The lines of the array along the given axis are filtered with a maximum filter of given size.
Parameters
input : array-like
input array to filter
size : int
length along which to calculate 1D maximum
axis : integer, optional
axis of input along which to calculate. Default is -1
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
median_filter(input, size=None, footprint=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculates a multi-dimensional median filter.
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Parameters
input : array-like
input array to filter
size : scalar or tuple, optional
See footprint, below
footprint : array, optional
Either size or footprint must be defined. size gives the shape that is taken
from the input array, at every element position, to define the input to the filter
function. footprint is a boolean array that specifies (implicitly) a shape, but
also which of the elements within this shape will get passed to the filter function.
Thus size=(n,m) is equivalent to footprint=np.ones((n,m)). We adjust
size to the number of dimensions of the input array, so that, if the input array is
shape (10,10,10), and size is 2, then the actual size used is (2,2,2).
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The origin parameter controls the placement of the filter. Default 0
minimum_filter(input, size=None, footprint=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculates a multi-dimensional minimum filter.
Parameters
input : array-like
input array to filter
size : scalar or tuple, optional
See footprint, below
footprint : array, optional
Either size or footprint must be defined. size gives the shape that is taken
from the input array, at every element position, to define the input to the filter
function. footprint is a boolean array that specifies (implicitly) a shape, but
also which of the elements within this shape will get passed to the filter function.
Thus size=(n,m) is equivalent to footprint=np.ones((n,m)). We adjust
size to the number of dimensions of the input array, so that, if the input array is
shape (10,10,10), and size is 2, then the actual size used is (2,2,2).
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
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cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
minimum_filter1d(input, size, axis=-1, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculate a one-dimensional minimum filter along the given axis.
The lines of the array along the given axis are filtered with a minimum filter of given size.
Parameters
input : array-like
input array to filter
size : int
length along which to calculate 1D minimum
axis : integer, optional
axis of input along which to calculate. Default is -1
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
percentile_filter(input, percentile, size=None, footprint=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculates a multi-dimensional percentile filter.
Parameters
input : array-like
input array to filter
percentile : scalar
The percentile parameter may be less then zero, i.e., percentile = -20 equals percentile = 80
size : scalar or tuple, optional
See footprint, below
footprint : array, optional
Either size or footprint must be defined. size gives the shape that is taken
from the input array, at every element position, to define the input to the filter
function. footprint is a boolean array that specifies (implicitly) a shape, but
also which of the elements within this shape will get passed to the filter function.
Thus size=(n,m) is equivalent to footprint=np.ones((n,m)). We adjust
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size to the number of dimensions of the input array, so that, if the input array is
shape (10,10,10), and size is 2, then the actual size used is (2,2,2).
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
prewitt(input, axis=-1, output=None, mode=’reflect’, cval=0.0)
Calculate a Prewitt filter.
Parameters
input : array-like
input array to filter
axis : integer, optional
axis of input along which to calculate. Default is -1
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
rank_filter(input, rank, size=None, footprint=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculates a multi-dimensional rank filter.
Parameters
input : array-like
input array to filter
rank : integer
The rank parameter may be less then zero, i.e., rank = -1 indicates the largest element.
size : scalar or tuple, optional
See footprint, below
footprint : array, optional
Either size or footprint must be defined. size gives the shape that is taken
from the input array, at every element position, to define the input to the filter
function. footprint is a boolean array that specifies (implicitly) a shape, but
also which of the elements within this shape will get passed to the filter function.
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Thus size=(n,m) is equivalent to footprint=np.ones((n,m)). We adjust
size to the number of dimensions of the input array, so that, if the input array is
shape (10,10,10), and size is 2, then the actual size used is (2,2,2).
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
sobel(input, axis=-1, output=None, mode=’reflect’, cval=0.0)
Calculate a Sobel filter.
Parameters
input : array-like
input array to filter
axis : integer, optional
axis of input along which to calculate. Default is -1
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
uniform_filter(input, size=3, output=None, mode=’reflect’, cval=0.0, origin=0)
Multi-dimensional uniform filter.
Parameters
input : array-like
input array to filter
size : int or sequence of ints
The sizes of the uniform filter are given for each axis as a sequence, or as a single
number, in which case the size is equal for all axes.
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
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cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
Notes
The multi-dimensional filter is implemented as a sequence of one-dimensional uniform filters. The intermediate
arrays are stored in the same data type as the output. Therefore, for output types with a limited precision, the
results may be imprecise because intermediate results may be stored with insufficient precision.
uniform_filter1d(input, size, axis=-1, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculate a one-dimensional uniform filter along the given axis.
The lines of the array along the given axis are filtered with a uniform filter of given size.
Parameters
input : array-like
input array to filter
size : integer
length of uniform filter
axis : integer, optional
axis of input along which to calculate. Default is -1
output : array, optional
The output parameter passes an array in which to store the filter output.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is
the value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0
origin : scalar, optional
The ‘‘origin‘‘ parameter controls the placement of the filter. Default 0 :
3.10.2 Fourier filters scipy.ndimage.fourier
fourier_ellipsoid (input, size[, n, axis, output])
fourier_gaussian (input, sigma[, n, axis, output])
fourier_shift (input, shift[, n, axis, output])
fourier_uniform (input, size[, n, axis, output])
Multi-dimensional ellipsoid fourier filter.
Multi-dimensional Gaussian fourier filter.
Multi-dimensional fourier shift filter.
Multi-dimensional uniform fourier filter.
fourier_ellipsoid(input, size, n=-1, axis=-1, output=None)
Multi-dimensional ellipsoid fourier filter.
The array is multiplied with the fourier transform of a ellipsoid of given sizes.
Parameters
input : array_like
The input array.
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size : float or sequence
The size of the box used for filtering. If a float, size is the same for all axes. If a
sequence, size has to contain one value for each axis.
n : int, optional
If n is negative (default), then the input is assumed to be the result of a complex fft.
If n is larger than or equal to zero, the input is assumed to be the result of a real fft,
and n gives the length of the array before transformation along the real transform
direction.
axis : int, optional
The axis of the real transform.
output : ndarray, optional
If given, the result of filtering the input is placed in this array. None is returned in
this case.
Returns
return_value : ndarray or None
The filtered input. If output is given as a parameter, None is returned.
Notes
This function is implemented for arrays of rank 1, 2, or 3.
fourier_gaussian(input, sigma, n=-1, axis=-1, output=None)
Multi-dimensional Gaussian fourier filter.
The array is multiplied with the fourier transform of a Gaussian kernel.
Parameters
input : array_like
The input array.
sigma : float or sequence
The sigma of the Gaussian kernel. If a float, sigma is the same for all axes. If a
sequence, sigma has to contain one value for each axis.
n : int, optional
If n is negative (default), then the input is assumed to be the result of a complex fft.
If n is larger than or equal to zero, the input is assumed to be the result of a real fft,
and n gives the length of the array before transformation along the real transform
direction.
axis : int, optional
The axis of the real transform.
output : ndarray, optional
If given, the result of filtering the input is placed in this array. None is returned in
this case.
Returns
return_value : ndarray or None
The filtered input. If output is given as a parameter, None is returned.
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fourier_shift(input, shift, n=-1, axis=-1, output=None)
Multi-dimensional fourier shift filter.
The array is multiplied with the fourier transform of a shift operation.
Parameters
input : array_like
The input array.
shift : float or sequence
The size of the box used for filtering. If a float, shift is the same for all axes. If a
sequence, shift has to contain one value for each axis.
n : int, optional
If n is negative (default), then the input is assumed to be the result of a complex fft.
If n is larger than or equal to zero, the input is assumed to be the result of a real fft,
and n gives the length of the array before transformation along the real transform
direction.
axis : int, optional
The axis of the real transform.
output : ndarray, optional
If given, the result of shifting the input is placed in this array. None is returned in
this case.
Returns
return_value : ndarray or None
The shifted input. If output is given as a parameter, None is returned.
fourier_uniform(input, size, n=-1, axis=-1, output=None)
Multi-dimensional uniform fourier filter.
The array is multiplied with the fourier transform of a box of given size.
Parameters
input : array_like
The input array.
size : float or sequence
The size of the box used for filtering. If a float, size is the same for all axes. If a
sequence, size has to contain one value for each axis.
n : int, optional
If n is negative (default), then the input is assumed to be the result of a complex fft.
If n is larger than or equal to zero, the input is assumed to be the result of a real fft,
and n gives the length of the array before transformation along the real transform
direction.
axis : int, optional
The axis of the real transform.
output : ndarray, optional
If given, the result of filtering the input is placed in this array. None is returned in
this case.
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Returns
return_value : ndarray or None
The filtered input. If output is given as a parameter, None is returned.
3.10.3 Interpolation scipy.ndimage.interpolation
affine_transform (input, matrix[, offset, output_shape, ...])
geometric_transform (input, mapping[, output_shape, output_type, ...])
map_coordinates (input, coordinates[, output_type, output, ...])
rotate (input, angle[, axes, 0), reshape, ...])
shift (input, shift[, output_type, output, ...])
spline_filter (input[, order, output, output_type])
spline_filter1d (input[, order, axis, output, ...])
zoom (input, zoom[, output_type, output, ...])
Apply an affine transformation.
Apply an arbritrary geometric transform.
Map the input array to new coordinates by
interpolation.
Rotate an array.
Shift an array.
Multi-dimensional spline filter.
Calculates a one-dimensional spline filter along
the given axis.
Zoom an array.
affine_transform(input, matrix, offset=0.0, output_shape=None, output_type=None, output=None, order=3,
mode=’constant’, cval=0.0, prefilter=True)
Apply an affine transformation.
The given matrix and offset are used to find for each point in the output the corresponding coordinates in the input
by an affine transformation. The value of the input at those coordinates is determined by spline interpolation of
the requested order. Points outside the boundaries of the input are filled according to the given mode.
Parameters
input : ndarray
The input array.
matrix : ndarray
The matrix must be two-dimensional or can also be given as a one-dimensional sequence or array. In the latter case, it is assumed that the matrix is diagonal. A more
efficient algorithms is then applied that exploits the separability of the problem.
offset : float or sequence, optional
The offset into the array where the transform is applied. If a float, offset is the same
for each axis. If a sequence, offset should contain one value for each axis.
output_shape : tuple of ints, optional
Shape tuple.
output : ndarray or dtype, optional
The array in which to place the output, or the dtype of the returned array.
output_type : dtype, optional
DEPRECATED, DO NOT USE. If used, a RuntimeError is raised.
order : int, optional
The order of the spline interpolation, default is 3. The order has to be in the range
0-5.
mode : str, optional
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Points outside the boundaries of the input are filled according to the given mode
(‘constant’, ‘nearest’, ‘reflect’ or ‘wrap’). Default is ‘constant’.
cval : scalar, optional
Value used for points outside the boundaries of the input if mode=’constant’.
Default is 0.0
prefilter : bool, optional
The parameter prefilter determines if the input is pre-filtered with spline_filter before
interpolation (necessary for spline interpolation of order > 1). If False, it is assumed
that the input is already filtered. Default is True.
Returns
return_value : ndarray or None
The transformed input. If output is given as a parameter, None is returned.
geometric_transform(input, mapping, output_shape=None, output_type=None, output=None, order=3,
mode=’constant’, cval=0.0, prefilter=True, extra_arguments=(), extra_keywords={})
Apply an arbritrary geometric transform.
The given mapping function is used to find, for each point in the output, the corresponding coordinates in the
input. The value of the input at those coordinates is determined by spline interpolation of the requested order.
Parameters
input : array_like
The input array.
mapping : callable
A callable object that accepts a tuple of length equal to the output array rank, and
returns the corresponding input coordinates as a tuple of length equal to the input
array rank.
output_shape : tuple of ints
Shape tuple.
output : ndarray or dtype, optional
The array in which to place the output, or the dtype of the returned array.
output_type : dtype, optional
DEPRECATED, DO NOT USE. If used, a RuntimeError is raised.
order : int, optional
The order of the spline interpolation, default is 3. The order has to be in the range
0-5.
mode : str, optional
Points outside the boundaries of the input are filled according to the given mode
(‘constant’, ‘nearest’, ‘reflect’ or ‘wrap’). Default is ‘constant’.
cval : scalar, optional
Value used for points outside the boundaries of the input if mode=’constant’.
Default is 0.0
prefilter : bool, optional
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The parameter prefilter determines if the input is pre-filtered with spline_filter before
interpolation (necessary for spline interpolation of order > 1). If False, it is assumed
that the input is already filtered. Default is True.
extra_arguments : tuple, optional
Extra arguments passed to mapping.
extra_keywords : dict, optional
Extra keywords passed to mapping.
Returns
return_value : ndarray or None
The filtered input. If output is given as a parameter, None is returned.
See Also:
map_coordinates, affine_transform, spline_filter1d
Examples
>>> a = np.arange(12.).reshape((4, 3))
>>> def shift_func(output_coords):
...
return (output_coords[0] - 0.5, output_coords[1] - 0.5)
...
>>> sp.ndimage.geometric_transform(a, shift_func)
array([[ 0.
, 0.
, 0.
],
[ 0.
, 1.362, 2.738],
[ 0.
, 4.812, 6.187],
[ 0.
, 8.263, 9.637]])
map_coordinates(input, coordinates, output_type=None, output=None, order=3, mode=’constant’, cval=0.0,
prefilter=True)
Map the input array to new coordinates by interpolation.
The array of coordinates is used to find, for each point in the output, the corresponding coordinates in the input.
The value of the input at those coordinates is determined by spline interpolation of the requested order.
The shape of the output is derived from that of the coordinate array by dropping the first axis. The values of the
array along the first axis are the coordinates in the input array at which the output value is found.
Parameters
input : ndarray
The input array.
coordinates : array_like
The coordinates at which input is evaluated.
output : ndarray or dtype, optional
The array in which to place the output, or the dtype of the returned array.
output_type : dtype, optional
DEPRECATED, DO NOT USE. If used, a RuntimeError is raised.
order : int, optional
The order of the spline interpolation, default is 3. The order has to be in the range
0-5.
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mode : str, optional
Points outside the boundaries of the input are filled according to the given mode
(‘constant’, ‘nearest’, ‘reflect’ or ‘wrap’). Default is ‘constant’.
cval : scalar, optional
Value used for points outside the boundaries of the input if mode=’constant’.
Default is 0.0
prefilter : bool, optional
The parameter prefilter determines if the input is pre-filtered with spline_filter before
interpolation (necessary for spline interpolation of order > 1). If False, it is assumed
that the input is already filtered. Default is True.
Returns
return_value : ndarray
The result of transforming the input. The shape of the output is derived from that of
coordinates by dropping the first axis.
See Also:
spline_filter, geometric_transform, scipy.interpolate
Examples
>>> import scipy.ndimage
>>> a = np.arange(12.).reshape((4, 3))
>>> a
array([[ 0.,
1.,
2.],
[ 3.,
4.,
5.],
[ 6.,
7.,
8.],
[ 9., 10., 11.]])
>>> sp.ndimage.map_coordinates(a, [[0.5, 2], [0.5, 1]], order=1)
[ 2. 7.]
Above, the interpolated value of a[0.5, 0.5] gives output[0], while a[2, 1] is output[1].
>>> inds = np.array([[0.5, 2], [0.5, 4]])
>>> sp.ndimage.map_coordinates(a, inds, order=1, cval=-33.3)
array([ 2. , -33.3])
>>> sp.ndimage.map_coordinates(a, inds, order=1, mode=’nearest’)
array([ 2., 8.])
>>> sp.ndimage.map_coordinates(a, inds, order=1, cval=0, output=bool)
array([ True, False], dtype=bool
rotate(input, angle, axes=(1, 0), reshape=True, output_type=None, output=None, order=3, mode=’constant’,
cval=0.0, prefilter=True)
Rotate an array.
The array is rotated in the plane defined by the two axes given by the axes parameter using spline interpolation
of the requested order.
Parameters
input : ndarray
The input array.
angle : float
The rotation angle in degrees.
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axes : tuple of 2 ints, optional
The two axes that define the plane of rotation. Default is the first two axes.
reshape : bool, optional
If reshape is true, the output shape is adapted so that the input array is contained
completely in the output. Default is True.
output : ndarray or dtype, optional
The array in which to place the output, or the dtype of the returned array.
output_type : dtype, optional
DEPRECATED, DO NOT USE. If used, a RuntimeError is raised.
order : int, optional
The order of the spline interpolation, default is 3. The order has to be in the range
0-5.
mode : str, optional
Points outside the boundaries of the input are filled according to the given mode
(‘constant’, ‘nearest’, ‘reflect’ or ‘wrap’). Default is ‘constant’.
cval : scalar, optional
Value used for points outside the boundaries of the input if mode=’constant’.
Default is 0.0
prefilter : bool, optional
The parameter prefilter determines if the input is pre-filtered with spline_filter before
interpolation (necessary for spline interpolation of order > 1). If False, it is assumed
that the input is already filtered. Default is True.
Returns
return_value : ndarray or None
The rotated input. If output is given as a parameter, None is returned.
shift(input, shift, output_type=None, output=None, order=3, mode=’constant’, cval=0.0, prefilter=True)
Shift an array.
The array is shifted using spline interpolation of the requested order. Points outside the boundaries of the input
are filled according to the given mode.
Parameters
input : ndarray
The input array.
shift : float or sequence, optional
The shift along the axes. If a float, shift is the same for each axis. If a sequence, shift
should contain one value for each axis.
output : ndarray or dtype, optional
The array in which to place the output, or the dtype of the returned array.
output_type : dtype, optional
DEPRECATED, DO NOT USE. If used, a RuntimeError is raised.
order : int, optional
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The order of the spline interpolation, default is 3. The order has to be in the range
0-5.
mode : str, optional
Points outside the boundaries of the input are filled according to the given mode
(‘constant’, ‘nearest’, ‘reflect’ or ‘wrap’). Default is ‘constant’.
cval : scalar, optional
Value used for points outside the boundaries of the input if mode=’constant’.
Default is 0.0
prefilter : bool, optional
The parameter prefilter determines if the input is pre-filtered with spline_filter before
interpolation (necessary for spline interpolation of order > 1). If False, it is assumed
that the input is already filtered. Default is True.
Returns
return_value : ndarray or None
The shifted input. If output is given as a parameter, None is returned.
spline_filter(input, order=3, output=<type ’numpy.float64’>, output_type=None)
Multi-dimensional spline filter.
For more details, see spline_filter1d.
See Also:
spline_filter1d
Notes
The multi-dimensional filter is implemented as a sequence of one-dimensional spline filters. The intermediate
arrays are stored in the same data type as the output. Therefore, for output types with a limited precision, the
results may be imprecise because intermediate results may be stored with insufficient precision.
spline_filter1d(input, order=3, axis=-1, output=<type ’numpy.float64’>, output_type=None)
Calculates a one-dimensional spline filter along the given axis.
The lines of the array along the given axis are filtered by a spline filter. The order of the spline must be >= 2 and
<= 5.
Parameters
input : array_like
The input array.
order : int, optional
The order of the spline, default is 3.
axis : int, optional
The axis along which the spline filter is applied. Default is the last axis.
output : ndarray or dtype, optional
The array in which to place the output, or the dtype of the returned array. Default is
numpy.float64.
output_type : dtype, optional
DEPRECATED, DO NOT USE. If used, a RuntimeError is raised.
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Returns
return_value : ndarray or None
The filtered input. If output is given as a parameter, None is returned.
zoom(input, zoom, output_type=None, output=None, order=3, mode=’constant’, cval=0.0, prefilter=True)
Zoom an array.
The array is zoomed using spline interpolation of the requested order.
Parameters
input : ndarray
The input array.
zoom : float or sequence, optional
The zoom factor along the axes. If a float, zoom is the same for each axis. If a
sequence, zoom should contain one value for each axis.
output : ndarray or dtype, optional
The array in which to place the output, or the dtype of the returned array.
output_type : dtype, optional
DEPRECATED, DO NOT USE. If used, a RuntimeError is raised.
order : int, optional
The order of the spline interpolation, default is 3. The order has to be in the range
0-5.
mode : str, optional
Points outside the boundaries of the input are filled according to the given mode
(‘constant’, ‘nearest’, ‘reflect’ or ‘wrap’). Default is ‘constant’.
cval : scalar, optional
Value used for points outside the boundaries of the input if mode=’constant’.
Default is 0.0
prefilter : bool, optional
The parameter prefilter determines if the input is pre-filtered with spline_filter before
interpolation (necessary for spline interpolation of order > 1). If False, it is assumed
that the input is already filtered. Default is True.
Returns
return_value : ndarray or None
The zoomed input. If output is given as a parameter, None is returned.
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center_of_mass (input[, labels, index])
extrema (input[, labels, index])
find_objects (input[, max_label])
histogram (input, min, max, bins[, labels, index])
label (input[, structure, output])
maximum (input[, labels, index])
maximum_position (input[, labels, index])
mean (input[, labels, index])
minimum (input[, labels, index])
minimum_position (input[, labels, index])
standard_deviation (input[, labels, index])
sum (input[, labels, index])
variance (input[, labels, index])
watershed_ift (input, markers[, structure, output])
Calculate the center of mass of the values of an array at labels.
Calculate the minimums and maximums of the values of an array at
labels, along with their positions.
Find objects in a labeled array.
Calculate the histogram of the values of an array at labels.
Label features in an array.
Calculate the maximum of the values of an array over labeled regions.
Find the positions of the maximums of the values of an array at labels.
Calculate the mean of the values of an array at labels.
Calculate the minimum of the values of an array over labeled regions.
Find the positions of the minimums of the values of an array at labels.
Calculate the standard deviation of the values of an array at labels.
Calculate the sum of the values of the array.
Calculate the variance of the values of an array at labels.
Apply watershed from markers using a iterative forest transform
algorithm.
center_of_mass(input, labels=None, index=None)
Calculate the center of mass of the values of an array at labels.
Labels must be None or an array of the same dimensions as the input.
Index must be None, a single label or sequence of labels. If none, all values where label is greater than zero are
used.
extrema(input, labels=None, index=None)
Calculate the minimums and maximums of the values of an array at labels, along with their positions.
Labels must be None or an array of the same dimensions as the input.
Index must be None, a single label or sequence of labels. If none, all values where label is greater than zero are
used.
Returns: minimums, maximums, min_positions, max_positions
find_objects(input, max_label=0)
Find objects in a labeled array.
The input must be an array with labeled objects. A list of slices into the array is returned that contain the objects.
The list represents a sequence of the numbered objects. If a number is missing, None is returned instead of a
slice. If max_label > 0, it gives the largest object number that is searched for, otherwise all are returned.
histogram(input, min, max, bins, labels=None, index=None)
Calculate the histogram of the values of an array at labels.
Labels must be None or an array of the same dimensions as the input.
The histograms are defined by the minimum and maximum values and the number of bins.
Index must be None, a single label or sequence of labels. If none, all values where label is greater than zero are
used.
label(input, structure=None, output=None)
Label features in an array.
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Parameters
input : array_like
An array-like object to be labeled. Any non-zero values in input are counted as
features and zero values are considered the background.
structure : array_like, optional
A structuring element that defines feature connections.
structure must be symmetric. If no structuring element is provided, one is automatically generated with a squared connectivity equal to one.
That is, for a 2D input array, the default structuring element is:
[[0,1,0],
[1,1,1],
[0,1,0]]
output : (None, data-type, array_like), optional
If output is a data type, it specifies the type of the resulting labeled feature array
If output is an array-like object, then output will be updated with the labeled features
from this function
Returns
labeled_array : array_like
An array-like object where each unique feature has a unique value
num_features : int
If ‘output‘ is None or a data type, this function returns a tuple, :
(‘labeled_array‘, ‘num_features‘). :
If ‘output‘ is an array, then it will be updated with values in :
‘labeled_array‘ and only ‘num_features‘ will be returned by this function. :
See Also:
find_objects
generate a list of slices for the labeled features (or objects); useful for finding features’ position or dimensions
Examples
Create an image with some features, then label it using the default (cross-shaped) structuring element:
>>> a = array([[0,0,1,1,0,0],
...
[0,0,0,1,0,0],
...
[1,1,0,0,1,0],
...
[0,0,0,1,0,0]])
>>> labeled_array, num_features = label(a)
Each of the 4 features are labeled with a different integer:
>>> print num_features
4
>>> print labeled_array
array([[0, 0, 1, 1, 0, 0],
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[0, 0, 0, 1, 0, 0],
[2, 2, 0, 0, 3, 0],
[0, 0, 0, 4, 0, 0]])
Generate a structuring element that will consider features connected even if they touch diagonally:
>>> s = generate_binary_structure(2,2)
or,
>>> s = [[1,1,1],
[1,1,1],
[1,1,1]]
Label the image using the new structuring element:
>>> labeled_array, num_features = label(a, structure=s)
Show the 2 labeled features (note that features 1, 3, and 4 from above are now considered a single feature):
>>> print num_features
2
>>> print labeled_array
array([[0, 0, 1, 1, 0, 0],
[0, 0, 0, 1, 0, 0],
[2, 2, 0, 0, 1, 0],
[0, 0, 0, 1, 0, 0]])
maximum(input, labels=None, index=None)
Calculate the maximum of the values of an array over labeled regions.
Parameters
input : array_like
Array-like of values. For each region specified by labels, the maximal values of
input over the region is computed.
labels : array_like, optional
An array of integers marking different regions over which the maximum value of
input is to be computed. labels must have the same shape as input. If labels is not
specified, the maximum over the whole array is returned.
index : array_like, optional
A list of region labels that are taken into account for computing the maxima. If index
is None, the maximum over all elements where labels is non-zero is returned.
Returns
output : float or list of floats
List of maxima of input over the regions determined by labels and whose index is
in index. If index or labels are not specified, a float is returned: the maximal value
of input if labels is None, and the maximal value of elements where labels is greater
than zero if index is None.
See Also:
label, minimum, maximum_position, extrema, sum, mean, variance, standard_deviation
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Notes
The function returns a Python list and not a Numpy array, use np.array to convert the list to an array.
Examples
>>> a = np.arange(16).reshape((4,4))
>>> a
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
>>> labels = np.zeros_like(a)
>>> labels[:2,:2] = 1
>>> labels[2:, 1:3] = 2
>>> labels
array([[1, 1, 0, 0],
[1, 1, 0, 0],
[0, 2, 2, 0],
[0, 2, 2, 0]])
>>> from scipy import ndimage
>>> ndimage.maximum(a)
15.0
>>> ndimage.maximum(a, labels=labels, index=[1,2])
[5.0, 14.0]
>>> ndimage.maximum(a, labels=labels)
14.0
>>> b = np.array([[1, 2, 0, 0],
[5, 3, 0, 4],
[0, 0, 0, 7],
[9, 3, 0, 0]])
>>> labels, labels_nb = ndimage.label(b)
>>> labels
array([[1, 1, 0, 0],
[1, 1, 0, 2],
[0, 0, 0, 2],
[3, 3, 0, 0]])
>>> ndimage.maximum(b, labels=labels, index=np.arange(1, labels_nb + 1))
[5.0, 7.0, 9.0]
maximum_position(input, labels=None, index=None)
Find the positions of the maximums of the values of an array at labels.
Labels must be None or an array of the same dimensions as the input.
Index must be None, a single label or sequence of labels. If none, all values where label is greater than zero are
used.
mean(input, labels=None, index=None)
Calculate the mean of the values of an array at labels.
Labels must be None or an array that can be broadcast to the input.
Index must be None, a single label or sequence of labels. If None, the mean for all values where label is greater
than 0 is calculated.
minimum(input, labels=None, index=None)
Calculate the minimum of the values of an array over labeled regions.
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Parameters
input: array-like :
Array-like of values. For each region specified by labels, the minimal values of input
over the region is computed.
labels: array-like, optional :
An array-like of integers marking different regions over which the minimum value
of input is to be computed. labels must have the same shape as input. If labels is not
specified, the minimum over the whole array is returned.
index: array-like, optional :
A list of region labels that are taken into account for computing the minima. If index
is None, the minimum over all elements where labels is non-zero is returned.
Returns
output : float or list of floats
List of minima of input over the regions determined by labels and whose index is
in index. If index or labels are not specified, a float is returned: the minimal value
of input if labels is None, and the minimal value of elements where labels is greater
than zero if index is None.
See Also:
label, maximum, minimum_position, extrema, sum, mean, variance, standard_deviation
Notes
The function returns a Python list and not a Numpy array, use np.array to convert the list to an array.
Examples
>>> a = np.array([[1, 2, 0, 0],
...
[5, 3, 0, 4],
...
[0, 0, 0, 7],
...
[9, 3, 0, 0]])
>>> labels, labels_nb = ndimage.label(a)
>>> labels
array([[1, 1, 0, 0],
[1, 1, 0, 2],
[0, 0, 0, 2],
[3, 3, 0, 0]])
>>> ndimage.minimum(a, labels=labels, index=np.arange(1, labels_nb + 1))
[1.0, 4.0, 3.0]
>>> ndimage.minimum(a)
0.0
>>> ndimage.minimum(a, labels=labels)
1.0
minimum_position(input, labels=None, index=None)
Find the positions of the minimums of the values of an array at labels.
Labels must be None or an array of the same dimensions as the input.
Index must be None, a single label or sequence of labels. If none, all values where label is greater than zero are
used.
standard_deviation(input, labels=None, index=None)
Calculate the standard deviation of the values of an array at labels.
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Labels must be None or an array of the same dimensions as the input.
Index must be None, a single label or sequence of labels. If none, all values where label is greater than zero are
used.
sum(input, labels=None, index=None)
Calculate the sum of the values of the array.
Parameters
input : array_like
Values of input inside the regions defined by labels are summed together.
labels : array of integers, same shape as input
Assign labels to the values of the array.
index : scalar or array
A single label number or a sequence of label numbers of the objects to be measured.
Returns
output : list
A list of the sums of the values of input inside the regions defined by labels.
See Also:
mean
Examples
>>> input = [0,1,2,3]
>>> labels = [1,1,2,2]
>>> sum(input, labels, index=[1,2])
[1.0, 5.0]
variance(input, labels=None, index=None)
Calculate the variance of the values of an array at labels.
Labels must be None or an array of the same dimensions as the input.
Index must be None, a single label or sequence of labels. If none, all values where label is greater than zero are
used.
watershed_ift(input, markers, structure=None, output=None)
Apply watershed from markers using a iterative forest transform algorithm.
Negative markers are considered background markers which are processed after the other markers. A structuring
element defining the connectivity of the object can be provided. If none is provided an element is generated iwth
a squared connecitiviy equal to one. An output array can optionally be provided.
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binary_closing (input[, strucMulti-dimensional binary closing with the given structuring element.
ture, iterations, ...])
binary_dilation (input[, strucMulti-dimensional binary dilation with the given structuring element.
ture, iterations, ...])
binary_erosion (input[, strucMulti-dimensional binary erosion with a given structuring element.
ture, iterations, ...])
binary_fill_holes (input[, strucFill the holes in binary objects.
ture, output, ...])
binary_hit_or_miss (input[, struc- Multi-dimensional binary hit-or-miss transform.
ture1, structure2, ...])
binary_opening (input[, strucMulti-dimensional binary opening with the given structuring element.
ture, iterations, ...])
binary_propagation (input[, struc- Multi-dimensional binary propagation with the given structuring
ture, mask, ...])
element.
black_tophat (input[, size, footMulti-dimensional black tophat filter.
print, ...])
distance_transform_bf (inDistance transform function by a brute force algorithm.
put[, metric, sampling, ...])
distance_transform_cdt (inDistance transform for chamfer type of transforms.
put[, metric, return_distances, ...])
distance_transform_edt (inExact euclidean distance transform.
put[, sampling, return_distances, ...])
generate_binary_structure (rank, conGenerate a binary structure for binary morphological operations.
nectivity)
grey_closing (input[, size, footMulti-dimensional greyscale closing.
print, ...])
grey_dilation (input[, size, footCalculate a greyscale dilation, using either a structuring element, or a
print, ...])
footprint corresponding to a flat structuring element.
grey_erosion (input[, size, footCalculate a greyscale erosion, using either a structuring element, or a
print, ...])
footprint corresponding to a flat structuring element.
grey_opening (input[, size, footMulti-dimensional greyscale opening.
print, ...])
iterate_structure (structure, iter- Iterate a structure by dilating it with itself.
ations[, origin])
morphological_gradient (inMulti-dimensional morphological gradient.
put[, size, footprint, ...])
morphological_laplace (inMulti-dimensional morphological laplace.
put[, size, footprint, ...])
white_tophat (input[, size, footMulti-dimensional white tophat filter.
print, ...])
binary_closing(input, structure=None, iterations=1, output=None, origin=0)
Multi-dimensional binary closing with the given structuring element.
The closing of an input image by a structuring element is the erosion of the dilation of the image by the structuring element.
Parameters
input : array_like
Binary array_like to be closed. Non-zero (True) elements form the subset to be
closed.
structure : array_like, optional
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Structuring element used for the closing. Non-zero elements are considered True. If
no structuring element is provided an element is generated with a square connectivity
equal to one (i.e., only nearest neighbors are connected to the center, diagonallyconnected elements are not considered neighbors).
iterations : {int, float}, optional
The dilation step of the closing, then the erosion step are each repeated iterations
times (one, by default). If iterations is less than 1, each operations is repeated until
the result does not change anymore.
output : ndarray, optional
Array of the same shape as input, into which the output is placed. By default, a new
array is created.
origin : int or tuple of ints, optional
Placement of the filter, by default 0.
Returns
out : ndarray of bools
Closing of the input by the structuring element.
See Also:
grey_closing,
binary_opening,
generate_binary_structure
binary_dilation,
binary_erosion,
Notes
Closing [R15] is a mathematical morphology operation [R16] that consists in the succession of a dilation and an
erosion of the input with the same structuring element. Closing therefore fills holes smaller than the structuring
element.
Together with opening (binary_opening), closing can be used for noise removal.
References
[R15], [R16]
Examples
>>> a = np.zeros((5,5), dtype=np.int)
>>> a[1:-1, 1:-1] = 1; a[2,2] = 0
>>> a
array([[0, 0, 0, 0, 0],
[0, 1, 1, 1, 0],
[0, 1, 0, 1, 0],
[0, 1, 1, 1, 0],
[0, 0, 0, 0, 0]])
>>> # Closing removes small holes
>>> ndimage.binary_closing(a).astype(np.int)
array([[0, 0, 0, 0, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 0, 0, 0, 0]])
>>> # Closing is the erosion of the dilation of the input
>>> ndimage.binary_dilation(a).astype(np.int)
array([[0, 1, 1, 1, 0],
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[1, 1, 1, 1, 1],
[1, 1, 1, 1, 1],
[1, 1, 1, 1, 1],
[0, 1, 1, 1, 0]])
>>> ndimage.binary_erosion(ndimage.binary_dilation(a)).astype(np.int)
array([[0, 0, 0, 0, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 0, 0, 0, 0]])
>>> a = np.zeros((7,7), dtype=np.int)
>>> a[1:6, 2:5] = 1; a[1:3,3] = 0
>>> a
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 1, 0, 0],
[0, 0, 1, 0, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> # In addition to removing holes, closing can also
>>> # coarsen boundaries with fine hollows.
>>> ndimage.binary_closing(a).astype(np.int)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> ndimage.binary_closing(a, structure=np.ones((2,2))).astype(np.int)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
binary_dilation(input, structure=None, iterations=1, mask=None, output=None, border_value=0, origin=0,
brute_force=False)
Multi-dimensional binary dilation with the given structuring element.
Parameters
input : array_like
Binary array_like to be dilated. Non-zero (True) elements form the subset to be
dilated.
structure : array_like, optional
Structuring element used for the dilation. Non-zero elements are considered True. If
no structuring element is provided an element is generated with a square connectivity
equal to one.
iterations : {int, float}, optional
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The dilation is repeated iterations times (one, by default). If iterations is less than 1,
the dilation is repeated until the result does not change anymore.
mask : array_like, optional
If a mask is given, only those elements with a True value at the corresponding mask
element are modified at each iteration.
output : ndarray, optional
Array of the same shape as input, into which the output is placed. By default, a new
array is created.
origin : int or tuple of ints, optional
Placement of the filter, by default 0.
border_value : int (cast to 0 or 1)
Value at the border in the output array.
Returns
out : ndarray of bools
Dilation of the input by the structuring element.
See Also:
grey_dilation,
binary_erosion,
generate_binary_structure
binary_closing,
binary_opening,
Notes
Dilation [R17] is a mathematical morphology operation [R18] that uses a structuring element for expanding the
shapes in an image. The binary dilation of an image by a structuring element is the locus of the points covered
by the structuring element, when its center lies within the non-zero points of the image.
References
[R17], [R18]
Examples
>>> a = np.zeros((5, 5))
>>> a[2, 2] = 1
>>> a
array([[ 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0.],
[ 0., 0., 1., 0., 0.],
[ 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0.]])
>>> ndimage.binary_dilation(a)
array([[False, False, False, False, False],
[False, False, True, False, False],
[False, True, True, True, False],
[False, False, True, False, False],
[False, False, False, False, False]], dtype=bool)
>>> ndimage.binary_dilation(a).astype(a.dtype)
array([[ 0., 0., 0., 0., 0.],
[ 0., 0., 1., 0., 0.],
[ 0., 1., 1., 1., 0.],
[ 0., 0., 1., 0., 0.],
[ 0., 0., 0., 0., 0.]])
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>>> # 3x3 structuring element with connectivity 1, used by default
>>> struct1 = ndimage.generate_binary_structure(2, 1)
>>> struct1
array([[False, True, False],
[ True, True, True],
[False, True, False]], dtype=bool)
>>> # 3x3 structuring element with connectivity 2
>>> struct2 = ndimage.generate_binary_structure(2, 2)
>>> struct2
array([[ True, True, True],
[ True, True, True],
[ True, True, True]], dtype=bool)
>>> ndimage.binary_dilation(a, structure=struct1).astype(a.dtype)
array([[ 0., 0., 0., 0., 0.],
[ 0., 0., 1., 0., 0.],
[ 0., 1., 1., 1., 0.],
[ 0., 0., 1., 0., 0.],
[ 0., 0., 0., 0., 0.]])
>>> ndimage.binary_dilation(a, structure=struct2).astype(a.dtype)
array([[ 0., 0., 0., 0., 0.],
[ 0., 1., 1., 1., 0.],
[ 0., 1., 1., 1., 0.],
[ 0., 1., 1., 1., 0.],
[ 0., 0., 0., 0., 0.]])
>>> ndimage.binary_dilation(a, structure=struct1,\
... iterations=2).astype(a.dtype)
array([[ 0., 0., 1., 0., 0.],
[ 0., 1., 1., 1., 0.],
[ 1., 1., 1., 1., 1.],
[ 0., 1., 1., 1., 0.],
[ 0., 0., 1., 0., 0.]])
binary_erosion(input, structure=None, iterations=1, mask=None, output=None, border_value=0, origin=0,
brute_force=False)
Multi-dimensional binary erosion with a given structuring element.
Binary erosion is a mathematical morphology operation used for image processing.
Parameters
input : array_like
Binary image to be eroded. Non-zero (True) elements form the subset to be eroded.
structure : array_like, optional
Structuring element used for the erosion. Non-zero elements are considered True. If
no structuring element is provided, an element is generated with a square connectivity equal to one.
iterations : {int, float}, optional
The erosion is repeated iterations times (one, by default). If iterations is less than 1,
the erosion is repeated until the result does not change anymore.
mask : array_like, optional
If a mask is given, only those elements with a True value at the corresponding mask
element are modified at each iteration.
output : ndarray, optional
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Array of the same shape as input, into which the output is placed. By default, a new
array is created.
origin: int or tuple of ints, optional :
Placement of the filter, by default 0.
border_value: int (cast to 0 or 1) :
Value at the border in the output array.
Returns
out: ndarray of bools :
Erosion of the input by the structuring element.
See Also:
grey_erosion,
binary_dilation,
generate_binary_structure
binary_closing,
binary_opening,
Notes
Erosion [R19] is a mathematical morphology operation [R20] that uses a structuring element for shrinking the
shapes in an image. The binary erosion of an image by a structuring element is the locus of the points where
a superimposition of the structuring element centered on the point is entirely contained in the set of non-zero
elements of the image.
References
[R19], [R20]
Examples
>>> a = np.zeros((7,7), dtype=np.int)
>>> a[1:6, 2:5] = 1
>>> a
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> ndimage.binary_erosion(a).astype(a.dtype)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> #Erosion removes objects smaller than the structure
>>> ndimage.binary_erosion(a, structure=np.ones((5,5))).astype(a.dtype)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
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binary_fill_holes(input, structure=None, output=None, origin=0)
Fill the holes in binary objects.
Parameters
input: array_like :
n-dimensional binary array with holes to be filled
structure: array_like, optional :
Structuring element used in the computation; large-size elements make computations
faster but may miss holes separated from the background by thin regions. The default
element (with a square connectivity equal to one) yields the intuitive result where all
holes in the input have been filled.
output: ndarray, optional :
Array of the same shape as input, into which the output is placed. By default, a new
array is created.
origin: int, tuple of ints, optional :
Position of the structuring element.
Returns
out: ndarray :
Transformation of the initial image input where holes have been filled.
See Also:
binary_dilation, binary_propagation, label
Notes
The algorithm used in this function consists in invading the complementary of the shapes in input from the outer
boundary of the image, using binary dilations. Holes are not connected to the boundary and are therefore not
invaded. The result is the complementary subset of the invaded region.
References
[R21]
Examples
>>> a = np.zeros((5, 5), dtype=int)
>>> a[1:4, 1:4] = 1
>>> a[2,2] = 0
>>> a
array([[0, 0, 0, 0, 0],
[0, 1, 1, 1, 0],
[0, 1, 0, 1, 0],
[0, 1, 1, 1, 0],
[0, 0, 0, 0, 0]])
>>> ndimage.binary_fill_holes(a).astype(int)
array([[0, 0, 0, 0, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 0, 0, 0, 0]])
>>> # Too big structuring element
>>> ndimage.binary_fill_holes(a, structure=np.ones((5,5))).astype(int)
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array([[0,
[0,
[0,
[0,
[0,
0,
1,
1,
1,
0,
0,
1,
0,
1,
0,
0,
1,
1,
1,
0,
0],
0],
0],
0],
0]])
binary_hit_or_miss(input, structure1=None, structure2=None, output=None, origin1=0, origin2=None)
Multi-dimensional binary hit-or-miss transform.
The hit-or-miss transform finds the locations of a given pattern inside the input image.
Parameters
input : array_like (cast to booleans)
Binary image where a pattern is to be detected.
structure1 : array_like (cast to booleans), optional
Part of the structuring element to be fitted to the foreground (non-zero elements) of
input. If no value is provided, a structure of square connectivity 1 is chosen.
structure2 : array_like (cast to booleans), optional
Second part of the structuring element that has to miss completely the foreground. If
no value is provided, the complementary of structure1 is taken.
output : ndarray, optional
Array of the same shape as input, into which the output is placed. By default, a new
array is created.
origin1 : int or tuple of ints, optional
Placement of the first part of the structuring element structure1, by default 0 for a
centered structure.
origin2 : int or tuple of ints, optional
Placement of the second part of the structuring element structure2, by default 0 for a
centered structure. If a value is provided for origin1 and not for origin2, then origin2
is set to origin1.
Returns
output : ndarray
Hit-or-miss transform of input with the given structuring element (structure1, structure2).
See Also:
ndimage.morphology, binary_erosion
References
[R22]
Examples
>>> a = np.zeros((7,7), dtype=np.int)
>>> a[1, 1] = 1; a[2:4, 2:4] = 1; a[4:6, 4:6] = 1
>>> a
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
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[0, 0, 1, 1, 0, 0, 0],
[0, 0, 1, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 1, 0],
[0, 0, 0, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> structure1 = np.array([[1, 0, 0], [0, 1, 1], [0, 1, 1]])
>>> structure1
array([[1, 0, 0],
[0, 1, 1],
[0, 1, 1]])
>>> # Find the matches of structure1 in the array a
>>> ndimage.binary_hit_or_miss(a, structure1=structure1).astype(np.int)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> # Change the origin of the filter
>>> # origin1=1 is equivalent to origin1=(1,1) here
>>> ndimage.binary_hit_or_miss(a, structure1=structure1,\
... origin1=1).astype(np.int)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0]])
binary_opening(input, structure=None, iterations=1, output=None, origin=0)
Multi-dimensional binary opening with the given structuring element.
The opening of an input image by a structuring element is the dilation of the erosion of the image by the
structuring element.
Parameters
input : array_like
Binary array_like to be opened. Non-zero (True) elements form the subset to be
opened.
structure : array_like, optional
Structuring element used for the opening. Non-zero elements are considered True. If
no structuring element is provided an element is generated with a square connectivity
equal to one (i.e., only nearest neighbors are connected to the center, diagonallyconnected elements are not considered neighbors).
iterations : {int, float}, optional
The erosion step of the opening, then the dilation step are each repeated iterations
times (one, by default). If iterations is less than 1, each operation is repeated until
the result does not change anymore.
output : ndarray, optional
Array of the same shape as input, into which the output is placed. By default, a new
array is created.
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origin : int or tuple of ints, optional
Placement of the filter, by default 0.
Returns
out : ndarray of bools
Opening of the input by the structuring element.
See Also:
grey_opening,
binary_closing,
generate_binary_structure
binary_erosion,
binary_dilation,
Notes
Opening [R23] is a mathematical morphology operation [R24] that consists in the succession of an erosion and
a dilation of the input with the same structuring element. Opening therefore removes objects smaller than the
structuring element.
Together with closing (binary_closing), opening can be used for noise removal.
References
[R23], [R24]
Examples
>>> a = np.zeros((5,5), dtype=np.int)
>>> a[1:4, 1:4] = 1; a[4, 4] = 1
>>> a
array([[0, 0, 0, 0, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 0, 0, 0, 1]])
>>> # Opening removes small objects
>>> ndimage.binary_opening(a, structure=np.ones((3,3))).astype(np.int)
array([[0, 0, 0, 0, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 1, 0],
[0, 0, 0, 0, 0]])
>>> # Opening can also smooth corners
>>> ndimage.binary_opening(a).astype(np.int)
array([[0, 0, 0, 0, 0],
[0, 0, 1, 0, 0],
[0, 1, 1, 1, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 0, 0]])
>>> # Opening is the dilation of the erosion of the input
>>> ndimage.binary_erosion(a).astype(np.int)
array([[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]])
>>> ndimage.binary_dilation(ndimage.binary_erosion(a)).astype(np.int)
array([[0, 0, 0, 0, 0],
[0, 0, 1, 0, 0],
[0, 1, 1, 1, 0],
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[0, 0, 1, 0, 0],
[0, 0, 0, 0, 0]])
binary_propagation(input, structure=None, mask=None, output=None, border_value=0, origin=0)
Multi-dimensional binary propagation with the given structuring element.
Parameters
input : array_like
Binary image to be propagated inside mask.
structure : array_like
Structuring element used in the successive dilations. The output may depend on
the structuring element, especially if mask has several connex components. If no
structuring element is provided, an element is generated with a squared connectivity
equal to one.
mask : array_like
Binary mask defining the region into which input is allowed to propagate.
output : ndarray, optional
Array of the same shape as input, into which the output is placed. By default, a new
array is created.
origin : int or tuple of ints, optional
Placement of the filter, by default 0.
Returns
ouput : ndarray
Binary propagation of input inside mask.
Notes
This function is functionally equivalent to calling binary_dilation with the number of iterations less then one:
iterative dilation until the result does not change anymore.
The succession of an erosion and propagation inside the original image can be used instead of an opening for
deleting small objects while keeping the contours of larger objects untouched.
References
[R25], [R26]
Examples
>>> input = np.zeros((8, 8), dtype=np.int)
>>> input[2, 2] = 1
>>> mask = np.zeros((8, 8), dtype=np.int)
>>> mask[1:4, 1:4] = mask[4, 4] = mask[6:8, 6:8] = 1
>>> input
array([[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0]])
>>> mask
array([[0, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 1],
[0, 0, 0, 0, 0, 0, 1, 1]])
>>> ndimage.binary_propagation(input, mask=mask).astype(np.int)
array([[0, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0]])
>>> ndimage.binary_propagation(input, mask=mask,\
... structure=np.ones((3,3))).astype(np.int)
array([[0, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0]])
>>> # Comparison between opening and erosion+propagation
>>> a = np.zeros((6,6), dtype=np.int)
>>> a[2:5, 2:5] = 1; a[0, 0] = 1; a[5, 5] = 1
>>> a
array([[1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 1, 1, 1, 0],
[0, 0, 1, 1, 1, 0],
[0, 0, 1, 1, 1, 0],
[0, 0, 0, 0, 0, 1]])
>>> ndimage.binary_opening(a).astype(np.int)
array([[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 1, 1, 1, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0]])
>>> b = ndimage.binary_erosion(a)
>>> b.astype(int)
array([[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0]])
>>> ndimage.binary_propagation(b, mask=a).astype(np.int)
array([[0, 0, 0, 0, 0, 0],
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[0,
[0,
[0,
[0,
[0,
0,
0,
0,
0,
0,
0,
1,
1,
1,
0,
0,
1,
1,
1,
0,
0,
1,
1,
1,
0,
0],
0],
0],
0],
0]])
black_tophat(input, size=None, footprint=None, structure=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Multi-dimensional black tophat filter.
Either a size or a footprint, or the structure must be provided. An output array can optionally be provided. The
origin parameter controls the placement of the filter. The mode parameter determines how the array borders are
handled, where cval is the value when mode is equal to ‘constant’.
See Also:
grey_opening, grey_closing
References
[R27], [R28]
distance_transform_bf(input,
metric=’euclidean’,
sampling=None,
return_distances=True,
turn_indices=False, distances=None, indices=None)
Distance transform function by a brute force algorithm.
re-
This function calculates the distance transform of the input, by replacing each background element (zero values),
with its shortest distance to the foreground (any element non-zero). Three types of distance metric are supported:
‘euclidean’, ‘taxicab’ and ‘chessboard’.
In addition to the distance transform, the feature transform can be calculated. In this case the index of the closest
background element is returned along the first axis of the result.
The return_distances, and return_indices flags can be used to indicate if the distance transform, the feature
transform, or both must be returned.
Optionally the sampling along each axis can be given by the sampling parameter which should be a sequence of
length equal to the input rank, or a single number in which the sampling is assumed to be equal along all axes.
This parameter is only used in the case of the euclidean distance transform.
This function employs a slow brute force algorithm, see also the function distance_transform_cdt for more
efficient taxicab and chessboard algorithms.
the distances and indices arguments can be used to give optional output arrays that must be of the correct size
and type (float64 and int32).
distance_transform_cdt(input, metric=’chessboard’, return_distances=True, return_indices=False, distances=None, indices=None)
Distance transform for chamfer type of transforms.
The metric determines the type of chamfering that is done. If the metric is equal to ‘taxicab’ a structure is generated using generate_binary_structure with a squared distance equal to 1. If the metric is equal to ‘chessboard’,
a metric is generated using generate_binary_structure with a squared distance equal to the rank of the array.
These choices correspond to the common interpretations of the taxicab and the chessboard distance metrics in
two dimensions.
In addition to the distance transform, the feature transform can be calculated. In this case the index of the closest
background element is returned along the first axis of the result.
The return_distances, and return_indices flags can be used to indicate if the distance transform, the feature
transform, or both must be returned.
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The distances and indices arguments can be used to give optional output arrays that must be of the correct size
and type (both int32).
distance_transform_edt(input, sampling=None, return_distances=True,
tances=None, indices=None)
Exact euclidean distance transform.
return_indices=False,
dis-
In addition to the distance transform, the feature transform can be calculated. In this case the index of the closest
background element is returned along the first axis of the result.
The return_distances, and return_indices flags can be used to indicate if the distance transform, the feature
transform, or both must be returned.
Optionally the sampling along each axis can be given by the sampling parameter which should be a sequence of
length equal to the input rank, or a single number in which the sampling is assumed to be equal along all axes.
the distances and indices arguments can be used to give optional output arrays that must be of the correct size
and type (float64 and int32).
generate_binary_structure(rank, connectivity)
Generate a binary structure for binary morphological operations.
Parameters
rank : int
Number of dimensions of the array to which the structuring element will be applied,
as returned by np.ndim.
connectivity : int
connectivity determines which elements of the output array belong to the structure,
i.e. are considered as neighbors of the central element. Elements up to a squared
distance of connectivity from the center are considered neighbors. connectivity may
range from 1 (no diagonal elements are neighbors) to rank (all elements are neighbors).
Returns
output : ndarray of bools
Structuring element which may be used for binary morphological operations, with
rank dimensions and all dimensions equal to 3.
See Also:
iterate_structure, binary_dilation, binary_erosion
Notes
generate_binary_structure can only create structuring elements with dimensions equal to 3, i.e. minimal dimensions. For larger structuring elements, that are useful e.g. for eroding large objects, one may either use
iterate_structure, or create directly custom arrays with numpy functions such as numpy.ones.
Examples
>>> struct = ndimage.generate_binary_structure(2, 1)
>>> struct
array([[False, True, False],
[ True, True, True],
[False, True, False]], dtype=bool)
>>> a = np.zeros((5,5))
>>> a[2, 2] = 1
>>> a
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array([[ 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0.],
[ 0., 0., 1., 0., 0.],
[ 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0.]])
>>> b = ndimage.binary_dilation(a, structure=struct).astype(a.dtype)
>>> b
array([[ 0., 0., 0., 0., 0.],
[ 0., 0., 1., 0., 0.],
[ 0., 1., 1., 1., 0.],
[ 0., 0., 1., 0., 0.],
[ 0., 0., 0., 0., 0.]])
>>> ndimage.binary_dilation(b, structure=struct).astype(a.dtype)
array([[ 0., 0., 1., 0., 0.],
[ 0., 1., 1., 1., 0.],
[ 1., 1., 1., 1., 1.],
[ 0., 1., 1., 1., 0.],
[ 0., 0., 1., 0., 0.]])
>>> struct = ndimage.generate_binary_structure(2, 2)
>>> struct
array([[ True, True, True],
[ True, True, True],
[ True, True, True]], dtype=bool)
>>> struct = ndimage.generate_binary_structure(3, 1)
>>> struct # no diagonal elements
array([[[False, False, False],
[False, True, False],
[False, False, False]],
[[False, True, False],
[ True, True, True],
[False, True, False]],
[[False, False, False],
[False, True, False],
[False, False, False]]], dtype=bool)
grey_closing(input, size=None, footprint=None, structure=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Multi-dimensional greyscale closing.
A greyscale closing consists in the succession of a greyscale dilation, and a greyscale erosion.
Parameters
input : array_like
Array over which the grayscale closing is to be computed.
size : tuple of ints
Shape of a flat and full structuring element used for the grayscale closing. Optional
if footprint is provided.
footprint : array of ints, optional
Positions of non-infinite elements of a flat structuring element used for the grayscale
closing.
structure : array of ints, optional
Structuring element used for the grayscale closing. structure may be a non-flat structuring element.
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output : array, optional
An array used for storing the ouput of the closing may be provided.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is the
value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0.
origin : scalar, optional
The origin parameter controls the placement of the filter. Default 0
Returns
output : ndarray
Result of the grayscale closing of input with structure.
See Also:
binary_closing,
grey_dilation,
generate_binary_structure
grey_erosion,
grey_opening,
Notes
The action of a grayscale closing with a flat structuring element amounts to smoothen deep local minima,
whereas binary closing fills small holes.
References
[R29]
Examples
>>> a = np.arange(36).reshape((6,6))
>>> a[3,3] = 0
>>> a
array([[ 0, 1, 2, 3, 4, 5],
[ 6, 7, 8, 9, 10, 11],
[12, 13, 14, 15, 16, 17],
[18, 19, 20, 0, 22, 23],
[24, 25, 26, 27, 28, 29],
[30, 31, 32, 33, 34, 35]])
>>> ndimage.grey_closing(a, size=(3,3))
array([[ 7, 7, 8, 9, 10, 11],
[ 7, 7, 8, 9, 10, 11],
[13, 13, 14, 15, 16, 17],
[19, 19, 20, 20, 22, 23],
[25, 25, 26, 27, 28, 29],
[31, 31, 32, 33, 34, 35]])
>>> # Note that the local minimum a[3,3] has disappeared
grey_dilation(input, size=None, footprint=None, structure=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculate a greyscale dilation, using either a structuring element, or a footprint corresponding to a flat structuring
element.
Grayscale dilation is a mathematical morphology operation. For the simple case of a full and flat structuring
element, it can be viewed as a maximum filter over a sliding window.
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Parameters
input : array_like
Array over which the grayscale dilation is to be computed.
size : tuple of ints
Shape of a flat and full structuring element used for the grayscale dilation. Optional
if footprint is provided.
footprint : array of ints, optional
Positions of non-infinite elements of a flat structuring element used for the grayscale
dilation. Non-zero values give the set of neighbors of the center over which the
maximum is chosen.
structure : array of ints, optional
Structuring element used for the grayscale dilation. structure may be a non-flat structuring element.
output : array, optional
An array used for storing the ouput of the dilation may be provided.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is the
value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0.
origin : scalar, optional
The origin parameter controls the placement of the filter. Default 0
Returns
output : ndarray
Grayscale dilation of input.
See Also:
binary_dilation,
grey_erosion,
grey_closing,
generate_binary_structure, ndimage.maximum_filter
grey_opening,
Notes
The grayscale dilation of an image input by a structuring element s defined over a domain E is given by:
(input+s)(x) = max {input(y) + s(x-y), for y in E}
In particular, for structuring elements defined as s(y) = 0 for y in E, the grayscale dilation computes the maximum
of the input image inside a sliding window defined by E.
Grayscale dilation [R30] is a mathematical morphology operation [R31].
References
[R30], [R31]
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Examples
>>> a = np.zeros((7,7), dtype=np.int)
>>> a[2:5, 2:5] = 1
>>> a[4,4] = 2; a[2,3] = 3
>>> a
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 3, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> ndimage.grey_dilation(a, size=(3,3))
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 3, 3, 3, 1, 0],
[0, 1, 3, 3, 3, 1, 0],
[0, 1, 3, 3, 3, 2, 0],
[0, 1, 1, 2, 2, 2, 0],
[0, 1, 1, 2, 2, 2, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> ndimage.grey_dilation(a, footprint=np.ones((3,3)))
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 3, 3, 3, 1, 0],
[0, 1, 3, 3, 3, 1, 0],
[0, 1, 3, 3, 3, 2, 0],
[0, 1, 1, 2, 2, 2, 0],
[0, 1, 1, 2, 2, 2, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> s = ndimage.generate_binary_structure(2,1)
>>> s
array([[False, True, False],
[ True, True, True],
[False, True, False]], dtype=bool)
>>> ndimage.grey_dilation(a, footprint=s)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 3, 1, 0, 0],
[0, 1, 3, 3, 3, 1, 0],
[0, 1, 1, 3, 2, 1, 0],
[0, 1, 1, 2, 2, 2, 0],
[0, 0, 1, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> ndimage.grey_dilation(a, size=(3,3), structure=np.ones((3,3)))
array([[1, 1, 1, 1, 1, 1, 1],
[1, 2, 4, 4, 4, 2, 1],
[1, 2, 4, 4, 4, 2, 1],
[1, 2, 4, 4, 4, 3, 1],
[1, 2, 2, 3, 3, 3, 1],
[1, 2, 2, 3, 3, 3, 1],
[1, 1, 1, 1, 1, 1, 1]])
grey_erosion(input, size=None, footprint=None, structure=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Calculate a greyscale erosion, using either a structuring element, or a footprint corresponding to a flat structuring
element.
Grayscale erosion is a mathematical morphology operation. For the simple case of a full and flat structuring
element, it can be viewed as a minimum filter over a sliding window.
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Parameters
input : array_like
Array over which the grayscale erosion is to be computed.
size : tuple of ints
Shape of a flat and full structuring element used for the grayscale erosion. Optional
if footprint is provided.
footprint : array of ints, optional
Positions of non-infinite elements of a flat structuring element used for the grayscale
erosion. Non-zero values give the set of neighbors of the center over which the
minimum is chosen.
structure : array of ints, optional
Structuring element used for the grayscale erosion. structure may be a non-flat structuring element.
output : array, optional
An array used for storing the ouput of the erosion may be provided.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is the
value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0.
origin : scalar, optional
The origin parameter controls the placement of the filter. Default 0
Returns
output : ndarray
Grayscale erosion of input.
See Also:
binary_erosion,
grey_dilation,
grey_opening,
generate_binary_structure, ndimage.minimum_filter
grey_closing,
Notes
The grayscale erosion of an image input by a structuring element s defined over a domain E is given by:
(input+s)(x) = min {input(y) - s(x-y), for y in E}
In particular, for structuring elements defined as s(y) = 0 for y in E, the grayscale erosion computes the minimum
of the input image inside a sliding window defined by E.
Grayscale erosion [R32] is a mathematical morphology operation [R33].
References
[R32], [R33]
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Examples
>>> a = np.zeros((7,7), dtype=np.int)
>>> a[1:6, 1:6] = 3
>>> a[4,4] = 2; a[2,3] = 1
>>> a
array([[0, 0, 0, 0, 0, 0, 0],
[0, 3, 3, 3, 3, 3, 0],
[0, 3, 3, 1, 3, 3, 0],
[0, 3, 3, 3, 3, 3, 0],
[0, 3, 3, 3, 2, 3, 0],
[0, 3, 3, 3, 3, 3, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> ndimage.grey_erosion(a, size=(3,3))
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 3, 2, 2, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> footprint = ndimage.generate_binary_structure(2, 1)
>>> footprint
array([[False, True, False],
[ True, True, True],
[False, True, False]], dtype=bool)
>>> # Diagonally-connected elements are not considered neighbors
>>> ndimage.grey_erosion(a, size=(3,3), footprint=footprint)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 3, 1, 2, 0, 0],
[0, 0, 3, 2, 2, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
grey_opening(input, size=None, footprint=None, structure=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Multi-dimensional greyscale opening.
A greyscale opening consists in the succession of a greyscale erosion, and a greyscale dilation.
Parameters
input : array_like
Array over which the grayscale opening is to be computed.
size : tuple of ints
Shape of a flat and full structuring element used for the grayscale opening. Optional
if footprint is provided.
footprint : array of ints, optional
Positions of non-infinite elements of a flat structuring element used for the grayscale
opening.
structure : array of ints, optional
Structuring element used for the grayscale opening. structure may be a non-flat
structuring element.
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output : array, optional
An array used for storing the ouput of the opening may be provided.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is the
value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0.
origin : scalar, optional
The origin parameter controls the placement of the filter. Default 0
Returns
output : ndarray
Result of the grayscale opening of input with structure.
See Also:
binary_opening,
grey_dilation,
generate_binary_structure
grey_erosion,
grey_closing,
Notes
The action of a grayscale opening with a flat structuring element amounts to smoothen high local maxima,
whereas binary opening erases small objects.
References
[R34]
Examples
>>> a = np.arange(36).reshape((6,6))
>>> a[3, 3] = 50
>>> a
array([[ 0, 1, 2, 3, 4, 5],
[ 6, 7, 8, 9, 10, 11],
[12, 13, 14, 15, 16, 17],
[18, 19, 20, 50, 22, 23],
[24, 25, 26, 27, 28, 29],
[30, 31, 32, 33, 34, 35]])
>>> ndimage.grey_opening(a, size=(3,3))
array([[ 0, 1, 2, 3, 4, 4],
[ 6, 7, 8, 9, 10, 10],
[12, 13, 14, 15, 16, 16],
[18, 19, 20, 22, 22, 22],
[24, 25, 26, 27, 28, 28],
[24, 25, 26, 27, 28, 28]])
>>> # Note that the local maximum a[3,3] has disappeared
iterate_structure(structure, iterations, origin=None)
Iterate a structure by dilating it with itself.
Parameters
structure : array_like
Structuring element (an array of bools, for example), to be dilated with itself.
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iterations : int
number of dilations performed on the structure with itself
origin : optional
If origin is None, only the iterated structure is returned. If not, a tuple of the iterated
structure and the modified origin is returned.
Returns
output: ndarray of bools :
A new structuring element obtained by dilating structure (iterations - 1) times with
itself.
See Also:
generate_binary_structure
Examples
>>> struct = ndimage.generate_binary_structure(2, 1)
>>> struct.astype(int)
array([[0, 1, 0],
[1, 1, 1],
[0, 1, 0]])
>>> ndimage.iterate_structure(struct, 2).astype(int)
array([[0, 0, 1, 0, 0],
[0, 1, 1, 1, 0],
[1, 1, 1, 1, 1],
[0, 1, 1, 1, 0],
[0, 0, 1, 0, 0]])
>>> ndimage.iterate_structure(struct, 3).astype(int)
array([[0, 0, 0, 1, 0, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 1, 1, 1, 1, 1, 0],
[1, 1, 1, 1, 1, 1, 1],
[0, 1, 1, 1, 1, 1, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 0, 1, 0, 0, 0]])
morphological_gradient(input, size=None, footprint=None, structure=None, output=None, mode=’reflect’,
cval=0.0, origin=0)
Multi-dimensional morphological gradient.
The morphological gradient is calculated as the difference between a dilation and an erosion of the input with a
given structuring element.
Parameters
input : array_like
Array over which to compute the morphlogical gradient.
size : tuple of ints
Shape of a flat and full structuring element used for the mathematical morphology
operations. Optional if footprint is provided. A larger size yields a more blurred
gradient.
footprint : array of ints, optional
Positions of non-infinite elements of a flat structuring element used for the morphology operations. Larger footprints give a more blurred morphological gradient.
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structure : array of ints, optional
Structuring element used for the morphology operations. structure may be a non-flat
structuring element.
output : array, optional
An array used for storing the ouput of the morphological gradient may be provided.
mode : {‘reflect’,’constant’,’nearest’,’mirror’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is the
value when mode is equal to ‘constant’. Default is ‘reflect’
cval : scalar, optional
Value to fill past edges of input if mode is ‘constant’. Default is 0.0.
origin : scalar, optional
The origin parameter controls the placement of the filter. Default 0
Returns
output : ndarray
Morphological gradient of input.
See Also:
grey_dilation, grey_erosion, ndimage.gaussian_gradient_magnitude
Notes
For a flat structuring element, the morphological gradient computed at a given point corresponds to the maximal
difference between elements of the input among the elements covered by the structuring element centered on
the point.
References
[R35]
Examples
>>> a = np.zeros((7,7), dtype=np.int)
>>> a[2:5, 2:5] = 1
>>> ndimage.morphological_gradient(a, size=(3,3))
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 0, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> # The morphological gradient is computed as the difference
>>> # between a dilation and an erosion
>>> ndimage.grey_dilation(a, size=(3,3)) -\
... ndimage.grey_erosion(a, size=(3,3))
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 0, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 0],
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[0, 0, 0, 0, 0, 0, 0]])
>>> a = np.zeros((7,7), dtype=np.int)
>>> a[2:5, 2:5] = 1
>>> a[4,4] = 2; a[2,3] = 3
>>> a
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 3, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> ndimage.morphological_gradient(a, size=(3,3))
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 3, 3, 3, 1, 0],
[0, 1, 3, 3, 3, 1, 0],
[0, 1, 3, 2, 3, 2, 0],
[0, 1, 1, 2, 2, 2, 0],
[0, 1, 1, 2, 2, 2, 0],
[0, 0, 0, 0, 0, 0, 0]])
morphological_laplace(input, size=None, footprint=None, structure=None, output=None, mode=’reflect’,
cval=0.0, origin=0)
Multi-dimensional morphological laplace.
Either a size or a footprint, or the structure must be provided. An output array can optionally be provided. The
origin parameter controls the placement of the filter. The mode parameter determines how the array borders are
handled, where cval is the value when mode is equal to ‘constant’.
white_tophat(input, size=None, footprint=None, structure=None, output=None, mode=’reflect’, cval=0.0, origin=0)
Multi-dimensional white tophat filter.
Either a size or a footprint, or the structure must be provided. An output array can optionally be provided. The
origin parameter controls the placement of the filter. The mode parameter determines how the array borders are
handled, where cval is the value when mode is equal to ‘constant’.
3.11 Orthogonal distance regression (scipy.odr)
Orthogonal Distance Regression
3.11.1 Introduction
Why Orthogonal Distance Regression (ODR)? Sometimes one has measurement errors in the explanatory variable,
not just the response variable. Ordinary Least Squares (OLS) fitting procedures treat the data for explanatory variables
as fixed. Furthermore, OLS procedures require that the response variable be an explicit function of the explanatory
variables; sometimes making the equation explicit is unwieldy and introduces errors. ODR can handle both of these
cases with ease and can even reduce to the OLS case if necessary.
ODRPACK is a FORTRAN-77 library for performing ODR with possibly non-linear fitting functions. It uses a modified trust-region Levenberg-Marquardt-type algorithm to estimate the function parameters. The fitting functions are
provided by Python functions operating on NumPy arrays. The required derivatives may be provided by Python functions as well or may be numerically estimated. ODRPACK can do explicit or implicit ODR fits or can do OLS. Input
and output variables may be multi-dimensional. Weights can be provided to account for different variances of the
observations (even covariances between dimensions of the variables).
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odr provides two interfaces: a single function and a set of high-level classes that wrap that function. Please refer to
their docstrings for more information. While the docstring of the function, odr, does not have a full explanation of its
arguments, the classes do, and the arguments with the same name usually have the same requirements. Furthermore,
it is highly suggested that one at least skim the ODRPACK User’s Guide. Know Thy Algorithm.
3.11.2 Use
See the docstrings of odr.odrpack and the functions and classes for usage instructions. The ODRPACK User’s Guide
is also quite helpful. It can be found on one of the ODRPACK’s original author’s website:
http://www.boulder.nist.gov/mcsd/Staff/JRogers/odrpack.html
Robert Kern [email protected]
class Data(x, y=None, we=None, wd=None, fix=None, meta={})
The Data class stores the data to fit.
Each argument is attached to the member of the instance of the same name. The structures of x and y are
described in the Model class docstring. If y is an integer, then the Data instance can only be used to fit with
implicit models where the dimensionality of the response is equal to the specified value of y. The structures of
wd and we are described below. meta is an freeform dictionary for application-specific use.
we weights the effect a deviation in the response variable has on the fit. wd weights the effect a deviation
in the input variable has on the fit. To handle multidimensional inputs and responses easily, the structure of
these arguments has the n’th dimensional axis first. These arguments heavily use the structured arguments
feature of ODRPACK to conveniently and flexibly support all options. See the ODRPACK User’s Guide for a
full explanation of how these weights are used in the algorithm. Basically, a higher value of the weight for a
particular data point makes a deviation at that point more detrimental to the fit.
we – if we is a scalar, then that value is used for all data points (and
all dimensions of the response variable).
If we is a rank-1 array of length q (the dimensionality of the response variable), then this
vector is the diagonal of the covariant weighting matrix for all data points.
If we is a rank-1 array of length n (the number of data points), then the i’th element is the
weight for the i’th response variable observation (single-dimensional only).
If we is a rank-2 array of shape (q, q), then this is the full covariant weighting matrix broadcast
to each observation.
If we is a rank-2 array of shape (q, n), then we[:,i] is the diagonal of the covariant weighting
matrix for the i’th observation.
If we is a rank-3 array of shape (q, q, n), then we[:,:,i] is the full specification of the covariant
weighting matrix for each observation.
If the fit is implicit, then only a positive scalar value is used.
wd – if wd is a scalar, then that value is used for all data points
(and all dimensions of the input variable). If wd = 0, then the covariant weighting matrix for
each observation is set to the identity matrix (so each dimension of each observation has the
same weight).
If wd is a rank-1 array of length m (the dimensionality of the input variable), then this vector
is the diagonal of the covariant weighting matrix for all data points.
If wd is a rank-1 array of length n (the number of data points), then the i’th element is the
weight for the i’th input variable observation (single-dimensional only).
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If wd is a rank-2 array of shape (m, m), then this is the full covariant weighting matrix broadcast to each observation.
If wd is a rank-2 array of shape (m, n), then wd[:,i] is the diagonal of the covariant weighting
matrix for the i’th observation.
If wd is a rank-3 array of shape (m, m, n), then wd[:,:,i] is the full specification of the covariant
weighting matrix for each observation.
fix – fix is the same as ifixx in the class ODR. It is an array of integers
with the same shape as data.x that determines which input observations are treated as fixed.
One can use a sequence of length m (the dimensionality of the input observations) to fix some
dimensions for all observations. A value of 0 fixes the observation, a value > 0 makes it free.
meta – optional, freeform dictionary for metadata
Methods
set_meta(**kwds)
Update the metadata dictionary with the keywords and data provided by keywords.
class Model(fcn, fjacb=None, fjacd=None, extra_args=None, estimate=None, implicit=0, meta=None)
The Model class stores information about the function you wish to fit.
It stores the function itself, at the least, and optionally stores functions which compute the Jacobians used
during fitting. Also, one can provide a function that will provide reasonable starting values for the fit parameters
possibly given the set of data.
The initialization method stores these into members of the same name.
fcn – fit function: fcn(beta, x) –> y
fjacb – Jacobian of fcn wrt the fit parameters beta:
fjacb(beta, x) –> @f_i(x,B)/@B_j
fjacd – Jacobian of fcn wrt the (possibly multidimensional) input variable:
fjacd(beta, x) –> @f_i(x,B)/@x_j
extra_args – if specified, extra_args should be a tuple of extra
arguments to pass to fcn, fjacb, and fjacd. Each will be called like the following: apply(fcn,
(beta, x) + extra_args)
estimate – provide estimates of the fit parameters from the data:
estimate(data) –> estbeta
implicit – boolean variable which, if TRUE, specifies that the model
is implicit; i.e fcn(beta, x) ~= 0 and there is no y data to fit against.
meta – an optional, freeform dictionary of metadata for the model
Note that the fcn, fjacb, and fjacd operate on NumPy arrays and return a NumPy array. estimate takes an instance
of the Data class.
Here are the rules for the shapes of the argument and return arrays:
x – if the input data is single-dimensional, then x is rank-1
array; i.e. x = array([1, 2, 3, ...]); x.shape = (n,) If the input data is multi-dimensional, then x
is a rank-2 array; i.e. x = array([[1, 2, ...], [2, 4, ...]]); x.shape = (m, n) In all cases, it has the
same shape as the input data array passed to odr(). m is the dimensionality of the input data,
n is the number of observations.
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y – if the response variable is single-dimensional, then y is a rank-1
array; i.e. y = array([2, 4, ...]); y.shape = (n,) If the response variable is multi-dimensional,
then y is a rank-2 array; i.e. y = array([[2, 4, ...], [3, 6, ...]]); y.shape = (q, n) where q is the
dimensionality of the response variable.
beta – rank-1 array of length p where p is the number of parameters;
i.e. beta = array([B_1, B_2, ..., B_p])
fjacb – if the response variable is multi-dimensional, then the return
array’s shape is (q, p, n) such that fjacb(x,beta)[l,k,i] = @f_l(X,B)/@B_k evaluated at the i’th
data point. If q == 1, then the return array is only rank-2 and with shape (p, n).
fjacd – as with fjacb, only the return array’s shape is (q, m, n) such that
fjacd(x,beta)[l,j,i] = @f_l(X,B)/@X_j at the i’th data point. If q == 1, then the return array’s
shape is (m, n). If m == 1, the shape is (q, n). If m == q == 1, the shape is (n,).
Methods
set_meta(**kwds)
Update the metadata dictionary with the keywords and data provided here.
class ODR(data, model, beta0=None, delta0=None, ifixb=None, ifixx=None, job=None, iprint=None, errfile=None,
rptfile=None, ndigit=None, taufac=None, sstol=None, partol=None, maxit=None, stpb=None,
stpd=None, sclb=None, scld=None, work=None, iwork=None)
The ODR class gathers all information and coordinates the running of the main fitting routine.
Members of instances of the ODR class have the same names as the arguments to the initialization routine.
Parameters
Required: :
data – instance of the Data class
model – instance of the Model class
beta0 – a rank-1 sequence of initial parameter values. Optional if
model provides an “estimate” function to estimate these values.
Optional:
delta0 – a (double-precision) float array to hold the initial values of
the errors in the input variables. Must be same shape as data.x .
ifixb – sequence of integers with the same length as beta0 that determines
which parameters are held fixed. A value of 0 fixes the parameter, a value > 0
makes the parameter free.
ifixx – an array of integers with the same shape as data.x that determines
which input observations are treated as fixed. One can use a sequence of
length m (the dimensionality of the input observations) to fix some dimensions for all observations. A value of 0 fixes the observation, a value > 0
makes it free.
job – an integer telling ODRPACK what tasks to perform. See p. 31 of the
ODRPACK User’s Guide if you absolutely must set the value here. Use the
method set_job post-initialization for a more readable interface.
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iprint – an integer telling ODRPACK what to print. See pp. 33-34 of the
ODRPACK User’s Guide if you absolutely must set the value here. Use the
method set_iprint post-initialization for a more readable interface.
errfile – string with the filename to print ODRPACK errors to. *Do Not Open
This File Yourself!*
rptfile – string with the filename to print ODRPACK summaries to. *Do Not
Open This File Yourself!*
ndigit – integer specifying the number of reliable digits in the computation
of the function.
taufac – float specifying the initial trust region. The default value is 1.
The initial trust region is equal to taufac times the length of the first computed
Gauss-Newton step. taufac must be less than 1.
sstol – float specifying the tolerance for convergence based on the relative
change in the sum-of-squares. The default value is eps**(1/2) where eps is
the smallest value such that 1 + eps > 1 for double precision computation on
the machine. sstol must be less than 1.
partol – float specifying the tolerance for convergence based on the relative
change in the estimated parameters. The default value is eps**(2/3) for
explicit models and eps**(1/3) for implicit models. partol must be less than
1.
maxit – integer specifying the maximum number of iterations to perform. For
first runs, maxit is the total number of iterations performed and defaults to
50. For restarts, maxit is the number of additional iterations to perform and
defaults to 10.
stpb – sequence (len(stpb) == len(beta0)) of relative step sizes to compute
finite difference derivatives wrt the parameters.
stpd – array (stpd.shape == data.x.shape or stpd.shape == (m,)) of relative
step sizes to compute finite difference derivatives wrt the input variable
errors. If stpd is a rank-1 array with length m (the dimensionality of the input
variable), then the values are broadcast to all observations.
sclb – sequence (len(stpb) == len(beta0)) of scaling factors for the
parameters. The purpose of these scaling factors are to scale all of the parameters to around unity. Normally appropriate scaling factors are computed if this
argument is not specified. Specify them yourself if the automatic procedure
goes awry.
scld – array (scld.shape == data.x.shape or scld.shape == (m,)) of scaling
factors for the errors in the input variables. Again, these factors are automatically computed if you do not provide them. If scld.shape == (m,), then the
scaling factors are broadcast to all observations.
work – array to hold the double-valued working data for ODRPACK. When
restarting, takes the value of self.output.work .
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iwork – array to hold the integer-valued working data for ODRPACK. When
restarting, takes the value of self.output.iwork .
Other Members (not supplied as initialization arguments):
output – an instance if the Output class containing all of the returned
data from an invocation of ODR.run() or ODR.restart()
Methods
restart(iter=None)
Restarts the run with iter more iterations.
Parameters
iter : int, optional
ODRPACK’s default for the number of new iterations is 10.
Returns
output : Output instance
This object is also assigned to the attribute .output .
run()
Run the fitting routine with all of the information given.
Returns
output : Output instance
This object is also assigned to the attribute .output .
set_iprint(init=None, so_init=None, iter=None, so_iter=None, iter_step=None, final=None, so_final=None)
Set the iprint parameter for the printing of computation reports.
If any of the arguments are specified here, then they are set in the iprint member. If iprint is not set
manually or with this method, then ODRPACK defaults to no printing. If no filename is specified with the
member rptfile, then ODRPACK prints to stdout. One can tell ODRPACK to print to stdout in addition
to the specified filename by setting the so_* arguments to this function, but one cannot specify to print to
stdout but not a file since one can do that by not specifying a rptfile filename.
There are three reports: initialization, iteration, and final reports. They are represented by the arguments
init, iter, and final respectively. The permissible values are 0, 1, and 2 representing “no report”, “short
report”, and “long report” respectively.
The argument iter_step (0 <= iter_step <= 9) specifies how often to make the iteration report; the report
will be made for every iter_step’th iteration starting with iteration one. If iter_step == 0, then no iteration
report is made, regardless of the other arguments.
If the rptfile is None, then any so_* arguments supplied will raise an exception.
set_job(fit_type=None, deriv=None, var_calc=None, del_init=None, restart=None)
Sets the “job” parameter is a hopefully comprehensible way.
If an argument is not specified, then the value is left as is. The default value from class initialization is for
all of these options set to 0.
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Parameter
fit_type
Value Meaning
01
2
deriv
01
2
3
var_calc 0
1
2
del_init 0 1
restart
01
explicit ODR implicit ODR ordinary least-squares
forward finite differences central finite differences user-supplied derivatives (Jacobians) with
results checked by ODRPACK user-supplied derivatives, no checking
calculate asymptotic covariance matrix and fit parameter uncertainties (V_B, s_B) using
derivatives recomputed at the final solution calculate V_B and s_B using derivatives from last
iteration do not calculate V_B and s_B
initial input variable offsets set to 0 initial offsets provided by user in variable “work”
fit is not a restart fit is a restart
The permissible values are different from those given on pg. 31 of the ODRPACK User’s Guide only in
that one cannot specify numbers greater than the last value for each variable.
If one does not supply functions to compute the Jacobians, the fitting procedure will change deriv to 0,
finite differences, as a default. To initialize the input variable offsets by yourself, set del_init to 1 and put
the offsets into the “work” variable correctly.
class Output(output)
The Output class stores the output of an ODR run.
Takes one argument for initialization: the return value from the function odr().
Attributes
Methods
pprint()
Pretty-print important results.
exception odr_error
exception odr_stop
odr(fcn, beta0, y, x, we=None, wd=None, fjacb=None, fjacd=None, extra_args=None, ifixx=None, ifixb=None,
job=0, iprint=0, errfile=None, rptfile=None, ndigit=0, taufac=0.0, sstol=-1.0, partol=-1.0, maxit=-1,
stpb=None, stpd=None, sclb=None, scld=None, work=None, iwork=None, full_output=0)
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3.12 Optimization and root finding (scipy.optimize)
3.12.1 Optimization
General-purpose
fmin (func, x0[, args=(), xtol, ftol, ...])
Minimize a function using the downhill simplex algorithm.
fmin_powell (func, x0[, args=(), xtol, ftol, ...]) Minimize a function using modified Powell’s method.
fmin_cg (f, x0[, fprime, args=(), ...])
Minimize a function using a nonlinear conjugate gradient
algorithm.
fmin_bfgs (f, x0[, fprime, args=(), ...])
Minimize a function using the BFGS algorithm.
fmin_ncg (f, x0, fprime[, fhess_p, fhess, ...])
Minimize a function using the Newton-CG method.
leastsq (func, x0[, args=(), Dfun, full_output, ...])Minimize the sum of squares of a set of equations.
fmin(func, x0, args=(), xtol=0.0001, ftol=0.0001, maxiter=None, maxfun=None, full_output=0, disp=1, retall=0,
callback=None)
Minimize a function using the downhill simplex algorithm.
Parameters
func : callable func(x,*args)
The objective function to be minimized.
x0 : ndarray
Initial guess.
args : tuple
Extra arguments passed to func, i.e. f(x,*args).
callback : callable
Called after each iteration, as callback(xk), where xk is the current parameter vector.
Returns
xopt : ndarray
Parameter that minimizes function.
fopt : float
Value of function at minimum: fopt = func(xopt).
iter : int
Number of iterations performed.
funcalls : int
Number of function calls made.
warnflag : int
1 : Maximum number of function evaluations made. 2 : Maximum number of iterations reached.
allvecs : list
Solution at each iteration.
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Notes
Uses a Nelder-Mead simplex algorithm to find the minimum of a function of one or more variables.
fmin_powell(func, x0, args=(), xtol=0.0001, ftol=0.0001, maxiter=None, maxfun=None, full_output=0, disp=1,
retall=0, callback=None, direc=None)
Minimize a function using modified Powell’s method.
Parameters
func : callable f(x,*args)
Objective function to be minimized.
x0 : ndarray
Initial guess.
args : tuple
Eextra arguments passed to func.
callback : callable
An optional user-supplied function, called after each iteration.
callback(xk), where xk is the current parameter vector.
Called as
direc : ndarray
Initial direction set.
Returns
xopt : ndarray
Parameter which minimizes func.
fopt : number
Value of function at minimum: fopt = func(xopt).
direc : ndarray
Current direction set.
iter : int
Number of iterations.
funcalls : int
Number of function calls made.
warnflag : int
Integer warning flag:
1 : Maximum number of function evaluations. 2 : Maximum number of iterations.
allvecs : list
List of solutions at each iteration.
Notes
Uses a modification of Powell’s method to find the minimum of a function of N variables.
fmin_cg(f, x0, fprime=None, args=(), gtol=1.0000000000000001e-05, norm=inf, epsilon=1.4901161193847656e08, maxiter=None, full_output=0, disp=1, retall=0, callback=None)
Minimize a function using a nonlinear conjugate gradient algorithm.
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Parameters
f : callable f(x,*args)
Objective function to be minimized.
x0 : ndarray
Initial guess.
fprime : callable f’(x,*args)
Function which computes the gradient of f.
args : tuple
Extra arguments passed to f and fprime.
gtol : float
Stop when norm of gradient is less than gtol.
norm : float
Order of vector norm to use. -Inf is min, Inf is max.
epsilon : float or ndarray
If fprime is approximated, use this value for the step size (can be scalar or vector).
callback : callable
An optional user-supplied function, called after each iteration. Called as callback(xk), where xk is the current parameter vector.
Returns
xopt : ndarray
Parameters which minimize f, i.e. f(xopt) == fopt.
fopt : float
Minimum value found, f(xopt).
func_calls : int
The number of function_calls made.
grad_calls : int
The number of gradient calls made.
warnflag : int
1 : Maximum number of iterations exceeded. 2 : Gradient and/or function calls not
changing.
allvecs : ndarray
If retall is True (see other parameters below), then this vector containing the result
at each iteration is returned.
Notes
Optimize the function, f, whose gradient is given by fprime using the nonlinear conjugate gradient algorithm of
Polak and Ribiere. See Wright & Nocedal, ‘Numerical Optimization’, 1999, pg. 120-122.
fmin_bfgs(f,
x0,
fprime=None,
args=(),
gtol=1.0000000000000001e-05,
norm=inf,
epsilon=1.4901161193847656e-08, maxiter=None, full_output=0, disp=1, retall=0, callback=None)
Minimize a function using the BFGS algorithm.
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Parameters
f : callable f(x,*args)
Objective function to be minimized.
x0 : ndarray
Initial guess.
fprime : callable f’(x,*args)
Gradient of f.
args : tuple
Extra arguments passed to f and fprime.
gtol : float
Gradient norm must be less than gtol before succesful termination.
norm : float
Order of norm (Inf is max, -Inf is min)
epsilon : int or ndarray
If fprime is approximated, use this value for the step size.
callback : callable
An optional user-supplied function to call after each iteration. Called as callback(xk), where xk is the current parameter vector.
Returns
xopt : ndarray
Parameters which minimize f, i.e. f(xopt) == fopt.
fopt : float
Minimum value.
gopt : ndarray
Value of gradient at minimum, f’(xopt), which should be near 0.
Bopt : ndarray
Value of 1/f’‘(xopt), i.e. the inverse hessian matrix.
func_calls : int
Number of function_calls made.
grad_calls : int
Number of gradient calls made.
warnflag : integer
1 : Maximum number of iterations exceeded. 2 : Gradient and/or function calls not
changing.
allvecs : list
Results at each iteration. Only returned if retall is True.
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Notes
Optimize the function, f, whose gradient is given by fprime using the quasi-Newton method of Broyden,
Fletcher, Goldfarb, and Shanno (BFGS) See Wright, and Nocedal ‘Numerical Optimization’, 1999, pg. 198.
fmin_ncg(f, x0, fprime, fhess_p=None, fhess=None, args=(), avextol=1.0000000000000001e-05,
epsilon=1.4901161193847656e-08, maxiter=None, full_output=0, disp=1, retall=0, callback=None)
Minimize a function using the Newton-CG method.
Parameters
f : callable f(x,*args)
Objective function to be minimized.
x0 : ndarray
Initial guess.
fprime : callable f’(x,*args)
Gradient of f.
fhess_p : callable fhess_p(x,p,*args)
Function which computes the Hessian of f times an arbitrary vector, p.
fhess : callable fhess(x,*args)
Function to compute the Hessian matrix of f.
args : tuple
Extra arguments passed to f, fprime, fhess_p, and fhess (the same set of extra arguments is supplied to all of these functions).
epsilon : float or ndarray
If fhess is approximated, use this value for the step size.
callback : callable
An optional user-supplied function which is called after each iteration. Called as
callback(xk), where xk is the current parameter vector.
Returns
xopt : ndarray
Parameters which minimizer f, i.e. f(xopt) == fopt.
fopt : float
Value of the function at xopt, i.e. fopt = f(xopt).
fcalls : int
Number of function calls made.
gcalls : int
Number of gradient calls made.
hcalls : int
Number of hessian calls made.
warnflag : int
Warnings generated by the algorithm. 1 : Maximum number of iterations exceeded.
allvecs : list
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The result at each iteration, if retall is True (see below).
Notes
Only one of fhess_p or fhess need to be given. If fhess is provided, then fhess_p will be ignored. If neither
fhess nor fhess_p is provided, then the hessian product will be approximated using finite differences on fprime.
fhess_p must compute the hessian times an arbitrary vector. If it is not given, finite-differences on fprime are
used to compute it. See Wright & Nocedal, ‘Numerical Optimization’, 1999, pg. 140.
leastsq(func, x0, args=(), Dfun=None, full_output=0, col_deriv=0, ftol=1.49012e-08, xtol=1.49012e-08,
gtol=0.0, maxfev=0, epsfcn=0.0, factor=100, diag=None, warning=True)
Minimize the sum of squares of a set of equations.
x = arg min(sum(func(y)**2,axis=0))
y
Parameters
func : callable
should take at least one (possibly length N vector) argument and returns M floating
point numbers.
x0 : ndarray
The starting estimate for the minimization.
args : tuple
Any extra arguments to func are placed in this tuple.
Dfun : callable
A function or method to compute the Jacobian of func with derivatives across the
rows. If this is None, the Jacobian will be estimated.
full_output : bool
non-zero to return all optional outputs.
col_deriv : bool
non-zero to specify that the Jacobian function computes derivatives down the
columns (faster, because there is no transpose operation).
ftol : float
Relative error desired in the sum of squares.
xtol : float
Relative error desired in the approximate solution.
gtol : float
Orthogonality desired between the function vector and the columns of the Jacobian.
maxfev : int
The maximum number of calls to the function. If zero, then 100*(N+1) is the maximum where N is the number of elements in x0.
epsfcn : float
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A suitable step length for the forward-difference approximation of the Jacobian (for
Dfun=None). If epsfcn is less than the machine precision, it is assumed that the
relative errors in the functions are of the order of the machine precision.
factor : float
A parameter determining the initial step bound (factor * || diag * x||).
Should be in interval (0.1, 100).
diag : sequence
N positive entries that serve as a scale factors for the variables.
warning : bool
Whether to print a warning message when the call is unsuccessful. Deprecated, use
the warnings module instead.
Returns
x : ndarray
The solution (or the result of the last iteration for an unsuccessful call).
cov_x : ndarray
Uses the fjac and ipvt optional outputs to construct an estimate of the jacobian around
the solution. None if a singular matrix encountered (indicates very flat curvature in
some direction). This matrix must be multiplied by the residual standard deviation
to get the covariance of the parameter estimates – see curve_fit.
infodict : dict
a dictionary of optional outputs with the keys:
- ’nfev’ : the number of function calls
- ’fvec’ : the function evaluated at the output
- ’fjac’ : A permutation of the R matrix of a QR
factorization of the final approximate
Jacobian matrix, stored column wise.
Together with ipvt, the covariance of the
estimate can be approximated.
- ’ipvt’ : an integer array of length N which defines
a permutation matrix, p, such that
fjac*p = q*r, where r is upper triangular
with diagonal elements of nonincreasing
magnitude. Column j of p is column ipvt(j)
of the identity matrix.
- ’qtf’ : the vector (transpose(q) * fvec).
mesg : str
A string message giving information about the cause of failure.
ier : int
An integer flag. If it is equal to 1, 2, 3 or 4, the solution was found. Otherwise,
the solution was not found. In either case, the optional output variable ‘mesg’ gives
more information.
Notes
“leastsq” is a wrapper around MINPACK’s lmdif and lmder algorithms.
From scipy 0.8.0 leastsq returns an array of size one instead of a scalar when solving for a single parameter.
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Constrained (multivariate)
fmin_l_bfgs_b (func, x0[, fprime, args=(), ...])
Minimize a function func using the L-BFGS-B algorithm.
fmin_tnc (func, x0[, fprime, args=(), ...])Minimize a function with variables subject to bounds, using gradient
information.
fmin_cobyla (func, x0, cons[, args=(), conMinimize a function using the Constrained Optimization BY Linear
sargs, ...])
Approximation (COBYLA) method.
fmin_slsqp (func, x0[, eqMinimize a function using Sequential Least SQuares Programming
cons, f_eqcons, ...])
nnls (A, b)
Solve argmin_x || Ax - b ||_2 for x>=0.
fmin_l_bfgs_b(func, x0, fprime=None, args=(), approx_grad=0, bounds=None, m=10, factr=10000000.0,
pgtol=1.0000000000000001e-05, epsilon=1e-08, iprint=-1, maxfun=15000)
Minimize a function func using the L-BFGS-B algorithm.
Parameters
func : callable f(x, *args)
Function to minimise.
x0
[ndarray] Initial guess.
fprime
[callable fprime(x, *args)] The gradient of func. If None, then func returns
the function value and the gradient (f, g = func(x, *args)), unless approx_grad is True in which case func returns only f.
args
[tuple] Arguments to pass to func and fprime.
approx_grad
[bool] Whether to approximate the gradient numerically (in which case func
returns only the function value).
bounds
[list] (min, max) pairs for each element in x, defining the bounds on that
parameter. Use None for one of min or max when there is no bound in that
direction.
m
[int] The maximum number of variable metric corrections used to define the
limited memory matrix. (The limited memory BFGS method does not store the
full hessian but uses this many terms in an approximation to it.)
factr
[float]
The
iteration
stops
when
(f^k f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps,
where
eps is the machine precision, which is automatically generated by the code.
Typical values for factr are: 1e12 for low accuracy; 1e7 for moderate accuracy;
10.0 for extremely high accuracy.
pgtol
[float] The iteration will stop when max{|proj g_i | i = 1, ...,
n} <= pgtol where pg_i is the i-th component of the projected gradient.
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epsilon
[float] Step size used when approx_grad is True, for numerically calculating the
gradient
iprint
[int] Controls the frequency of output. iprint < 0 means no output.
maxfun
[int] Maximum number of function evaluations.
Returns
x : ndarray
Estimated position of the minimum.
f
[float] Value of func at the minimum.
d
[dict] Information dictionary.
d[’warnflag’] is
0 if converged, 1 if too many function evaluations, 2 if stopped for another
reason, given in d[’task’]
d[’grad’] is the gradient at the minimum (should be 0 ish) d[’funcalls’] is the
number of function calls made.
Notes
License of L-BFGS-B (Fortran code):
The version included here (in fortran code) is 2.1 (released in 1997). It was written by Ciyou Zhu, Richard Byrd,
and Jorge Nocedal <[email protected]>. It carries the following condition for use:
This software is freely available, but we expect that all publications describing work using this software, or all
commercial products using it, quote at least one of the references given below.
References
• R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound Constrained Optimization, (1995), SIAM Journal on Scientific and Statistical Computing , 16, 5, pp. 1190-1208.
• C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for
large scale bound constrained optimization (1997), ACM Transactions on Mathematical Software,
Vol 23, Num. 4, pp. 550 - 560.
fmin_tnc(func, x0, fprime=None, args=(), approx_grad=0, bounds=None, epsilon=1e-08, scale=None, offset=None, messages=15, maxCGit=-1, maxfun=None, eta=-1, stepmx=0, accuracy=0, fmin=0, ftol=-1,
xtol=-1, pgtol=-1, rescale=-1)
Minimize a function with variables subject to bounds, using gradient information.
Parameters
func : callable func(x, *args)
Function to minimize. Should return f and g, where f is the value of the function
and g its gradient (a list of floats). If the function returns None, the minimization is
aborted.
x0 : list of floats
Initial estimate of minimum.
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fprime : callable fprime(x, *args)
Gradient of func. If None, then func must return the function value and the gradient
(f,g = func(x, *args)).
args : tuple
Arguments to pass to function.
approx_grad : bool
If true, approximate the gradient numerically.
bounds : list
(min, max) pairs for each element in x, defining the bounds on that parameter. Use
None or +/-inf for one of min or max when there is no bound in that direction.
scale : list of floats
Scaling factors to apply to each variable. If None, the factors are up-low for interval
bounded variables and 1+|x] fo the others. Defaults to None
offset : float
Value to substract from each variable. If None, the offsets are (up+low)/2 for interval
bounded variables and x for the others.
messages : :
Bit mask used to select messages display during minimization values defined in the
MSGS dict. Defaults to MGS_ALL.
maxCGit : int
Maximum number of hessian*vector evaluations per main iteration. If maxCGit == 0, the direction chosen is -gradient if maxCGit < 0, maxCGit is set to
max(1,min(50,n/2)). Defaults to -1.
maxfun : int
Maximum number of function evaluation. if None, maxfun is set to max(100,
10*len(x0)). Defaults to None.
eta : float
Severity of the line search. if < 0 or > 1, set to 0.25. Defaults to -1.
stepmx : float
Maximum step for the line search. May be increased during call. If too small, it will
be set to 10.0. Defaults to 0.
accuracy : float
Relative precision for finite difference calculations. If <= machine_precision, set to
sqrt(machine_precision). Defaults to 0.
fmin : float
Minimum function value estimate. Defaults to 0.
ftol : float
Precision goal for the value of f in the stoping criterion. If ftol < 0.0, ftol is set to 0.0
defaults to -1.
xtol : float
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Precision goal for the value of x in the stopping criterion (after applying x scaling
factors). If xtol < 0.0, xtol is set to sqrt(machine_precision). Defaults to -1.
pgtol : float
Precision goal for the value of the projected gradient in the stopping criterion (after
applying x scaling factors). If pgtol < 0.0, pgtol is set to 1e-2 * sqrt(accuracy).
Setting it to 0.0 is not recommended. Defaults to -1.
rescale : float
Scaling factor (in log10) used to trigger f value rescaling. If 0, rescale at each iteration. If a large value, never rescale. If < 0, rescale is set to 1.3.
Returns
x : list of floats
The solution.
nfeval : int
The number of function evaluations.
rc : :
Return code as defined in the RCSTRINGS dict.
fmin_cobyla(func, x0, cons, args=(), consargs=None, rhobeg=1.0, rhoend=0.0001, iprint=1, maxfun=1000)
Minimize a function using the Constrained Optimization BY Linear Approximation (COBYLA) method.
Parameters
func : callable f(x, *args)
Function to minimize.
x0 : ndarray
Initial guess.
cons : sequence
Constraint functions; must all be >=0 (a single function if only 1 constraint).
args : tuple
Extra arguments to pass to function.
consargs : tuple
Extra arguments to pass to constraint functions (default of None means use same
extra arguments as those passed to func). Use () for no extra arguments.
rhobeg : :
Reasonable initial changes to the variables.
rhoend : :
Final accuracy in the optimization (not precisely guaranteed).
iprint : {0, 1, 2, 3}
Controls the frequency of output; 0 implies no output.
maxfun : int
Maximum number of function evaluations.
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Returns
x : ndarray
The argument that minimises f.
fmin_slsqp(func, x0, eqcons=, [], f_eqcons=None, ieqcons=, [], f_ieqcons=None, bounds=, [], fprime=None,
fprime_eqcons=None, fprime_ieqcons=None, args=(), iter=100, acc=9.9999999999999995e-07,
iprint=1, full_output=0, epsilon=1.4901161193847656e-08)
Minimize a function using Sequential Least SQuares Programming
Python interface function for the SLSQP Optimization subroutine originally implemented by Dieter Kraft.
Parameters
func : callable f(x,*args)
Objective function.
x0 : ndarray of float
Initial guess for the independent variable(s).
eqcons : list
A list of functions of length n such that eqcons[j](x0,*args) == 0.0 in a successfully
optimized problem.
f_eqcons : callable f(x,*args)
Returns an array in which each element must equal 0.0 in a successfully optimized
problem. If f_eqcons is specified, eqcons is ignored.
ieqcons : list
A list of functions of length n such that ieqcons[j](x0,*args) >= 0.0 in a successfully
optimized problem.
f_ieqcons : callable f(x0,*args)
Returns an array in which each element must be greater or equal to 0.0 in a successfully optimized problem. If f_ieqcons is specified, ieqcons is ignored.
bounds : list
A list of tuples specifying the lower and upper bound for each independent variable
[(xl0, xu0),(xl1, xu1),...]
fprime : callable f(x,*args)
A function that evaluates the partial derivatives of func.
fprime_eqcons : callable f(x,*args)
A function of the form f(x, *args) that returns the m by n array of equality constraint
normals. If not provided, the normals will be approximated. The array returned by
fprime_eqcons should be sized as ( len(eqcons), len(x0) ).
fprime_ieqcons : callable f(x,*args)
A function of the form f(x, *args) that returns the m by n array of inequality constraint normals. If not provided, the normals will be approximated. The array returned by fprime_ieqcons should be sized as ( len(ieqcons), len(x0) ).
args : sequence
Additional arguments passed to func and fprime.
iter : int
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The maximum number of iterations.
acc : float
Requested accuracy.
iprint : int
The verbosity of fmin_slsqp :
• iprint <= 0 : Silent operation
• iprint == 1 : Print summary upon completion (default)
• iprint >= 2 : Print status of each iterate and summary
full_output : bool
If False, return only the minimizer of func (default). Otherwise, output final objective
function and summary information.
epsilon : float
The step size for finite-difference derivative estimates.
Returns
x : ndarray of float
The final minimizer of func.
fx : ndarray of float, if full_output is true
The final value of the objective function.
its : int, if full_output is true
The number of iterations.
imode : int, if full_output is true
The exit mode from the optimizer (see below).
smode : string, if full_output is true
Message describing the exit mode from the optimizer.
Notes
Exit modes are defined as follows
-1
0
1
2
3
4
5
6
7
8
9
:
:
:
:
:
:
:
:
:
:
:
Gradient evaluation required (g & a)
Optimization terminated successfully.
Function evaluation required (f & c)
More equality constraints than independent variables
More than 3*n iterations in LSQ subproblem
Inequality constraints incompatible
Singular matrix E in LSQ subproblem
Singular matrix C in LSQ subproblem
Rank-deficient equality constraint subproblem HFTI
Positive directional derivative for linesearch
Iteration limit exceeded
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Examples
Examples are given in the tutorial.
nnls(A, b)
Solve argmin_x || Ax - b ||_2 for x>=0.
Parameters
A : ndarray
Matrix A as shown above.
b : ndarray
Right-hand side vector.
Returns
x : ndarray
Solution vector.
rnorm : float
The residual, || Ax-b ||_2.
Notes
This is a wrapper for NNLS.F.
Global
anneal (func, x0[, args=(), schedule, ...])
brute (func, ranges[, args=(), Ns, full_output, ...])
Minimize a function using simulated annealing.
Minimize a function over a given range by brute force.
anneal(func, x0, args=(), schedule=’fast’, full_output=0, T0=None, Tf=9.9999999999999998e-13, maxeval=None, maxaccept=None, maxiter=400, boltzmann=1.0, learn_rate=0.5, feps=9.9999999999999995e07, quench=1.0, m=1.0, n=1.0, lower=-100, upper=100, dwell=50)
Minimize a function using simulated annealing.
Schedule is a schedule class implementing the annealing schedule. Available ones are ‘fast’, ‘cauchy’, ‘boltzmann’
Parameters
func : callable f(x, *args)
Function to be optimized.
x0 : ndarray
Initial guess.
args : tuple
Extra parameters to func.
schedule : base_schedule
Annealing schedule to use (a class).
full_output : bool
Whether to return optional outputs.
T0 : float
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Initial Temperature (estimated as 1.2 times the largest cost-function deviation over
random points in the range).
Tf : float
Final goal temperature.
maxeval : int
Maximum function evaluations.
maxaccept : int
Maximum changes to accept.
maxiter : int
Maximum cooling iterations.
learn_rate : float
Scale constant for adjusting guesses.
boltzmann : float
Boltzmann constant in acceptance test (increase for less stringent test at each temperature).
feps : float
Stopping relative error tolerance for the function value in last four coolings.
quench, m, n : float
Parameters to alter fast_sa schedule.
lower, upper : float or ndarray
Lower and upper bounds on x.
dwell : int
The number of times to search the space at each temperature.
brute(func, ranges, args=(), Ns=20, full_output=0, finish=<function fmin at 0x54e9430>)
Minimize a function over a given range by brute force.
Parameters
func : callable f(x,*args)
Objective function to be minimized.
ranges : tuple
Each element is a tuple of parameters or a slice object to be handed to
numpy.mgrid.
args : tuple
Extra arguments passed to function.
Ns : int
Default number of samples, if those are not provided.
full_output : bool
If True, return the evaluation grid.
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Returns
x0 : ndarray
Value of arguments to func, giving minimum over the grid.
fval : int
Function value at minimum.
grid : tuple
Representation of the evaluation grid. It has the same length as x0.
Jout : ndarray
Function values over grid: Jout = func(*grid).
Notes
Find the minimum of a function evaluated on a grid given by the tuple ranges.
Scalar function minimizers
fminbound (func, x1, x2[, args=(),
Bounded
xtol,minimization
maxfor scalar functions.
fun, ...])
golden (func[, args=(), brack,Given
...]) a function of one-variable and a possible bracketing interval, return the
minimum of the function isolated to a fractional precision of tol.
bracket (func[, xa, xb, args=(),
Given
...])a function and distinct initial points, search in the downhill direction (as
defined by the initital points) and return new points xa, xb, xc that bracket the
minimum of the function f(xa) > f(xb) < f(xc). It doesn’t always mean that obtained
solution will satisfy xa<=x<=xb
brent (func[, args=(), brack, ...])
Given a function of one-variable and a possible bracketing interval, return the
minimum of the function isolated to a fractional precision of tol.
fminbound(func, x1, x2, args=(), xtol=1.0000000000000001e-05, maxfun=500, full_output=0, disp=1)
Bounded minimization for scalar functions.
Parameters
func : callable f(x,*args)
Objective function to be minimized (must accept and return scalars).
x1, x2 : float or array scalar
The optimization bounds.
args : tuple
Extra arguments passed to function.
xtol : float
The convergence tolerance.
maxfun : int
Maximum number of function evaluations allowed.
full_output : bool
If True, return optional outputs.
disp : int
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If non-zero, print messages.
0 : no message printing. 1 : non-convergence notification messages only. 2 :
print a message on convergence too. 3 : print iteration results.
Returns
xopt : ndarray
Parameters (over given interval) which minimize the objective function.
fval : number
The function value at the minimum point.
ierr : int
An error flag (0 if converged, 1 if maximum number of function calls reached).
numfunc : int
The number of function calls made.
Notes
Finds a local minimizer of the scalar function func in the interval x1 < xopt < x2 using Brent’s method. (See
brent for auto-bracketing).
golden(func, args=(), brack=None, tol=1.4901161193847656e-08, full_output=0)
Given a function of one-variable and a possible bracketing interval, return the minimum of the function isolated
to a fractional precision of tol.
Parameters
func : callable func(x,*args)
Objective function to minimize.
args : tuple
Additional arguments (if present), passed to func.
brack : tuple
Triple (a,b,c), where (a<b<c) and func(b) < func(a),func(c). If bracket consists of
two numbers (a, c), then they are assumed to be a starting interval for a downhill
bracket search (see bracket); it doesn’t always mean that obtained solution will satisfy a<=x<=c.
tol : float
x tolerance stop criterion
full_output : bool
If True, return optional outputs.
Notes
Uses analog of bisection method to decrease the bracketed interval.
bracket(func, xa=0.0, xb=1.0, args=(), grow_limit=110.0, maxiter=1000)
Given a function and distinct initial points, search in the downhill direction (as defined by the initital points)
and return new points xa, xb, xc that bracket the minimum of the function f(xa) > f(xb) < f(xc). It doesn’t
always mean that obtained solution will satisfy xa<=x<=xb
Parameters
func : callable f(x,*args)
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Objective function to minimize.
xa, xb : float
Bracketing interval.
args : tuple
Additional arguments (if present), passed to func.
grow_limit : float
Maximum grow limit.
maxiter : int
Maximum number of iterations to perform.
Returns
xa, xb, xc : float
Bracket.
fa, fb, fc : float
Objective function values in bracket.
funcalls : int
Number of function evaluations made.
brent(func, args=(), brack=None, tol=1.48e-08, full_output=0, maxiter=500)
Given a function of one-variable and a possible bracketing interval, return the minimum of the function isolated
to a fractional precision of tol.
Parameters
func : callable f(x,*args)
Objective function.
args :
Additional arguments (if present).
brack : tuple
Triple (a,b,c) where (a<b<c) and func(b) < func(a),func(c). If bracket consists of two
numbers (a,c) then they are assumed to be a starting interval for a downhill bracket
search (see bracket); it doesn’t always mean that the obtained solution will satisfy
a<=x<=c.
full_output : bool
If True, return all output args (xmin, fval, iter, funcalls).
Returns
xmin : ndarray
Optimum point.
fval : float
Optimum value.
iter : int
Number of iterations.
funcalls : int
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Number of objective function evaluations made.
Notes
Uses inverse parabolic interpolation when possible to speed up convergence of golden section method.
3.12.2 Fitting
curve_fit (f, xdata, ydata[, p0, sigma, **kw)
Use non-linear least squares to fit a function, f, to data.
curve_fit(f, xdata, ydata, p0=None, sigma=None, **kw)
Use non-linear least squares to fit a function, f, to data.
Assumes ydata = f(xdata, *params) + eps
Parameters
f : callable
The model function, f(x, ...). It must take the independent variable as the first argument and the parameters to fit as separate remaining arguments.
xdata : An N-length sequence or an (k,N)-shaped array
for functions with k predictors. The independent variable where the data is measured.
ydata : N-length sequence
The dependent data — nominally f(xdata, ...)
p0 : None, scalar, or M-length sequence
Initial guess for the parameters. If None, then the initial values will all be 1 (if
the number of parameters for the function can be determined using introspection,
otherwise a ValueError is raised).
sigma : None or N-length sequence
If not None, it represents the standard-deviation of ydata. This vector, if given, will
be used as weights in the least-squares problem.
Returns
popt : array
Optimal values for the parameters so that the sum of the squared error of f(xdata,
*popt) - ydata is minimized
pcov : 2d array
The estimated covariance of popt. The diagonals provide the variance of the parameter estimate.
Notes
The algorithm uses the Levenburg-Marquardt algorithm: scipy.optimize.leastsq. Additional keyword arguments
are passed directly to that algorithm.
Examples
>>> import numpy as np
>>> from scipy.optimize import curve_fit
>>> def func(x, a, b, c):
...
return a*np.exp(-b*x) + c
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>>> x = np.linspace(0,4,50)
>>> y = func(x, 2.5, 1.3, 0.5)
>>> yn = y + 0.2*np.random.normal(size=len(x))
>>> popt, pcov = curve_fit(func, x, yn)
3.12.3 Root finding
fsolve (func, x0[, args=(), fprime, ...])
Find the roots of a function.
fsolve(func, x0, args=(), fprime=None, full_output=0, col_deriv=0, xtol=1.49012e-08, maxfev=0, band=None,
epsfcn=0.0, factor=100, diag=None, warning=True)
Find the roots of a function.
Return the roots of the (non-linear) equations defined by func(x) = 0 given a starting estimate.
Parameters
func : callable f(x, *args)
A function that takes at least one (possibly vector) argument.
x0 : ndarray
The starting estimate for the roots of func(x) = 0.
args : tuple
Any extra arguments to func.
fprime : callable(x)
A function to compute the Jacobian of func with derivatives across the rows. By
default, the Jacobian will be estimated.
full_output : bool
If True, return optional outputs.
col_deriv : bool
Specify whether the Jacobian function computes derivatives down the columns
(faster, because there is no transpose operation).
warning : bool
Whether to print a warning message when the call is unsuccessful. This option is
deprecated, use the warnings module instead.
Returns
x : ndarray
The solution (or the result of the last iteration for an unsuccessful call).
infodict : dict
A dictionary of optional outputs with the keys:
*
*
*
*
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’nfev’:
’njev’:
’fvec’:
’fjac’:
number of function calls
number of Jacobian calls
function evaluated at the output
the orthogonal matrix, q, produced by the QR
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factorization of the final approximate Jacobian
matrix, stored column wise
* ’r’: upper triangular matrix produced by QR factorization of same
matrix
* ’qtf’: the vector (transpose(q) * fvec)
ier : int
An integer flag. Set to 1 if a solution was found, otherwise refer to mesg for more
information.
mesg : str
If no solution is found, mesg details the cause of failure.
Notes
fsolve is a wrapper around MINPACK’s hybrd and hybrj algorithms.
From scipy 0.8.0 fsolve returns an array of size one instead of a scalar when solving for a single parameter.
Scalar function solvers
brentq (f, a, b[, args=(), xtol, rtol, ...])
brenth (f, a, b[, args=(), xtol, rtol, ...])
ridder (f, a, b[, args=(), xtol, rtol, ...])
bisect (f, a, b[, args=(), xtol, rtol, ...])
newton (func, x0[, fprime, args=(), ...])
Find a root of a function in given interval.
Find root of f in [a,b].
Find a root of a function in an interval.
Find root of f in [a,b].
Find a zero using the Newton-Raphson or secant method.
brentq(f, a, b, args=(), xtol=9.9999999999999998e-13,
full_output=False, disp=True)
Find a root of a function in given interval.
rtol=4.4408920985006262e-16,
maxiter=100,
Return float, a zero of f between a and b. f must be a continuous function, and [a,b] must be a sign changing
interval.
Description: Uses the classic Brent (1973) method to find a zero of the function f on the sign changing interval [a
, b]. Generally considered the best of the rootfinding routines here. It is a safe version of the secant method that
uses inverse quadratic extrapolation. Brent’s method combines root bracketing, interval bisection, and inverse
quadratic interpolation. It is sometimes known as the van Wijngaarden-Deker-Brent method. Brent (1973)
claims convergence is guaranteed for functions computable within [a,b].
[Brent1973] provides the classic description of the algorithm.
Another description can be found
in a recent edition of Numerical Recipes, including [PressEtal1992].
Another description is at
http://mathworld.wolfram.com/BrentsMethod.html. It should be easy to understand the algorithm just by reading our code. Our code diverges a bit from standard presentations: we choose a different formula for the
extrapolation step.
Parameters
f : function
Python function returning a number. f must be continuous, and f(a) and f(b) must
have opposite signs.
a : number
One end of the bracketing interval [a,b].
b : number
The other end of the bracketing interval [a,b].
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xtol : number, optional
The routine converges when a root is known to lie within xtol of the value return.
Should be >= 0. The routine modifies this to take into account the relative precision
of doubles.
maxiter : number, optional
if convergence is not achieved in maxiter iterations, and error is raised. Must be >=
0.
args : tuple, optional
containing extra arguments for the function f.
(x)+args).
f is called by apply(f,
full_output : bool, optional
If full_output is False, the root is returned. If full_output is True, the return value is
(x, r), where x is the root, and r is a RootResults object.
disp : {True, bool} optional
If True, raise RuntimeError if the algorithm didn’t converge.
Returns
x0 : float
Zero of f between a and b.
r : RootResults (present if full_output = True)
Object containing information about the convergence. In particular, r.converged
is True if the routine converged.
See Also:
multivariate
fmin, fmin_powell, fmin_cg, fmin_bfgs, fmin_ncg
nonlinear
leastsq
constrained
fmin_l_bfgs_b, fmin_tnc, fmin_cobyla
global
anneal, brute
local
fminbound, brent, golden, bracket
n-dimensional
fsolve
one-dimensional
brentq, brenth, ridder, bisect, newton
scalar
fixed_point
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Notes
f must be continuous. f(a) and f(b) must have opposite signs.
brenth(f, a, b, args=(), xtol=9.9999999999999998e-13,
full_output=False, disp=True)
Find root of f in [a,b].
rtol=4.4408920985006262e-16,
maxiter=100,
A variation on the classic Brent routine to find a zero of the function f between the arguments a and b that uses
hyperbolic extrapolation instead of inverse quadratic extrapolation. There was a paper back in the 1980’s ... f(a)
and f(b) can not have the same signs. Generally on a par with the brent routine, but not as heavily tested. It is a
safe version of the secant method that uses hyperbolic extrapolation. The version here is by Chuck Harris.
Parameters
f : function
Python function returning a number. f must be continuous, and f(a) and f(b) must
have opposite signs.
a : number
One end of the bracketing interval [a,b].
b : number
The other end of the bracketing interval [a,b].
xtol : number, optional
The routine converges when a root is known to lie within xtol of the value return.
Should be >= 0. The routine modifies this to take into account the relative precision
of doubles.
maxiter : number, optional
if convergence is not achieved in maxiter iterations, and error is raised. Must be >=
0.
args : tuple, optional
containing extra arguments for the function f.
(x)+args).
f is called by apply(f,
full_output : bool, optional
If full_output is False, the root is returned. If full_output is True, the return value is
(x, r), where x is the root, and r is a RootResults object.
disp : {True, bool} optional
If True, raise RuntimeError if the algorithm didn’t converge.
Returns
x0 : float
Zero of f between a and b.
r : RootResults (present if full_output = True)
Object containing information about the convergence. In particular, r.converged
is True if the routine converged.
ridder(f, a, b, args=(), xtol=9.9999999999999998e-13,
full_output=False, disp=True)
Find a root of a function in an interval.
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maxiter=100,
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Parameters
f : function
Python function returning a number. f must be continuous, and f(a) and f(b) must
have opposite signs.
a : number
One end of the bracketing interval [a,b].
b : number
The other end of the bracketing interval [a,b].
xtol : number, optional
The routine converges when a root is known to lie within xtol of the value return.
Should be >= 0. The routine modifies this to take into account the relative precision
of doubles.
maxiter : number, optional
if convergence is not achieved in maxiter iterations, and error is raised. Must be >=
0.
args : tuple, optional
containing extra arguments for the function f.
(x)+args).
f is called by apply(f,
full_output : bool, optional
If full_output is False, the root is returned. If full_output is True, the return value is
(x, r), where x is the root, and r is a RootResults object.
disp : {True, bool} optional
If True, raise RuntimeError if the algorithm didn’t converge.
Returns
x0 : float
Zero of f between a and b.
r : RootResults (present if full_output = True)
Object containing information about the convergence. In particular, r.converged
is True if the routine converged.
See Also:
brentq, brenth, bisect, newton
fixed_point
scalar fixed-point finder
Notes
Uses [Ridders1979] method to find a zero of the function f between the arguments a and b. Ridders’ method
is faster than bisection, but not generally as fast as the Brent rountines. [Ridders1979] provides the classic
description and source of the algorithm. A description can also be found in any recent edition of Numerical
Recipes.
The routine used here diverges slightly from standard presentations in order to be a bit more careful of tolerance.
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References
[Ridders1979]
bisect(f, a, b, args=(), xtol=9.9999999999999998e-13,
full_output=False, disp=True)
Find root of f in [a,b].
rtol=4.4408920985006262e-16,
maxiter=100,
Basic bisection routine to find a zero of the function f between the arguments a and b. f(a) and f(b) can not have
the same signs. Slow but sure.
Parameters
f : function
Python function returning a number. f must be continuous, and f(a) and f(b) must
have opposite signs.
a : number
One end of the bracketing interval [a,b].
b : number
The other end of the bracketing interval [a,b].
xtol : number, optional
The routine converges when a root is known to lie within xtol of the value return.
Should be >= 0. The routine modifies this to take into account the relative precision
of doubles.
maxiter : number, optional
if convergence is not achieved in maxiter iterations, and error is raised. Must be >=
0.
args : tuple, optional
containing extra arguments for the function f.
(x)+args).
f is called by apply(f,
full_output : bool, optional
If full_output is False, the root is returned. If full_output is True, the return value is
(x, r), where x is the root, and r is a RootResults object.
disp : {True, bool} optional
If True, raise RuntimeError if the algorithm didn’t converge.
Returns
x0 : float
Zero of f between a and b.
r : RootResults (present if full_output = True)
Object containing information about the convergence. In particular, r.converged
is True if the routine converged.
See Also:
fixed_point
scalar fixed-point finder fsolve – n-dimensional root-finding
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newton(func, x0, fprime=None, args=(), tol=1.48e-08, maxiter=50)
Find a zero using the Newton-Raphson or secant method.
Find a zero of the function func given a nearby starting point x0. The Newton-Rapheson method is used if the
derivative fprime of func is provided, otherwise the secant method is used.
Parameters
func : function
The function whose zero is wanted. It must be a function of a single variable of the
form f(x,a,b,c...), where a,b,c... are extra arguments that can be passed in the args
parameter.
x0 : float
An initial estimate of the zero that should be somewhere near the actual zero.
fprime : {None, function}, optional
The derivative of the function when available and convenient. If it is None, then the
secant method is used. The default is None.
args : tuple, optional
Extra arguments to be used in the function call.
tol : float, optional
The allowable error of the zero value.
maxiter : int, optional
Maximum number of iterations.
Returns
zero : float
Estimated location where function is zero.
See Also:
brentq, brenth, ridder, bisect, fsolve
Notes
The convergence rate of the Newton-Rapheson method is quadratic while that of the secant method is somewhat
less. This means that if the function is well behaved the actual error in the estimated zero is approximatly the
square of the requested tolerance up to roundoff error. However, the stopping criterion used here is the step
size and there is no quarantee that a zero has been found. Consequently the result should be verified. Safer
algorithms are brentq, brenth, ridder, and bisect, but they all require that the root first be bracketed in an interval
where the function changes sign. The brentq algorithm is recommended for general use in one dimemsional
problems when such an interval has been found.
Fixed point finding:
fixed_point (func, x0[, args=(), xtol, maxiter])
Find the point where func(x) == x
fixed_point(func, x0, args=(), xtol=1e-08, maxiter=500)
Find the point where func(x) == x
Given a function of one or more variables and a starting point, find a fixed-point of the function: i.e. where
func(x)=x.
Uses Steffensen’s Method using Aitken’s Del^2 convergence acceleration. See Burden, Faires, “Numerical
Analysis”, 5th edition, pg. 80
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General-purpose nonlinear (multidimensional)
broyden1 (F, xin[, iter, alpha, verbose])
broyden2 (F, xin[, iter, alpha, verbose])
broyden3 (F, xin[, iter, alpha, verbose])
broyden_generalized (F, xin[, iter, alpha, M, ...])
anderson (F, xin[, iter, alpha, M, ...])
anderson2 (F, xin[, iter, alpha, M, ...])
Broyden’s first method.
Broyden’s second method.
Broyden’s second method.
Generalized Broyden’s method.
Extended Anderson method.
Anderson method.
broyden1(F, xin, iter=10, alpha=0.10000000000000001, verbose=False)
Broyden’s first method.
Updates Jacobian and computes inv(J) by a matrix inversion at every iteration. It’s very slow.
The best norm |F(x)|=0.005 achieved in ~45 iterations.
broyden2(F, xin, iter=10, alpha=0.40000000000000002, verbose=False)
Broyden’s second method.
Updates inverse Jacobian by an optimal formula. There is NxN matrix multiplication in every iteration.
The best norm |F(x)|=0.003 achieved in ~20 iterations.
Recommended.
broyden3(F, xin, iter=10, alpha=0.40000000000000002, verbose=False)
Broyden’s second method.
Updates inverse Jacobian by an optimal formula. The NxN matrix multiplication is avoided.
The best norm |F(x)|=0.003 achieved in ~20 iterations.
Recommended.
broyden_generalized(F, xin, iter=10, alpha=0.10000000000000001, M=5, verbose=False)
Generalized Broyden’s method.
Computes an approximation to the inverse Jacobian from the last M interations. Avoids NxN matrix multiplication, it only has MxM matrix multiplication and inversion.
M=0 .... linear mixing M=1 .... Anderson mixing with 2 iterations M=2 .... Anderson mixing with 3 iterations
etc. optimal is M=5
anderson(F, xin, iter=10, alpha=0.10000000000000001, M=5, w0=0.01, verbose=False)
Extended Anderson method.
Computes an approximation to the inverse Jacobian from the last M interations. Avoids NxN matrix multiplication, it only has MxM matrix multiplication and inversion.
M=0 .... linear mixing M=1 .... Anderson mixing with 2 iterations M=2 .... Anderson mixing with 3 iterations
etc. optimal is M=5
anderson2(F, xin, iter=10, alpha=0.10000000000000001, M=5, w0=0.01, verbose=False)
Anderson method.
M=0 .... linear mixing M=1 .... Anderson mixing with 2 iterations M=2 .... Anderson mixing with 3 iterations
etc. optimal is M=5
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3.12.4 Utility Functions
line_search (f, myfprime, xk, pk, gfk, old_fval, old_old_fval[, args=(), c1, c2, ...])
check_grad (func, grad, x0, *args)
Find alpha that satisfies strong Wolfe
conditions.
line_search(f, myfprime, xk, pk, gfk, old_fval, old_old_fval, args=(), c1=0.0001, c2=0.90000000000000002,
amax=50)
Find alpha that satisfies strong Wolfe conditions.
Parameters
f : callable f(x,*args)
Objective function.
myfprime : callable f’(x,*args)
Objective function gradient (can be None).
xk : ndarray
Starting point.
pk : ndarray
Search direction.
gfk : ndarray
Gradient value for x=xk (xk being the current parameter estimate).
args : tuple
Additional arguments passed to objective function.
c1 : float
Parameter for Armijo condition rule.
c2 : float
Parameter for curvature condition rule.
Returns
alpha0 : float
Alpha for which x_new = x0 + alpha * pk.
fc : int
Number of function evaluations made.
gc : int
Number of gradient evaluations made.
Notes
Uses the line search algorithm to enforce strong Wolfe conditions. See Wright and Nocedal, ‘Numerical Optimization’, 1999, pg. 59-60.
For the zoom phase it uses an algorithm by [...].
check_grad(func, grad, x0, *args)
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3.13 Signal processing (scipy.signal)
3.13.1 Convolution
convolve (in1, in2[, mode, old_behavior])
correlate (in1, in2[, mode, old_behavior])
fftconvolve (in1, in2[, mode])
convolve2d (in1, in2[, mode, boundary, ...])
correlate2d (in1, in2[, mode, boundary, ...])
sepfir2d ()
Convolve two N-dimensional arrays.
Cross-correlate two N-dimensional arrays.
Convolve two N-dimensional arrays using FFT. See convolve.
Convolve two 2-dimensional arrays.
Cross-correlate two 2-dimensional arrays.
sepfir2d(input, hrow, hcol) -> output
convolve(in1, in2, mode=’full’, old_behavior=True)
Convolve two N-dimensional arrays.
Convolve in1 and in2 with output size determined by mode.
Parameters
in1: array :
first input.
in2: array :
second input. Should have the same number of dimensions as in1.
mode: str {‘valid’, ‘same’, ‘full’} :
a string indicating the size of the output:
valid
[the output consists only of those elements that do not] rely on the zero-padding.
same
[the output is the same size as the largest input centered] with respect to the ‘full’
output.
full
[the output is the full discrete linear cross-correlation] of the inputs. (Default)
Returns
out: array :
an N-dimensional array containing a subset of the discrete linear cross-correlation of
in1 with in2.
correlate(in1, in2, mode=’full’, old_behavior=True)
Cross-correlate two N-dimensional arrays.
Cross-correlate in1 and in2 with the output size determined by the mode argument.
Parameters
in1: array :
first input.
in2: array :
second input. Should have the same number of dimensions as in1.
mode: str {‘valid’, ‘same’, ‘full’} :
a string indicating the size of the output:
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• ‘valid’: the output consists only of those elements that do not
rely on the zero-padding. - ‘same’: the output is the same size as the largest
input centered
with respect to the ‘full’ output.
• ‘full’: the output is the full discrete linear cross-correlation of the inputs.
(Default)
old_behavior: bool :
If True (default), the old behavior of correlate is implemented:
• if in1.size < in2.size, in1 and in2 are swapped (correlate(in1, in2) == correlate(in2, in1))
• For complex inputs, the conjugate is not taken for in2
If False, the new, conventional definition of correlate is implemented.
Returns
out: array :
an N-dimensional array containing a subset of the discrete linear cross-correlation of
in1 with in2.
Notes
The correlation z of two arrays x and y of rank d is defined as
z[...,k,...] = sum[..., i_l, ...]
x[..., i_l,...] * conj(y[..., i_l + k,...])
fftconvolve(in1, in2, mode=’full’)
Convolve two N-dimensional arrays using FFT. See convolve.
convolve2d(in1, in2, mode=’full’, boundary=’fill’, fillvalue=0, old_behavior=True)
Convolve two 2-dimensional arrays.
Description:
Convolve in1 and in2 with output size determined by mode and boundary conditions determined by
boundary and fillvalue.
Inputs:
in1 – a 2-dimensional array. in2 – a 2-dimensional array. mode – a flag indicating the size of the
output
‘valid’ (0): The output consists only of those elements that
do not rely on the zero-padding.
‘same’ (1): The output is the same size as the input centered
with respect to the ‘full’ output.
‘full’ (2): The output is the full discrete linear convolution
of the inputs. (Default)
boundary – a flag indicating how to handle boundaries
‘fill’ : pad input arrays with fillvalue. (Default) ‘wrap’ : circular boundary conditions. ‘symm’
: symmetrical boundary conditions.
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fillvalue – value to fill pad input arrays with (Default = 0)
Outputs: (out,)
out – a 2-dimensional array containing a subset of the discrete linear
convolution of in1 with in2.
correlate2d(in1, in2, mode=’full’, boundary=’fill’, fillvalue=0, old_behavior=True)
Cross-correlate two 2-dimensional arrays.
Description:
Cross correlate in1 and in2 with output size determined by mode and boundary conditions determined by boundary and fillvalue.
Inputs:
in1 – a 2-dimensional array. in2 – a 2-dimensional array. mode – a flag indicating the size of the
output
‘valid’ (0): The output consists only of those elements that
do not rely on the zero-padding.
‘same’ (1): The output is the same size as the input centered
with respect to the ‘full’ output.
‘full’ (2): The output is the full discrete linear convolution
of the inputs. (Default)
boundary – a flag indicating how to handle boundaries
‘fill’ : pad input arrays with fillvalue. (Default) ‘wrap’ : circular boundary conditions. ‘symm’
: symmetrical boundary conditions.
fillvalue – value to fill pad input arrays with (Default = 0)
Outputs: (out,)
out – a 2-dimensional array containing a subset of the discrete linear
cross-correlation of in1 with in2.
sepfir2d()
sepfir2d(input, hrow, hcol) -> output
Description:
Convolve the rank-2 input array with the separable filter defined by the rank-1 arrays hrow, and
hcol. Mirror symmetric boundary conditions are assumed. This function can be used to find an
image given its B-spline representation.
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3.13.2 B-splines
bspline (x, n)
bspline(x,n): B-spline basis function of order n. uses numpy.piecewise and
automatic function-generator.
Gaussian approximation to B-spline basis function of order n.
Compute cubic spline coefficients for rank-1 array.
gauss_spline (x, n)
cspline1d (signal[, lamb])
qspline1d (sigCompute quadratic spline coefficients for rank-1 array.
nal[, lamb])
cspline2d ()
cspline2d(input {, lambda, precision}) -> ck
qspline2d ()
qspline2d(input {, lambda, precision}) -> qk
spline_filter (Iin[, lmbda])
Smoothing spline (cubic) filtering of a rank-2 array.
bspline(x, n)
bspline(x,n): B-spline basis function of order n. uses numpy.piecewise and automatic function-generator.
gauss_spline(x, n)
Gaussian approximation to B-spline basis function of order n.
cspline1d(signal, lamb=0.0)
Compute cubic spline coefficients for rank-1 array.
Description:
Find the cubic spline coefficients for a 1-D signal assuming mirror-symmetric boundary conditions. To obtain the signal back from the spline representation mirror-symmetric-convolve these
coefficients with a length 3 FIR window [1.0, 4.0, 1.0]/ 6.0 .
Inputs:
signal – a rank-1 array representing samples of a signal. lamb – smoothing coefficient (default =
0.0)
Output:
c – cubic spline coefficients.
qspline1d(signal, lamb=0.0)
Compute quadratic spline coefficients for rank-1 array.
Description:
Find the quadratic spline coefficients for a 1-D signal assuming mirror-symmetric boundary conditions. To obtain the signal back from the spline representation mirror-symmetric-convolve these
coefficients with a length 3 FIR window [1.0, 6.0, 1.0]/ 8.0 .
Inputs:
signal – a rank-1 array representing samples of a signal. lamb – smoothing coefficient (must be zero
for now.)
Output:
c – cubic spline coefficients.
cspline2d()
cspline2d(input {, lambda, precision}) -> ck
Description:
Return the third-order B-spline coefficients over a regularly spacedi input grid for the twodimensional input image. The lambda argument specifies the amount of smoothing. The precision
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argument allows specifying the precision used when computing the infinite sum needed to apply
mirror- symmetric boundary conditions.
qspline2d()
qspline2d(input {, lambda, precision}) -> qk
Description:
Return the second-order B-spline coefficients over a regularly spaced input grid for the twodimensional input image. The lambda argument specifies the amount of smoothing. The precision
argument allows specifying the precision used when computing the infinite sum needed to apply
mirror- symmetric boundary conditions.
spline_filter(Iin, lmbda=5.0)
Smoothing spline (cubic) filtering of a rank-2 array.
Filter an input data set, Iin, using a (cubic) smoothing spline of fall-off lmbda.
3.13.3 Filtering
order_filter (a, domain, rank)
medfilt (volume[, kernel_size])
medfilt2d (input[, kernel_size])
wiener (im[, mysize, noise])
symiirorder1 ()
symiirorder2 ()
lfilter (b, a, x[, axis, zi])
lfiltic (b, a, y[, x])
deconvolve (signal, divisor)
hilbert (x[, N, axis])
get_window (window, Nx[, fftbins])
decimate (x, q[, n, ftype, axis])
detrend (data[, axis, type, bp])
resample (x, num[, t, axis, window])
Perform an order filter on an N-dimensional array.
Perform a median filter on an N-dimensional array.
Median filter two 2-dimensional arrays.
Perform a Wiener filter on an N-dimensional array.
symiirorder1(input, c0, z1 {, precision}) -> output
symiirorder2(input, r, omega {, precision}) -> output
Filter data along one-dimension with an IIR or FIR filter.
Construct initial conditions for lfilter
Deconvolves divisor out of signal.
Compute the analytic signal.
Return a window of length Nx and type window.
downsample the signal x by an integer factor q, using an order n filter
Remove linear trend along axis from data.
Resample to num samples using Fourier method along the given axis.
order_filter(a, domain, rank)
Perform an order filter on an N-dimensional array.
Description:
Perform an order filter on the array in. The domain argument acts as a mask centered over each
pixel. The non-zero elements of domain are used to select elements surrounding each input pixel
which are placed in a list. The list is sorted, and the output for that pixel is the element corresponding to rank in the sorted list.
Parameters
in – an N-dimensional input array. :
domain – a mask array with the same number of dimensions as in. Each
dimension should have an odd number of elements.
rank – an non-negative integer which selects the element from the
sorted list (0 corresponds to the largest element, 1 is the next largest element,
etc.)
Returns
out – the results of the order filter in an array with the same :
shape as in.
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medfilt(volume, kernel_size=None)
Perform a median filter on an N-dimensional array.
Description:
Apply a median filter to the input array using a local window-size given by kernel_size.
Inputs:
in – An N-dimensional input array. kernel_size – A scalar or an N-length list giving the size of the
median filter window in each dimension. Elements of kernel_size should be odd. If
kernel_size is a scalar, then this scalar is used as the size in each dimension.
Outputs: (out,)
out – An array the same size as input containing the median filtered
result.
medfilt2d(input, kernel_size=3)
Median filter two 2-dimensional arrays.
Description:
Apply a median filter to the input array using a local window-size given by kernel_size (must be
odd).
Inputs:
in – An 2 dimensional input array. kernel_size – A scalar or an length-2 list giving the size of the
median filter window in each dimension. Elements of kernel_size should be odd. If
kernel_size is a scalar, then this scalar is used as the size in each dimension.
Outputs: (out,)
out – An array the same size as input containing the median filtered
result.
wiener(im, mysize=None, noise=None)
Perform a Wiener filter on an N-dimensional array.
Description:
Apply a Wiener filter to the N-dimensional array in.
Inputs:
in – an N-dimensional array. kernel_size – A scalar or an N-length list giving the size of the
Wiener filter window in each dimension. Elements of kernel_size should be odd. If
kernel_size is a scalar, then this scalar is used as the size in each dimension.
noise – The noise-power to use. If None, then noise is estimated as
the average of the local variance of the input.
Outputs: (out,)
out – Wiener filtered result with the same shape as in.
symiirorder1()
symiirorder1(input, c0, z1 {, precision}) -> output
Description:
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Implement a smoothing IIR filter with mirror-symmetric boundary conditions using a cascade of
first-order sections. The second section uses a reversed sequence. This implements a system with
the following transfer function and mirror-symmetric boundary conditions.
c0
H(z) = ———————
(1-z1/z) (1 - z1 z)
The resulting signal will have mirror symmetric boundary conditions as well.
Inputs:
input – the input signal. c0, z1 – parameters in the transfer function. precision – specifies the
precision for calculating initial conditions
of the recursive filter based on mirror-symmetric input.
Output:
output – filtered signal.
symiirorder2()
symiirorder2(input, r, omega {, precision}) -> output
Description:
Implement a smoothing IIR filter with mirror-symmetric boundary conditions using a cascade of
second-order sections. The second section uses a reversed sequence. This implements the following
transfer function:
cs^2
H(z) = —————————————
(1 - a2/z - a3/z^2) (1 - a2 z - a3 z^2 )
where a2 = (2 r cos omega)
a3 = - r^2 cs = 1 - 2 r cos omega + r^2
Inputs:
input – the input signal. r, omega – parameters in the transfer function. precision – specifies the
precision for calculating initial conditions
of the recursive filter based on mirror-symmetric input.
Output:
output – filtered signal.
lfilter(b, a, x, axis=-1, zi=None)
Filter data along one-dimension with an IIR or FIR filter.
Filter a data sequence, x, using a digital filter. This works for many fundamental data types (including Object
type). The filter is a direct form II transposed implementation of the standard difference equation (see Notes).
Parameters
b : array_like
The numerator coefficient vector in a 1-D sequence.
a : array_like
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The denominator coefficient vector in a 1-D sequence. If a[0] is not 1, then both a
and b are normalized by a[0].
x : array_like
An N-dimensional input array.
axis : int
The axis of the input data array along which to apply the linear filter. The filter is
applied to each subarray along this axis (Default = -1)
zi : array_like (optional)
Initial conditions for the filter delays. It is a vector (or array of vectors for an Ndimensional input) of length max(len(a),len(b))-1. If zi=None or is not given then
initial rest is assumed. SEE signal.lfiltic for more information.
Returns
y : array
The output of the digital filter.
zf : array (optional)
If zi is None, this is not returned, otherwise, zf holds the final filter delay values.
Notes
The filter function is implemented as a direct II transposed structure. This means that the filter implements
a[0]*y[n] = b[0]*x[n] + b[1]*x[n-1] + ... + b[nb]*x[n-nb]
- a[1]*y[n-1] - ... - a[na]*y[n-na]
using the following difference equations:
y[m] = b[0]*x[m] + z[0,m-1]
z[0,m] = b[1]*x[m] + z[1,m-1] - a[1]*y[m]
...
z[n-3,m] = b[n-2]*x[m] + z[n-2,m-1] - a[n-2]*y[m]
z[n-2,m] = b[n-1]*x[m] - a[n-1]*y[m]
where m is the output sample number and n=max(len(a),len(b)) is the model order.
The rational transfer function describing this filter in the z-transform domain is:
-1
-nb
b[0] + b[1]z + ... + b[nb] z
Y(z) = ---------------------------------- X(z)
-1
-na
a[0] + a[1]z + ... + a[na] z
lfiltic(b, a, y, x=None)
Construct initial conditions for lfilter
Given a linear filter (b,a) and initial conditions on the output y and the input x, return the inital conditions on the
state vector zi which is used by lfilter to generate the output given the input.
If M=len(b)-1 and N=len(a)-1. Then, the initial conditions are given in the vectors x and y as:
x = {x[-1],x[-2],...,x[-M]}
y = {y[-1],y[-2],...,y[-N]}
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If x is not given, its inital conditions are assumed zero. If either vector is too short, then zeros are added to
achieve the proper length.
The output vector zi contains:
zi = {z_0[-1], z_1[-1], ..., z_K-1[-1]}
where K=max(M,N).
deconvolve(signal, divisor)
Deconvolves divisor out of signal.
hilbert(x, N=None, axis=-1)
Compute the analytic signal.
The transformation is done along the last axis by default.
Parameters
x : array-like
Signal data
N : int, optional
Number of Fourier components. Default: x.shape[axis]
axis : int, optional
Returns
xa : ndarray
Analytic signal of x, of each 1d array along axis
Notes
The analytic signal x_a(t) of x(t) is:
x_a = F^{-1}(F(x) 2U) = x + i y
where F is the Fourier transform, U the unit step function, and y the Hilbert transform of x. [1]
changes in scipy 0.8.0: new axis argument, new default axis=-1
References
[R36]
get_window(window, Nx, fftbins=True)
Return a window of length Nx and type window.
If fftbins is True, create a “periodic” window ready to use with ifftshift and be multiplied by the result of an fft
(SEE ALSO fftfreq).
Window types: boxcar, triang, blackman, hamming, hanning, bartlett,
parzen, bohman, blackmanharris, nuttall, barthann, kaiser (needs beta), gaussian (needs std), general_gaussian (needs power, width), slepian (needs width), chebwin (needs attenuation)
If the window requires no parameters, then it can be a string. If the window requires parameters, the window
argument should be a tuple
with the first argument the string name of the window, and the next arguments the needed parameters.
If window is a floating point number, it is interpreted as the beta
parameter of the kaiser window.
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decimate(x, q, n=None, ftype=’iir’, axis=-1)
downsample the signal x by an integer factor q, using an order n filter
By default an order 8 Chebyshev type I filter is used or a 30 point FIR filter with hamming window if ftype is
‘fir’.
Parameters
x : N-d array
the signal to be downsampled
q : int
the downsampling factor
n : int or None
the order of the filter (1 less than the length for ‘fir’)
ftype : {‘iir’ or ‘fir’}
the type of the lowpass filter
axis : int
the axis along which to decimate
Returns
y : N-d array
the down-sampled signal
See also: resample :
detrend(data, axis=-1, type=’linear’, bp=0)
Remove linear trend along axis from data.
If type is ‘constant’ then remove mean only.
If bp is given, then it is a sequence of points at which to
break a piecewise-linear fit to the data.
resample(x, num, t=None, axis=0, window=None)
Resample to num samples using Fourier method along the given axis.
The resampled signal starts at the same value of x but is sampled with a spacing of len(x) / num * (spacing of
x). Because a Fourier method is used, the signal is assumed periodic.
Window controls a Fourier-domain window that tapers the Fourier spectrum before zero-padding to alleviate
ringing in the resampled values for sampled signals you didn’t intend to be interpreted as band-limited.
If window is a function, then it is called with a vector of inputs indicating the frequency bins (i.e. fftfreq(x.shape[axis]) )
If window is an array of the same length as x.shape[axis] it is assumed to be the window to be applied directly
in the Fourier domain (with dc and low-frequency first).
If window is a string then use the named window. If window is a float, then it represents a value of beta for a
kaiser window. If window is a tuple, then the first component is a string representing the window, and the next
arguments are parameters for that window.
Possible windows are:
‘flattop’ – ‘flat’, ‘flt’ ‘boxcar’ – ‘ones’, ‘box’ ‘triang’ – ‘traing’, ‘tri’ ‘parzen’ – ‘parz’, ‘par’ ‘bohman’ –
‘bman’, ‘bmn’ ‘blackmanharris’ – ‘blackharr’, ‘bkh’ ‘nuttall’, – ‘nutl’, ‘nut’ ‘barthann’ – ‘brthan’, ‘bth’
‘blackman’ – ‘black’, ‘blk’ ‘hamming’ – ‘hamm’, ‘ham’ ‘bartlett’ – ‘bart’, ‘brt’ ‘hanning’ – ‘hann’, ‘han’
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(‘kaiser’, beta) – ‘ksr’ (‘gaussian’, std) – ‘gauss’, ‘gss’ (‘general gauss’, power, width) – ‘general’, ‘ggs’
(‘slepian’, width) – ‘slep’, ‘optimal’, ‘dss’
The first sample of the returned vector is the same as the first sample of the input vector, the spacing between
samples is changed from dx to
dx * len(x) / num
If t is not None, then it represents the old sample positions, and the new sample positions will be returned as
well as the new samples.
3.13.4 Filter design
bilinear (b, a[, fs])
firwin (N, cutoff[, width, window])
freqs (b, a[, worN, plot])
freqz (b[, a, worN, whole, ...])
iirdesign (wp, ws, gpass, gstop[, analog, ftype, output])
iirfilter (N, Wn[, rp, rs, btype, analog, ...])
kaiserord (ripple, width)
remez (numtaps, bands, desired[, weight, Hz, type, ...])
unique_roots (p[, tol, rtype])
residue (b, a[, tol, rtype])
residuez (b, a[, tol, rtype])
invres (r, p, k[, tol, rtype])
Return a digital filter from an analog filter using the bilinear
transform.
FIR Filter Design using windowed ideal filter method.
Compute frequency response of analog filter.
Compute the frequency response of a digital filter.
Complete IIR digital and analog filter design.
IIR digital and analog filter design given order and critical
points.
Design a Kaiser window to limit ripple and width of
transition region.
Calculate the minimax optimal filter using Remez exchange
algorithm.
Determine the unique roots and their multiplicities in two
lists
Compute partial-fraction expansion of b(s) / a(s).
Compute partial-fraction expansion of b(z) / a(z).
Compute b(s) and a(s) from partial fraction expansion: r,p,k
bilinear(b, a, fs=1.0)
Return a digital filter from an analog filter using the bilinear transform.
The bilinear transform substitutes (z-1) / (z+1) for s
firwin(N, cutoff, width=None, window=’hamming’)
FIR Filter Design using windowed ideal filter method.
Parameters
N – order of filter (number of taps) :
cutoff – cutoff frequency of filter (normalized so that 1 corresponds to :
Nyquist or pi radians / sample)
width – if width is not None, then assume it is the approximate width of :
the transition region (normalized so that 1 corresonds to pi) for use in kaiser FIR
filter design.
window – desired window to use. See get_window for a list :
of windows and required parameters.
Returns
h – coefficients of length N fir filter. :
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freqs(b, a, worN=None, plot=None)
Compute frequency response of analog filter.
Given the numerator (b) and denominator (a) of a filter compute its frequency response.
b[0]*(jw)**(nb-1) + b[1]*(jw)**(nb-2) + ... + b[nb-1]
H(w) = ——————————————————–
a[0]*(jw)**(na-1) + a[1]*(jw)**(na-2) + ... + a[na-1]
Parameters
b : ndarray
numerator of a linear filter
a : ndarray
numerator of a linear filter
worN : {None, int}, optional
If None, then compute at 200 frequencies around the interesting parts of the response
curve (determined by pole-zero locations). If a single integer, the compute at that
many frequencies. Otherwise, compute the response at frequencies given in worN.
Returns
w : ndarray
The frequencies at which h was computed.
h : ndarray
The frequency response.
freqz(b, a=1, worN=None, whole=0, plot=None)
Compute the frequency response of a digital filter.
Given the numerator b and denominator a of a digital filter compute its frequency response:
jw
-jw
-jmw
jw B(e)
b[0] + b[1]e + .... + b[m]e
H(e) = ---- = -----------------------------------jw
-jw
-jnw
A(e)
a[0] + a[1]e + .... + a[n]e
Parameters
b : ndarray
numerator of a linear filter
a : ndarray
numerator of a linear filter
worN : {None, int}, optional
If None, then compute at 512 frequencies around the unit circle. If a single integer,
the compute at that many frequencies. Otherwise, compute the response at frequencies given in worN
whole : {0,1}, optional
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Normally, frequencies are computed from 0 to pi (upper-half of unit-circle. If whole
is non-zero compute frequencies from 0 to 2*pi.
Returns
w : ndarray
The frequencies at which h was computed.
h : ndarray
The frequency response.
Examples
>>> b = firwin(80, 0.5, window=(’kaiser’, 8))
>>> h, w = freqz(b)
>>>
>>>
>>>
>>>
import matplotlib.pyplot as plt
fig = plt.figure()
plt.title(’Digital filter frequency response’)
ax1 = fig.add_subplot(111)
>>>
>>>
>>>
>>>
>>>
plt.semilogy(h, np.abs(w), ’b’)
plt.ylabel(’Amplitude (dB)’, color=’b’)
plt.xlabel(’Frequency (rad/sample)’)
plt.grid()
plt.legend()
>>>
>>>
>>>
>>>
>>>
ax2 = ax1.twinx()
angles = np.unwrap(np.angle(w))
plt.plot(h, angles, ’g’)
plt.ylabel(’Angle (radians)’, color=’g’)
plt.show()
iirdesign(wp, ws, gpass, gstop, analog=0, ftype=’ellip’, output=’ba’)
Complete IIR digital and analog filter design.
Given passband and stopband frequencies and gains construct an analog or digital IIR filter of minimum order
for a given basic type. Return the output in numerator, denominator (‘ba’) or pole-zero (‘zpk’) form.
Parameters
wp, ws – Passband and stopband edge frequencies, normalized from 0 :
to 1 (1 corresponds to pi radians / sample). For example:
Lowpass: wp = 0.2, ws = 0.3 Highpass: wp = 0.3, ws = 0.2 Bandpass: wp =
[0.2, 0.5], ws = [0.1, 0.6] Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
gpass – The maximum loss in the passband (dB). :
gstop – The minimum attenuation in the stopband (dB). :
analog – Non-zero to design an analog filter (in this case wp and :
ws are in radians / second).
ftype – The type of iir filter to design: :
elliptic : ‘ellip’ Butterworth : ‘butter’, Chebyshev I : ‘cheby1’, Chebyshev II:
‘cheby2’, Bessel : ‘bessel’
output – Type of output: numerator/denominator (‘ba’) or pole-zero (‘zpk’) :
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Returns
b,a – Numerator and denominator of the iir filter. :
z,p,k – Zeros, poles, and gain of the iir filter.
iirfilter(N, Wn, rp=None, rs=None, btype=’band’, analog=0, ftype=’butter’, output=’ba’)
IIR digital and analog filter design given order and critical points.
Design an Nth order lowpass digital or analog filter and return the filter coefficients in (B,A) (numerator, denominator) or (Z,P,K) form.
Parameters
N – the order of the filter. :
Wn – a scalar or length-2 sequence giving the critical frequencies. :
rp, rs – For chebyshev and elliptic filters provides the maximum ripple :
in the passband and the minimum attenuation in the stop band.
btype – the type of filter (lowpass, highpass, bandpass, or bandstop). :
analog – non-zero to return an analog filter, otherwise :
a digital filter is returned.
ftype – the type of IIR filter (Butterworth, Cauer (Elliptic), :
Bessel, Chebyshev1, Chebyshev2)
output – ‘ba’ for (b,a) output, ‘zpk’ for (z,p,k) output. :
SEE ALSO butterord, cheb1ord, cheb2ord, ellipord :
kaiserord(ripple, width)
Design a Kaiser window to limit ripple and width of transition region.
Parameters
ripple – positive number specifying maximum ripple in passband (dB) :
and minimum ripple in stopband
width – width of transition region (normalized so that 1 corresponds :
to pi radians / sample)
Returns
N, beta – the order and beta parameter for the kaiser window. :
signal.kaiser(N,beta,sym=0) returns the window as does signal.get_window(beta,N)
signal.get_window((‘kaiser’,beta),N)
Uses the empirical equations discovered by Kaiser. :
Oppenheim, Schafer, “Discrete-Time Signal Processing,”, p.475-476. :
remez(numtaps, bands, desired, weight=None, Hz=1, type=’bandpass’, maxiter=25, grid_density=16)
Calculate the minimax optimal filter using Remez exchange algorithm.
Description:
Calculate the filter-coefficients for the finite impulse response (FIR) filter whose transfer function
minimizes the maximum error between the desired gain and the realized gain in the specified bands
using the remez exchange algorithm.
Inputs:
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numtaps – The desired number of taps in the filter. bands – A montonic sequence containing the
band edges. All elements
must be non-negative and less than 1/2 the sampling frequency as given by Hz.
desired – A sequency half the size of bands containing the desired gain
in each of the specified bands
weight – A relative weighting to give to each band region. type — The type of filter:
‘bandpass’ : flat response in bands. ‘differentiator’ : frequency proportional response in
bands.
Outputs: (out,)
out – A rank-1 array containing the coefficients of the optimal
(in a minimax sense) filter.
unique_roots(p, tol=0.001, rtype=’min’)
Determine the unique roots and their multiplicities in two lists
Inputs:
p – The list of roots tol — The tolerance for two roots to be considered equal. rtype — How to
determine the returned root from the close
ones: ‘max’: pick the maximum
‘min’: pick the minimum ‘avg’: average roots
Outputs: (pout, mult)
pout – The list of sorted roots mult – The multiplicity of each root
residue(b, a, tol=0.001, rtype=’avg’)
Compute partial-fraction expansion of b(s) / a(s).
If M = len(b) and N = len(a)
b(s) b[0] s**(M-1) + b[1] s**(M-2) + ... + b[M-1]
H(s) = —— = ———————————————a(s) a[0] s**(N-1) + a[1] s**(N-2) + ... + a[N-1]
r[0] r[1] r[-1]
= ——– + ——– + ... + ——— + k(s)
(s-p[0]) (s-p[1]) (s-p[-1])
If there are any repeated roots (closer than tol), then the partial fraction expansion has terms like
r[i] r[i+1] r[i+n-1]
——– + ———– + ... + ———– (s-p[i]) (s-p[i])**2 (s-p[i])**n
Returns
r : ndarray
Residues
p : ndarray
Poles
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k : ndarray
Coefficients of the direct polynomial term.
See Also:
invres, poly, polyval, unique_roots
residuez(b, a, tol=0.001, rtype=’avg’)
Compute partial-fraction expansion of b(z) / a(z).
If M = len(b) and N = len(a)
b(z) b[0] + b[1] z**(-1) + ... + b[M-1] z**(-M+1)
H(z) = —— = ———————————————a(z) a[0] + a[1] z**(-1) + ... + a[N-1] z**(-N+1)
r[0] r[-1]
= ————— + ... + —————- + k[0] + k[1]z**(-1) ...
(1-p[0]z**(-1)) (1-p[-1]z**(-1))
If there are any repeated roots (closer than tol), then the partial fraction expansion has terms like
r[i] r[i+1] r[i+n-1]
————– + —————— + ... + —————— (1-p[i]z**(-1)) (1-p[i]z**(-1))**2 (1-p[i]z**(1))**n
See also: invresz, poly, polyval, unique_roots
invres(r, p, k, tol=0.001, rtype=’avg’)
Compute b(s) and a(s) from partial fraction expansion: r,p,k
If M = len(b) and N = len(a)
b(s) b[0] x**(M-1) + b[1] x**(M-2) + ... + b[M-1]
H(s) = —— = ———————————————a(s) a[0] x**(N-1) + a[1] x**(N-2) + ... + a[N-1]
r[0] r[1] r[-1]
= ——– + ——– + ... + ——— + k(s)
(s-p[0]) (s-p[1]) (s-p[-1])
If there are any repeated roots (closer than tol), then the partial fraction expansion has terms like
r[i] r[i+1] r[i+n-1]
——– + ———– + ... + ———– (s-p[i]) (s-p[i])**2 (s-p[i])**n
See Also:
residue, poly, polyval, unique_roots
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3.13.5 Matlab-style IIR filter design
butter (N, Wn[, btype, analog, output])
buttord (wp, ws, gpass, gstop[, analog])
cheby1 (N, rp, Wn[, btype, analog, output])
cheb1ord (wp, ws, gpass, gstop[, analog])
cheby2 (N, rs, Wn[, btype, analog, output])
cheb2ord (wp, ws, gpass, gstop[, analog])
ellip (N, rp, rs, Wn[, btype, analog, output])
ellipord (wp, ws, gpass, gstop[, analog])
bessel (N, Wn[, btype, analog, output])
Butterworth digital and analog filter design.
Butterworth filter order selection.
Chebyshev type I digital and analog filter design.
Chebyshev type I filter order selection.
Chebyshev type I digital and analog filter design.
Chebyshev type II filter order selection.
Elliptic (Cauer) digital and analog filter design.
Elliptic (Cauer) filter order selection.
Bessel digital and analog filter design.
butter(N, Wn, btype=’low’, analog=0, output=’ba’)
Butterworth digital and analog filter design.
Description:
Design an Nth order lowpass digital or analog Butterworth filter and return the filter coefficients in
(B,A) or (Z,P,K) form.
See also buttord.
buttord(wp, ws, gpass, gstop, analog=0)
Butterworth filter order selection.
Return the order of the lowest order digital Butterworth filter that loses no more than gpass dB in the passband
and has at least gstop dB attenuation in the stopband.
Parameters
wp, ws – Passband and stopband edge frequencies, normalized from 0 :
to 1 (1 corresponds to pi radians / sample). For example:
Lowpass: wp = 0.2, ws = 0.3 Highpass: wp = 0.3, ws = 0.2 Bandpass: wp =
[0.2, 0.5], ws = [0.1, 0.6] Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
gpass – The maximum loss in the passband (dB). :
gstop – The minimum attenuation in the stopband (dB). :
analog – Non-zero to design an analog filter (in this case wp and :
ws are in radians / second).
Returns
ord – The lowest order for a Butterworth filter which meets specs. :
Wn – The Butterworth natural frequency (i.e. the “3dB frequency”). :
Should be used with scipy.signal.butter to give filter results.
cheby1(N, rp, Wn, btype=’low’, analog=0, output=’ba’)
Chebyshev type I digital and analog filter design.
Description:
Design an Nth order lowpass digital or analog Chebyshev type I filter and return the filter coefficients in (B,A) or (Z,P,K) form.
See also cheb1ord.
cheb1ord(wp, ws, gpass, gstop, analog=0)
Chebyshev type I filter order selection.
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Return the order of the lowest order digital Chebyshev Type I filter that loses no more than gpass dB in the
passband and has at least gstop dB attenuation in the stopband.
Parameters
wp, ws – Passband and stopband edge frequencies, normalized from 0 :
to 1 (1 corresponds to pi radians / sample). For example:
Lowpass: wp = 0.2, ws = 0.3 Highpass: wp = 0.3, ws = 0.2 Bandpass: wp =
[0.2, 0.5], ws = [0.1, 0.6] Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
gpass – The maximum loss in the passband (dB). :
gstop – The minimum attenuation in the stopband (dB). :
analog – Non-zero to design an analog filter (in this case wp and :
ws are in radians / second).
Returns
ord – The lowest order for a Chebyshev type I filter that meets specs. :
Wn – The Chebyshev natural frequency (the “3dB frequency”) for :
use with scipy.signal.cheby1 to give filter results.
cheby2(N, rs, Wn, btype=’low’, analog=0, output=’ba’)
Chebyshev type I digital and analog filter design.
Description:
Design an Nth order lowpass digital or analog Chebyshev type I filter and return the filter coefficients in (B,A) or (Z,P,K) form.
See also cheb2ord.
cheb2ord(wp, ws, gpass, gstop, analog=0)
Chebyshev type II filter order selection.
Description:
Return the order of the lowest order digital Chebyshev Type II filter that loses no more than gpass
dB in the passband and has at least gstop dB attenuation in the stopband.
Parameters
wp, ws – Passband and stopband edge frequencies, normalized from 0 :
to 1 (1 corresponds to pi radians / sample). For example:
Lowpass: wp = 0.2, ws = 0.3 Highpass: wp = 0.3, ws = 0.2 Bandpass: wp =
[0.2, 0.5], ws = [0.1, 0.6] Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
gpass – The maximum loss in the passband (dB). :
gstop – The minimum attenuation in the stopband (dB). :
analog – Non-zero to design an analog filter (in this case wp and :
ws are in radians / second).
Returns
ord – The lowest order for a Chebyshev type II filter that meets specs. :
Wn – The Chebyshev natural frequency for :
use with scipy.signal.cheby2 to give the filter.
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ellip(N, rp, rs, Wn, btype=’low’, analog=0, output=’ba’)
Elliptic (Cauer) digital and analog filter design.
Description:
Design an Nth order lowpass digital or analog elliptic filter and return the filter coefficients in (B,A)
or (Z,P,K) form.
See also ellipord.
ellipord(wp, ws, gpass, gstop, analog=0)
Elliptic (Cauer) filter order selection.
Return the order of the lowest order digital elliptic filter that loses no more than gpass dB in the passband and
has at least gstop dB attenuation in the stopband.
Parameters
wp, ws – Passband and stopband edge frequencies, normalized from 0 :
to 1 (1 corresponds to pi radians / sample). For example:
Lowpass: wp = 0.2, ws = 0.3 Highpass: wp = 0.3, ws = 0.2 Bandpass: wp =
[0.2, 0.5], ws = [0.1, 0.6] Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
gpass – The maximum loss in the passband (dB). :
gstop – The minimum attenuation in the stopband (dB). :
analog – Non-zero to design an analog filter (in this case wp and :
ws are in radians / second).
Returns
ord – The lowest order for an Elliptic (Cauer) filter that meets specs. :
Wn – The natural frequency for use with scipy.signal.ellip :
to give the filter.
bessel(N, Wn, btype=’low’, analog=0, output=’ba’)
Bessel digital and analog filter design.
Description:
Design an Nth order lowpass digital or analog Bessel filter and return the filter coefficients in (B,A)
or (Z,P,K) form.
3.13.6 Linear Systems
lti
lsim (system, U, T[, X0, interp])
lsim2 (system[, U, T, X0, **kwargs)
impulse (system[, X0, T, N])
impulse2 (system[, X0, T, N, **kwargs)
step (system[, X0, T, N])
step2 (system[, X0, T, N, **kwargs)
Linear Time Invariant class which simplifies representation.
Simulate output of a continuous-time linear system.
Simulate output of a continuous-time linear system, by using the ODE solver
scipy.integrate.odeint.
Impulse response of continuous-time system.
Impulse response of a single-input continuous-time linear system.
Step response of continuous-time system.
Step response of continuous-time system.
class lti(*args, **kwords)
Linear Time Invariant class which simplifies representation.
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Methods
lsim(system, U, T, X0=None, interp=1)
Simulate output of a continuous-time linear system.
Parameters
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and the interpretation:
• 2: (num, den)
• 3: (zeros, poles, gain)
• 4: (A, B, C, D)
U : array_like
An input array describing the input at each time T (interpolation is assumed between
given times). If there are multiple inputs, then each column of the rank-2 array
represents an input.
T : array_like
The time steps at which the input is defined and at which the output is desired.
X0 : :
The initial conditions on the state vector (zero by default).
interp : {1, 0}
Whether to use linear (1) or zero-order hold (0) interpolation.
Returns
T : 1D ndarray
Time values for the output.
yout : 1D ndarray
System response.
xout : ndarray
Time-evolution of the state-vector.
lsim2(system, U=None, T=None, X0=None, **kwargs)
Simulate output of a continuous-time linear system, by using the ODE solver scipy.integrate.odeint.
Parameters
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and the interpretation:
• 2: (num, den)
• 3: (zeros, poles, gain)
• 4: (A, B, C, D)
U : ndarray or array-like (1D or 2D), optional
An input array describing the input at each time T. Linear interpolation is used between given times. If there are multiple inputs, then each column of the rank-2 array
represents an input. If U is not given, the input is assumed to be zero.
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T : ndarray or array-like (1D or 2D), optional
The time steps at which the input is defined and at which the output is desired. The
default is 101 evenly spaced points on the interval [0,10.0].
X0 : ndarray or array-like (1D), optional
The initial condition of the state vector. If X0 is not given, the initial conditions are
assumed to be 0.
kwargs : dict
Additional keyword arguments are passed on to the function odeint. See the notes
below for more details.
Returns
T : 1D ndarray
The time values for the output.
yout : ndarray
The response of the system.
xout : ndarray
The time-evolution of the state-vector.
Notes
This function uses scipy.integrate.odeint to solve the system’s differential equations. Additional keyword arguments given to lsim2 are passed on to odeint.
See the documentation for
scipy.integrate.odeint for the full list of arguments.
impulse(system, X0=None, T=None, N=None)
Impulse response of continuous-time system.
Parameters
system : LTI class or tuple
If specified as a tuple, the system is described as (num, den), (zero, pole,
gain), or (A, B, C, D).
X0 : array_like, optional
Initial state-vector. Defaults to zero.
T : array_like, optional
Time points. Computed if not given.
N : int, optional
The number of time points to compute (if T is not given).
Returns
T : 1D ndarray
Time points.
yout : 1D ndarray
Impulse response of the system (except for singularities at zero).
impulse2(system, X0=None, T=None, N=None, **kwargs)
Impulse response of a single-input continuous-time linear system.
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The solution is generated by calling scipy.signal.lsim2, which uses the differential equation solver
scipy.integrate.odeint.
Parameters
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and the interpretation.
2 (num, den) 3 (zeros, poles, gain) 4 (A, B, C, D)
T : 1D ndarray or array-like, optional
The time steps at which the input is defined and at which the output is desired. If T
is not given, the function will generate a set of time samples automatically.
X0 : 1D ndarray or array-like, optional
The initial condition of the state vector. If X0 is None, the initial conditions are
assumed to be 0.
N : int, optional
Number of time points to compute. If N is not given, 100 points are used.
**kwargs : :
Additional
keyword
arguments
are
passed
on
the
function
scipy.signal.lsim2,
which
in
turn
passes
them
on
to
scipy.integrate.odeint.
See
the
documentation
for
scipy.integrate.odeint for information about these arguments.
Returns
T : 1D ndarray
The time values for the output.
yout : ndarray
The output response of the system.
See Also:
scipy.signal.impulse
Notes
New in version 0.8.0.
step(system, X0=None, T=None, N=None)
Step response of continuous-time system.
Parameters
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and the interpretation.
2 (num, den) 3 (zeros, poles, gain) 4 (A, B, C, D)
X0 : array_like, optional
Initial state-vector (default is zero).
T : array_like, optional
Time points (computed if not given).
N : int
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Number of time points to compute if T is not given.
Returns
T : 1D ndarray
Output time points.
yout : 1D ndarray
Step response of system.
See Also:
scipy.signal.step2
step2(system, X0=None, T=None, N=None, **kwargs)
Step response of continuous-time system.
This function is functionally the same as scipy.signal.step,
scipy.signal.lsim2 to compute the step response.
but it uses the function
Parameters
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and the interpretation.
2 (num, den) 3 (zeros, poles, gain) 4 (A, B, C, D)
X0 : array_like, optional
Initial state-vector (default is zero).
T : array_like, optional
Time points (computed if not given).
N : int
Number of time points to compute if T is not given.
**kwargs : :
Additional
keyword
arguments
are
passed
on
the
function
scipy.signal.lsim2,
which
in
turn
passes
them
on
to
scipy.integrate.odeint.
See
the
documentation
for
scipy.integrate.odeint for information about these arguments.
Returns
T : 1D ndarray
Output time points.
yout : 1D ndarray
Step response of system.
See Also:
scipy.signal.step
Notes
New in version 0.8.0.
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3.13.7 LTI Representations
tf2zpk (b, a)
Return zero, pole, gain (z,p,k) representation from a numerator, denominator
representation of a linear filter.
Return polynomial transfer function representation from zeros and poles
Transfer function to state-space representation.
State-space to transfer function.
zpk2tf (z, p, k)
tf2ss (num, den)
ss2tf (A, B, C, D[, input])
zpk2ss (z, p, k)
Zero-pole-gain representation to state-space representation
ss2zpk (A, B, C, D[, in- State-space representation to zero-pole-gain representation.
put])
tf2zpk(b, a)
Return zero, pole, gain (z,p,k) representation from a numerator, denominator representation of a linear filter.
Parameters
b : ndarray
numerator polynomial.
a : ndarray
numerator and denominator polynomials.
Returns
z : ndarray
zeros of the transfer function.
p : ndarray
poles of the transfer function.
k : float
system gain.
If some values of b are too close to 0, they are removed. In that case, a :
BadCoefficients warning is emitted. :
zpk2tf(z, p, k)
Return polynomial transfer function representation from zeros and poles
Parameters
z : ndarray
zeros of the transfer function.
p : ndarray
poles of the transfer function.
k : float
system gain.
Returns
b : ndarray
numerator polynomial.
a : ndarray
numerator and denominator polynomials.
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tf2ss(num, den)
Transfer function to state-space representation.
Parameters
num, den : array_like
Sequences representing the numerator and denominator polynomials.
Returns
A, B, C, D : ndarray
State space representation of the system.
ss2tf(A, B, C, D, input=0)
State-space to transfer function.
Parameters
A, B, C, D : ndarray
State-space representation of linear system.
input : int
For multiple-input systems, the input to use.
Returns
num, den : 1D ndarray
Numerator and denominator polynomials (as sequences) respectively.
zpk2ss(z, p, k)
Zero-pole-gain representation to state-space representation
Parameters
z, p : sequence
Zeros and poles.
k : float
System gain.
Returns
A, B, C, D : ndarray
State-space matrices.
ss2zpk(A, B, C, D, input=0)
State-space representation to zero-pole-gain representation.
Inputs:
A, B, C, D – state-space matrices. input – for multiple-input systems, the input to use.
Outputs:
z, p, k – zeros and poles in sequences and gain constant.
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3.13.8 Waveforms
chirp (t, f0, t1, f1[, method, phi, vertex_zero, ...])
gausspulse (t[, fc, bw, bwr, tpr, ...])
sawtooth (t[, width])
square (t[, duty])
sweep_poly (t, poly[, phi])
Frequency-swept cosine generator.
Return a gaussian modulated sinusoid: exp(-a t^2) exp(1j*2*pi*fc).
Return a periodic sawtooth waveform.
Return a periodic square-wave waveform.
Frequency-swept cosine generator, with a time-dependent frequency
specified as a polynomial.
chirp(t, f0, t1, f1, method=’linear’, phi=0, vertex_zero=True, qshape=None)
Frequency-swept cosine generator.
In the following, ‘Hz’ should be interpreted as ‘cycles per time unit’; there is no assumption here that the time
unit is one second. The important distinction is that the units of rotation are cycles, not radians.
Parameters
t : ndarray
Times at which to evaluate the waveform.
f0 : float
Frequency (in Hz) at time t=0.
t1 : float
Time at which f1 is specified.
f1 : float
Frequency (in Hz) of the waveform at time t1.
method : {‘linear’, ‘quadratic’, ‘logarithmic’, ‘hyperbolic’}, optional
Kind of frequency sweep. If not given, linear is assumed. See Notes below for more
details.
phi : float, optional
Phase offset, in degrees. Default is 0.
vertex_zero : bool, optional
This parameter is only used when method is ‘quadratic’. It determines whether the
vertex of the parabola that is the graph of the frequency is at t=0 or t=t1.
qshape : str (deprecated)
If method is quadratic and qshape is not None, chirp() will use
scipy.signal.waveforms.old_chirp to compute the wave form. This parameter
is deprecated, and will be removed in SciPy 0.9.
Returns
A numpy array containing the signal evaluated at ‘t’ with the requested :
time-varying frequency. More precisely, the function returns: :
cos(phase + (pi/180)*phi)
where ‘phase‘ is the integral (from 0 to t) of ‘‘2*pi*f(t)‘‘. :
‘‘f(t)‘‘ is defined below. :
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See Also:
scipy.signal.waveforms.sweep_poly
Notes
There are four options for the method. The following formulas give the instantaneous frequency (in Hz) of the
signal generated by chirp(). For convenience, the shorter names shown below may also be used.
linear, lin, li:
f(t) = f0 + (f1 - f0) * t / t1
quadratic, quad, q:
The graph of the frequency f(t) is a parabola through (0, f0) and (t1, f1). By default, the vertex of
the parabola is at (0, f0). If vertex_zero is False, then the vertex is at (t1, f1). The formula is:
if vertex_zero is True:
f(t) = f0 + (f1 - f0) * t**2 / t1**2
else:
f(t) = f1 - (f1 - f0) * (t1 - t)**2 / t1**2
To use a more general quadratic function, or an arbitrary polynomial, use the function
scipy.signal.waveforms.sweep_poly.
logarithmic, log, lo:
f(t) = f0 * (f1/f0)**(t/t1)
f0 and f1 must be nonzero and have the same sign.
This signal is also known as a geometric or exponential chirp.
hyperbolic, hyp:
f(t) = f0*f1*t1 / ((f0 - f1)*t + f1*t1)
f1 must be positive, and f0 must be greater than f1.
gausspulse(t, fc=1000, bw=0.5, bwr=-6, tpr=-60, retquad=0, retenv=0)
Return a gaussian modulated sinusoid: exp(-a t^2) exp(1j*2*pi*fc).
If retquad is non-zero, then return the real and imaginary parts (in-phase and quadrature) If retenv is non-zero,
then return the envelope (unmodulated signal). Otherwise, return the real part of the modulated sinusoid.
Parameters
t : ndarray
Input array.
fc : int, optional
Center frequency (Hz).
bw : float, optional
Fractional bandwidth in frequency domain of pulse (Hz).
bwr: float, optional :
Reference level at which fractional bandwidth is calculated (dB).
tpr : float, optional
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If t is ‘cutoff’, then the function returns the cutoff time for when the pulse amplitude
falls below tpr (in dB).
retquad : int, optional
Return the quadrature (imaginary) as well as the real part of the signal.
retenv : int, optional
Return the envelope of the signal.
sawtooth(t, width=1)
Return a periodic sawtooth waveform.
The sawtooth waveform has a period 2*pi, rises from -1 to 1 on the interval 0 to width*2*pi and drops from 1
to -1 on the interval width*2*pi to 2*pi. width must be in the interval [0,1].
Parameters
t : array_like
Time.
width : float, optional
Width of the waveform. Default is 1.
Returns
y : ndarray
Output array containing the sawtooth waveform.
Examples
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 20*np.pi, 500)
>>> plt.plot(x, sp.signal.sawtooth(x))
square(t, duty=0.5)
Return a periodic square-wave waveform.
The square wave has a period 2*pi, has value +1 from 0 to 2*pi*duty and -1 from 2*pi*duty to 2*pi. duty must
be in the interval [0,1].
Parameters
t : array_like
The input time array.
duty : float, optional
Duty cycle.
Returns
y : array_like
The output square wave.
sweep_poly(t, poly, phi=0)
Frequency-swept cosine generator, with a time-dependent frequency specified as a polynomial.
This function generates a sinusoidal function whose instantaneous frequency varies with time. The frequency at
time t is given by the polynomial poly.
Parameters
t : ndarray
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Times at which to evaluate the waveform.
poly : 1D ndarray (or array-like), or instance of numpy.poly1d
The desired frequency expressed as a polynomial. If poly is a list or ndarray of
length n, then the elements of poly are the coefficients of the polynomial, and the
instantaneous frequency is
f(t) = poly[0]*t**(n-1) + poly[1]*t**(n-2) + ...
+ poly[n-1]
If poly is an instance of numpy.poly1d, then the instantaneous frequency is
f(t) = poly(t)
phi : float, optional
Phase offset, in degrees. Default is 0.
Returns
A numpy array containing the signal evaluated at ‘t’ with the requested :
time-varying frequency. More precisely, the function returns :
cos(phase + (pi/180)*phi)
where ‘phase‘ is the integral (from 0 to t) of ‘‘2 * pi * f(t)‘‘; :
‘‘f(t)‘‘ is defined above. :
See Also:
scipy.signal.waveforms.chirp
Notes
New in version 0.8.0.
3.13.9 Window functions
get_window (window, Nx[, fftbins])
barthann (M[, sym])
bartlett (M[, sym])
blackman (M[, sym])
blackmanharris (M[, sym])
bohman (M[, sym])
boxcar (M[, sym])
chebwin (M, at[, sym])
flattop (M[, sym])
gaussian (M, std[, sym])
general_gaussian (M, p, sig[, sym])
hamming (M[, sym])
hann (M[, sym])
kaiser (M, beta[, sym])
nuttall (M[, sym])
parzen (M[, sym])
slepian (M, width[, sym])
triang (M[, sym])
Return a window of length Nx and type window.
Return the M-point modified Bartlett-Hann window.
The M-point Bartlett window.
The M-point Blackman window.
The M-point minimum 4-term Blackman-Harris window.
The M-point Bohman window.
The M-point boxcar window.
Dolph-Chebyshev window.
The M-point Flat top window.
Return a Gaussian window of length M with standard-deviation std.
Return a window with a generalized Gaussian shape.
The M-point Hamming window.
The M-point Hanning window.
Return a Kaiser window of length M with shape parameter beta.
A minimum 4-term Blackman-Harris window according to Nuttall.
The M-point Parzen window.
Return the M-point slepian window.
The M-point triangular window.
get_window(window, Nx, fftbins=True)
Return a window of length Nx and type window.
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If fftbins is True, create a “periodic” window ready to use with ifftshift and be multiplied by the result of an fft
(SEE ALSO fftfreq).
Window types: boxcar, triang, blackman, hamming, hanning, bartlett,
parzen, bohman, blackmanharris, nuttall, barthann, kaiser (needs beta), gaussian (needs std), general_gaussian (needs power, width), slepian (needs width), chebwin (needs attenuation)
If the window requires no parameters, then it can be a string. If the window requires parameters, the window
argument should be a tuple
with the first argument the string name of the window, and the next arguments the needed parameters.
If window is a floating point number, it is interpreted as the beta
parameter of the kaiser window.
barthann(M, sym=True)
Return the M-point modified Bartlett-Hann window.
bartlett(M, sym=True)
The M-point Bartlett window.
blackman(M, sym=True)
The M-point Blackman window.
blackmanharris(M, sym=True)
The M-point minimum 4-term Blackman-Harris window.
bohman(M, sym=True)
The M-point Bohman window.
boxcar(M, sym=True)
The M-point boxcar window.
chebwin(M, at, sym=True)
Dolph-Chebyshev window.
INPUTS:
M
[int] Window size
at
[float] Attenuation (in dB)
sym
[bool] Generates symmetric window if True.
flattop(M, sym=True)
The M-point Flat top window.
gaussian(M, std, sym=True)
Return a Gaussian window of length M with standard-deviation std.
general_gaussian(M, p, sig, sym=True)
Return a window with a generalized Gaussian shape.
exp(-0.5*(x/sig)**(2*p))
half power point is at (2*log(2)))**(1/(2*p))*sig
hamming(M, sym=True)
The M-point Hamming window.
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hann(M, sym=True)
The M-point Hanning window.
kaiser(M, beta, sym=True)
Return a Kaiser window of length M with shape parameter beta.
nuttall(M, sym=True)
A minimum 4-term Blackman-Harris window according to Nuttall.
parzen(M, sym=True)
The M-point Parzen window.
slepian(M, width, sym=True)
Return the M-point slepian window.
triang(M, sym=True)
The M-point triangular window.
3.13.10 Wavelets
cascade (hk[, J])
daub (p)
morlet (M[, w, s, complete])
qmf (hk)
(x,phi,psi) at dyadic points K/2**J from filter coefficients.
The coefficients for the FIR low-pass filter producing Daubechies wavelets.
Complex Morlet wavelet.
Return high-pass qmf filter from low-pass
cascade(hk, J=7)
(x,phi,psi) at dyadic points K/2**J from filter coefficients.
Inputs:
hk – coefficients of low-pass filter J – values will be computed at grid points $K/2^J$
Outputs:
x – the dyadic points $K/2^J$ for $K=0...N*(2^J)-1$
where len(hk)=len(gk)=N+1
phi – the scaling function phi(x) at x
$phi(x) = sum_{k=0}^{N} h_k phi(2x-k)$
psi – the wavelet function psi(x) at x
$psi(x) = sum_{k=0}^N g_k phi(2x-k)$
Only returned if gk is not None
Algorithm:
Uses the vector cascade algorithm described by Strang and Nguyen in “Wavelets and Filter Banks”
Builds a dictionary of values and slices for quick reuse. Then inserts vectors into final vector at then
end
daub(p)
The coefficients for the FIR low-pass filter producing Daubechies wavelets.
p>=1 gives the order of the zero at f=1/2. There are 2p filter coefficients.
Parameters
p : int
Order of the zero at f=1/2, can have values from 1 to 34.
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morlet(M, w=5.0, s=1.0, complete=True)
Complex Morlet wavelet.
Parameters
M : int
Length of the wavelet.
w : float
Omega0
s : float
Scaling factor, windowed from -s*2*pi to +s*2*pi.
complete : bool
Whether to use the complete or the standard version.
Notes
The standard version:
pi**-0.25 * exp(1j*w*x) * exp(-0.5*(x**2))
This commonly used wavelet is often referred to simply as the Morlet wavelet. Note that, this simplified
version can cause admissibility problems at low values of w.
The complete version:
pi**-0.25 * (exp(1j*w*x) - exp(-0.5*(w**2))) * exp(-0.5*(x**2))
The complete version of the Morlet wavelet, with a correction term to improve admissibility. For w greater
than 5, the correction term is negligible.
Note that the energy of the return wavelet is not normalised according to s.
The fundamental frequency of this wavelet in Hz is given by f = 2*s*w*r / M where r is the sampling rate.
qmf(hk)
Return high-pass qmf filter from low-pass
3.14 Sparse matrices (scipy.sparse)
3.14.1 Sparse Matrices
Scipy 2D sparse matrix module.
Original code by Travis Oliphant. Modified and extended by Ed Schofield, Robert Cimrman, and Nathan Bell.
There are seven available sparse matrix types:
1. csc_matrix: Compressed Sparse Column format
2. csr_matrix: Compressed Sparse Row format
3. bsr_matrix: Block Sparse Row format
4. lil_matrix: List of Lists format
5. dok_matrix: Dictionary of Keys format
6. coo_matrix: COOrdinate format (aka IJV, triplet format)
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7. dia_matrix: DIAgonal format
To construct a matrix efficiently, use either lil_matrix (recommended) or dok_matrix. The lil_matrix class supports
basic slicing and fancy indexing with a similar syntax to NumPy arrays. As illustrated below, the COO format may
also be used to efficiently construct matrices.
To perform manipulations such as multiplication or inversion, first convert the matrix to either CSC or CSR format.
The lil_matrix format is row-based, so conversion to CSR is efficient, whereas conversion to CSC is less so.
All conversions among the CSR, CSC, and COO formats are efficient, linear-time operations.
3.14.2 Example 1
Construct a 1000x1000 lil_matrix and add some values to it:
>>>
>>>
>>>
>>>
>>>
>>>
>>>
from scipy import sparse, linsolve
from numpy import linalg
from numpy.random import rand
A = sparse.lil_matrix((1000, 1000))
A[0, :100] = rand(100)
A[1, 100:200] = A[0, :100]
A.setdiag(rand(1000))
Now convert it to CSR format and solve A x = b for x:
>>> A = A.tocsr()
>>> b = rand(1000)
>>> x = linsolve.spsolve(A, b)
Convert it to a dense matrix and solve, and check that the result is the same:
>>> x_ = linalg.solve(A.todense(), b)
Now we can compute norm of the error with:
>>> err = linalg.norm(x-x_)
>>> err < 1e-10
True
It should be small :)
3.14.3 Example 2
Construct a matrix in COO format:
>>>
>>>
>>>
>>>
>>>
>>>
from scipy import sparse
from numpy import array
I = array([0,3,1,0])
J = array([0,3,1,2])
V = array([4,5,7,9])
A = sparse.coo_matrix((V,(I,J)),shape=(4,4))
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Notice that the indices do not need to be sorted.
Duplicate (i,j) entries are summed when converting to CSR or CSC.
>>>
>>>
>>>
>>>
I
J
V
B
=
=
=
=
array([0,0,1,3,1,0,0])
array([0,2,1,3,1,0,0])
array([1,1,1,1,1,1,1])
sparse.coo_matrix((V,(I,J)),shape=(4,4)).tocsr()
This is useful for constructing finite-element stiffness and mass matrices.
3.14.4 Further Details
CSR column indices are not necessarily sorted. Likewise for CSC row indices. Use the .sorted_indices() and
.sort_indices() methods when sorted indices are required (e.g. when passing data to other libraries).
3.14.5 Sparse matrix classes
csc_matrix
csr_matrix
bsr_matrix
lil_matrix
dok_matrix
coo_matrix
dia_matrix
Compressed Sparse Column matrix
Compressed Sparse Row matrix
Block Sparse Row matrix
Row-based linked list sparse matrix
Dictionary Of Keys based sparse matrix.
A sparse matrix in COOrdinate format.
Sparse matrix with DIAgonal storage
class csc_matrix(arg1, shape=None, dtype=None, copy=False, dims=None, nzmax=None)
Compressed Sparse Column matrix
This can be instantiated in several ways:
csc_matrix(D)
with a dense matrix or rank-2 ndarray D
csc_matrix(S)
with another sparse matrix S (equivalent to S.tocsc())
csc_matrix((M, N), [dtype])
to construct an empty matrix with shape (M, N) dtype is optional, defaulting to dtype=’d’.
csc_matrix((data, ij), [shape=(M, N)])
where data and ij satisfy the relationship a[ij[0, k], ij[1, k]] = data[k]
csc_matrix((data, indices, indptr), [shape=(M, N)])
is the standard CSC representation where the row indices for column i are stored in
indices[indptr[i]:indices[i+1]] and their corresponding values are stored in
data[indptr[i]:indptr[i+1]]. If the shape parameter is not supplied, the matrix dimensions are inferred from the index arrays.
Notes
Advantages of the CSC format
• efficient arithmetic operations CSC + CSC, CSC * CSC, etc.
• efficient column slicing
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• fast matrix vector products (CSR, BSR may be faster)
Disadvantages of the CSC format
• slow row slicing operations (consider CSR)
• changes to the sparsity structure are expensive (consider LIL or DOK)
Examples
>>> from scipy.sparse import *
>>> from scipy import *
>>> csc_matrix( (3,4), dtype=int8 ).todense()
matrix([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
>>> row = array([0,2,2,0,1,2])
>>> col = array([0,0,1,2,2,2])
>>> data = array([1,2,3,4,5,6])
>>> csc_matrix( (data,(row,col)), shape=(3,3) ).todense()
matrix([[1, 0, 4],
[0, 0, 5],
[2, 3, 6]])
>>> indptr = array([0,2,3,6])
>>> indices = array([0,2,2,0,1,2])
>>> data = array([1,2,3,4,5,6])
>>> csc_matrix( (data,indices,indptr), shape=(3,3) ).todense()
matrix([[1, 0, 4],
[0, 0, 5],
[2, 3, 6]])
Methods
class csr_matrix(arg1, shape=None, dtype=None, copy=False, dims=None, nzmax=None)
Compressed Sparse Row matrix
This can be instantiated in several ways:
csr_matrix(D)
with a dense matrix or rank-2 ndarray D
csr_matrix(S)
with another sparse matrix S (equivalent to S.tocsr())
csr_matrix((M, N), [dtype])
to construct an empty matrix with shape (M, N) dtype is optional, defaulting to dtype=’d’.
csr_matrix((data, ij), [shape=(M, N)])
where data and ij satisfy the relationship a[ij[0, k], ij[1, k]] = data[k]
csr_matrix((data, indices, indptr), [shape=(M, N)])
is the standard CSR representation where the column indices for row i are stored in
indices[indptr[i]:indices[i+1]] and their corresponding values are stored in
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data[indptr[i]:indptr[i+1]]. If the shape parameter is not supplied, the matrix dimensions are inferred from the index arrays.
Notes
Advantages of the CSR format
• efficient arithmetic operations CSR + CSR, CSR * CSR, etc.
• efficient row slicing
• fast matrix vector products
Disadvantages of the CSR format
• slow column slicing operations (consider CSC)
• changes to the sparsity structure are expensive (consider LIL or DOK)
Examples
>>> from scipy.sparse import *
>>> from scipy import *
>>> csr_matrix( (3,4), dtype=int8 ).todense()
matrix([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
>>> row = array([0,0,1,2,2,2])
>>> col = array([0,2,2,0,1,2])
>>> data = array([1,2,3,4,5,6])
>>> csr_matrix( (data,(row,col)), shape=(3,3) ).todense()
matrix([[1, 0, 2],
[0, 0, 3],
[4, 5, 6]])
>>> indptr = array([0,2,3,6])
>>> indices = array([0,2,2,0,1,2])
>>> data = array([1,2,3,4,5,6])
>>> csr_matrix( (data,indices,indptr), shape=(3,3) ).todense()
matrix([[1, 0, 2],
[0, 0, 3],
[4, 5, 6]])
Methods
class bsr_matrix(arg1, shape=None, dtype=None, copy=False, blocksize=None)
Block Sparse Row matrix
This can be instantiated in several ways:
bsr_matrix(D, [blocksize=(R,C)])
with a dense matrix or rank-2 ndarray D
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bsr_matrix(S, [blocksize=(R,C)])
with another sparse matrix S (equivalent to S.tobsr())
bsr_matrix((M, N), [blocksize=(R,C), dtype])
to construct an empty matrix with shape (M, N) dtype is optional, defaulting to dtype=’d’.
bsr_matrix((data, ij), [blocksize=(R,C), shape=(M, N)])
where data and ij satisfy a[ij[0, k], ij[1, k]] = data[k]
bsr_matrix((data, indices, indptr), [shape=(M, N)])
is the standard BSR representation where the block column indices for row i are stored in
indices[indptr[i]:indices[i+1]] and their corresponding block values are stored in
data[ indptr[i]: indptr[i+1] ]. If the shape parameter is not supplied, the matrix
dimensions are inferred from the index arrays.
Notes
Summary
• The Block Compressed Row (BSR) format is very similar to the Compressed Sparse Row (CSR)
format. BSR is appropriate for sparse matrices with dense sub matrices like the last example below.
Block matrices often arise in vector-valued finite element discretizations. In such cases, BSR is
considerably more efficient than CSR and CSC for many sparse arithmetic operations.
Blocksize
• The blocksize (R,C) must evenly divide the shape of the matrix (M,N). That is, R and C must satisfy
the relationship M % R = 0 and N % C = 0.
• If no blocksize is specified, a simple heuristic is applied to determine an appropriate blocksize.
Examples
>>> from scipy.sparse import *
>>> from scipy import *
>>> bsr_matrix( (3,4), dtype=int8 ).todense()
matrix([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
>>> row = array([0,0,1,2,2,2])
>>> col = array([0,2,2,0,1,2])
>>> data = array([1,2,3,4,5,6])
>>> bsr_matrix( (data,(row,col)), shape=(3,3) ).todense()
matrix([[1, 0, 2],
[0, 0, 3],
[4, 5, 6]])
>>> indptr = array([0,2,3,6])
>>> indices = array([0,2,2,0,1,2])
>>> data
= array([1,2,3,4,5,6]).repeat(4).reshape(6,2,2)
>>> bsr_matrix( (data,indices,indptr), shape=(6,6) ).todense()
matrix([[1, 1, 0, 0, 2, 2],
[1, 1, 0, 0, 2, 2],
[0, 0, 0, 0, 3, 3],
[0, 0, 0, 0, 3, 3],
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[4, 4, 5, 5, 6, 6],
[4, 4, 5, 5, 6, 6]])
Methods
class lil_matrix(arg1, shape=None, dtype=None, copy=False)
Row-based linked list sparse matrix
This is an efficient structure for constructing sparse matrices incrementally.
This can be instantiated in several ways:
lil_matrix(D)
with a dense matrix or rank-2 ndarray D
lil_matrix(S)
with another sparse matrix S (equivalent to S.tocsc())
lil_matrix((M, N), [dtype])
to construct an empty matrix with shape (M, N) dtype is optional, defaulting to dtype=’d’.
Notes
Advantages of the LIL format
• supports flexible slicing
• changes to the matrix sparsity structure are efficient
Disadvantages of the LIL format
• arithmetic operations LIL + LIL are slow (consider CSR or CSC)
• slow column slicing (consider CSC)
• slow matrix vector products (consider CSR or CSC)
Intended Usage
• LIL is a convenient format for constructing sparse matrices
• once a matrix has been constructed, convert to CSR or CSC format for fast arithmetic and matrix
vector operations
• consider using the COO format when constructing large matrices
Data Structure
• An array (self.rows) of rows, each of which is a sorted list of column indices of non-zero elements.
• The corresponding nonzero values are stored in similar fashion in self.data.
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Methods
class dok_matrix(arg1, shape=None, dtype=None, copy=False)
Dictionary Of Keys based sparse matrix.
This is an efficient structure for constructing sparse matrices incrementally.
This can be instantiated in several ways:
dok_matrix(D)
with a dense matrix, D
dok_matrix(S)
with a sparse matrix, S
dok_matrix((M,N), [dtype])
create the matrix with initial shape (M,N) dtype is optional, defaulting to dtype=’d’
Notes
Allows for efficient O(1) access of individual elements. Duplicates are not allowed. Can be efficiently converted
to a coo_matrix once constructed.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
from scipy.sparse import *
from scipy import *
S = dok_matrix((5,5), dtype=float32)
for i in range(5):
for j in range(5):
S[i,j] = i+j # Update element
Methods
class coo_matrix(arg1, shape=None, dtype=None, copy=False, dims=None)
A sparse matrix in COOrdinate format.
Also known as the ‘ijv’ or ‘triplet’ format.
This can be instantiated in several ways:
coo_matrix(D)
with a dense matrix D
coo_matrix(S)
with another sparse matrix S (equivalent to S.tocoo())
coo_matrix((M, N), [dtype])
to construct an empty matrix with shape (M, N) dtype is optional, defaulting to dtype=’d’.
coo_matrix((data, ij), [shape=(M, N)])
The arguments ‘data’ and ‘ij’ represent three arrays:
1. data[:] the entries of the matrix, in any order
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2. ij[0][:] the row indices of the matrix entries
3. ij[1][:] the column indices of the matrix entries
Where A[ij[0][k], ij[1][k] = data[k]. When shape is not specified, it is inferred from
the index arrays
Notes
Advantages of the COO format
• facilitates fast conversion among sparse formats
• permits duplicate entries (see example)
• very fast conversion to and from CSR/CSC formats
Disadvantages of the COO format
• does not directly support:
– arithmetic operations
– slicing
Intended Usage
• COO is a fast format for constructing sparse matrices
• Once a matrix has been constructed, convert to CSR or CSC format for fast arithmetic and matrix
vector operations
• By default when converting to CSR or CSC format, duplicate (i,j) entries will be summed together.
This facilitates efficient construction of finite element matrices and the like. (see example)
Examples
>>> from scipy.sparse import *
>>> from scipy import *
>>> coo_matrix( (3,4), dtype=int8 ).todense()
matrix([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
>>> row = array([0,3,1,0])
>>> col = array([0,3,1,2])
>>> data = array([4,5,7,9])
>>> coo_matrix( (data,(row,col)), shape=(4,4) ).todense()
matrix([[4, 0, 9, 0],
[0, 7, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 5]])
>>>
>>>
>>>
>>>
398
# example with duplicates
row = array([0,0,1,3,1,0,0])
col = array([0,2,1,3,1,0,0])
data = array([1,1,1,1,1,1,1])
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>>> coo_matrix( (data,(row,col)), shape=(4,4)).todense()
matrix([[3, 0, 1, 0],
[0, 2, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 1]])
Methods
class dia_matrix(arg1, shape=None, dtype=None, copy=False)
Sparse matrix with DIAgonal storage
This can be instantiated in several ways:
dia_matrix(D)
with a dense matrix
dia_matrix(S)
with another sparse matrix S (equivalent to S.todia())
dia_matrix((M, N), [dtype])
to construct an empty matrix with shape (M, N), dtype is optional, defaulting to dtype=’d’.
dia_matrix((data, offsets), shape=(M, N))
where the data[k,:] stores the diagonal entries for diagonal offsets[k] (See example below)
Examples
>>> from scipy.sparse import *
>>> from scipy import *
>>> dia_matrix( (3,4), dtype=int8).todense()
matrix([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
>>> data = array([[1,2,3,4]]).repeat(3,axis=0)
>>> offsets = array([0,-1,2])
>>> dia_matrix( (data,offsets), shape=(4,4)).todense()
matrix([[1, 0, 3, 0],
[1, 2, 0, 4],
[0, 2, 3, 0],
[0, 0, 3, 4]])
Methods
3.14.6 Functions
Building sparse matrices:
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eye (m, n[, k, dtype, format])
identity (n[, dtype, format])
kron (A, B[, format])
kronsum (A, B[, format])
lil_eye ((r, c)[, k, dtype])
lil_diags (diags, offsets, (m, n)[, dtype])
spdiags (data, diags, m, n[, format])
tril (A[, k, format])
triu (A[, k, format])
bmat (blocks[, format, dtype])
hstack (blocks[, format, dtype])
vstack (blocks[, format, dtype])
eye(m, n) returns a sparse (m x n) matrix where the k-th diagonal is all ones
and everything else is zeros.
Identity matrix in sparse format
kronecker product of sparse matrices A and B
kronecker sum of sparse matrices A and B
Generate a lil_matrix of dimensions (r,c) with the k-th diagonal set to 1.
Generate a lil_matrix with the given diagonals.
Return a sparse matrix from diagonals.
Return the lower triangular portion of a matrix in sparse format
Return the upper triangular portion of a matrix in sparse format
Build a sparse matrix from sparse sub-blocks
Stack sparse matrices horizontally (column wise)
Stack sparse matrices vertically (row wise)
eye(m, n, k=0, dtype=’d’, format=None)
eye(m, n) returns a sparse (m x n) matrix where the k-th diagonal is all ones and everything else is zeros.
identity(n, dtype=’d’, format=None)
Identity matrix in sparse format
Returns an identity matrix with shape (n,n) using a given sparse format and dtype.
Parameters
n : integer
Shape of the identity matrix.
dtype : :
Data type of the matrix
format : string
Sparse format of the result, e.g. format=”csr”, etc.
Examples
>>> identity(3).todense()
matrix([[ 1., 0., 0.],
[ 0., 1., 0.],
[ 0., 0., 1.]])
>>> identity(3, dtype=’int8’, format=’dia’)
<3x3 sparse matrix of type ’<type ’numpy.int8’>’
with 3 stored elements (1 diagonals) in DIAgonal format>
kron(A, B, format=None)
kronecker product of sparse matrices A and B
Parameters
A : sparse or dense matrix
first matrix of the product
B : sparse or dense matrix
second matrix of the product
format : string
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format of the result (e.g. “csr”)
Returns
kronecker product in a sparse matrix format :
Examples
>>> A = csr_matrix(array([[0,2],[5,0]]))
>>> B = csr_matrix(array([[1,2],[3,4]]))
>>> kron(A,B).todense()
matrix([[ 0, 0, 2, 4],
[ 0, 0, 6, 8],
[ 5, 10, 0, 0],
[15, 20, 0, 0]])
>>> kron(A,[[1,2],[3,4]]).todense()
matrix([[ 0, 0, 2, 4],
[ 0, 0, 6, 8],
[ 5, 10, 0, 0],
[15, 20, 0, 0]])
kronsum(A, B, format=None)
kronecker sum of sparse matrices A and B
Kronecker sum of two sparse matrices is a sum of two Kronecker products kron(I_n,A) + kron(B,I_m) where
A has shape (m,m) and B has shape (n,n) and I_m and I_n are identity matrices of shape (m,m) and (n,n)
respectively.
Parameters
A:
square matrix
B:
square matrix
format : string
format of the result (e.g. “csr”)
Returns
kronecker sum in a sparse matrix format :
lil_eye((r, c), k=0, dtype=’d’)
Generate a lil_matrix of dimensions (r,c) with the k-th diagonal set to 1.
Parameters
r,c : int
row and column-dimensions of the output.
k : int
• diagonal offset. In the output matrix,
• out[m,m+k] == 1 for all m.
dtype : dtype
data-type of the output array.
lil_diags(diags, offsets, (m, n), dtype=’d’)
Generate a lil_matrix with the given diagonals.
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Parameters
diags : list of list of values e.g. [[1,2,3],[4,5]]
values to be placed on each indicated diagonal.
offsets : list of ints
diagonal offsets. This indicates the diagonal on which the given values should be
placed.
(r,c) : tuple of ints
row and column dimensions of the output.
dtype : dtype
output data-type.
Examples
>>> lil_diags([[1,2,3],[4,5],[6]],[0,1,2],(3,3)).todense()
matrix([[ 1., 4., 6.],
[ 0., 2., 5.],
[ 0., 0., 3.]])
spdiags(data, diags, m, n, format=None)
Return a sparse matrix from diagonals.
Parameters
data : array_like
matrix diagonals stored row-wise
diags : diagonals to set
• k = 0 the main diagonal
• k > 0 the k-th upper diagonal
• k < 0 the k-th lower diagonal
m, n : int
shape of the result
format : format of the result (e.g. “csr”)
By default (format=None) an appropriate sparse matrix format is returned. This
choice is subject to change.
See Also:
dia_matrix
the sparse DIAgonal format.
Examples
>>> data = array([[1,2,3,4],[1,2,3,4],[1,2,3,4]])
>>> diags = array([0,-1,2])
>>> spdiags(data, diags, 4, 4).todense()
matrix([[1, 0, 3, 0],
[1, 2, 0, 4],
[0, 2, 3, 0],
[0, 0, 3, 4]])
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tril(A, k=0, format=None)
Return the lower triangular portion of a matrix in sparse format
Returns the elements on or below the k-th diagonal of the matrix A.
• k = 0 corresponds to the main diagonal
• k > 0 is above the main diagonal
• k < 0 is below the main diagonal
Parameters
A : dense or sparse matrix
Matrix whose lower trianglar portion is desired.
k : integer
The top-most diagonal of the lower triangle.
format : string
Sparse format of the result, e.g. format=”csr”, etc.
Returns
L : sparse matrix
Lower triangular portion of A in sparse format.
See Also:
triu
upper triangle in sparse format
Examples
>>> from scipy.sparse import csr_matrix
>>> A = csr_matrix( [[1,2,0,0,3],[4,5,0,6,7],[0,0,8,9,0]], dtype=’int32’ )
>>> A.todense()
matrix([[1, 2, 0, 0, 3],
[4, 5, 0, 6, 7],
[0, 0, 8, 9, 0]])
>>> tril(A).todense()
matrix([[1, 0, 0, 0, 0],
[4, 5, 0, 0, 0],
[0, 0, 8, 0, 0]])
>>> tril(A).nnz
4
>>> tril(A, k=1).todense()
matrix([[1, 2, 0, 0, 0],
[4, 5, 0, 0, 0],
[0, 0, 8, 9, 0]])
>>> tril(A, k=-1).todense()
matrix([[0, 0, 0, 0, 0],
[4, 0, 0, 0, 0],
[0, 0, 0, 0, 0]])
>>> tril(A, format=’csc’)
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<3x5 sparse matrix of type ’<type ’numpy.int32’>’
with 4 stored elements in Compressed Sparse Column format>
triu(A, k=0, format=None)
Return the upper triangular portion of a matrix in sparse format
Returns the elements on or above the k-th diagonal of the matrix A.
• k = 0 corresponds to the main diagonal
• k > 0 is above the main diagonal
• k < 0 is below the main diagonal
Parameters
A : dense or sparse matrix
Matrix whose upper trianglar portion is desired.
k : integer
The bottom-most diagonal of the upper triangle.
format : string
Sparse format of the result, e.g. format=”csr”, etc.
Returns
L : sparse matrix
Upper triangular portion of A in sparse format.
See Also:
tril
lower triangle in sparse format
Examples
>>> from scipy.sparse import csr_matrix
>>> A = csr_matrix( [[1,2,0,0,3],[4,5,0,6,7],[0,0,8,9,0]], dtype=’int32’ )
>>> A.todense()
matrix([[1, 2, 0, 0, 3],
[4, 5, 0, 6, 7],
[0, 0, 8, 9, 0]])
>>> triu(A).todense()
matrix([[1, 2, 0, 0, 3],
[0, 5, 0, 6, 7],
[0, 0, 8, 9, 0]])
>>> triu(A).nnz
8
>>> triu(A, k=1).todense()
matrix([[0, 2, 0, 0, 3],
[0, 0, 0, 6, 7],
[0, 0, 0, 9, 0]])
>>> triu(A, k=-1).todense()
matrix([[1, 2, 0, 0, 3],
[4, 5, 0, 6, 7],
[0, 0, 8, 9, 0]])
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>>> triu(A, format=’csc’)
<3x5 sparse matrix of type ’<type ’numpy.int32’>’
with 8 stored elements in Compressed Sparse Column format>
bmat(blocks, format=None, dtype=None)
Build a sparse matrix from sparse sub-blocks
Parameters
blocks :
grid of sparse matrices with compatible shapes an entry of None implies an all-zero
matrix
format : sparse format of the result (e.g. “csr”)
by default an appropriate sparse matrix format is returned. This choice is subject to
change.
Examples
>>> from scipy.sparse import coo_matrix, bmat
>>> A = coo_matrix([[1,2],[3,4]])
>>> B = coo_matrix([[5],[6]])
>>> C = coo_matrix([[7]])
>>> bmat( [[A,B],[None,C]] ).todense()
matrix([[1, 2, 5],
[3, 4, 6],
[0, 0, 7]])
>>> bmat( [[A,None],[None,C]] ).todense()
matrix([[1, 2, 0],
[3, 4, 0],
[0, 0, 7]])
hstack(blocks, format=None, dtype=None)
Stack sparse matrices horizontally (column wise)
Parameters
blocks :
sequence of sparse matrices with compatible shapes
format : string
sparse format of the result (e.g. “csr”) by default an appropriate sparse matrix format
is returned. This choice is subject to change.
See Also:
vstack
stack sparse matrices vertically (row wise)
Examples
>>>
>>>
>>>
>>>
from scipy.sparse import coo_matrix, vstack
A = coo_matrix([[1,2],[3,4]])
B = coo_matrix([[5],[6]])
hstack( [A,B] ).todense()
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matrix([[1, 2, 5],
[3, 4, 6]])
vstack(blocks, format=None, dtype=None)
Stack sparse matrices vertically (row wise)
Parameters
blocks :
sequence of sparse matrices with compatible shapes
format : string
sparse format of the result (e.g. “csr”) by default an appropriate sparse matrix format
is returned. This choice is subject to change.
See Also:
hstack
stack sparse matrices horizontally (column wise)
Examples
>>> from scipy.sparse import coo_matrix, vstack
>>> A = coo_matrix([[1,2],[3,4]])
>>> B = coo_matrix([[5,6]])
>>> vstack( [A,B] ).todense()
matrix([[1, 2],
[3, 4],
[5, 6]])
Identifying sparse matrices:
issparse (x)
isspmatrix (x)
isspmatrix_csc (x)
isspmatrix_csr (x)
isspmatrix_bsr (x)
isspmatrix_lil (x)
isspmatrix_dok (x)
isspmatrix_coo (x)
isspmatrix_dia (x)
issparse(x)
isspmatrix(x)
isspmatrix_csc(x)
isspmatrix_csr(x)
isspmatrix_bsr(x)
isspmatrix_lil(x)
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isspmatrix_dok(x)
isspmatrix_coo(x)
isspmatrix_dia(x)
3.14.7 Exceptions
exception SparseEfficiencyWarning
exception SparseWarning
3.15 Sparse linear algebra (scipy.sparse.linalg)
Warning: This documentation is work-in-progress and unorganized.
3.15.1 Sparse Linear Algebra
The submodules of sparse.linalg:
1. eigen: sparse eigenvalue problem solvers
2. isolve: iterative methods for solving linear systems
3. dsolve: direct factorization methods for solving linear systems
3.15.2 Examples
class LinearOperator(shape, matvec, rmatvec=None, matmat=None, dtype=None)
Common interface for performing matrix vector products
Many iterative methods (e.g. cg, gmres) do not need to know the individual entries of a matrix to solve a linear
system A*x=b. Such solvers only require the computation of matrix vector products, A*v where v is a dense
vector. This class serves as an abstract interface between iterative solvers and matrix-like objects.
Parameters
shape : tuple
Matrix dimensions (M,N)
matvec : callable f(v)
Returns returns A * v.
See Also:
aslinearoperator
Construct LinearOperators
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Notes
The user-defined matvec() function must properly handle the case where v has shape (N,) as well as the (N,1)
case. The shape of the return type is handled internally by LinearOperator.
Examples
>>> from scipy.sparse.linalg import LinearOperator
>>> from scipy import *
>>> def mv(v):
...
return array([ 2*v[0], 3*v[1]])
...
>>> A = LinearOperator( (2,2), matvec=mv )
>>> A
<2x2 LinearOperator with unspecified dtype>
>>> A.matvec( ones(2) )
array([ 2., 3.])
>>> A * ones(2)
array([ 2., 3.])
Methods
matmat(X)
Matrix-matrix multiplication
Performs the operation y=A*X where A is an MxN linear operator and X dense N*K matrix or ndarray.
Parameters
X : {matrix, ndarray}
An array with shape (N,K).
Returns
Y : {matrix, ndarray}
A matrix or ndarray with shape (M,K) depending on the type of the X argument.
Notes
This matmat wraps any user-specified matmat routine to ensure that y has the correct type.
matvec(x)
Matrix-vector multiplication
Performs the operation y=A*x where A is an MxN linear operator and x is a column vector or rank-1
array.
Parameters
x : {matrix, ndarray}
An array with shape (N,) or (N,1).
Returns
y : {matrix, ndarray}
A matrix or ndarray with shape (M,) or (M,1) depending on the type and shape of
the x argument.
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Notes
This matvec wraps the user-specified matvec routine to ensure that y has the correct shape and type.
Tester
alias of NoseTester
aslinearoperator(A)
Return A as a LinearOperator.
‘A’ may be any of the following types:
• ndarray
• matrix
• sparse matrix (e.g. csr_matrix, lil_matrix, etc.)
• LinearOperator
• An object with .shape and .matvec attributes
See the LinearOperator documentation for additonal information.
Examples
>>> from scipy import matrix
>>> M = matrix( [[1,2,3],[4,5,6]], dtype=’int32’ )
>>> aslinearoperator( M )
<2x3 LinearOperator with dtype=int32>
bicg(A, b, x0=None, tol=1.0000000000000001e-05, maxiter=None, xtype=None, M=None, callback=None)
Use BIConjugate Gradient iteration to solve A x = b
Parameters
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
bicgstab(A, b, x0=None, tol=1.0000000000000001e-05, maxiter=None, xtype=None, M=None, callback=None)
Use BIConjugate Gradient STABilized iteration to solve A x = b
Parameters
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
cg(A, b, x0=None, tol=1.0000000000000001e-05, maxiter=None, xtype=None, M=None, callback=None)
Use Conjugate Gradient iteration to solve A x = b
Parameters
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
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Right hand side of the linear system. Has shape (N,) or (N,1).
cgs(A, b, x0=None, tol=1.0000000000000001e-05, maxiter=None, xtype=None, M=None, callback=None)
Use Conjugate Gradient Squared iteration to solve A x = b
Parameters
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
class csc_matrix(arg1, shape=None, dtype=None, copy=False, dims=None, nzmax=None)
Compressed Sparse Column matrix
This can be instantiated in several ways:
csc_matrix(D)
with a dense matrix or rank-2 ndarray D
csc_matrix(S)
with another sparse matrix S (equivalent to S.tocsc())
csc_matrix((M, N), [dtype])
to construct an empty matrix with shape (M, N) dtype is optional, defaulting to dtype=’d’.
csc_matrix((data, ij), [shape=(M, N)])
where data and ij satisfy the relationship a[ij[0, k], ij[1, k]] = data[k]
csc_matrix((data, indices, indptr), [shape=(M, N)])
is the standard CSC representation where the row indices for column i are stored in
indices[indptr[i]:indices[i+1]] and their corresponding values are stored in
data[indptr[i]:indptr[i+1]]. If the shape parameter is not supplied, the matrix dimensions are inferred from the index arrays.
Notes
Advantages of the CSC format
• efficient arithmetic operations CSC + CSC, CSC * CSC, etc.
• efficient column slicing
• fast matrix vector products (CSR, BSR may be faster)
Disadvantages of the CSC format
• slow row slicing operations (consider CSR)
• changes to the sparsity structure are expensive (consider LIL or DOK)
Examples
>>> from scipy.sparse import *
>>> from scipy import *
>>> csc_matrix( (3,4), dtype=int8 ).todense()
matrix([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
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>>> row = array([0,2,2,0,1,2])
>>> col = array([0,0,1,2,2,2])
>>> data = array([1,2,3,4,5,6])
>>> csc_matrix( (data,(row,col)), shape=(3,3) ).todense()
matrix([[1, 0, 4],
[0, 0, 5],
[2, 3, 6]])
>>> indptr = array([0,2,3,6])
>>> indices = array([0,2,2,0,1,2])
>>> data = array([1,2,3,4,5,6])
>>> csc_matrix( (data,indices,indptr), shape=(3,3) ).todense()
matrix([[1, 0, 4],
[0, 0, 5],
[2, 3, 6]])
Methods
rowcol(*args, **kwds)
rowcol is deprecated!
class csr_matrix(arg1, shape=None, dtype=None, copy=False, dims=None, nzmax=None)
Compressed Sparse Row matrix
This can be instantiated in several ways:
csr_matrix(D)
with a dense matrix or rank-2 ndarray D
csr_matrix(S)
with another sparse matrix S (equivalent to S.tocsr())
csr_matrix((M, N), [dtype])
to construct an empty matrix with shape (M, N) dtype is optional, defaulting to dtype=’d’.
csr_matrix((data, ij), [shape=(M, N)])
where data and ij satisfy the relationship a[ij[0, k], ij[1, k]] = data[k]
csr_matrix((data, indices, indptr), [shape=(M, N)])
is the standard CSR representation where the column indices for row i are stored in
indices[indptr[i]:indices[i+1]] and their corresponding values are stored in
data[indptr[i]:indptr[i+1]]. If the shape parameter is not supplied, the matrix dimensions are inferred from the index arrays.
Notes
Advantages of the CSR format
• efficient arithmetic operations CSR + CSR, CSR * CSR, etc.
• efficient row slicing
• fast matrix vector products
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Disadvantages of the CSR format
• slow column slicing operations (consider CSC)
• changes to the sparsity structure are expensive (consider LIL or DOK)
Examples
>>> from scipy.sparse import *
>>> from scipy import *
>>> csr_matrix( (3,4), dtype=int8 ).todense()
matrix([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]], dtype=int8)
>>> row = array([0,0,1,2,2,2])
>>> col = array([0,2,2,0,1,2])
>>> data = array([1,2,3,4,5,6])
>>> csr_matrix( (data,(row,col)), shape=(3,3) ).todense()
matrix([[1, 0, 2],
[0, 0, 3],
[4, 5, 6]])
>>> indptr = array([0,2,3,6])
>>> indices = array([0,2,2,0,1,2])
>>> data = array([1,2,3,4,5,6])
>>> csr_matrix( (data,indices,indptr), shape=(3,3) ).todense()
matrix([[1, 0, 2],
[0, 0, 3],
[4, 5, 6]])
Methods
rowcol(*args, **kwds)
rowcol is deprecated!
eigen(A, k=6, M=None, sigma=None, which=’LM’, v0=None, ncv=None, maxiter=None, tol=0, return_eigenvectors=True)
Find k eigenvalues and eigenvectors of the square matrix A.
Solves A * x[i] = w[i] * x[i], the standard eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i].
Parameters
A : matrix, array, or object with matvec(x) method
An N x N matrix, array, or an object with matvec(x) method to perform the matrix
vector product A * x. The sparse matrix formats in scipy.sparse are appropriate for
A.
k : integer
The number of eigenvalues and eigenvectors desired
Returns
w : array
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Array of k eigenvalues
v : array
An array of k eigenvectors The v[i] is the eigenvector corresponding to the eigenvector w[i]
See Also:
eigen_symmetric
eigenvalues and eigenvectors for symmetric matrix A
eigen_symmetric(A, k=6, M=None, sigma=None, which=’LM’, v0=None, ncv=None, maxiter=None, tol=0,
return_eigenvectors=True)
Find k eigenvalues and eigenvectors of the real symmetric square matrix A.
Solves A * x[i] = w[i] * x[i], the standard eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i].
Parameters
A : matrix or array with real entries or object with matvec(x) method
An N x N real symmetric matrix or array or an object with matvec(x) method to
perform the matrix vector product A * x. The sparse matrix formats in scipy.sparse
are appropriate for A.
k : integer
The number of eigenvalues and eigenvectors desired
Returns
w : array
Array of k eigenvalues
v : array
An array of k eigenvectors The v[i] is the eigenvector corresponding to the eigenvector w[i]
See Also:
eigen
eigenvalues and eigenvectors for a general (nonsymmetric) matrix A
factorized(A)
Return a fuction for solving a sparse linear system, with A pre-factorized.
Example:
solve = factorized( A ) # Makes LU decomposition. x1 = solve( rhs1 ) # Uses the LU factors. x2 = solve(
rhs2 ) # Uses again the LU factors.
gmres(A, b, x0=None, tol=1.0000000000000001e-05, restart=None, maxiter=None, xtype=None, M=None, callback=None, restrt=None)
Use Generalized Minimal RESidual iteration to solve A x = b
Parameters
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
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See Also:
LinearOperator
lgmres(A, b, x0=None, tol=1.0000000000000001e-05, maxiter=1000, M=None, callback=None, inner_m=30,
outer_k=3, outer_v=None, store_outer_Av=True)
Solve a matrix equation using the LGMRES algorithm.
The LGMRES algorithm [BJM] [BPh] is designed to avoid some problems in the convergence in restarted
GMRES, and often converges in fewer iterations.
Parameters
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
x0 : {array, matrix}
Starting guess for the solution.
tol : float
Tolerance to achieve. The algorithm terminates when either the relative or the absolute residual is below tol.
maxiter : integer
Maximum number of iterations. Iteration will stop after maxiter steps even if the
specified tolerance has not been achieved.
M : {sparse matrix, dense matrix, LinearOperator}
Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies
that fewer iterations are needed to reach a given error tolerance.
callback : function
User-supplied function to call after each iteration. It is called as callback(xk), where
xk is the current solution vector.
Returns
x : array or matrix
The converged solution.
info : integer
Provides convergence information:
0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown
Notes
The LGMRES algorithm [BJM] [BPh] is designed to avoid the slowing of convergence in restarted GMRES, due
to alternating residual vectors. Typically, it often outperforms GMRES(m) of comparable memory requirements
by some measure, or at least is not much worse.
Another advantage in this algorithm is that you can supply it with ‘guess’ vectors in the outer_v argument that
augment the Krylov subspace. If the solution lies close to the span of these vectors, the algorithm converges
faster. This can be useful if several very similar matrices need to be inverted one after another, such as in
Newton-Krylov iteration where the Jacobian matrix often changes little in the nonlinear steps.
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References
[BJM], [BPh]
lobpcg(A, X, B=None, M=None, Y=None, tol=None, maxiter=20, largest=True, verbosityLevel=0, retLambdaHistory=False, retResidualNormsHistory=False)
Solve symmetric partial eigenproblems with optional preconditioning
This function implements the Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).
Parameters
A : {sparse matrix, dense matrix, LinearOperator}
The symmetric linear operator of the problem, usually a sparse matrix. Often called
the “stiffness matrix”.
X : array_like
Initial approximation to the k eigenvectors. If A has shape=(n,n) then X should have
shape shape=(n,k).
Returns
w : array
Array of k eigenvalues
v : array
An array of k eigenvectors. V has the same shape as X.
Notes
If both retLambdaHistory and retResidualNormsHistory are True, the return tuple has the following format
(lambda, V, lambda history, residual norms history)
lsqr(A, b, damp=0.0, atol=1e-08, btol=1e-08, conlim=100000000.0,
calc_var=False)
Find the least-squares solution to a large, sparse, linear system of equations.
iter_lim=None,
show=False,
The function solves Ax = b or min ||b - Ax||^2 or ‘‘min ||Ax - b||^2 + d^2 ||x||^2.
The matrix A may be square or rectangular (over-determined or under-determined), and may have any rank.
1. Unsymmetric equations --
solve
A*x = b
2. Linear least squares
--
solve A*x = b
in the least-squares sense
3. Damped least squares
--
solve
(
A
)*x = ( b )
( damp*I )
( 0 )
in the least-squares sense
Parameters
A : {sparse matrix, ndarray, LinearOperatorLinear}
Representation of an mxn matrix. It is required that the linear operator can produce
Ax and A^T x.
b : (m,) ndarray
Right-hand side vector b.
damp : float
Damping coefficient.
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atol, btol : float
Stopping tolerances. If both are 1.0e-9 (say), the final residual norm should be accurate to about 9 digits. (The final x will usually have fewer correct digits, depending
on cond(A) and the size of damp.)
conlim : float
Another stopping tolerance. lsqr terminates if an estimate of cond(A) exceeds
conlim. For compatible systems Ax = b, conlim could be as large as 1.0e+12 (say).
For least-squares problems, conlim should be less than 1.0e+8. Maximum precision
can be obtained by setting atol = btol = conlim = zero, but the number
of iterations may then be excessive.
iter_lim : int
Explicit limitation on number of iterations (for safety).
show : bool
Display an iteration log.
calc_var : bool
Whether to estimate diagonals of (A’A + damp^2*I)^{-1}.
Returns
x : ndarray of float
The final solution.
istop : int
Gives the reason for termination. 1 means x is an approximate solution to Ax = b. 2
means x approximately solves the least-squares problem.
itn : int
Iteration number upon termination.
r1norm : float
norm(r), where r = b - Ax.
r2norm : float
sqrt( norm(r)^2 + damp^2 * norm(x)^2 ). Equal to r1norm if damp
== 0.
anorm : float
Estimate of Frobenius norm of Abar = [[A]; [damp*I]].
acond : float
Estimate of cond(Abar).
arnorm : float
Estimate of norm(A’*r - damp^2*x).
xnorm : float
norm(x)
var : ndarray of float
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If calc_var is True, estimates all diagonals of (A’A)^{-1} (if damp == 0)
or more generally (A’A + damp^2*I)^{-1}. This is well defined if A has full
column rank or damp > 0. (Not sure what var means if rank(A) < n and damp
= 0.)
Notes
LSQR uses an iterative method to approximate the solution. The number of iterations required to reach a certain
accuracy depends strongly on the scaling of the problem. Poor scaling of the rows or columns of A should
therefore be avoided where possible.
For example, in problem 1 the solution is unaltered by row-scaling. If a row of A is very small or large compared
to the other rows of A, the corresponding row of ( A b ) should be scaled up or down.
In problems 1 and 2, the solution x is easily recovered following column-scaling. Unless better information is
known, the nonzero columns of A should be scaled so that they all have the same Euclidean norm (e.g., 1.0).
In problem 3, there is no freedom to re-scale if damp is nonzero. However, the value of damp should be assigned
only after attention has been paid to the scaling of A.
The parameter damp is intended to help regularize ill-conditioned systems, by preventing the true solution from
being very large. Another aid to regularization is provided by the parameter acond, which may be used to
terminate iterations before the computed solution becomes very large.
If some initial estimate x0 is known and if damp == 0, one could proceed as follows:
1.Compute a residual vector r0 = b - A*x0.
2.Use LSQR to solve the system A*dx = r0.
3.Add the correction dx to obtain a final solution x = x0 + dx.
This requires that x0 be available before and after the call to LSQR. To judge the benefits, suppose LSQR
takes k1 iterations to solve A*x = b and k2 iterations to solve A*dx = r0. If x0 is “good”, norm(r0) will be
smaller than norm(b). If the same stopping tolerances atol and btol are used for each system, k1 and k2 will be
similar, but the final solution x0 + dx should be more accurate. The only way to reduce the total work is to use
a larger stopping tolerance for the second system. If some value btol is suitable for A*x = b, the larger value
btol*norm(b)/norm(r0) should be suitable for A*dx = r0.
Preconditioning is another way to reduce the number of iterations. If it is possible to solve a related system M*x
= b efficiently, where M approximates A in some helpful way (e.g. M - A has low rank or its elements are
small relative to those of A), LSQR may converge more rapidly on the system A*M(inverse)*z = b, after
which x can be recovered by solving M*x = z.
If A is symmetric, LSQR should not be used!
Alternatives are the symmetric conjugate-gradient method (cg) and/or SYMMLQ. SYMMLQ is an implementation of symmetric cg that applies to any symmetric A and will converge more rapidly than LSQR. If A is
positive definite, there are other implementations of symmetric cg that require slightly less work per iteration
than SYMMLQ (but will take the same number of iterations).
References
[R68], [R69], [R70]
minres(A, b, x0=None, shift=0.0, tol=1.0000000000000001e-05, maxiter=None, xtype=None, M=None, callback=None, show=False, check=False)
Use MINimum RESidual iteration to solve Ax=b
MINRES minimizes norm(A*x - b) for the symmetric matrix A. Unlike the Conjugate Gradient method, A can
be indefinite or singular.
If shift != 0 then the method solves (A - shift*I)x = b
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Parameters
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
Notes
THIS FUNCTION IS EXPERIMENTAL AND SUBJECT TO CHANGE!
References
Solution of sparse indefinite systems of linear equations,
C. C. Paige and M. A. Saunders (1975), SIAM J. Numer.
http://www.stanford.edu/group/SOL/software/minres.html
Anal.
12(4), pp.
617-629.
This file is a translation of the following MATLAB implementation:
http://www.stanford.edu/group/SOL/software/minres/matlab/
qmr(A, b, x0=None, tol=1.0000000000000001e-05, maxiter=None, xtype=None, M1=None, M2=None, callback=None)
Use Quasi-Minimal Residual iteration to solve A x = b
Parameters
A : {sparse matrix, dense matrix, LinearOperator}
The N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
See Also:
LinearOperator
spilu(A, drop_tol=None, fill_factor=None, drop_rule=None, permc_spec=None, diag_pivot_thresh=None, relax=None, panel_size=None, options=None)
Compute an incomplete LU decomposition for a sparse, square matrix A.
The resulting object is an approximation to the inverse of A.
Parameters
A:
Sparse matrix to factorize
drop_tol : float, optional
Drop tolerance (0 <= tol <= 1) for an incomplete LU decomposition. (default: 1e-4)
fill_factor : float, optional
Specifies the fill ratio upper bound (>= 1.0) for ILU. (default: 10)
drop_rule : str, optional
Comma-separated string of drop rules to use. Available rules: basic, prows,
column, area, secondary, dynamic, interp. (Default: basic,area)
See SuperLU documentation for details.
milu : str, optional
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Which version of modified ILU to use. (Choices: silu, smilu_1, smilu_2
(default), smilu_3.)
Remaining other options :
Same as for splu
Returns
invA_approx : scipy.sparse.linalg.dsolve._superlu.SciPyLUType
Object, which has a solve method.
See Also:
splu
complete LU decomposition
Notes
To improve the better approximation to the inverse, you may need to increase fill_factor AND decrease
drop_tol.
This function uses the SuperLU library.
References
[SLU]
splu(A, permc_spec=None, diag_pivot_thresh=None, drop_tol=None, relax=None, panel_size=None, options={})
Compute the LU decomposition of a sparse, square matrix.
Parameters
A:
Sparse matrix to factorize. Should be in CSR or CSC format.
permc_spec : str, optional
How to permute the columns of the matrix for sparsity preservation. (default: ‘COLAMD’)
• NATURAL: natural ordering.
• MMD_ATA: minimum degree ordering on the structure of A^T A.
• MMD_AT_PLUS_A: minimum degree ordering on the structure of A^T+A.
• COLAMD: approximate minimum degree column ordering
diag_pivot_thresh : float, optional
Threshold used for a diagonal entry to be an acceptable pivot. See SuperLU user’s
guide for details [SLU]
drop_tol : float, optional
(deprecated) No effect.
relax : int, optional
Expert option for customizing the degree of relaxing supernodes. See SuperLU
user’s guide for details [SLU]
panel_size : int, optional
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Expert option for customizing the panel size. See SuperLU user’s guide for details
[SLU]
options : dict, optional
Dictionary containing additional expert options to SuperLU. See SuperLU user guide
[SLU] (section 2.4 on the ‘Options’ argument) for more details. For example, you
can specify options=dict(Equil=False, IterRefine=’SINGLE’))
to turn equilibration off and perform a single iterative refinement.
Returns
invA : scipy.sparse.linalg.dsolve._superlu.SciPyLUType
Object, which has a solve method.
See Also:
spilu
incomplete LU decomposition
Notes
This function uses the SuperLU library.
References
[SLU]
spsolve(A, b, permc_spec=None, use_umfpack=True)
Solve the sparse linear system Ax=b
svd(A, k=6)
Compute a few singular values/vectors for a sparse matrix using ARPACK.
Parameters
A: sparse matrix :
Array to compute the SVD on.
k: int :
Number of singular values and vectors to compute.
use_solver(**kwargs)
Valid keyword arguments with defaults (other ignored):
useUmfpack = True assumeSortedIndices = False
The default sparse solver is umfpack when available. This can be changed by passing useUmfpack = False,
which then causes the always present SuperLU based solver to be used.
Umfpack requires a CSR/CSC matrix to have sorted column/row indices. If sure that the matrix fulfills this,
pass assumeSortedIndices=True to gain some speed.
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Warning: This documentation is work-in-progress and unorganized.
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3.16.1 Distance computations (scipy.spatial.distance)
Function Reference
Distance matrix computation from a collection of raw observation vectors stored in a rectangular array.
Function
pdist
cdist
squareform
Description
pairwise distances between observation vectors.
distances between between two collections of observation vectors.
converts a square distance matrix to a condensed one and vice versa.
Predicates for checking the validity of distance matrices, both condensed and redundant. Also contained in this module
are functions for computing the number of observations in a distance matrix.
Function
is_valid_dm
is_valid_y
num_obs_dm
num_obs_y
Description
checks for a valid distance matrix.
checks for a valid condensed distance matrix.
# of observations in a distance matrix.
# of observations in a condensed distance matrix.
Distance functions between two vectors u and v. Computing distances over a large collection of vectors is inefficient
for these functions. Use pdist for this purpose.
Function
braycurtis
canberra
chebyshev
cityblock
correlation
cosine
dice
euclidean
hamming
jaccard
kulsinski
mahalanobis
matching
minkowski
rogerstanimoto
russellrao
seuclidean
sokalmichener
sokalsneath
sqeuclidean
yule
Description
the Bray-Curtis distance.
the Canberra distance.
the Chebyshev distance.
the Manhattan distance.
the Correlation distance.
the Cosine distance.
the Dice dissimilarity (boolean).
the Euclidean distance.
the Hamming distance (boolean).
the Jaccard distance (boolean).
the Kulsinski distance (boolean).
the Mahalanobis distance.
the matching dissimilarity (boolean).
the Minkowski distance.
the Rogers-Tanimoto dissimilarity (boolean).
the Russell-Rao dissimilarity (boolean).
the normalized Euclidean distance.
the Sokal-Michener dissimilarity (boolean).
the Sokal-Sneath dissimilarity (boolean).
the squared Euclidean distance.
the Yule dissimilarity (boolean).
References
Copyright Notice
Copyright (C) Damian Eads, 2007-2008. New BSD License.
braycurtis(u, v)
Computes the Bray-Curtis distance between two n-vectors u and v, which is defined as
X
X
|ui − vi |/
|ui + vi |.
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Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Bray-Curtis distance between vectors u and v.
canberra(u, v)
Computes the Canberra distance between two n-vectors u and v, which is defined as
P
|u − vi |
Pi i
.
|u
i | + |vi |
i
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Canberra distance between vectors u and v.
cdist(XA, XB, metric=’euclidean’, p=2, V=None, VI=None, w=None)
Computes distance between each pair of observation vectors in the Cartesian product of two collections of
vectors. XA is a mA by n array while XB is a mB by n array. A mA by mB array is returned. An exception is
thrown if XA and XB do not have the same number of columns.
A rectangular distance matrix Y is returned. For each i and j, the metric dist(u=XA[i], v=XB[j]) is
computed and stored in the ij th entry.
The following are common calling conventions:
1.Y = cdist(XA, XB, ’euclidean’)
Computes the distance between m points using Euclidean distance (2-norm) as the distance metric between the points. The points are arranged as m n-dimensional row vectors in the matrix X.
2.Y = cdist(XA, XB, ’minkowski’, p)
Computes the distances using the Minkowski distance ||u − v||p (p-norm) where p ≥ 1.
3.Y = cdist(XA, XB, ’cityblock’)
Computes the city block or Manhattan distance between the points.
4.Y = cdist(XA, XB, ’seuclidean’, V=None)
Computes the standardized Euclidean distance. The standardized Euclidean distance between two nvectors u and v is
qX
(ui − vi )2 /V [xi ].
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V is the variance vector; V[i] is the variance computed over all
the i’th components of the points. If not passed, it is automatically computed.
5.Y = cdist(XA, XB, ’sqeuclidean’)
Computes the squared Euclidean distance ||u − v||22 between the vectors.
6.Y = cdist(XA, XB, ’cosine’)
Computes the cosine distance between vectors u and v,
1 − uv T
|u|2 |v|2
where | ∗ |2 is the 2-norm of its argument *.
7.Y = cdist(XA, XB, ’correlation’)
Computes the correlation distance between vectors u and v. This is
1 − (u − n|u|1 )(v − n|v|1 )
T
T
|(u − n|u|1 )|2 |(v − n|v|1 )|
where | ∗ |1 is the Manhattan (or 1-norm) of its argument, and n is the common dimensionality of the
vectors.
8.Y = cdist(XA, XB, ’hamming’)
Computes the normalized Hamming distance, or the proportion of those vector elements between two
n-vectors u and v which disagree. To save memory, the matrix X can be of type boolean.
9.Y = cdist(XA, XB, ’jaccard’)
Computes the Jaccard distance between the points. Given two vectors, u and v, the Jaccard distance is
the proportion of those elements u[i] and v[i] that disagree where at least one of them is non-zero.
10.Y = cdist(XA, XB, ’chebyshev’)
Computes the Chebyshev distance between the points. The Chebyshev distance between two nvectors u and v is the maximum norm-1 distance between their respective elements. More precisely,
the distance is given by
d(u, v) = max |ui − vi |.
i
11.Y = cdist(XA, XB, ’canberra’)
Computes the Canberra distance between the points. The Canberra distance between two points u
and v is
X |ui − vi |
d(u, v) =
(|ui | + |vi |)
u
12.Y = cdist(XA, XB, ’braycurtis’)
Computes the Bray-Curtis distance between the points. The Bray-Curtis distance between two
points u and v is
P
(ui − vi )
d(u, v) = Pi
i (ui + vi )
13.Y = cdist(XA, XB, ’mahalanobis’, VI=None)
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Computes the Mahalanobis distance between the points. The Mahalanobis distance between two
points u and v is (u − v)(1/V )(u − v)T where (1/V ) (the VI variable) is the inverse covariance.
If VI is not None, VI will be used as the inverse covariance matrix.
14.Y = cdist(XA, XB, ’yule’)
Computes the Yule distance between the boolean vectors. (see yule function documentation)
15.Y = cdist(XA, XB, ’matching’)
Computes the matching distance between the boolean vectors. (see matching function documentation)
16.Y = cdist(XA, XB, ’dice’)
Computes the Dice distance between the boolean vectors. (see dice function documentation)
17.Y = cdist(XA, XB, ’kulsinski’)
Computes the Kulsinski distance between the boolean vectors. (see kulsinski function documentation)
18.Y = cdist(XA, XB, ’rogerstanimoto’)
Computes the Rogers-Tanimoto distance between the boolean vectors. (see rogerstanimoto function
documentation)
19.Y = cdist(XA, XB, ’russellrao’)
Computes the Russell-Rao distance between the boolean vectors. (see russellrao function documentation)
20.Y = cdist(XA, XB, ’sokalmichener’)
Computes the Sokal-Michener distance between the boolean vectors. (see sokalmichener function
documentation)
21.Y = cdist(XA, XB, ’sokalsneath’)
Computes the Sokal-Sneath distance between the vectors. (see sokalsneath function documentation)
22.Y = cdist(XA, XB, ’wminkowski’)
Computes the weighted Minkowski distance between the vectors. (see sokalsneath function documentation)
23.Y = cdist(XA, XB, f)
Computes the distance between all pairs of vectors in X using the user supplied 2-arity function f.
For example, Euclidean distance between the vectors could be computed as follows:
dm = cdist(XA, XB, (lambda u, v: np.sqrt(((u-v)*(u-v).T).sum())))
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Note that you should avoid passing a reference to one of the distance functions defined in this
library. For example,:
dm = cdist(XA, XB, sokalsneath)
would calculate the pair-wise distances between the vectors
in X using the Python function
sokalsneath. This would result in sokalsneath being called n2 times, which is inefficient. Instead,
the optimized C version is more efficient, and we call it using the following syntax.:
dm = cdist(XA, XB, ’sokalsneath’)
Parameters
XA
[ndarray] An mA by n array of mA original observations in an n-dimensional space.
XB
[ndarray] An mB by n array of mB original observations in an n-dimensional space.
metric
[string or function] The distance metric to use. The distance function can be ‘braycurtis’,
‘canberra’, ‘chebyshev’, ‘cityblock’, ‘correlation’, ‘cosine’, ‘dice’, ‘euclidean’, ‘hamming’, ‘jaccard’, ‘kulsinski’, ‘mahalanobis’, ‘matching’, ‘minkowski’, ‘rogerstanimoto’,
‘russellrao’, ‘seuclidean’, ‘sokalmichener’, ‘sokalsneath’, ‘sqeuclidean’, ‘wminkowski’,
‘yule’.
w
[ndarray] The weight vector (for weighted Minkowski).
p
[double] The p-norm to apply (for Minkowski, weighted and unweighted)
V
[ndarray] The variance vector (for standardized Euclidean).
VI
[ndarray] The inverse of the covariance matrix (for Mahalanobis).
Returns
Y
[ndarray] A mA by mB distance matrix.
chebyshev(u, v)
Computes the Chebyshev distance between two n-vectors u and v, which is defined as
max |ui − vi |.
i
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
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Returns
d
[double] The Chebyshev distance between vectors u and v.
cityblock(u, v)
Computes the Manhattan distance between two n-vectors u and v, which is defined as
X
(ui − vi ).
i
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The City Block distance between vectors u and v.
correlation(u, v)
Computes the correlation distance between two n-vectors u and v, which is defined as
T
1 − (u − ū)(v − v̄)
T
||(u − ū)||2 ||(v − v̄)||2
where ū is the mean of a vectors elements and n is the common dimensionality of u and v.
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The correlation distance between vectors u and v.
cosine(u, v)
Computes the Cosine distance between two n-vectors u and v, which is defined as
1 − uv T
.
||u||2 ||v||2
Parameters
u
[ndarray] An n-dimensional vector.
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v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Cosine distance between vectors u and v.
dice(u, v)
Computes the Dice dissimilarity between two boolean n-vectors u and v, which is
cT F + cF T
2cT T + cF T + cT F
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n.
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Dice dissimilarity between vectors u and v.
euclidean(u, v)
Computes the Euclidean distance between two n-vectors u and v, which is defined as
||u − v||2
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Euclidean distance between vectors u and v.
hamming(u, v)
Computes the Hamming distance between two n-vectors u and v, which is simply the proportion of disagreeing
components in u and v. If u and v are boolean vectors, the Hamming distance is
c01 + c10
n
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n.
Parameters
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u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Hamming distance between vectors u and v.
is_valid_dm(D, tol=0.0, throw=False, name=’D’, warning=False)
Returns True if the variable D passed is a valid distance matrix. Distance matrices must be 2-dimensional
numpy arrays containing doubles. They must have a zero-diagonal, and they must be symmetric.
Parameters
D
[ndarray] The candidate object to test for validity.
tol
[double] The distance matrix should be symmetric. tol is the maximum difference between the :math:‘ij‘th entry and the :math:‘ji‘th entry for the distance metric to be considered symmetric.
throw
[bool] An exception is thrown if the distance matrix passed is not valid.
name
[string] the name of the variable to checked. This is useful ifa throw is set to True so
the offending variable can be identified in the exception message when an exception is
thrown.
warning
[boolx] Instead of throwing an exception, a warning message is raised.
Returns
Returns True if the variable D passed is a valid distance matrix. Small numerical differences
in D and D.T and non-zeroness of the diagonal are ignored if they are within the tolerance
specified by tol.
is_valid_y(y, warning=False, throw=False, name=None)
Returns True if the variable y passed is a valid condensed distance matrix. Condensed distance matrices
must
be 1-dimensional numpy arrays containing doubles. Their length must be a binomial coefficient n2 for some
positive integer n.
Parameters
y
[ndarray] The condensed distance matrix.
warning
[bool] Invokes a warning if the variable passed is not a valid condensed distance matrix.
The warning message explains why the distance matrix is not valid. ‘name’ is used when
referencing the offending variable.
throws
[throw] Throws an exception if the variable passed is not a valid condensed distance
matrix.
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name
[bool] Used when referencing the offending variable in the warning or exception message.
jaccard(u, v)
Computes the Jaccard-Needham dissimilarity between two boolean n-vectors u and v, which is
raccT F + cF T cT T + cF T + cT F
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n.
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Jaccard distance between vectors u and v.
kulsinski(u, v)
Computes the Kulsinski dissimilarity between two boolean n-vectors u and v, which is defined as
raccT F + cF T − cT T + ncF T + cT F + n
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n.
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Kulsinski distance between vectors u and v.
mahalanobis(u, v, VI)
Computes the Mahalanobis distance between two n-vectors u and v, which is defiend as
(u − v)V −1 (u − v)T
where VI is the inverse covariance matrix V −1 .
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
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Returns
d
[double] The Mahalanobis distance between vectors u and v.
matching(u, v)
Computes the Matching dissimilarity between two boolean n-vectors u and v, which is defined as
cT F + cF T
n
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n.
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Matching dissimilarity between vectors u and v.
minkowski(u, v, p)
Computes the Minkowski distance between two vectors u and v, defined as
X
||u − v||p = (
|ui − vi |p )1/p .
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
p
[ndarray] The norm of the difference ||u − v||p .
Returns
d
[double] The Minkowski distance between vectors u and v.
num_obs_dm(d)
Returns the number of original observations that correspond to a square, redudant distance matrix D.
Parameters
d
[ndarray] The target distance matrix.
Returns
The number of observations in the redundant distance matrix.
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num_obs_y(Y)
Returns the number of original observations that correspond to a condensed distance matrix Y.
Parameters
Y
[ndarray] The number of original observations in the condensed observation Y.
Returns
n
[int] The number of observations in the condensed distance matrix passed.
pdist(X, metric=’euclidean’, p=2, V=None, VI=None)
Computes the pairwise distances between m original observations in n-dimensional space. Returns a condensed
distance matrix Y. For each i and j (where i < j < n), the metric dist(u=X[i], v=X[j]) is computed
and stored in the :math:‘ij‘th entry.
See squareform for information on how to calculate the index of this entry or to convert the condensed
distance matrix to a redundant square matrix.
The following are common calling conventions.
1.Y = pdist(X, ’euclidean’)
Computes the distance between m points using Euclidean distance (2-norm) as the distance metric between the points. The points are arranged as m n-dimensional row vectors in the matrix X.
2.Y = pdist(X, ’minkowski’, p)
Computes the distances using the Minkowski distance ||u − v||p (p-norm) where p ≥ 1.
3.Y = pdist(X, ’cityblock’)
Computes the city block or Manhattan distance between the points.
4.Y = pdist(X, ’seuclidean’, V=None)
Computes the standardized Euclidean distance. The standardized Euclidean distance between two nvectors u and v is
qX
(ui − vi )2 /V [xi ].
V is the variance vector; V[i] is the variance computed over all
the i’th components of the points. If not passed, it is automatically computed.
5.Y = pdist(X, ’sqeuclidean’)
Computes the squared Euclidean distance ||u − v||22 between the vectors.
6.Y = pdist(X, ’cosine’)
Computes the cosine distance between vectors u and v,
1 − uv T
|u|2 |v|2
where |*|_2 is the 2 norm of its argument *.
7.Y = pdist(X, ’correlation’)
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Computes the correlation distance between vectors u and v. This is
1 − (u − ū)(v − v̄)T
T
|(u − ū)||(v − v̄)|
where v̄ is the mean of the elements of vector v.
8.Y = pdist(X, ’hamming’)
Computes the normalized Hamming distance, or the proportion of those vector elements between two
n-vectors u and v which disagree. To save memory, the matrix X can be of type boolean.
9.Y = pdist(X, ’jaccard’)
Computes the Jaccard distance between the points. Given two vectors, u and v, the Jaccard distance is
the proportion of those elements u[i] and v[i] that disagree where at least one of them is non-zero.
10.Y = pdist(X, ’chebyshev’)
Computes the Chebyshev distance between the points. The Chebyshev distance between two nvectors u and v is the maximum norm-1 distance between their respective elements. More precisely,
the distance is given by
d(u, v) = max |ui − vi |.
i
11.Y = pdist(X, ’canberra’)
Computes the Canberra distance between the points. The Canberra distance between two points u
and v is
d(u, v) =
X |ui − vi |
(|ui | + |vi |)
u
12.Y = pdist(X, ’braycurtis’)
Computes the Bray-Curtis distance between the points. The Bray-Curtis distance between two
points u and v is
P
u i − vi
d(u, v) = Pi
i u i + vi
13.Y = pdist(X, ’mahalanobis’, VI=None)
Computes the Mahalanobis distance between the points. The Mahalanobis distance between two
points u and v is (u − v)(1/V )(u − v)T where (1/V ) (the VI variable) is the inverse covariance.
If VI is not None, VI will be used as the inverse covariance matrix.
14.Y = pdist(X, ’yule’)
Computes the Yule distance between each pair of boolean vectors. (see yule function documentation)
15.Y = pdist(X, ’matching’)
Computes the matching distance between each pair of boolean vectors. (see matching function
documentation)
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16.Y = pdist(X, ’dice’)
Computes the Dice distance between each pair of boolean vectors. (see dice function documentation)
17.Y = pdist(X, ’kulsinski’)
Computes the Kulsinski distance between each pair of boolean vectors. (see kulsinski function
documentation)
18.Y = pdist(X, ’rogerstanimoto’)
Computes the Rogers-Tanimoto distance between each pair of boolean vectors. (see rogerstanimoto
function documentation)
19.Y = pdist(X, ’russellrao’)
Computes the Russell-Rao distance between each pair of boolean vectors. (see russellrao function
documentation)
20.Y = pdist(X, ’sokalmichener’)
Computes the Sokal-Michener distance between each pair of boolean vectors. (see sokalmichener
function documentation)
21.Y = pdist(X, ’sokalsneath’)
Computes the Sokal-Sneath distance between each pair of boolean vectors. (see sokalsneath function documentation)
22.Y = pdist(X, ’wminkowski’)
Computes the weighted Minkowski distance between each pair of vectors. (see wminkowski function documentation)
22.Y = pdist(X, f)
Computes the distance between all pairs of vectors in X using the user supplied 2-arity function f.
For example, Euclidean distance between the vectors could be computed as follows:
dm = pdist(X, (lambda u, v: np.sqrt(((u-v)*(u-v).T).sum())))
Note that you should avoid passing a reference to one of the distance functions defined in this
library. For example,:
dm = pdist(X, sokalsneath)
would calculate the pair-wise distances between the vectors
in X using the Python function
sokalsneath. This would result in sokalsneath being called n2 times, which is inefficient. Instead,
the optimized C version is more efficient, and we call it using the following syntax.:
dm = pdist(X, ’sokalsneath’)
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Parameters
X
[ndarray] An m by n array of m original observations in an n-dimensional space.
metric
[string or function] The distance metric to use. The distance function can be ‘braycurtis’,
‘canberra’, ‘chebyshev’, ‘cityblock’, ‘correlation’, ‘cosine’, ‘dice’, ‘euclidean’, ‘hamming’, ‘jaccard’, ‘kulsinski’, ‘mahalanobis’, ‘matching’, ‘minkowski’, ‘rogerstanimoto’,
‘russellrao’, ‘seuclidean’, ‘sokalmichener’, ‘sokalsneath’, ‘sqeuclidean’, ‘yule’.
w
[ndarray] The weight vector (for weighted Minkowski).
p
[double] The p-norm to apply (for Minkowski, weighted and unweighted)
V
[ndarray] The variance vector (for standardized Euclidean).
VI
[ndarray] The inverse of the covariance matrix (for Mahalanobis).
Returns
Y
[ndarray] A condensed distance matrix.
Seealso
squareform
[converts between condensed distance matrices and] square distance matrices.
rogerstanimoto(u, v)
Computes the Rogers-Tanimoto dissimilarity between two boolean n-vectors u and v, which is defined as
R
cT T + cF F + R
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n and R = 2(cT F + cF T ).
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Rogers-Tanimoto dissimilarity between vectors u and v.
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russellrao(u, v)
Computes the Russell-Rao dissimilarity between two boolean n-vectors u and v, which is defined as
n − cT T
n
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n.
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Russell-Rao dissimilarity between vectors u and v.
seuclidean(u, v, V)
Returns the standardized Euclidean distance between two n-vectors u and v. V is an m-dimensional vector of
component variances. It is usually computed among a larger collection vectors.
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The standardized Euclidean distance between vectors u and v.
sokalmichener(u, v)
Computes the Sokal-Michener dissimilarity between two boolean vectors u and v, which is defined as
2R
S + 2R
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n, R = 2 ∗ (cT F + cF T ) and
S = cF F + cT T .
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Sokal-Michener dissimilarity between vectors u and v.
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sokalsneath(u, v)
Computes the Sokal-Sneath dissimilarity between two boolean vectors u and v,
2R
cT T + 2R
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n and R = 2(cT F + cF T ).
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Sokal-Sneath dissimilarity between vectors u and v.
sqeuclidean(u, v)
Computes the squared Euclidean distance between two n-vectors u and v, which is defined as
2
||u − v||2 .
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The squared Euclidean distance between vectors u and v.
squareform(X, force=’no’, checks=True)
Converts a vector-form distance vector to a square-form distance matrix, and vice-versa.
Parameters
X
[ndarray] Either a condensed or redundant distance matrix.
Returns
Y
[ndarray] If a condensed distance matrix is passed, a redundant one is returned, or if a
redundant one is passed, a condensed distance matrix is returned.
force
[string] As with MATLAB(TM), if force is equal to ‘tovector’ or ‘tomatrix’, the input
will be treated as a distance matrix or distance vector respectively.
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checks
[bool] If checks is set to False, no checks will be made for matrix symmetry nor
zero diagonals. This is useful if it is known that X - X.T1 is small and diag(X) is
close to zero. These values are ignored any way so they do not disrupt the squareform
transformation.
wminkowski(u, v, p, w)
Computes the weighted Minkowski distance between two vectors u and v, defined as
X
1/p
(wi |ui − vi |p )
.
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
p
[ndarray] The norm of the difference ||u − v||p .
w
[ndarray] The weight vector.
Returns
d
[double] The Minkowski distance between vectors u and v.
yule(u, v)
Computes the Yule dissimilarity between two boolean n-vectors u and v, which is defined as
cT T
R
+ cF F +
R
2
where cij is the number of occurrences of u[k] = i and v[k] = j for k < n and R = 2.0 ∗ (cT F + cF T ).
Parameters
u
[ndarray] An n-dimensional vector.
v
[ndarray] An n-dimensional vector.
Returns
d
[double] The Yule dissimilarity between vectors u and v.
3.16.2 Spatial data structures and algorithms
Nearest-neighbor queries:
KDTree – class for efficient nearest-neighbor queries distance – module containing many different distance measures
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class KDTree(data, leafsize=10)
kd-tree for quick nearest-neighbor lookup
This class provides an index into a set of k-dimensional points which can be used to rapidly look up the nearest
neighbors of any point.
The algorithm used is described in Maneewongvatana and Mount 1999. The general idea is that the kd-tree is
a binary tree, each of whose nodes represents an axis-aligned hyperrectangle. Each node specifies an axis and
splits the set of points based on whether their coordinate along that axis is greater than or less than a particular
value.
During construction, the axis and splitting point are chosen by the “sliding midpoint” rule, which ensures that
the cells do not all become long and thin.
The tree can be queried for the r closest neighbors of any given point (optionally returning only those within
some maximum distance of the point). It can also be queried, with a substantial gain in efficiency, for the r
approximate closest neighbors.
For large dimensions (20 is already large) do not expect this to run significantly faster than brute force. Highdimensional nearest-neighbor queries are a substantial open problem in computer science.
The tree also supports all-neighbors queries, both with arrays of points and with other kd-trees. These do use a
reasonably efficient algorithm, but the kd-tree is not necessarily the best data structure for this sort of calculation.
Methods
count_neighbors(other, r, p=2.0)
Count how many nearby pairs can be formed.
Count the number of pairs (x1,x2) can be formed, with x1 drawn from self and x2 drawn from other,
and where distance(x1,x2,p)<=r. This is the “two-point correlation” described in Gray and Moore 2000,
“N-body problems in statistical learning”, and the code here is based on their algorithm.
Parameters
other : KDTree
r : float or one-dimensional array of floats
The radius to produce a count for. Multiple radii are searched with a single tree
traversal.
p : float, 1<=p<=infinity
Which Minkowski p-norm to use
Returns
result : integer or one-dimensional array of integers
The number of pairs. Note that this is internally stored in a numpy int, and so
may overflow if very large (two billion).
query(x, k=1, eps=0, p=2, distance_upper_bound=inf )
query the kd-tree for nearest neighbors
Parameters
x : array-like, last dimension self.m
An array of points to query.
k : integer
The number of nearest neighbors to return.
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eps : nonnegative float
Return approximate nearest neighbors; the kth returned value is guaranteed to be
no further than (1+eps) times the distance to the real kth nearest neighbor.
p : float, 1<=p<=infinity
Which Minkowski p-norm to use. 1 is the sum-of-absolute-values “Manhattan”
distance 2 is the usual Euclidean distance infinity is the maximum-coordinatedifference distance
distance_upper_bound : nonnegative float
Return only neighbors within this distance. This is used to prune tree searches,
so if you are doing a series of nearest-neighbor queries, it may help to supply the
distance to the nearest neighbor of the most recent point.
Returns
d : array of floats
The distances to the nearest neighbors. If x has shape tuple+(self.m,), then d has
shape tuple if k is one, or tuple+(k,) if k is larger than one. Missing neighbors are
indicated with infinite distances. If k is None, then d is an object array of shape
tuple, containing lists of distances. In either case the hits are sorted by distance
(nearest first).
i : array of integers
The locations of the neighbors in self.data. i is the same shape as d.
Examples
>>> from scipy.spatial import KDTree
>>> x, y = np.mgrid[0:5, 2:8]
>>> tree = KDTree(zip(x.ravel(), y.ravel()))
>>> tree.data
array([[0, 2],
[0, 3],
[0, 4],
[0, 5],
[0, 6],
[0, 7],
[1, 2],
[1, 3],
[1, 4],
[1, 5],
[1, 6],
[1, 7],
[2, 2],
[2, 3],
[2, 4],
[2, 5],
[2, 6],
[2, 7],
[3, 2],
[3, 3],
[3, 4],
[3, 5],
[3, 6],
[3, 7],
[4, 2],
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[4, 3],
[4, 4],
[4, 5],
[4, 6],
[4, 7]])
>>> pts = np.array([[0, 0], [2.1, 2.9]])
>>> tree.query(pts)
(array([ 2.
, 0.14142136]), array([ 0, 13]))
query_ball_point(x, r, p=2.0, eps=0)
Find all points within r of x
Parameters
x : array_like, shape tuple + (self.m,)
The point or points to search for neighbors of
r : positive float
The radius of points to return
p : float 1<=p<=infinity
Which Minkowski p-norm to use
eps : nonnegative float
Approximate search. Branches of the tree are not explored if their nearest points
are further than r/(1+eps), and branches are added in bulk if their furthest points
are nearer than r*(1+eps).
Returns
results : list or array of lists
If x is a single point, returns a list of the indices of the neighbors of x. If x is an
array of points, returns an object array of shape tuple containing lists of neighbors.
Note: if you have many points whose neighbors you want to find, you may save :
substantial amounts of time by putting them in a KDTree and using query_ball_tree.
:
query_ball_tree(other, r, p=2.0, eps=0)
Find all pairs of points whose distance is at most r
Parameters
other : KDTree
The tree containing points to search against
r : positive float
The maximum distance
p : float 1<=p<=infinity
Which Minkowski norm to use
eps : nonnegative float
Approximate search. Branches of the tree are not explored if their nearest points
are further than r/(1+eps), and branches are added in bulk if their furthest points
are nearer than r*(1+eps).
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Returns
results : list of lists
For each element self.data[i] of this tree, results[i] is a list of the indices of its
neighbors in other.data.
query_pairs(r, p=2.0, eps=0)
Find all pairs of points whose distance is at most r
Parameters
r : positive float
The maximum distance
p : float 1<=p<=infinity
Which Minkowski norm to use
eps : nonnegative float
Approximate search. Branches of the tree are not explored if their nearest points
are further than r/(1+eps), and branches are added in bulk if their furthest points
are nearer than r*(1+eps).
Returns
results : set
set of pairs (i,j), i<j, for which the corresponing positions are close.
sparse_distance_matrix(other, max_distance, p=2.0)
Compute a sparse distance matrix
Computes a distance matrix between two KDTrees, leaving as zero any distance greater than
max_distance.
Parameters
other : KDTree
max_distance : positive float
Returns
result : dok_matrix
Sparse matrix representing the results in “dictionary of keys” format.
class Rectangle(maxes, mins)
Hyperrectangle class.
Represents a Cartesian product of intervals.
Methods
max_distance_point(x, p=2.0)
Compute the maximum distance between x and a point in the hyperrectangle.
max_distance_rectangle(other, p=2.0)
Compute the maximum distance between points in the two hyperrectangles.
min_distance_point(x, p=2.0)
Compute the minimum distance between x and a point in the hyperrectangle.
min_distance_rectangle(other, p=2.0)
Compute the minimum distance between points in the two hyperrectangles.
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split(d, split)
Produce two hyperrectangles by splitting along axis d.
In general, if you need to compute maximum and minimum distances to the children, it can be done more
efficiently by updating the maximum and minimum distances to the parent.
volume()
Total volume.
class cKDTree()
kd-tree for quick nearest-neighbor lookup
This class provides an index into a set of k-dimensional points which can be used to rapidly look up the nearest
neighbors of any point.
The algorithm used is described in Maneewongvatana and Mount 1999. The general idea is that the kd-tree is
a binary trie, each of whose nodes represents an axis-aligned hyperrectangle. Each node specifies an axis and
splits the set of points based on whether their coordinate along that axis is greater than or less than a particular
value.
During construction, the axis and splitting point are chosen by the “sliding midpoint” rule, which ensures that
the cells do not all become long and thin.
The tree can be queried for the r closest neighbors of any given point (optionally returning only those within
some maximum distance of the point). It can also be queried, with a substantial gain in efficiency, for the r
approximate closest neighbors.
For large dimensions (20 is already large) do not expect this to run significantly faster than brute force. Highdimensional nearest-neighbor queries are a substantial open problem in computer science.
Methods
query
query the kd-tree for nearest neighbors
distance_matrix(x, y, p=2, threshold=1000000)
Compute the distance matrix.
Computes the matrix of all pairwise distances.
Parameters
x : array-like, m by k
y : array-like, n by k
p : float 1<=p<=infinity
Which Minkowski p-norm to use.
threshold : positive integer
If m*n*k>threshold use a python loop instead of creating a very large temporary.
Returns
result : array-like, m by n
heappop()
Pop the smallest item off the heap, maintaining the heap invariant.
heappush()
Push item onto heap, maintaining the heap invariant.
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minkowski_distance(x, y, p=2)
Compute the L**p distance between x and y
minkowski_distance_p(x, y, p=2)
Compute the pth power of the L**p distance between x and y
For efficiency, this function computes the L**p distance but does not extract the pth root. If p is 1 or infinity,
this is equal to the actual L**p distance.
3.17 Special functions (scipy.special)
Nearly all of the functions below are universal functions and follow broadcasting and automatic array-looping rules.
Exceptions are noted.
3.17.1 Error handling
Errors are handled by returning nans, or other appropriate values. Some of the special function routines will print an
error message when an error occurs. By default this printing is disabled. To enable such messages use errprint(1) To
disable such messages use errprint(0).
Example:
>>> print scipy.special.bdtr(-1,10,0.3)
>>> scipy.special.errprint(1)
>>> print scipy.special.bdtr(-1,10,0.3)
errprint ()
errprint({flag}) sets the error printing flag for special functions (from the cephesmodule). The
output is the previous state. With errprint(0) no error messages are shown; the default is
errprint(1). If no argument is given the current state of the flag is returned and no change occurs.
errstate (**kwargs)
Context manager for floating-point error handling.
errprint()
errprint({flag}) sets the error printing flag for special functions (from the cephesmodule). The output is the
previous state. With errprint(0) no error messages are shown; the default is errprint(1). If no argument is given
the current state of the flag is returned and no change occurs.
class errstate(**kwargs)
Context manager for floating-point error handling.
Using an instance of errstate as a context manager allows statements in that context to execute with a known
error handling behavior. Upon entering the context the error handling is set with seterr and seterrcall, and upon
exiting it is reset to what it was before.
Parameters
kwargs : {divide, over, under, invalid}
Keyword arguments. The valid keywords are the possible floating-point exceptions.
Each keyword should have a string value that defines the treatment for the particular
error. Possible values are {‘ignore’, ‘warn’, ‘raise’, ‘call’, ‘print’, ‘log’}.
See Also:
seterr, geterr, seterrcall, geterrcall
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Notes
The with statement was introduced in Python 2.5, and can only be used there by importing it: from
__future__ import with_statement. In earlier Python versions the with statement is not available.
For complete documentation of the types of floating-point exceptions and treatment options, see seterr.
Examples
>>> from __future__ import with_statement # use ’with’ in Python 2.5
>>> olderr = np.seterr(all=’ignore’) # Set error handling to known state.
>>> np.arange(3) / 0.
array([ NaN, Inf, Inf])
>>> with np.errstate(divide=’warn’):
...
np.arange(3) / 0.
...
__main__:2: RuntimeWarning: divide by zero encountered in divide
array([ NaN, Inf, Inf])
>>> np.sqrt(-1)
nan
>>> with np.errstate(invalid=’raise’):
...
np.sqrt(-1)
...
Traceback (most recent call last):
File "<stdin>", line 2, in <module>
FloatingPointError: invalid value encountered in sqrt
Outside the context the error handling behavior has not changed:
>>> np.geterr()
{’over’: ’ignore’, ’divide’: ’ignore’, ’invalid’: ’ignore’,
’under’: ’ignore’}
3.17.2 Available functions
Airy functions
airy (x[, out1, out2,(Ai,Aip,Bi,Bip)=airy(z)
out3, ...)
calculates the Airy functions and their derivatives evaluated at real or
complex number z. The Airy functions Ai and Bi are two independent solutions of y’‘(x)=xy.
Aip and Bip are the first derivatives evaluated at x of Ai and Bi respectively.
airye (x[, out1, out2,
(Aie,Aipe,Bie,Bipe)=airye(z)
out3, ...)
calculates the exponentially scaled Airy functions and their
derivatives evaluated at real or complex number z. airye(z)[0:1] = airy(z)[0:1] *
exp(2.0/3.0*z*sqrt(z)) airye(z)[2:3] = airy(z)[2:3] * exp(-abs((2.0/3.0*z*sqrt(z)).real))
ai_zeros (nt)
Compute the zeros of Airy Functions Ai(x) and Ai’(x), a and a’ respectively, and the
associated values of Ai(a’) and Ai’(a).
bi_zeros (nt)
Compute the zeros of Airy Functions Bi(x) and Bi’(x), b and b’ respectively, and the
associated values of Ai(b’) and Ai’(b).
airy(x, [out1, out2, out3, out4])
(Ai,Aip,Bi,Bip)=airy(z) calculates the Airy functions and their derivatives evaluated at real or complex number
z. The Airy functions Ai and Bi are two independent solutions of y’‘(x)=xy. Aip and Bip are the first derivatives
evaluated at x of Ai and Bi respectively.
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airye(x, [out1, out2, out3, out4])
(Aie,Aipe,Bie,Bipe)=airye(z) calculates the exponentially scaled Airy functions and their derivatives evaluated
at real or complex number z. airye(z)[0:1] = airy(z)[0:1] * exp(2.0/3.0*z*sqrt(z)) airye(z)[2:3] = airy(z)[2:3] *
exp(-abs((2.0/3.0*z*sqrt(z)).real))
ai_zeros(nt)
Compute the zeros of Airy Functions Ai(x) and Ai’(x), a and a’ respectively, and the associated values of Ai(a’)
and Ai’(a).
Outputs:
a[l-1] – the lth zero of Ai(x) ap[l-1] – the lth zero of Ai’(x) ai[l-1] – Ai(ap[l-1]) aip[l-1] – Ai’(a[l-1])
bi_zeros(nt)
Compute the zeros of Airy Functions Bi(x) and Bi’(x), b and b’ respectively, and the associated values of Ai(b’)
and Ai’(b).
Outputs:
b[l-1] – the lth zero of Bi(x) bp[l-1] – the lth zero of Bi’(x) bi[l-1] – Bi(bp[l-1]) bip[l-1] – Bi’(b[l-1])
Elliptic Functions and Integrals
ellipj (x1, x2[, out1,
(sn,cn,dn,ph)=ellipj(u,m)
out2, ...)
calculates the Jacobian elliptic functions of parameter m between 0
and 1, and real u. The returned functions are often written sn(u|m), cn(u|m), and dn(u|m).
The value of ph is such that if u = ellik(ph,m), then sn(u|m) = sin(ph) and cn(u|m) = cos(ph).
ellipk (x[, out]) y=ellipk(m) returns the complete integral of the first kind:
integral(1/sqrt(1-m*sin(t)**2),t=0..pi/2)
ellipkinc (x1, x2[,y=ellipkinc(phi,m)
out])
returns the incomplete elliptic integral of the first kind:
integral(1/sqrt(1-m*sin(t)**2),t=0..phi)
ellipe (x[, out]) y=ellipe(m) returns the complete integral of the second kind:
integral(sqrt(1-m*sin(t)**2),t=0..pi/2)
ellipeinc (x1, x2[,y=ellipeinc(phi,m)
out])
returns the incomplete elliptic integral of the second kind:
integral(sqrt(1-m*sin(t)**2),t=0..phi)
ellipj(x1, x2, [out1, out2, out3, out4])
(sn,cn,dn,ph)=ellipj(u,m) calculates the Jacobian elliptic functions of parameter m between 0 and 1, and real
u. The returned functions are often written sn(u|m), cn(u|m), and dn(u|m). The value of ph is such that if u =
ellik(ph,m), then sn(u|m) = sin(ph) and cn(u|m) = cos(ph).
ellipk(x, [out])
y=ellipk(m) returns the complete integral of the first kind: integral(1/sqrt(1-m*sin(t)**2),t=0..pi/2)
ellipkinc(x1, x2, [out])
y=ellipkinc(phi,m) returns the incomplete elliptic integral of the first kind:
m*sin(t)**2),t=0..phi)
integral(1/sqrt(1-
ellipe(x, [out])
y=ellipe(m) returns the complete integral of the second kind: integral(sqrt(1-m*sin(t)**2),t=0..pi/2)
ellipeinc(x1, x2, [out])
y=ellipeinc(phi,m) returns the incomplete elliptic integral of the second kind:
m*sin(t)**2),t=0..phi)
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Bessel Functions
jn (x1, x2[, out]) y=jv(v,z) returns the Bessel function of real order v at complex z.
jv (x1, x2[, out]) y=jv(v,z) returns the Bessel function of real order v at complex z.
jve (x1, x2[, out])y=jve(v,z) returns the exponentially scaled Bessel function of real order v at complex z: jve(v,z)
= jv(v,z) * exp(-abs(z.imag))
yn (x1, x2[, out]) y=yn(n,x) returns the Bessel function of the second kind of integer order n at x.
yv (x1, x2[, out]) y=yv(v,z) returns the Bessel function of the second kind of real order v at complex z.
yve (x1, x2[, out])y=yve(v,z) returns the exponentially scaled Bessel function of the second kind of real order v at
complex z: yve(v,z) = yv(v,z) * exp(-abs(z.imag))
kn (x1, x2[, out]) y=kn(n,x) returns the modified Bessel function of the second kind (sometimes called the third
kind) for integer order n at x.
kv (x1, x2[, out]) y=kv(v,z) returns the modified Bessel function of the second kind (sometimes called the third
kind) for real order v at complex z.
kve (x1, x2[, out])y=kve(v,z) returns the exponentially scaled, modified Bessel function of the second kind
(sometimes called the third kind) for real order v at complex z: kve(v,z) = kv(v,z) * exp(z)
iv (x1, x2[, out]) y=iv(v,z) returns the modified Bessel function of real order v of z. If z is of real type and
negative, v must be integer valued.
ive (x1, x2[, out])y=ive(v,z) returns the exponentially scaled modified Bessel function of real order v and
complex z: ive(v,z) = iv(v,z) * exp(-abs(z.real))
hankel1 (x1, x2[,y=hankel1(v,z)
out])
returns the Hankel function of the first kind for real order v and complex
argument z.
hankel1e (x1, x2[,
y=hankel1e(v,z)
out])
returns the exponentially scaled Hankel function of the first kind for real order
v and complex argument z: hankel1e(v,z) = hankel1(v,z) * exp(-1j * z)
hankel2 (x1, x2[,y=hankel2(v,z)
out])
returns the Hankel function of the second kind for real order v and complex
argument z.
hankel2e (x1, x2[,
y=hankel2e(v,z)
out])
returns the exponentially scaled Hankel function of the second kind for real
order v and complex argument z: hankel1e(v,z) = hankel1(v,z) * exp(1j * z)
jn(x1, x2, [out])
y=jv(v,z) returns the Bessel function of real order v at complex z.
jv(x1, x2, [out])
y=jv(v,z) returns the Bessel function of real order v at complex z.
jve(x1, x2, [out])
y=jve(v,z) returns the exponentially scaled Bessel function of real order v at complex z: jve(v,z) = jv(v,z) *
exp(-abs(z.imag))
yn(x1, x2, [out])
y=yn(n,x) returns the Bessel function of the second kind of integer order n at x.
yv(x1, x2, [out])
y=yv(v,z) returns the Bessel function of the second kind of real order v at complex z.
yve(x1, x2, [out])
y=yve(v,z) returns the exponentially scaled Bessel function of the second kind of real order v at complex z:
yve(v,z) = yv(v,z) * exp(-abs(z.imag))
kn(x1, x2, [out])
y=kn(n,x) returns the modified Bessel function of the second kind (sometimes called the third kind) for integer
order n at x.
kv(x1, x2, [out])
y=kv(v,z) returns the modified Bessel function of the second kind (sometimes called the third kind) for real
order v at complex z.
kve(x1, x2, [out])
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y=kve(v,z) returns the exponentially scaled, modified Bessel function of the second kind (sometimes called the
third kind) for real order v at complex z: kve(v,z) = kv(v,z) * exp(z)
iv(x1, x2, [out])
y=iv(v,z) returns the modified Bessel function of real order v of z. If z is of real type and negative, v must be
integer valued.
ive(x1, x2, [out])
y=ive(v,z) returns the exponentially scaled modified Bessel function of real order v and complex z: ive(v,z) =
iv(v,z) * exp(-abs(z.real))
hankel1(x1, x2, [out])
y=hankel1(v,z) returns the Hankel function of the first kind for real order v and complex argument z.
hankel1e(x1, x2, [out])
y=hankel1e(v,z) returns the exponentially scaled Hankel function of the first kind for real order v and complex
argument z: hankel1e(v,z) = hankel1(v,z) * exp(-1j * z)
hankel2(x1, x2, [out])
y=hankel2(v,z) returns the Hankel function of the second kind for real order v and complex argument z.
hankel2e(x1, x2, [out])
y=hankel2e(v,z) returns the exponentially scaled Hankel function of the second kind for real order v and complex
argument z: hankel1e(v,z) = hankel1(v,z) * exp(1j * z)
The following is not an universal function:
lmbda (v, x) Compute sequence of lambda functions with arbitrary order v and their derivatives. Lv0(x)..Lv(x)
are computed with v0=v-int(v).
lmbda(v, x)
Compute sequence of lambda functions with arbitrary order v and their derivatives. Lv0(x)..Lv(x) are computed
with v0=v-int(v).
Zeros of Bessel Functions
These are not universal functions:
jnjnp_zeros (nt)
Compute nt (<=1200) zeros of the bessel functions Jn and Jn’ and arange them in order
of their magnitudes.
jnyn_zeros (n, nt) Compute nt zeros of the Bessel functions Jn(x), Jn’(x), Yn(x), and Yn’(x), respectively.
Returns 4 arrays of length nt.
jn_zeros (n, nt)
Compute nt zeros of the Bessel function Jn(x).
jnp_zeros (n, nt)
Compute nt zeros of the Bessel function Jn’(x).
yn_zeros (n, nt)
Compute nt zeros of the Bessel function Yn(x).
ynp_zeros (n, nt)
Compute nt zeros of the Bessel function Yn’(x).
y0_zeros (nt[, com- Returns nt (complex or real) zeros of Y0(z), z0, and the value of Y0’(z0) = -Y1(z0) at
plex])
each zero.
y1_zeros (nt[, com- Returns nt (complex or real) zeros of Y1(z), z1, and the value of Y1’(z1) = Y0(z1) at
plex])
each zero.
y1p_zeros (nt[, com- Returns nt (complex or real) zeros of Y1’(z), z1’, and the value of Y1(z1’) at each zero.
plex])
jnjnp_zeros(nt)
Compute nt (<=1200) zeros of the bessel functions Jn and Jn’ and arange them in order of their magnitudes.
Outputs (all are arrays of length nt):
zo[l-1] – Value of the lth zero of of Jn(x) and Jn’(x) n[l-1] – Order of the Jn(x) or Jn’(x) associated
with lth zero m[l-1] – Serial number of the zeros of Jn(x) or Jn’(x) associated
with lth zero.
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t[l-1] – 0 if lth zero in zo is zero of Jn(x), 1 if it is a zero
of Jn’(x)
See jn_zeros, jnp_zeros to get separated arrays of zeros.
jnyn_zeros(n, nt)
Compute nt zeros of the Bessel functions Jn(x), Jn’(x), Yn(x), and Yn’(x), respectively. Returns 4 arrays of
length nt.
See jn_zeros, jnp_zeros, yn_zeros, ynp_zeros to get separate arrays.
jn_zeros(n, nt)
Compute nt zeros of the Bessel function Jn(x).
jnp_zeros(n, nt)
Compute nt zeros of the Bessel function Jn’(x).
yn_zeros(n, nt)
Compute nt zeros of the Bessel function Yn(x).
ynp_zeros(n, nt)
Compute nt zeros of the Bessel function Yn’(x).
y0_zeros(nt, complex=0)
Returns nt (complex or real) zeros of Y0(z), z0, and the value of Y0’(z0) = -Y1(z0) at each zero.
y1_zeros(nt, complex=0)
Returns nt (complex or real) zeros of Y1(z), z1, and the value of Y1’(z1) = Y0(z1) at each zero.
y1p_zeros(nt, complex=0)
Returns nt (complex or real) zeros of Y1’(z), z1’, and the value of Y1(z1’) at each zero.
Faster versions of common Bessel Functions
j0 (x[, out])y=j0(x) returns the Bessel function of order 0 at x.
j1 (x[, out])y=j1(x) returns the Bessel function of order 1 at x.
y0 (x[, out])y=y0(x) returns the Bessel function of the second kind of order 0 at x.
y1 (x[, out])y=y1(x) returns the Bessel function of the second kind of order 1 at x.
i0 (x[, out])y=i0(x) returns the modified Bessel function of order 0 at x.
i0e (x[, out])
y=i0e(x) returns the exponentially scaled modified Bessel function of order 0 at x. i0e(x) = exp(-|x|) *
i0(x).
i1 (x[, out])y=i1(x) returns the modified Bessel function of order 1 at x.
i1e (x[, out])
y=i1e(x) returns the exponentially scaled modified Bessel function of order 0 at x. i1e(x) = exp(-|x|) *
i1(x).
k0 (x[, out])y=k0(x) returns the modified Bessel function of the second kind (sometimes called the third kind) of
order 0 at x.
k0e (x[, out])
y=k0e(x) returns the exponentially scaled modified Bessel function of the second kind (sometimes
called the third kind) of order 0 at x. k0e(x) = exp(x) * k0(x).
k1 (x[, out])y=i1(x) returns the modified Bessel function of the second kind (sometimes called the third kind) of
order 1 at x.
k1e (x[, out])
y=k1e(x) returns the exponentially scaled modified Bessel function of the second kind (sometimes
called the third kind) of order 1 at x. k1e(x) = exp(x) * k1(x)
j0(x, [out])
y=j0(x) returns the Bessel function of order 0 at x.
j1(x, [out])
y=j1(x) returns the Bessel function of order 1 at x.
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y0(x, [out])
y=y0(x) returns the Bessel function of the second kind of order 0 at x.
y1(x, [out])
y=y1(x) returns the Bessel function of the second kind of order 1 at x.
i0(x, [out])
y=i0(x) returns the modified Bessel function of order 0 at x.
i0e(x, [out])
y=i0e(x) returns the exponentially scaled modified Bessel function of order 0 at x. i0e(x) = exp(-|x|) * i0(x).
i1(x, [out])
y=i1(x) returns the modified Bessel function of order 1 at x.
i1e(x, [out])
y=i1e(x) returns the exponentially scaled modified Bessel function of order 0 at x. i1e(x) = exp(-|x|) * i1(x).
k0(x, [out])
y=k0(x) returns the modified Bessel function of the second kind (sometimes called the third kind) of order 0 at
x.
k0e(x, [out])
y=k0e(x) returns the exponentially scaled modified Bessel function of the second kind (sometimes called the
third kind) of order 0 at x. k0e(x) = exp(x) * k0(x).
k1(x, [out])
y=i1(x) returns the modified Bessel function of the second kind (sometimes called the third kind) of order 1 at
x.
k1e(x, [out])
y=k1e(x) returns the exponentially scaled modified Bessel function of the second kind (sometimes called the
third kind) of order 1 at x. k1e(x) = exp(x) * k1(x)
Integrals of Bessel Functions
itj0y0 (x[, out1, out2])
(ij0,iy0)=itj0y0(x) returns simple integrals from 0 to x of the zeroth order bessel
functions j0 and y0.
it2j0y0 (x[, out1, out2]) (ij0,iy0)=it2j0y0(x) returns the integrals int((1-j0(t))/t,t=0..x) and
int(y0(t)/t,t=x..infinitity).
iti0k0 (x[, out1, out2]) (ii0,ik0)=iti0k0(x) returns simple integrals from 0 to x of the zeroth order modified
bessel functions i0 and k0.
it2i0k0 (x[, out1, out2]) (ii0,ik0)=it2i0k0(x) returns the integrals int((i0(t)-1)/t,t=0..x) and
int(k0(t)/t,t=x..infinitity).
besselpoly (x1, x2, x3[, out])
y=besselpoly(a,lam,nu) returns the value of the integral: integral(x**lam *
jv(nu,2*a*x),x=0..1).
itj0y0(x, [out1, out2])
(ij0,iy0)=itj0y0(x) returns simple integrals from 0 to x of the zeroth order bessel functions j0 and y0.
it2j0y0(x, [out1, out2])
(ij0,iy0)=it2j0y0(x) returns the integrals int((1-j0(t))/t,t=0..x) and int(y0(t)/t,t=x..infinitity).
iti0k0(x, [out1, out2])
(ii0,ik0)=iti0k0(x) returns simple integrals from 0 to x of the zeroth order modified bessel functions i0 and k0.
it2i0k0(x, [out1, out2])
(ii0,ik0)=it2i0k0(x) returns the integrals int((i0(t)-1)/t,t=0..x) and int(k0(t)/t,t=x..infinitity).
besselpoly(x1, x2, x3, [out])
y=besselpoly(a,lam,nu) returns the value of the integral: integral(x**lam * jv(nu,2*a*x),x=0..1).
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Derivatives of Bessel Functions
jvp (v, z[, n])
yvp (v, z[, n])
kvp (v, z[, n])
ivp (v, z[, n])
h1vp (v, z[, n])
h2vp (v, z[, n])
Return the nth derivative of Jv(z) with respect to z.
Return the nth derivative of Yv(z) with respect to z.
Return the nth derivative of Kv(z) with respect to z.
Return the nth derivative of Iv(z) with respect to z.
Return the nth derivative of H1v(z) with respect to z.
Return the nth derivative of H2v(z) with respect to z.
jvp(v, z, n=1)
Return the nth derivative of Jv(z) with respect to z.
yvp(v, z, n=1)
Return the nth derivative of Yv(z) with respect to z.
kvp(v, z, n=1)
Return the nth derivative of Kv(z) with respect to z.
ivp(v, z, n=1)
Return the nth derivative of Iv(z) with respect to z.
h1vp(v, z, n=1)
Return the nth derivative of H1v(z) with respect to z.
h2vp(v, z, n=1)
Return the nth derivative of H2v(z) with respect to z.
Spherical Bessel Functions
These are not universal functions:
sph_jn (n, z)
Compute the spherical Bessel function jn(z) and its derivative for all orders up to and including
n.
sph_yn (n, z) Compute the spherical Bessel function yn(z) and its derivative for all orders up to and including
n.
sph_jnyn (n, z) Compute the spherical Bessel functions, jn(z) and yn(z) and their derivatives for all orders up to
and including n.
sph_in (n, z) Compute the spherical Bessel function in(z) and its derivative for all orders up to and including
n.
sph_kn (n, z) Compute the spherical Bessel function kn(z) and its derivative for all orders up to and including
n.
sph_inkn (n, z) Compute the spherical Bessel functions, in(z) and kn(z) and their derivatives for all orders up to
and including n.
sph_jn(n, z)
Compute the spherical Bessel function jn(z) and its derivative for all orders up to and including n.
sph_yn(n, z)
Compute the spherical Bessel function yn(z) and its derivative for all orders up to and including n.
sph_jnyn(n, z)
Compute the spherical Bessel functions, jn(z) and yn(z) and their derivatives for all orders up to and including
n.
sph_in(n, z)
Compute the spherical Bessel function in(z) and its derivative for all orders up to and including n.
sph_kn(n, z)
Compute the spherical Bessel function kn(z) and its derivative for all orders up to and including n.
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sph_inkn(n, z)
Compute the spherical Bessel functions, in(z) and kn(z) and their derivatives for all orders up to and including
n.
Riccati-Bessel Functions
These are not universal functions:
riccati_jn (n, x) Compute the Ricatti-Bessel function of the first kind and its derivative for all orders up to and
including n.
riccati_yn (n, x) Compute the Ricatti-Bessel function of the second kind and its derivative for all orders up to
and including n.
riccati_jn(n, x)
Compute the Ricatti-Bessel function of the first kind and its derivative for all orders up to and including n.
riccati_yn(n, x)
Compute the Ricatti-Bessel function of the second kind and its derivative for all orders up to and including n.
Struve Functions
struve (x1, x2[, out])
y=struve(v,x) returns the Struve function Hv(x) of order v at x, x must be positive unless v is
an integer.
modstruve (x1, x2[,
y=modstruve(v,x)
out])
returns the modified Struve function Lv(x) of order v at x, x must be
positive unless v is an integer and it is recommended that |v|<=20.
itstruve0 (x[, out])
y=itstruve0(x) returns the integral of the Struve function of order 0 from 0 to x:
integral(H0(t), t=0..x).
it2struve0 (x[, out])
y=it2struve0(x) returns the integral of the Struve function of order 0 divided by t from x to
infinity: integral(H0(t)/t, t=x..inf).
itmodstruve0 (x[,
y=itmodstruve0(x)
out])
returns the integral of the modified Struve function of order 0 from 0 to x:
integral(L0(t), t=0..x).
struve(x1, x2, [out])
y=struve(v,x) returns the Struve function Hv(x) of order v at x, x must be positive unless v is an integer.
modstruve(x1, x2, [out])
y=modstruve(v,x) returns the modified Struve function Lv(x) of order v at x, x must be positive unless v is an
integer and it is recommended that |v|<=20.
itstruve0(x, [out])
y=itstruve0(x) returns the integral of the Struve function of order 0 from 0 to x: integral(H0(t), t=0..x).
it2struve0(x, [out])
y=it2struve0(x) returns the integral of the Struve function of order 0 divided by t from x to infinity: integral(H0(t)/t, t=x..inf).
itmodstruve0(x, [out])
y=itmodstruve0(x) returns the integral of the modified Struve function of order 0 from 0 to x: integral(L0(t),
t=0..x).
Raw Statistical Functions
See Also:
scipy.stats: Friendly versions of these functions.
bdtr (x1, x2, x3[, out])
y=bdtr(k,n,p) returns the sum of the terms 0 through k of the Binomial probability density: sum(nCj p**
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bdtrc (x1, x2, x3[, out])
bdtri (x1, x2, x3[, out])
btdtr (x1, x2, x3[, out])
btdtri (x1, x2, x3[, out])
fdtr (x1, x2, x3[, out])
fdtrc (x1, x2, x3[, out])
fdtri (x1, x2, x3[, out])
gdtr (x1, x2, x3[, out])
gdtrc (x1, x2, x3[, out])
gdtria (x1, x2, x3[, out])
gdtrib (x1, x2, x3[, out])
gdtrix (x1, x2, x3[, out])
nbdtr (x1, x2, x3[, out])
nbdtrc (x1, x2, x3[, out])
nbdtri (x1, x2, x3[, out])
pdtr (x1, x2[, out])
pdtrc (x1, x2[, out])
pdtri (x1, x2[, out])
stdtr (x1, x2[, out])
stdtridf (x1, x2[, out])
stdtrit (x1, x2[, out])
chdtr (x1, x2[, out])
chdtrc (x1, x2[, out])
chdtri (x1, x2[, out])
ndtr (x[, out])
ndtri (x[, out])
smirnov (x1, x2[, out])
smirnovi (x1, x2[, out])
kolmogorov (x[, out])
kolmogi (x[, out])
tklmbda (x1, x2[, out])
y=bdtrc(k,n,p) returns the sum of the terms k+1 through n of the Binomial probability density: sum(nCj
p=bdtri(k,n,y) finds the probability p such that the sum of the terms 0 through k of the Binomial probabi
y=btdtr(a,b,x) returns the area from zero to x under the beta density function: gamma(a+b)/(gamma(a)*
x=btdtri(a,b,p) returns the pth quantile of the beta distribution. It is effectively the inverse of btdtr return
y=fdtr(dfn,dfd,x) returns the area from zero to x under the F density function (also known as Snedcor’s
y=fdtrc(dfn,dfd,x) returns the complemented F distribution function.
x=fdtri(dfn,dfd,p) finds the F density argument x such that fdtr(dfn,dfd,x)=p.
y=gdtr(a,b,x) returns the integral from zero to x of the gamma probability density function: a**b / gamm
y=gdtrc(a,b,x) returns the integral from x to infinity of the gamma probability density function. SEE gdt
y=nbdtr(k,n,p) returns the sum of the terms 0 through k of the negative binomial distribution: sum((n+jy=nbdtrc(k,n,p) returns the sum of the terms k+1 to infinity of the negative binomial distribution.
p=nbdtri(k,n,y) finds the argument p such that nbdtr(k,n,p)=y.
y=pdtr(k,m) returns the sum of the first k terms of the Poisson distribution: sum(exp(-m) * m**j / j!, j=0
y=pdtrc(k,m) returns the sum of the terms from k+1 to infinity of the Poisson distribution: sum(exp(-m)
m=pdtri(k,y) returns the Poisson variable m such that the sum from 0 to k of the Poisson density is equa
p=stdtr(df,t) returns the integral from minus infinity to t of the Student t distribution with df > 0 degrees
t=stdtridf(p,t) returns the argument df such that stdtr(df,t) is equal to p.
t=stdtrit(df,p) returns the argument t such that stdtr(df,t) is equal to p.
p=chdtr(v,x) Returns the area under the left hand tail (from 0 to x) of the Chi square probability density
p=chdtrc(v,x) returns the area under the right hand tail (from x to infinity) of the Chi square probability
x=chdtri(v,p) returns the argument x such that chdtrc(v,x) is equal to p.
y=ndtr(x) returns the area under the standard Gaussian probability density function, integrated from min
x=ndtri(y) returns the argument x for which the area udnder the Gaussian probability density function (i
y=smirnov(n,e) returns the exact Kolmogorov-Smirnov complementary cumulative distribution function
e=smirnovi(n,y) returns e such that smirnov(n,e) = y.
p=kolmogorov(y) returns the complementary cumulative distribution function of Kolmogorov’s limiting
y=kolmogi(p) returns y such that kolmogorov(y) = p
bdtr(x1, x2, x3, [out])
y=bdtr(k,n,p) returns the sum of the terms 0 through k of the Binomial probability density: sum(nCj p**j
(1-p)**(n-j),j=0..k)
bdtrc(x1, x2, x3, [out])
y=bdtrc(k,n,p) returns the sum of the terms k+1 through n of the Binomial probability density: sum(nCj p**j
(1-p)**(n-j), j=k+1..n)
bdtri(x1, x2, x3, [out])
p=bdtri(k,n,y) finds the probability p such that the sum of the terms 0 through k of the Binomial probability
density is equal to the given cumulative probability y.
btdtr(x1, x2, x3, [out])
y=btdtr(a,b,x) returns the area from zero to x under the beta density
gamma(a+b)/(gamma(a)*gamma(b)))*integral(t**(a-1) (1-t)**(b-1), t=0..x). SEE ALSO betainc
function:
btdtri(x1, x2, x3, [out])
x=btdtri(a,b,p) returns the pth quantile of the beta distribution. It is effectively the inverse of btdtr returning the
value of x for which btdtr(a,b,x) = p. SEE ALSO betaincinv
fdtr(x1, x2, x3, [out])
y=fdtr(dfn,dfd,x) returns the area from zero to x under the F density function (also known as Snedcor’s density
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or the variance ratio density). This is the density of X = (unum/dfn)/(uden/dfd), where unum and uden are
random variables having Chi square distributions with dfn and dfd degrees of freedom, respectively.
fdtrc(x1, x2, x3, [out])
y=fdtrc(dfn,dfd,x) returns the complemented F distribution function.
fdtri(x1, x2, x3, [out])
x=fdtri(dfn,dfd,p) finds the F density argument x such that fdtr(dfn,dfd,x)=p.
gdtr(x1, x2, x3, [out])
y=gdtr(a,b,x) returns the integral from zero to x of the gamma probability density function: a**b / gamma(b) *
integral(t**(b-1) exp(-at),t=0..x). The arguments a and b are used differently here than in other definitions.
gdtrc(x1, x2, x3, [out])
y=gdtrc(a,b,x) returns the integral from x to infinity of the gamma probability density function. SEE gdtr, gdtri
gdtria(x1, x2, x3, [out])
gdtrib(x1, x2, x3, [out])
gdtrix(x1, x2, x3, [out])
nbdtr(x1, x2, x3, [out])
y=nbdtr(k,n,p) returns the sum of the terms 0 through k of the negative binomial distribution: sum((n+j-1)Cj
p**n (1-p)**j,j=0..k). In a sequence of Bernoulli trials this is the probability that k or fewer failures precede the
nth success.
nbdtrc(x1, x2, x3, [out])
y=nbdtrc(k,n,p) returns the sum of the terms k+1 to infinity of the negative binomial distribution.
nbdtri(x1, x2, x3, [out])
p=nbdtri(k,n,y) finds the argument p such that nbdtr(k,n,p)=y.
pdtr(x1, x2, [out])
y=pdtr(k,m) returns the sum of the first k terms of the Poisson distribution: sum(exp(-m) * m**j / j!, j=0..k) =
gammaincc( k+1, m). Arguments must both be positive and k an integer.
pdtrc(x1, x2, [out])
y=pdtrc(k,m) returns the sum of the terms from k+1 to infinity of the Poisson distribution: sum(exp(-m) * m**j
/ j!, j=k+1..inf) = gammainc( k+1, m). Arguments must both be positive and k an integer.
pdtri(x1, x2, [out])
m=pdtri(k,y) returns the Poisson variable m such that the sum from 0 to k of the Poisson density is equal to the
given probability y: calculated by gammaincinv( k+1, y). k must be a nonnegative integer and y between 0 and
1.
stdtr(x1, x2, [out])
p=stdtr(df,t) returns the integral from minus infinity to t of the Student t distribution with df > 0 degrees of
freedom: gamma((df+1)/2)/(sqrt(df*pi)*gamma(df/2)) * integral((1+x**2/df)**(-df/2-1/2), x=-inf..t)
stdtridf(x1, x2, [out])
t=stdtridf(p,t) returns the argument df such that stdtr(df,t) is equal to p.
stdtrit(x1, x2, [out])
t=stdtrit(df,p) returns the argument t such that stdtr(df,t) is equal to p.
chdtr(x1, x2, [out])
p=chdtr(v,x) Returns the area under the left hand tail (from 0 to x) of the Chi square probability density function
with v degrees of freedom: 1/(2**(v/2) * gamma(v/2)) * integral(t**(v/2-1) * exp(-t/2), t=0..x)
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chdtrc(x1, x2, [out])
p=chdtrc(v,x) returns the area under the right hand tail (from x to infinity) of the Chi square probability density
function with v degrees of freedom: 1/(2**(v/2) * gamma(v/2)) * integral(t**(v/2-1) * exp(-t/2), t=x..inf)
chdtri(x1, x2, [out])
x=chdtri(v,p) returns the argument x such that chdtrc(v,x) is equal to p.
ndtr(x, [out])
y=ndtr(x) returns the area under the standard Gaussian probability density function, integrated from minus
infinity to x: 1/sqrt(2*pi) * integral(exp(-t**2 / 2),t=-inf..x)
ndtri(x, [out])
x=ndtri(y) returns the argument x for which the area udnder the Gaussian probability density function (integrated
from minus infinity to x) is equal to y.
smirnov(x1, x2, [out])
y=smirnov(n,e) returns the exact Kolmogorov-Smirnov complementary cumulative distribution function (Dn+
or Dn-) for a one-sided test of equality between an empirical and a theoretical distribution. It is equal to the
probability that the maximum difference between a theoretical distribution and an empirical one based on n
samples is greater than e.
smirnovi(x1, x2, [out])
e=smirnovi(n,y) returns e such that smirnov(n,e) = y.
kolmogorov(x, [out])
p=kolmogorov(y) returns the complementary cumulative distribution function of Kolmogorov’s limiting distribution (Kn* for large n) of a two-sided test for equality between an empirical and a theoretical distribution. It is
equal to the (limit as n->infinity of the) probability that sqrt(n) * max absolute deviation > y.
kolmogi(x, [out])
y=kolmogi(p) returns y such that kolmogorov(y) = p
tklmbda(x1, x2, [out])
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Gamma and Related Functions
gamma (x[, out])
y=gamma(z) returns the gamma function of the argument. The gamma function is often
referred to as the generalized factorial since z*gamma(z) = gamma(z+1) and gamma(n+1) =
n! for natural number n.
gammaln (x[, out]) y=gammaln(z) returns the base e logarithm of the absolute value of the gamma function of z:
ln(|gamma(z)|)
gammainc (x1, x2[, out])
y=gammainc(a,x) returns the incomplete gamma integral defined as 1 / gamma(a) *
integral(exp(-t) * t**(a-1), t=0..x). Both arguments must be positive.
gammaincinv (x1, x2[,
gammaincinv(a,
out])
y) returns x such that gammainc(a, x) = y.
gammaincc (x1, x2[,y=gammaincc(a,x)
out])
returns the complemented incomplete gamma integral defined as 1 /
gamma(a) * integral(exp(-t) * t**(a-1), t=x..inf) = 1 - gammainc(a,x). Both arguments must
be positive.
gammainccinv (x1,x=gammainccinv(a,y)
x2[, out])
returns x such that gammaincc(a,x) = y.
beta (x1, x2[, out]) y=beta(a,b) returns gamma(a) * gamma(b) / gamma(a+b)
betaln (x1, x2[, out])
y=betaln(a,b) returns the natural logarithm of the absolute value of beta: ln(|beta(x)|).
betainc (x1, x2, x3[,
y=betainc(a,b,x)
out])
returns the incomplete beta integral of the arguments, evaluated from zero
to x:
betaincinv (x1, x2,
x=betaincinv(a,b,y)
x3[, out])
returns x such that betainc(a,b,x) = y.
psi (x[, out])
y=psi(z) is the derivative of the logarithm of the gamma function evaluated at z (also called
the digamma function).
rgamma (x[, out]) y=rgamma(z) returns one divided by the gamma function of x.
polygamma (n, x) Polygamma function which is the nth derivative of the digamma (psi) function.
multigammaln (a, returns
d)
the log of multivariate gamma, also sometimes called the generalized gamma.
gamma(x, [out])
y=gamma(z) returns the gamma function of the argument. The gamma function is often referred to as the
generalized factorial since z*gamma(z) = gamma(z+1) and gamma(n+1) = n! for natural number n.
gammaln(x, [out])
y=gammaln(z) returns the base e logarithm of the absolute value of the gamma function of z: ln(|gamma(z)|)
gammainc(x1, x2, [out])
y=gammainc(a,x) returns the incomplete gamma integral defined as 1 / gamma(a) * integral(exp(-t) * t**(a-1),
t=0..x). Both arguments must be positive.
gammaincinv(x1, x2, [out])
gammaincinv(a, y) returns x such that gammainc(a, x) = y.
gammaincc(x1, x2, [out])
y=gammaincc(a,x) returns the complemented incomplete gamma integral defined as 1 / gamma(a) *
integral(exp(-t) * t**(a-1), t=x..inf) = 1 - gammainc(a,x). Both arguments must be positive.
gammainccinv(x1, x2, [out])
x=gammainccinv(a,y) returns x such that gammaincc(a,x) = y.
beta(x1, x2, [out])
y=beta(a,b) returns gamma(a) * gamma(b) / gamma(a+b)
betaln(x1, x2, [out])
y=betaln(a,b) returns the natural logarithm of the absolute value of beta: ln(|beta(x)|).
betainc(x1, x2, x3, [out])
y=betainc(a,b,x) returns the incomplete beta integral of the arguments, evaluated from zero to x:
gamma(a+b) / (gamma(a)*gamma(b)) * integral(t**(a-1) (1-t)**(b-1), t=0..x).
betaincinv(x1, x2, x3, [out])
x=betaincinv(a,b,y) returns x such that betainc(a,b,x) = y.
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psi(x, [out])
y=psi(z) is the derivative of the logarithm of the gamma function evaluated at z (also called the digamma
function).
rgamma(x, [out])
y=rgamma(z) returns one divided by the gamma function of x.
polygamma(n, x)
Polygamma function which is the nth derivative of the digamma (psi) function.
multigammaln(a, d)
returns the log of multivariate gamma, also sometimes called the generalized gamma.
Parameters
a : ndarray
the multivariate gamma is computed for each item of a
d : int
the dimension of the space of integration.
Returns
res : ndarray
the values of the log multivariate gamma at the given points a.
Notes
Reference:
R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in probability and mathematical statistics).
Error Function and Fresnel Integrals
erf (x[, out])
y=erf(z) returns the error function of complex argument defined as as
2/sqrt(pi)*integral(exp(-t**2),t=0..z)
y=erfc(x) returns 1 - erf(x).
erfc (x[, out])
erfinv (y)
erfcinv (y)
erf_zeros (nt)
Compute nt complex zeros of the error function erf(z).
fresnel (x[, out1, out2])
(ssa,cca)=fresnel(z) returns the fresnel sin and cos integrals: integral(sin(pi/2 * t**2),t=0..z)
and integral(cos(pi/2 * t**2),t=0..z) for real or complex z.
fresnel_zeros (nt)
Compute nt complex zeros of the sine and cosine fresnel integrals S(z) and C(z).
modfresnelp (x[, out1,
(fp,kp)=modfresnelp(x)
out2])
returns the modified fresnel integrals F_+(x) and K_+(x) as
fp=integral(exp(1j*t*t),t=x..inf) and kp=1/sqrt(pi)*exp(-1j*(x*x+pi/4))*fp
modfresnelm (x[, out1,
(fm,km)=modfresnelp(x)
out2])
returns the modified fresnel integrals F_-(x) amd K_-(x) as
fp=integral(exp(-1j*t*t),t=x..inf) and kp=1/sqrt(pi)*exp(1j*(x*x+pi/4))*fp
erf(x, [out])
y=erf(z) returns the error function of complex argument defined as as 2/sqrt(pi)*integral(exp(-t**2),t=0..z)
erfc(x, [out])
y=erfc(x) returns 1 - erf(x).
erfinv(y)
erfcinv(y)
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erf_zeros(nt)
Compute nt complex zeros of the error function erf(z).
fresnel(x, [out1, out2])
(ssa,cca)=fresnel(z) returns the fresnel sin and cos integrals: integral(sin(pi/2 * t**2),t=0..z) and integral(cos(pi/2 * t**2),t=0..z) for real or complex z.
fresnel_zeros(nt)
Compute nt complex zeros of the sine and cosine fresnel integrals S(z) and C(z).
modfresnelp(x, [out1, out2])
(fp,kp)=modfresnelp(x) returns the modified fresnel integrals
fp=integral(exp(1j*t*t),t=x..inf) and kp=1/sqrt(pi)*exp(-1j*(x*x+pi/4))*fp
F_+(x)
and
K_+(x)
as
modfresnelm(x, [out1, out2])
(fm,km)=modfresnelp(x) returns the modified fresnel integrals F_-(x) amd K_-(x) as fp=integral(exp(1j*t*t),t=x..inf) and kp=1/sqrt(pi)*exp(1j*(x*x+pi/4))*fp
These are not universal functions:
fresnelc_zeros (nt)
fresnels_zeros (nt)
Compute nt complex zeros of the cosine fresnel integral C(z).
Compute nt complex zeros of the sine fresnel integral S(z).
fresnelc_zeros(nt)
Compute nt complex zeros of the cosine fresnel integral C(z).
fresnels_zeros(nt)
Compute nt complex zeros of the sine fresnel integral S(z).
Legendre Functions
lpmv (x1, x2, x3[, out])y=lpmv(m,v,x) returns the associated legendre function of integer order m and nonnegative
degree v: |x|<=1.
sph_harm ()
Compute spherical harmonics.
lpmv(x1, x2, x3, [out])
y=lpmv(m,v,x) returns the associated legendre function of integer order m and nonnegative degree v: |x|<=1.
sph_harm(pyfunc, otypes=”, doc=None)
Compute spherical harmonics.
This is a ufunc and may take scalar or array arguments like any other ufunc. The inputs will be broadcasted
against each other.
Parameters
m : int
|m| <= n; the order of the harmonic.
n : int
where n >= 0; the degree of the harmonic. This is often called l (lower case L) in
descriptions of spherical harmonics.
theta : float
[0, 2*pi]; the azimuthal (longitudinal) coordinate.
phi : float
[0, pi]; the polar (colatitudinal) coordinate.
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Returns
y_mn : complex float
The harmonic $Y^m_n$ sampled at theta and phi
Notes
There are different conventions for the meaning of input arguments theta and phi. We take theta to be the
azimuthal angle and phi to be the polar angle. It is common to see the opposite convention - that is theta as the
polar angle and phi as the azimuthal angle.
These are not universal functions:
lpn (n, z) Compute sequence of Legendre functions of the first kind (polynomials), Pn(z) and derivatives for all
degrees from 0 to n (inclusive).
lqn (n, z) Compute sequence of Legendre functions of the second kind, Qn(z) and derivatives for all degrees
from 0 to n (inclusive).
lpmn (m, n,Associated
z)
Legendre functions of the first kind, Pmn(z) and its derivative, Pmn’(z) of order m and
degree n. Returns two arrays of size (m+1,n+1) containing Pmn(z) and Pmn’(z) for all orders from
0..m and degrees from 0..n.
lqmn (m, n,Associated
z)
Legendre functions of the second kind, Qmn(z) and its derivative, Qmn’(z) of order m and
degree n. Returns two arrays of size (m+1,n+1) containing Qmn(z) and Qmn’(z) for all orders from
0..m and degrees from 0..n.
lpn(n, z)
Compute sequence of Legendre functions of the first kind (polynomials), Pn(z) and derivatives for all degrees
from 0 to n (inclusive).
See also special.legendre for polynomial class.
lqn(n, z)
Compute sequence of Legendre functions of the second kind, Qn(z) and derivatives for all degrees from 0 to n
(inclusive).
lpmn(m, n, z)
Associated Legendre functions of the first kind, Pmn(z) and its derivative, Pmn’(z) of order m and degree n.
Returns two arrays of size (m+1,n+1) containing Pmn(z) and Pmn’(z) for all orders from 0..m and degrees from
0..n.
z can be complex.
Parameters
m : int
|m| <= n; the order of the Legendre function
n : int
where n >= 0; the degree of the Legendre function. Often called l (lower case L) in
descriptions of the associated Legendre function
z : float or complex
input value
Returns
Pmn_z : (m+1, n+1) array
Values for all orders 0..m and degrees 0..n
Pmn_d_z : (m+1, n+1) array
Derivatives for all orders 0..m and degrees 0..n
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lqmn(m, n, z)
Associated Legendre functions of the second kind, Qmn(z) and its derivative, Qmn’(z) of order m and degree
n. Returns two arrays of size (m+1,n+1) containing Qmn(z) and Qmn’(z) for all orders from 0..m and degrees
from 0..n.
z can be complex.
Orthogonal polynomials
The following functions evaluate values of orthogonal polynomials:
eval_legendre ()
eval_chebyt ()
eval_chebyu ()
eval_chebyc ()
eval_chebys ()
eval_jacobi ()
eval_laguerre ()
eval_genlaguerre ()
eval_hermite ()
eval_hermitenorm ()
eval_gegenbauer ()
eval_sh_legendre ()
eval_sh_chebyt ()
eval_sh_chebyu ()
eval_sh_jacobi ()
Evaluate Legendre polynomial at a point.
Evaluate Chebyshev T polynomial at a point.
Evaluate Chebyshev U polynomial at a point.
Evaluate Chebyshev C polynomial at a point.
Evaluate Chebyshev S polynomial at a point.
Evaluate Jacobi polynomial at a point.
Evaluate Laguerre polynomial at a point.
Evaluate generalized Laguerre polynomial at a point.
Evaluate Hermite polynomial at a point.
Evaluate normalized Hermite polynomial at a point.
Evaluate Gegenbauer polynomial at a point.
Evaluate shifted Legendre polynomial at a point.
Evaluate shifted Chebyshev T polynomial at a point.
Evaluate shifted Chebyshev U polynomial at a point.
Evaluate shifted Jacobi polynomial at a point.
eval_legendre()
Evaluate Legendre polynomial at a point.
eval_chebyt()
Evaluate Chebyshev T polynomial at a point.
This routine is numerically stable for x in [-1, 1] at least up to order 10000.
eval_chebyu()
Evaluate Chebyshev U polynomial at a point.
eval_chebyc()
Evaluate Chebyshev C polynomial at a point.
eval_chebys()
Evaluate Chebyshev S polynomial at a point.
eval_jacobi()
Evaluate Jacobi polynomial at a point.
eval_laguerre()
Evaluate Laguerre polynomial at a point.
eval_genlaguerre()
Evaluate generalized Laguerre polynomial at a point.
eval_hermite()
Evaluate Hermite polynomial at a point.
eval_hermitenorm()
Evaluate normalized Hermite polynomial at a point.
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eval_gegenbauer()
Evaluate Gegenbauer polynomial at a point.
eval_sh_legendre()
Evaluate shifted Legendre polynomial at a point.
eval_sh_chebyt()
Evaluate shifted Chebyshev T polynomial at a point.
eval_sh_chebyu()
Evaluate shifted Chebyshev U polynomial at a point.
eval_sh_jacobi()
Evaluate shifted Jacobi polynomial at a point.
The functions below, in turn, return orthopoly1d objects, which functions similarly as numpy.poly1d. The orthopoly1d
class also has an attribute weights which returns the roots, weights, and total weights for the appropriate form of
Gaussian quadrature. These are returned in an n x 3 array with roots in the first column, weights in the second
column, and total weights in the final column.
legendre (n[, monic])
Returns the nth order Legendre polynomial, P_n(x), orthogonal over [-1,1] with weight
function 1.
chebyt (n[, monic]) Return nth order Chebyshev polynomial of first kind, Tn(x). Orthogonal over [-1,1] with
weight function (1-x**2)**(-1/2).
chebyu (n[, monic]) Return nth order Chebyshev polynomial of second kind, Un(x). Orthogonal over [-1,1]
with weight function (1-x**2)**(1/2).
chebyc (n[, monic]) Return nth order Chebyshev polynomial of first kind, Cn(x). Orthogonal over [-2,2] with
weight function (1-(x/2)**2)**(-1/2).
chebys (n[, monic]) Return nth order Chebyshev polynomial of second kind, Sn(x). Orthogonal over [-2,2] with
weight function (1-(x/)**2)**(1/2).
jacobi (n, alReturns the nth order Jacobi polynomial, P^(alpha,beta)_n(x) orthogonal over [-1,1] with
pha, beta[, monic])
weighting function (1-x)**alpha (1+x)**beta with alpha,beta > -1.
laguerre (n[, monic])
Return the nth order Laguerre polynoimal, L_n(x), orthogonal over [0,inf) with weighting
function exp(-x)
genlaguerre (n, al-Returns the nth order generalized (associated) Laguerre polynomial, L^(alpha)_n(x),
pha[, monic])
orthogonal over [0,inf) with weighting function exp(-x) x**alpha with alpha > -1
hermite (n[, monic])Return the nth order Hermite polynomial, H_n(x), orthogonal over (-inf,inf) with weighting
function exp(-x**2)
hermitenorm (n[, monic])
Return the nth order normalized Hermite polynomial, He_n(x), orthogonal over (-inf,inf)
with weighting function exp(-(x/2)**2)
gegenbauer (n, al- Return the nth order Gegenbauer (ultraspherical) polynomial, C^(alpha)_n(x), orthogonal
pha[, monic])
over [-1,1] with weighting function (1-x**2)**(alpha-1/2) with alpha > -1/2
sh_legendre (n[, monic])
Returns the nth order shifted Legendre polynomial, P^*_n(x), orthogonal over [0,1] with
weighting function 1.
sh_chebyt (n[, monic])
Return nth order shifted Chebyshev polynomial of first kind, Tn(x). Orthogonal over [0,1]
with weight function (x-x**2)**(-1/2).
sh_chebyu (n[, monic])
Return nth order shifted Chebyshev polynomial of second kind, Un(x). Orthogonal over
[0,1] with weight function (x-x**2)**(1/2).
sh_jacobi (n, p, q[, monic])
Returns the nth order Jacobi polynomial, G_n(p,q,x) orthogonal over [0,1] with weighting
function (1-x)**(p-q) (x)**(q-1) with p>q-1 and q > 0.
legendre(n, monic=0)
Returns the nth order Legendre polynomial, P_n(x), orthogonal over [-1,1] with weight function 1.
chebyt(n, monic=0)
Return nth order Chebyshev polynomial of first kind, Tn(x). Orthogonal over [-1,1] with weight function
(1-x**2)**(-1/2).
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chebyu(n, monic=0)
Return nth order Chebyshev polynomial of second kind, Un(x). Orthogonal over [-1,1] with weight function
(1-x**2)**(1/2).
chebyc(n, monic=0)
Return nth order Chebyshev polynomial of first kind, Cn(x). Orthogonal over [-2,2] with weight function
(1-(x/2)**2)**(-1/2).
chebys(n, monic=0)
Return nth order Chebyshev polynomial of second kind, Sn(x). Orthogonal over [-2,2] with weight function
(1-(x/)**2)**(1/2).
jacobi(n, alpha, beta, monic=0)
Returns the nth order Jacobi polynomial, P^(alpha,beta)_n(x) orthogonal over [-1,1] with weighting function
(1-x)**alpha (1+x)**beta with alpha,beta > -1.
laguerre(n, monic=0)
Return the nth order Laguerre polynoimal, L_n(x), orthogonal over [0,inf) with weighting function exp(-x)
genlaguerre(n, alpha, monic=0)
Returns the nth order generalized (associated) Laguerre polynomial, L^(alpha)_n(x), orthogonal over [0,inf)
with weighting function exp(-x) x**alpha with alpha > -1
hermite(n, monic=0)
Return the nth order Hermite polynomial, H_n(x), orthogonal over (-inf,inf) with weighting function exp(-x**2)
hermitenorm(n, monic=0)
Return the nth order normalized Hermite polynomial, He_n(x), orthogonal over (-inf,inf) with weighting function exp(-(x/2)**2)
gegenbauer(n, alpha, monic=0)
Return the nth order Gegenbauer (ultraspherical) polynomial, C^(alpha)_n(x), orthogonal over [-1,1] with
weighting function (1-x**2)**(alpha-1/2) with alpha > -1/2
sh_legendre(n, monic=0)
Returns the nth order shifted Legendre polynomial, P^*_n(x), orthogonal over [0,1] with weighting function 1.
sh_chebyt(n, monic=0)
Return nth order shifted Chebyshev polynomial of first kind, Tn(x). Orthogonal over [0,1] with weight function
(x-x**2)**(-1/2).
sh_chebyu(n, monic=0)
Return nth order shifted Chebyshev polynomial of second kind, Un(x). Orthogonal over [0,1] with weight
function (x-x**2)**(1/2).
sh_jacobi(n, p, q, monic=0)
Returns the nth order Jacobi polynomial, G_n(p,q,x) orthogonal over [0,1] with weighting function (1-x)**(p-q)
(x)**(q-1) with p>q-1 and q > 0.
Warning: Large-order polynomials obtained from these functions are numerically unstable.
orthopoly1d objects are converted to poly1d, when doing arithmetic. numpy.poly1d works in power
basis and cannot represent high-order polynomials accurately, which can cause significant inaccuracy.
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Hypergeometric Functions
hyp2f1 (x1, x2, x3, x4[,
y=hyp2f1(a,b,c,z)
out])
returns the gauss hypergeometric function ( 2F1(a,b;c;z) ).
hyp1f1 (x1, x2, x3[, out])
y=hyp1f1(a,b,x) returns the confluent hypergeometeric function ( 1F1(a,b;x) ) evaluated at
the values a, b, and x.
hyperu (x1, x2, x3[, out])
y=hyperu(a,b,x) returns the confluent hypergeometric function of the second kind U(a,b,x).
hyp0f1 (v, z)
Confluent hypergeometric limit function 0F1. Limit as q->infinity of 1F1(q;a;z/q)
hyp2f0 (x1, x2, x3, x4[,
(y,err)=hyp2f0(a,b,x,type)
out1, ...)
returns (y,err) with the hypergeometric function 2F0 in y and an
error estimate in err. The input type determines a convergence factor and can be either 1 or
2.
hyp1f2 (x1, x2, x3, x4[,
(y,err)=hyp1f2(a,b,c,x)
out1, ...)
returns (y,err) with the hypergeometric function 1F2 in y and an
error estimate in err.
hyp3f0 (x1, x2, x3, x4[,
(y,err)=hyp3f0(a,b,c,x)
out1, ...)
returns (y,err) with the hypergeometric function 3F0 in y and an
error estimate in err.
hyp2f1(x1, x2, x3, x4, [out])
y=hyp2f1(a,b,c,z) returns the gauss hypergeometric function ( 2F1(a,b;c;z) ).
hyp1f1(x1, x2, x3, [out])
y=hyp1f1(a,b,x) returns the confluent hypergeometeric function ( 1F1(a,b;x) ) evaluated at the values a, b, and
x.
hyperu(x1, x2, x3, [out])
y=hyperu(a,b,x) returns the confluent hypergeometric function of the second kind U(a,b,x).
hyp0f1(v, z)
Confluent hypergeometric limit function 0F1. Limit as q->infinity of 1F1(q;a;z/q)
hyp2f0(x1, x2, x3, x4, [out1, out2])
(y,err)=hyp2f0(a,b,x,type) returns (y,err) with the hypergeometric function 2F0 in y and an error estimate in err.
The input type determines a convergence factor and can be either 1 or 2.
hyp1f2(x1, x2, x3, x4, [out1, out2])
(y,err)=hyp1f2(a,b,c,x) returns (y,err) with the hypergeometric function 1F2 in y and an error estimate in err.
hyp3f0(x1, x2, x3, x4, [out1, out2])
(y,err)=hyp3f0(a,b,c,x) returns (y,err) with the hypergeometric function 3F0 in y and an error estimate in err.
Parabolic Cylinder Functions
pbdv (x1, x2[, out1,
(d,dp)=pbdv(v,x)
out2])
returns (d,dp) with the parabolic cylinder function Dv(x) in d and the
derivative, Dv’(x) in dp.
pbvv (x1, x2[, out1,
(v,vp)=pbvv(v,x)
out2])
returns (v,vp) with the parabolic cylinder function Vv(x) in v and the
derivative, Vv’(x) in vp.
pbwa (x1, x2[, out1,
(w,wp)=pbwa(a,x)
out2])
returns (w,wp) with the parabolic cylinder function W(a,x) in w and the
derivative, W’(a,x) in wp. May not be accurate for large (>5) arguments in a and/or x.
pbdv(x1, x2, [out1, out2])
(d,dp)=pbdv(v,x) returns (d,dp) with the parabolic cylinder function Dv(x) in d and the derivative, Dv’(x) in dp.
pbvv(x1, x2, [out1, out2])
(v,vp)=pbvv(v,x) returns (v,vp) with the parabolic cylinder function Vv(x) in v and the derivative, Vv’(x) in vp.
pbwa(x1, x2, [out1, out2])
(w,wp)=pbwa(a,x) returns (w,wp) with the parabolic cylinder function W(a,x) in w and the derivative, W’(a,x)
in wp. May not be accurate for large (>5) arguments in a and/or x.
These are not universal functions:
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pbdv_seq (v, x) Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for
Dv0(x)..Dv(x) with v0=v-int(v).
pbvv_seq (v, x) Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for
Dv0(x)..Dv(x) with v0=v-int(v).
pbdn_seq (n, z) Compute sequence of parabolic cylinder functions Dn(z) and their derivatives for D0(z)..Dn(z).
pbdv_seq(v, x)
Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for Dv0(x)..Dv(x) with v0=vint(v).
pbvv_seq(v, x)
Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for Dv0(x)..Dv(x) with v0=vint(v).
pbdn_seq(n, z)
Compute sequence of parabolic cylinder functions Dn(z) and their derivatives for D0(z)..Dn(z).
Mathieu and Related Functions
mathieu_a (x1, x2[, out])
lmbda=mathieu_a(m,q) returns the characteristic value for the even solution, ce_m(z,q),
of Mathieu’s equation
mathieu_b (x1, x2[, out])
lmbda=mathieu_b(m,q) returns the characteristic value for the odd solution, se_m(z,q),
of Mathieu’s equation
mathieu_a(x1, x2, [out])
lmbda=mathieu_a(m,q) returns the characteristic value for the even solution, ce_m(z,q), of Mathieu’s equation
mathieu_b(x1, x2, [out])
lmbda=mathieu_b(m,q) returns the characteristic value for the odd solution, se_m(z,q), of Mathieu’s equation
These are not universal functions:
mathieu_even_coef (m, q)Compute expansion coefficients for even mathieu functions and modified mathieu
functions.
mathieu_odd_coef (m, q) Compute expansion coefficients for even mathieu functions and modified mathieu
functions.
mathieu_even_coef(m, q)
Compute expansion coefficients for even mathieu functions and modified mathieu functions.
mathieu_odd_coef(m, q)
Compute expansion coefficients for even mathieu functions and modified mathieu functions.
The following return both function and first derivative:
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mathieu_cem (x1, x2, x3[,
(y,yp)=mathieu_cem(m,q,x)
out1, ...)
returns the even Mathieu function, ce_m(x,q), of order m
and parameter q evaluated at x (given in degrees). Also returns the derivative with
respect to x of ce_m(x,q)
mathieu_sem (x1, x2, x3[,
(y,yp)=mathieu_sem(m,q,x)
out1, ...)
returns the odd Mathieu function, se_m(x,q), of order m
and parameter q evaluated at x (given in degrees). Also returns the derivative with
respect to x of se_m(x,q).
mathieu_modcem1 (x1,(y,yp)=mathieu_modcem1(m,q,x)
x2, x3[, out1, ...)
evaluates the even modified Matheiu function of the
first kind, Mc1m(x,q), and its derivative at x for order m and parameter q.
mathieu_modcem2 (x1,(y,yp)=mathieu_modcem2(m,q,x)
x2, x3[, out1, ...)
evaluates the even modified Matheiu function of the
second kind, Mc2m(x,q), and its derivative at x (given in degrees) for order m and
parameter q.
mathieu_modsem1 (x1,(y,yp)=mathieu_modsem1(m,q,x)
x2, x3[, out1, ...)
evaluates the odd modified Matheiu function of the
first kind, Ms1m(x,q), and its derivative at x (given in degrees) for order m and
parameter q.
mathieu_modsem2 (x1,(y,yp)=mathieu_modsem2(m,q,x)
x2, x3[, out1, ...)
evaluates the odd modified Matheiu function of the
second kind, Ms2m(x,q), and its derivative at x (given in degrees) for order m and
parameter q.
mathieu_cem(x1, x2, x3, [out1, out2])
(y,yp)=mathieu_cem(m,q,x) returns the even Mathieu function, ce_m(x,q), of order m and parameter q evaluated
at x (given in degrees). Also returns the derivative with respect to x of ce_m(x,q)
mathieu_sem(x1, x2, x3, [out1, out2])
(y,yp)=mathieu_sem(m,q,x) returns the odd Mathieu function, se_m(x,q), of order m and parameter q evaluated
at x (given in degrees). Also returns the derivative with respect to x of se_m(x,q).
mathieu_modcem1(x1, x2, x3, [out1, out2])
(y,yp)=mathieu_modcem1(m,q,x) evaluates the even modified Matheiu function of the first kind, Mc1m(x,q),
and its derivative at x for order m and parameter q.
mathieu_modcem2(x1, x2, x3, [out1, out2])
(y,yp)=mathieu_modcem2(m,q,x) evaluates the even modified Matheiu function of the second kind, Mc2m(x,q),
and its derivative at x (given in degrees) for order m and parameter q.
mathieu_modsem1(x1, x2, x3, [out1, out2])
(y,yp)=mathieu_modsem1(m,q,x) evaluates the odd modified Matheiu function of the first kind, Ms1m(x,q),
and its derivative at x (given in degrees) for order m and parameter q.
mathieu_modsem2(x1, x2, x3, [out1, out2])
(y,yp)=mathieu_modsem2(m,q,x) evaluates the odd modified Matheiu function of the second kind, Ms2m(x,q),
and its derivative at x (given in degrees) for order m and parameter q.
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Spheroidal Wave Functions
pro_ang1 (x1, x2, x3,(s,sp)=pro_ang1(m,n,c,x)
x4[, out1, ...)
computes the prolate sheroidal angular function of the first kind
and its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal
parameter c and |x|<1.0.
pro_rad1 (x1, x2, x3,(s,sp)=pro_rad1(m,n,c,x)
x4[, out1, ...)
computes the prolate sheroidal radial function of the first kind
and its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal
parameter c and |x|<1.0.
pro_rad2 (x1, x2, x3,(s,sp)=pro_rad2(m,n,c,x)
x4[, out1, ...)
computes the prolate sheroidal radial function of the second kind
and its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal
parameter c and |x|<1.0.
obl_ang1 (x1, x2, x3,(s,sp)=obl_ang1(m,n,c,x)
x4[, out1, ...)
computes the oblate sheroidal angular function of the first kind
and its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal
parameter c and |x|<1.0.
obl_rad1 (x1, x2, x3,(s,sp)=obl_rad1(m,n,c,x)
x4[, out1, ...)
computes the oblate sheroidal radial function of the first kind and
its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal
parameter c and |x|<1.0.
obl_rad2 (x1, x2, x3,(s,sp)=obl_rad2(m,n,c,x)
x4[, out1, ...)
computes the oblate sheroidal radial function of the second kind
and its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal
parameter c and |x|<1.0.
pro_cv (x1, x2, x3[, out])
cv=pro_cv(m,n,c) computes the characteristic value of prolate spheroidal wave functions
of order m,n (n>=m) and spheroidal parameter c.
obl_cv (x1, x2, x3[, out])
cv=obl_cv(m,n,c) computes the characteristic value of oblate spheroidal wave functions of
order m,n (n>=m) and spheroidal parameter c.
pro_cv_seq (m, n, c)Compute a sequence of characteristic values for the prolate spheroidal wave functions for
mode m and n’=m..n and spheroidal parameter c.
obl_cv_seq (m, n, c)Compute a sequence of characteristic values for the oblate spheroidal wave functions for
mode m and n’=m..n and spheroidal parameter c.
pro_ang1(x1, x2, x3, x4, [out1, out2])
(s,sp)=pro_ang1(m,n,c,x) computes the prolate sheroidal angular function of the first kind and its derivative
(with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
pro_rad1(x1, x2, x3, x4, [out1, out2])
(s,sp)=pro_rad1(m,n,c,x) computes the prolate sheroidal radial function of the first kind and its derivative (with
respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
pro_rad2(x1, x2, x3, x4, [out1, out2])
(s,sp)=pro_rad2(m,n,c,x) computes the prolate sheroidal radial function of the second kind and its derivative
(with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
obl_ang1(x1, x2, x3, x4, [out1, out2])
(s,sp)=obl_ang1(m,n,c,x) computes the oblate sheroidal angular function of the first kind and its derivative
(with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
obl_rad1(x1, x2, x3, x4, [out1, out2])
(s,sp)=obl_rad1(m,n,c,x) computes the oblate sheroidal radial function of the first kind and its derivative (with
respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
obl_rad2(x1, x2, x3, x4, [out1, out2])
(s,sp)=obl_rad2(m,n,c,x) computes the oblate sheroidal radial function of the second kind and its derivative
(with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
pro_cv(x1, x2, x3, [out])
cv=pro_cv(m,n,c) computes the characteristic value of prolate spheroidal wave functions of order m,n (n>=m)
and spheroidal parameter c.
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obl_cv(x1, x2, x3, [out])
cv=obl_cv(m,n,c) computes the characteristic value of oblate spheroidal wave functions of order m,n (n>=m)
and spheroidal parameter c.
pro_cv_seq(m, n, c)
Compute a sequence of characteristic values for the prolate spheroidal wave functions for mode m and n’=m..n
and spheroidal parameter c.
obl_cv_seq(m, n, c)
Compute a sequence of characteristic values for the oblate spheroidal wave functions for mode m and n’=m..n
and spheroidal parameter c.
The following functions require pre-computed characteristic value:
pro_ang1_cv (x1, x2,(s,sp)=pro_ang1_cv(m,n,c,cv,x)
x3, x4, x5[, ...)
computes the prolate sheroidal angular function of the
first kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m,
spheroidal parameter c and |x|<1.0. Requires pre-computed characteristic value.
pro_rad1_cv (x1, x2,(s,sp)=pro_rad1_cv(m,n,c,cv,x)
x3, x4, x5[, ...)
computes the prolate sheroidal radial function of the first
kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m,
spheroidal parameter c and |x|<1.0. Requires pre-computed characteristic value.
pro_rad2_cv (x1, x2,(s,sp)=pro_rad2_cv(m,n,c,cv,x)
x3, x4, x5[, ...)
computes the prolate sheroidal radial function of the
second kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m,
spheroidal parameter c and |x|<1.0. Requires pre-computed characteristic value.
obl_ang1_cv (x1, x2,(s,sp)=obl_ang1_cv(m,n,c,cv,x)
x3, x4, x5[, ...)
computes the oblate sheroidal angular function of the
first kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m,
spheroidal parameter c and |x|<1.0. Requires pre-computed characteristic value.
obl_rad1_cv (x1, x2,(s,sp)=obl_rad1_cv(m,n,c,cv,x)
x3, x4, x5[, ...)
computes the oblate sheroidal radial function of the first
kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m,
spheroidal parameter c and |x|<1.0. Requires pre-computed characteristic value.
obl_rad2_cv (x1, x2,(s,sp)=obl_rad2_cv(m,n,c,cv,x)
x3, x4, x5[, ...)
computes the oblate sheroidal radial function of the
second kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m,
spheroidal parameter c and |x|<1.0. Requires pre-computed characteristic value.
pro_ang1_cv(x1, x2, x3, x4, x5, [out1, out2])
(s,sp)=pro_ang1_cv(m,n,c,cv,x) computes the prolate sheroidal angular function of the first kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires
pre-computed characteristic value.
pro_rad1_cv(x1, x2, x3, x4, x5, [out1, out2])
(s,sp)=pro_rad1_cv(m,n,c,cv,x) computes the prolate sheroidal radial function of the first kind and its derivative
(with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires
pre-computed characteristic value.
pro_rad2_cv(x1, x2, x3, x4, x5, [out1, out2])
(s,sp)=pro_rad2_cv(m,n,c,cv,x) computes the prolate sheroidal radial function of the second kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires
pre-computed characteristic value.
obl_ang1_cv(x1, x2, x3, x4, x5, [out1, out2])
(s,sp)=obl_ang1_cv(m,n,c,cv,x) computes the oblate sheroidal angular function of the first kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires
pre-computed characteristic value.
obl_rad1_cv(x1, x2, x3, x4, x5, [out1, out2])
(s,sp)=obl_rad1_cv(m,n,c,cv,x) computes the oblate sheroidal radial function of the first kind and its derivative
(with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires
pre-computed characteristic value.
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obl_rad2_cv(x1, x2, x3, x4, x5, [out1, out2])
(s,sp)=obl_rad2_cv(m,n,c,cv,x) computes the oblate sheroidal radial function of the second kind and its derivative (with respect to x) for mode paramters m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires
pre-computed characteristic value.
Kelvin Functions
kelvin (x[, out1, out2,
(Be,out3,
Ke, ...)
Bep, Kep)=kelvin(x) returns the tuple (Be, Ke, Bep, Kep) which containes complex
numbers representing the real and imaginary Kelvin functions and their derivatives evaluated
at x. For example, kelvin(x)[0].real = ber x and kelvin(x)[0].imag = bei x with similar
relationships for ker and kei.
kelvin_zeros (nt)Compute nt zeros of all the kelvin functions returned in a length 8 tuple of arrays of length
nt. The tuple containse the arrays of zeros of (ber, bei, ker, kei, ber’, bei’, ker’, kei’)
ber (x[, out])
y=ber(x) returns the Kelvin function ber x
bei (x[, out])
y=bei(x) returns the Kelvin function bei x
berp (x[, out])
y=berp(x) returns the derivative of the Kelvin function ber x
beip (x[, out])
y=beip(x) returns the derivative of the Kelvin function bei x
ker (x[, out])
y=ker(x) returns the Kelvin function ker x
kei (x[, out])
y=kei(x) returns the Kelvin function ker x
kerp (x[, out])
y=kerp(x) returns the derivative of the Kelvin function ker x
keip (x[, out])
y=keip(x) returns the derivative of the Kelvin function kei x
kelvin(x, [out1, out2, out3, out4])
(Be, Ke, Bep, Kep)=kelvin(x) returns the tuple (Be, Ke, Bep, Kep) which containes complex numbers representing the real and imaginary Kelvin functions and their derivatives evaluated at x. For example, kelvin(x)[0].real
= ber x and kelvin(x)[0].imag = bei x with similar relationships for ker and kei.
kelvin_zeros(nt)
Compute nt zeros of all the kelvin functions returned in a length 8 tuple of arrays of length nt. The tuple
containse the arrays of zeros of (ber, bei, ker, kei, ber’, bei’, ker’, kei’)
ber(x, [out])
y=ber(x) returns the Kelvin function ber x
bei(x, [out])
y=bei(x) returns the Kelvin function bei x
berp(x, [out])
y=berp(x) returns the derivative of the Kelvin function ber x
beip(x, [out])
y=beip(x) returns the derivative of the Kelvin function bei x
ker(x, [out])
y=ker(x) returns the Kelvin function ker x
kei(x, [out])
y=kei(x) returns the Kelvin function ker x
kerp(x, [out])
y=kerp(x) returns the derivative of the Kelvin function ker x
keip(x, [out])
y=keip(x) returns the derivative of the Kelvin function kei x
These are not universal functions:
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ber_zeros (nt)
bei_zeros (nt)
berp_zeros (nt)
beip_zeros (nt)
ker_zeros (nt)
kei_zeros (nt)
kerp_zeros (nt)
keip_zeros (nt)
Compute nt zeros of the kelvin function ber x
Compute nt zeros of the kelvin function bei x
Compute nt zeros of the kelvin function ber’ x
Compute nt zeros of the kelvin function bei’ x
Compute nt zeros of the kelvin function ker x
Compute nt zeros of the kelvin function kei x
Compute nt zeros of the kelvin function ker’ x
Compute nt zeros of the kelvin function kei’ x
ber_zeros(nt)
Compute nt zeros of the kelvin function ber x
bei_zeros(nt)
Compute nt zeros of the kelvin function bei x
berp_zeros(nt)
Compute nt zeros of the kelvin function ber’ x
beip_zeros(nt)
Compute nt zeros of the kelvin function bei’ x
ker_zeros(nt)
Compute nt zeros of the kelvin function ker x
kei_zeros(nt)
Compute nt zeros of the kelvin function kei x
kerp_zeros(nt)
Compute nt zeros of the kelvin function ker’ x
keip_zeros(nt)
Compute nt zeros of the kelvin function kei’ x
Other Special Functions
expn (x1, x2[, out])
y=expn(n,x) returns the exponential integral for integer n and non-negative x and n:
integral(exp(-x*t) / t**n, t=1..inf).
exp1 (x[, out]) y=exp1(z) returns the exponential integral (n=1) of complex argument z:
integral(exp(-z*t)/t,t=1..inf).
expi (x[, out]) y=expi(x) returns an exponential integral of argument x defined as integral(exp(t)/t,t=-inf..x).
See expn for a different exponential integral.
wofz (x[, out]) y=wofz(z) returns the value of the fadeeva function for complex argument z:
exp(-z**2)*erfc(-i*z)
dawsn (x[, out]) y=dawsn(x) returns dawson’s integral: exp(-x**2) * integral(exp(t**2),t=0..x).
shichi (x[, out1,(shi,chi)=shichi(x)
out2])
returns the hyperbolic sine and cosine integrals: integral(sinh(t)/t,t=0..x) and
eul + ln x + integral((cosh(t)-1)/t,t=0..x) where eul is Euler’s Constant.
sici (x[, out1, out2])
(si,ci)=sici(x) returns in si the integral of the sinc function from 0 to x: integral(sin(t)/t,t=0..x). It
returns in ci the cosine integral: eul + ln x + integral((cos(t) - 1)/t,t=0..x).
spence (x[, out])y=spence(x) returns the dilogarithm integral: -integral(log t / (t-1),t=1..x)
lambertw (z[, k,Lambert
tol])
W function.
zeta (x1, x2[, out])
y=zeta(x,q) returns the Riemann zeta function of two arguments: sum((k+q)**(-x),k=0..inf)
zetac (x[, out]) y=zetac(x) returns 1.0 - the Riemann zeta function: sum(k**(-x), k=2..inf)
expn(x1, x2, [out])
y=expn(n,x) returns the exponential integral for integer n and non-negative x and n: integral(exp(-x*t) / t**n,
t=1..inf).
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exp1(x, [out])
y=exp1(z) returns the exponential integral (n=1) of complex argument z: integral(exp(-z*t)/t,t=1..inf).
expi(x, [out])
y=expi(x) returns an exponential integral of argument x defined as integral(exp(t)/t,t=-inf..x). See expn for a
different exponential integral.
wofz(x, [out])
y=wofz(z) returns the value of the fadeeva function for complex argument z: exp(-z**2)*erfc(-i*z)
dawsn(x, [out])
y=dawsn(x) returns dawson’s integral: exp(-x**2) * integral(exp(t**2),t=0..x).
shichi(x, [out1, out2])
(shi,chi)=shichi(x) returns the hyperbolic sine and cosine integrals: integral(sinh(t)/t,t=0..x) and eul + ln x +
integral((cosh(t)-1)/t,t=0..x) where eul is Euler’s Constant.
sici(x, [out1, out2])
(si,ci)=sici(x) returns in si the integral of the sinc function from 0 to x: integral(sin(t)/t,t=0..x). It returns in ci
the cosine integral: eul + ln x + integral((cos(t) - 1)/t,t=0..x).
spence(x, [out])
y=spence(x) returns the dilogarithm integral: -integral(log t / (t-1),t=1..x)
lambertw(z, k=0, tol=1e-8)
Lambert W function.
The Lambert W function W(z) is defined as the inverse function of w exp(w). In other words, the value of W (z)
is such that z = W (z) exp(W (z)) for any complex number z.
The Lambert W function is a multivalued function with infinitely many branches. Each branch gives a separate
solution of the equation w exp(w). Here, the branches are indexed by the integer k.
Parameters
z : array_like
Input argument
k : integer, optional
Branch index
tol : float
Evaluation tolerance
Notes
All branches are supported by lambertw:
•lambertw(z) gives the principal solution (branch 0)
•lambertw(z, k) gives the solution on branch k
The Lambert W function has two partially real branches: the principal branch (k = 0) is real for real z > -1/e,
and the k = -1 branch is real for -1/e . All branches except k = 0 have a logarithmic singularity at z = 0.
Possible issues
The evaluation can become inaccurate very close to the branch point at -1/e. In some corner cases, lambertw
might currently fail to converge, or can end up on the wrong branch.
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Algorithm
Halley’s iteration is used to invert w exp(w), using a first-order asymptotic approximation (O(log(w)) or O(w))
as the initial estimate.
The definition, implementation and choice of branches is based on Corless et al, “On the Lambert W function”,
Adv. Comp. Math. 5 (1996) 329-359, available online here: http://www.apmaths.uwo.ca/~djeffrey/Offprints/Wadv-cm.pdf
TODO: use a series expansion when extremely close to the branch point at -1/e and make sure that the proper
branch is chosen there
Examples
The Lambert W function is the inverse of w exp(w):
>>> from scipy.special import lambertw
>>> w = lambertw(1)
>>> w
0.56714329040978387299996866221035555 >>> w*exp(w) 1.0
Any branch gives a valid inverse:
>>> w = lambertw(1, k=3)
>>> w
(-2.8535817554090378072068187234910812 + 17.113535539412145912607826671159289j) >>> w*exp(w)
(1.0 + 3.5075477124212226194278700785075126e-36j)
Applications to equation-solving
The Lambert W function may be used to solve various kinds of equations, such as finding the value of the infinite
power tower z^{z^{z^{ldots}}}:
>>> def tower(z, n):
... if n == 0: ... return z ... return z ** tower(z, n-1) ... >>> tower(0.5, 100) 0.641185744504986 >>> -lambertw(log(0.5))/log(0.5) 0.6411857445049859844862004821148236665628209571911
Properties
The Lambert W function grows roughly like the natural logarithm for large arguments:
>>> lambertw(1000)
5.2496028524016 >>> log(1000) 6.90775527898214 >>> lambertw(10**100) 224.843106445119 >>>
log(10**100) 230.258509299405
The principal branch of the Lambert W function has a rational Taylor series expansion around z = 0:
>>> nprint(taylor(lambertw, 0, 6), 10)
[0.0, 1.0, -1.0, 1.5, -2.666666667, 5.208333333, -10.8]
Some special values and limits are:
>>> lambertw(0)
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0.0 >>> lambertw(1) 0.567143290409784 >>> lambertw(e) 1.0 >>> lambertw(inf) +inf >>> lambertw(0, k=-1)
-inf >>> lambertw(0, k=3) -inf >>> lambertw(inf, k=3) (+inf + 18.8495559215388j)
The k = 0 and k = -1 branches join at z = -1/e where W(z) = -1 for both branches. Since -1/e can only be
represented approximately with mpmath numbers, evaluating the Lambert W function at this point only gives -1
approximately:
>>> lambertw(-1/e, 0)
-0.999999999999837133022867 >>> lambertw(-1/e, -1) -1.00000000000016286697718
If -1/e happens to round in the negative direction, there might be a small imaginary part:
>>> lambertw(-1/e)
(-1.0 + 8.22007971511612e-9j)
zeta(x1, x2, [out])
y=zeta(x,q) returns the Riemann zeta function of two arguments: sum((k+q)**(-x),k=0..inf)
zetac(x, [out])
y=zetac(x) returns 1.0 - the Riemann zeta function: sum(k**(-x), k=2..inf)
Convenience Functions
cbrt (x[, out])
y=cbrt(x) returns the real cube root of x.
exp10 (x[, out])
y=exp10(x) returns 10 raised to the x power.
exp2 (x[, out])
y=exp2(x) returns 2 raised to the x power.
radian (x1, x2, x3[, y=radian(d,m,s)
out])
returns the angle given in (d)egrees, (m)inutes, and (s)econds in radians.
cosdg (x[, out])
y=cosdg(x) calculates the cosine of the angle x given in degrees.
sindg (x[, out])
y=sindg(x) calculates the sine of the angle x given in degrees.
tandg (x[, out])
y=tandg(x) calculates the tangent of the angle x given in degrees.
cotdg (x[, out])
y=cotdg(x) calculates the cotangent of the angle x given in degrees.
log1p (x[, out])
y=log1p(x) calculates log(1+x) for use when x is near zero.
expm1 (x[, out])
y=expm1(x) calculates exp(x) - 1 for use when x is near zero.
cosm1 (x[, out])
y=calculates cos(x) - 1 for use when x is near zero.
round (x[, out])
y=Returns the nearest integer to x as a double precision floating point result. If x ends in 0.5
exactly, the nearest even integer is chosen.
cbrt(x, [out])
y=cbrt(x) returns the real cube root of x.
exp10(x, [out])
y=exp10(x) returns 10 raised to the x power.
exp2(x, [out])
y=exp2(x) returns 2 raised to the x power.
radian(x1, x2, x3, [out])
y=radian(d,m,s) returns the angle given in (d)egrees, (m)inutes, and (s)econds in radians.
cosdg(x, [out])
y=cosdg(x) calculates the cosine of the angle x given in degrees.
sindg(x, [out])
y=sindg(x) calculates the sine of the angle x given in degrees.
tandg(x, [out])
y=tandg(x) calculates the tangent of the angle x given in degrees.
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cotdg(x, [out])
y=cotdg(x) calculates the cotangent of the angle x given in degrees.
log1p(x, [out])
y=log1p(x) calculates log(1+x) for use when x is near zero.
expm1(x, [out])
y=expm1(x) calculates exp(x) - 1 for use when x is near zero.
cosm1(x, [out])
y=calculates cos(x) - 1 for use when x is near zero.
round(x, [out])
y=Returns the nearest integer to x as a double precision floating point result. If x ends in 0.5 exactly, the nearest
even integer is chosen.
3.18 Statistical functions (scipy.stats)
This module contains a large number of probability distributions as well as a growing library of statistical functions.
Each included continuous distribution is an instance of the class rv_continous:
rv_continuous
rv_continuous.pdf (self, x, *args, **kwds)
rv_continuous.cdf (self, x, *args, **kwds)
rv_continuous.sf (self, x, *args, **kwds)
rv_continuous.ppf (self, q, *args, **kwds)
rv_continuous.isf (self, q, *args, **kwds)
rv_continuous.stats (self, *args, **kwds)
A generic continuous random variable class meant for subclassing.
Probability density function at x of the given RV.
Cumulative distribution function at x of the given RV.
Survival function (1-cdf) at x of the given RV.
Percent point function (inverse of cdf) at q of the given RV.
Inverse survival function at q of the given RV.
Some statistics of the given RV
class rv_continuous(momtype=1, a=None, b=None, xa=-10.0, xb=10.0, xtol=1e-14, badvalue=None,
name=None, longname=None, shapes=None, extradoc=None)
A generic continuous random variable class meant for subclassing.
rv_continuous is a base class to construct specific distribution classes and instances from for continuous random
variables. It cannot be used directly as a distribution.
Parameters
momtype : int, optional
The type of generic moment calculation to use (check this).
a : float, optional
Lower bound of the support of the distribution, default is minus infinity.
b : float, optional
Upper bound of the support of the distribution, default is plus infinity.
xa : float, optional
Lower bound for fixed point calculation for generic ppf.
xb : float, optional
Upper bound for fixed point calculation for generic ppf.
xtol : float, optional
The tolerance for fixed point calculation for generic ppf.
badvalue : object, optional
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The value in a result arrays that indicates a value that for which some argument
restriction is violated, default is np.nan.
name : str, optional
The name of the instance. This string is used to construct the default example for
distributions.
longname : str, optional
This string is used as part of the first line of the docstring returned when a subclass
has no docstring of its own. Note: longname exists for backwards compatibility, do
not use for new subclasses.
shapes : str, optional
The shape of the distribution. For example "m, n" for a distribution that takes two
integers as the two shape arguments for all its methods.
extradoc : str, optional
This string is used as the last part of the docstring returned when a subclass has no
docstring of its own. Note: extradoc exists for backwards compatibility, do not use
for new subclasses.
Notes
Frozen Distribution
Alternatively, the object may be called (as a function) to fix the shape, location, and scale parameters returning
a “frozen” continuous RV object:
rv = generic(<shape(s)>, loc=0, scale=1)
frozen RV object with the same methods but holding the given shape, location, and scale fixed
Subclassing
New random variables can be defined by subclassing rv_continuous class and re-defining at least the
_pdf or the cdf method which will be given clean arguments (in between a and b) and passing the argument
check method
If postive argument checking is not correct for your RV then you will also need to re-define
_argcheck
Correct, but potentially slow defaults exist for the remaining methods but for speed and/or accuracy you can
over-ride
_cdf, _ppf, _rvs, _isf, _sf
Rarely would you override _isf and _sf but you could.
Statistics are computed using numerical integration by default. For speed you can redefine this using
_stats
• take shape parameters and return mu, mu2, g1, g2
• If you can’t compute one of these, return it as None
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• Can also be defined with a keyword argument moments=<str> where <str> is a string composed of
‘m’, ‘v’, ‘s’, and/or ‘k’. Only the components appearing in string should be computed and returned
in the order ‘m’, ‘v’, ‘s’, or ‘k’ with missing values returned as None
OR
You can override
_munp
takes n and shape parameters and returns the nth non-central moment of the distribution.
Examples
To create a new Gaussian distribution, we would do the following:
class gaussian_gen(rv_continuous):
"Gaussian distribution"
def _pdf:
...
...
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Methods
rvs(<shape(s)>, loc=0,
scale=1, size=1)
pdf(x, <shape(s)>, loc=0,
scale=1)
cdf(x, <shape(s)>, loc=0,
scale=1)
sf(x, <shape(s)>, loc=0,
scale=1)
ppf(q, <shape(s)>, loc=0,
scale=1)
isf(q, <shape(s)>, loc=0,
scale=1)
moments(n, <shape(s)>)
stats(<shape(s)>, loc=0,
scale=1, moments=’mv’)
entropy(<shape(s)>, loc=0,
scale=1)
fit(data, <shape(s)>, loc=0,
scale=1)
__call__(<shape(s)>, loc=0,
scale=1)
Parameters for Methods
x
q
<shape(s)>
loc
scale
size
moments
n
Methods that can be
overwritten by subclasses
random variates
probability density function
cumulative density function
survival function (1-cdf — sometimes more accurate)
percent point function (inverse of cdf — quantiles)
inverse survival function (inverse of sf)
non-central n-th moment of the standard distribution (oc=0, scale=1)
mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’)
(differential) entropy of the RV.
Parameter estimates for generic data
calling a distribution instance creates a frozen RV object with the
same methods but holding the given shape, location, and scale fixed.
see Notes section
array-like
array-like
array-like
array-like,
optional
array-like,
optional
int or tuple
of ints,
optional
string,
optional
int
quantiles
lower or upper tail probability
shape parameters
location parameter (default=0)
scale parameter (default=1)
shape of random variates (default computed from input arguments )
composed of letters [’mvsk’] specifying which moments to compute
where ‘m’ = mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ =
(Fisher’s) kurtosis. (default=’mv’)
order of moment to calculate in method moments
_rvs _pdf _cdf _sf _ppf _isf _stats _munp _entropy _argcheck
There are additional
(internal and private)
generic methods that can
be useful for cross-checking
and for debugging, but
might work in all
cases when directly called.
pdf(x, *args, **kwds)
Probability density function at x of the given RV.
Parameters
x : array-like
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quantiles
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
scale : array-like, optional
scale parameter (default=1)
Returns
pdf : array-like
Probability density function evaluated at x
cdf(x, *args, **kwds)
Cumulative distribution function at x of the given RV.
Parameters
x : array-like
quantiles
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
scale : array-like, optional
scale parameter (default=1)
Returns
cdf : array-like
Cumulative distribution function evaluated at x
sf(x, *args, **kwds)
Survival function (1-cdf) at x of the given RV.
Parameters
x : array-like
quantiles
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
scale : array-like, optional
scale parameter (default=1)
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Returns
sf : array-like
Survival function evaluated at x
ppf(q, *args, **kwds)
Percent point function (inverse of cdf) at q of the given RV.
Parameters
q : array-like
lower tail probability
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
scale : array-like, optional
scale parameter (default=1)
Returns
x : array-like
quantile corresponding to the lower tail probability q.
isf(q, *args, **kwds)
Inverse survival function at q of the given RV.
Parameters
q : array-like
upper tail probability
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
scale : array-like, optional
scale parameter (default=1)
Returns
x : array-like
quantile corresponding to the upper tail probability q.
stats(*args, **kwds)
Some statistics of the given RV
Parameters
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
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location parameter (default=0)
scale : array-like, optional
scale parameter (default=1)
moments : string, optional
composed of letters [’mvsk’] defining which moments to compute: ‘m’ = mean, ‘v’
= variance, ‘s’ = (Fisher’s) skew, ‘k’ = (Fisher’s) kurtosis. (default=’mv’)
Returns
stats : sequence
of requested moments.
Each discrete distribution is an instance of the class rv_discrete:
rv_discrete
rv_discrete.pmf (self, k, *args, **kwds)
rv_discrete.cdf (self, k, *args, **kwds)
rv_discrete.sf (self, k, *args, **kwds)
rv_discrete.ppf (self, q, *args, **kwds)
rv_discrete.isf (self, q, *args, **kwds)
rv_discrete.stats (self, *args, **kwds)
A generic discrete random variable class meant for subclassing.
Probability mass function at k of the given RV.
Cumulative distribution function at k of the given RV
Survival function (1-cdf) at k of the given RV
Percent point function (inverse of cdf) at q of the given RV
Inverse survival function (1-sf) at q of the given RV
Some statistics of the given discrete RV
class rv_discrete(a=0, b=inf, name=None, badvalue=None, moment_tol=1e-08, values=None, inc=1, longname=None, shapes=None, extradoc=None)
A generic discrete random variable class meant for subclassing.
rv_discrete is a base class to construct specific distribution classes and instances from for discrete random
variables. rv_discrete can be used to construct an arbitrary distribution with defined by a list of support points
and the corresponding probabilities.
Parameters
a : float, optional
Lower bound of the support of the distribution, default: 0
b : float, optional
Upper bound of the support of the distribution, default: plus infinity
moment_tol : float, optional
The tolerance for the generic calculation of moments
values : tuple of two array_like
(xk, pk) where xk are points (integers) with positive probability pk with sum(pk) = 1
inc : integer
increment for the support of the distribution, default: 1 other values have not been
tested
badvalue : object, optional
The value in (masked) arrays that indicates a value that should be ignored.
name : str, optional
The name of the instance. This string is used to construct the default example for
distributions.
longname : str, optional
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This string is used as part of the first line of the docstring returned when a subclass
has no docstring of its own. Note: longname exists for backwards compatibility, do
not use for new subclasses.
shapes : str, optional
The shape of the distribution. For example "m, n" for a distribution that takes two
integers as the first two arguments for all its methods.
extradoc : str, optional
This string is used as the last part of the docstring returned when a subclass has no
docstring of its own. Note: extradoc exists for backwards compatibility, do not use
for new subclasses.
Notes
Alternatively, the object may be called (as a function) to fix the shape and location parameters returning a
“frozen” discrete RV object:
myrv = generic(<shape(s)>, loc=0)
• frozen RV object with the same methods but holding the given shape and location fixed.
You can construct an aribtrary discrete rv where P{X=xk} = pk by passing to the rv_discrete initialization
method (through the values=keyword) a tuple of sequences (xk, pk) which describes only those values of X (xk)
that occur with nonzero probability (pk).
To create a new discrete distribution, we would do the following:
class poisson_gen(rv_continuous):
#"Poisson distribution"
def _pmf(self, k, mu):
...
and create an instance
poisson = poisson_gen(name=”poisson”, shapes=”mu”, longname=’A Poisson’)
The docstring can be created from a template.
Examples
>>> import matplotlib.pyplot as plt
>>> numargs = generic.numargs
>>> [ <shape(s)> ] = [’Replace with resonable value’, ]*numargs
Display frozen pmf:
>>> rv = generic(<shape(s)>)
>>> x = np.arange(0, np.min(rv.dist.b, 3)+1)
>>> h = plt.plot(x, rv.pmf(x))
Check accuracy of cdf and ppf:
>>> prb = generic.cdf(x, <shape(s)>)
>>> h = plt.semilogy(np.abs(x-generic.ppf(prb, <shape(s)>))+1e-20)
Random number generation:
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>>> R = generic.rvs(<shape(s)>, size=100)
Custom made discrete distribution:
>>> vals = [arange(7), (0.1, 0.2, 0.3, 0.1, 0.1, 0.1, 0.1)]
>>> custm = rv_discrete(name=’custm’, values=vals)
>>> h = plt.plot(vals[0], custm.pmf(vals[0]))
Methods
generic.rvs(<shape(s)>, loc=0, size=1)
generic.pmf(x, <shape(s)>, loc=0)
generic.cdf(x, <shape(s)>, loc=0)
generic.sf(x, <shape(s)>, loc=0)
generic.ppf(q, <shape(s)>, loc=0)
generic.isf(q, <shape(s)>, loc=0)
generic.stats(<shape(s)>, loc=0,
moments=’mv’)
generic.entropy(<shape(s)>, loc=0)
generic(<shape(s)>, loc=0)
random variates
probability mass function
cumulative density function
survival function (1-cdf — sometimes more accurate)
percent point function (inverse of cdf — percentiles)
inverse survival function (inverse of sf)
mean(‘m’, axis=0), variance(‘v’), skew(‘s’), and/or
kurtosis(‘k’)
entropy of the RV
calling a distribution instance returns a frozen distribution
pmf(k, *args, **kwds)
Probability mass function at k of the given RV.
Parameters
k : array-like
quantiles
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
Returns
pmf : array-like
Probability mass function evaluated at k
cdf(k, *args, **kwds)
Cumulative distribution function at k of the given RV
Parameters
k : array-like, int
quantiles
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
Returns
cdf : array-like
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Cumulative distribution function evaluated at k
sf(k, *args, **kwds)
Survival function (1-cdf) at k of the given RV
Parameters
k : array-like
quantiles
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
Returns
sf : array-like
Survival function evaluated at k
ppf(q, *args, **kwds)
Percent point function (inverse of cdf) at q of the given RV
Parameters
q : array-like
lower tail probability
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
Returns
k : array-like
quantile corresponding to the lower tail probability, q.
isf(q, *args, **kwds)
Inverse survival function (1-sf) at q of the given RV
Parameters
q : array-like
upper tail probability
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
Returns
k : array-like
quantile corresponding to the upper tail probability, q.
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stats(*args, **kwds)
Some statistics of the given discrete RV
Parameters
arg1, arg2, arg3,... : array-like
The shape parameter(s) for the distribution (see docstring of the instance object for
more information)
loc : array-like, optional
location parameter (default=0)
moments : string, optional
composed of letters [’mvsk’] defining which moments to compute: ‘m’ = mean, ‘v’
= variance, ‘s’ = (Fisher’s) skew, ‘k’ = (Fisher’s) kurtosis. (default=’mv’)
Returns
stats : sequence
of requested moments.
3.18.1 Continuous distributions
norm ()
alpha ()
anglit ()
arcsine ()
beta ()
betaprime ()
bradford ()
burr ()
fisk ()
cauchy ()
chi ()
chi2 ()
cosine ()
dgamma ()
dweibull ()
erlang ()
expon ()
exponweib ()
exponpow ()
fatiguelife ()
foldcauchy ()
f ()
foldnorm ()
fretchet_r
fretcher_l
genlogistic ()
genpareto ()
genexpon ()
genextreme ()
gausshyper ()
gamma ()
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A normal continuous random variable.
A alpha continuous random variable.
A anglit continuous random variable.
A arcsine continuous random variable.
A beta continuous random variable.
A betaprime continuous random variable.
A Bradford continuous random variable.
Burr continuous random variable.
A funk continuous random variable.
Cauchy continuous random variable.
A chi continuous random variable.
A chi-squared continuous random variable.
A cosine continuous random variable.
A double gamma continuous random variable.
A double Weibull continuous random variable.
An Erlang continuous random variable.
An exponential continuous random variable.
An exponentiated Weibull continuous random variable.
An exponential power continuous random variable.
A fatigue-life (Birnbaum-Sanders) continuous random variable.
A folded Cauchy continuous random variable.
An F continuous random variable.
A folded normal continuous random variable.
A generalized logistic continuous random variable.
A generalized Pareto continuous random variable.
A generalized exponential continuous random variable.
A generalized extreme value continuous random variable.
A Gauss hypergeometric continuous random variable.
A gamma continuous random variable.
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gengamma ()
genhalflogistic ()
gompertz ()
gumbel_r ()
gumbel_l ()
halfcauchy ()
halflogistic ()
halfnorm ()
hypsecant ()
invgamma ()
invnorm ()
invweibull ()
johnsonsb ()
johnsonsu ()
laplace ()
logistic ()
loggamma ()
loglaplace ()
lognorm ()
gilbrat ()
lomax ()
maxwell ()
mielke ()
nakagami ()
ncx2 ()
ncf ()
t ()
nct ()
pareto ()
powerlaw ()
powerlognorm ()
powernorm ()
rdist ()
reciprocal ()
rayleigh ()
rice ()
recipinvgauss ()
semicircular ()
triang ()
truncexpon ()
truncnorm ()
tukeylambda ()
uniform ()
von_mises
wald ()
weibull_min ()
weibull_max ()
wrapcauchy ()
ksone ()
kstwobign ()
Table 3.3 – continued from previous page
A generalized gamma continuous random variable.
A generalized half-logistic continuous random variable.
A Gompertz (truncated Gumbel) distribution continuous random variable.
A (right-skewed) Gumbel continuous random variable.
A left-skewed Gumbel continuous random variable.
A Half-Cauchy continuous random variable.
A half-logistic continuous random variable.
A half-normal continuous random variable.
A hyperbolic secant continuous random variable.
An inverted gamma continuous random variable.
An inverse normal continuous random variable.
An inverted Weibull continuous random variable.
A Johnson SB continuous random variable.
A Johnson SU continuous random variable.
A Laplace continuous random variable.
A logistic continuous random variable.
A log gamma continuous random variable.
A log-Laplace continuous random variable.
A lognormal continuous random variable.
A Gilbrat continuous random variable.
A Lomax (Pareto of the second kind) continuous random variable.
A Maxwell continuous random variable.
A Mielke’s Beta-Kappa continuous random variable.
A Nakagami continuous random variable.
A non-central chi-squared continuous random variable.
A non-central F distribution continuous random variable.
Student’s T continuous random variable.
A Noncentral T continuous random variable.
A Pareto continuous random variable.
A power-function continuous random variable.
A power log-normal continuous random variable.
A power normal continuous random variable.
An R-distributed continuous random variable.
A reciprocal continuous random variable.
A Rayleigh continuous random variable.
A Rice continuous random variable.
A reciprocal inverse Gaussian continuous random variable.
A semicircular continuous random variable.
A Triangular continuous random variable.
A truncated exponential continuous random variable.
A truncated normal continuous random variable.
A Tukey-Lambda continuous random variable.
A uniform continuous random variable.
A Wald continuous random variable.
A Weibull minimum continuous random variable.
A Weibull maximum continuous random variable.
A wrapped Cauchy continuous random variable.
Kolmogorov-Smirnov A one-sided test statistic. continuous random variable.
Kolmogorov-Smirnov two-sided (for large N) continuous random variable.
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norm(momtype=1, a=None, b=None, xa=-10.0, xb=10.0, xtol=1e-14, badvalue=None, name=None, longname=None, shapes=None, extradoc=None)
A normal continuous random variable.
Continuous random variables are defined from a standard form and may require some shape parameters to
complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as
given below:
Parameters
x : array-like
quantiles
q : array-like
lower or upper tail probability
loc : array-like, optional
location parameter (default=0)
scale : array-like, optional
scale parameter (default=1)
size : int or tuple of ints, optional
shape of random variates (default computed from input arguments )
moments : str, optional
composed of letters [’mvsk’] specifying which moments to compute where ‘m’
= mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)
Alternatively, the object may be called (as a function) to fix the shape, :
location, and scale parameters returning a “frozen” continuous RV object: :
rv = norm(loc=0, scale=1) :
• Frozen RV object with the same methods but holding the given shape, location, and scale
fixed.
Notes
Normal distribution
The location (loc) keyword specifies the mean. The scale (scale) keyword specifies the standard deviation.
normal.pdf(x) = exp(-x**2/2)/sqrt(2*pi)
Examples
>>>
>>>
>>>
>>>
import matplotlib.pyplot as plt
numargs = norm.numargs
[ ] = [0.9,] * numargs
rv = norm()
Display frozen pdf
>>> x = np.linspace(0, np.minimum(rv.dist.b, 3))
>>> h = plt.plot(x, rv.pdf(x))
Check accuracy of cdf and ppf
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>>> prb = norm.cdf(x, )
>>> h = plt.semilogy(np.abs(x - norm.ppf(prb, )) + 1e-20)
Random number generation
>>> R = norm.rvs(size=100)
Methods
rvs(loc=0, scale=1, size=1)
pdf(x, loc=0, scale=1)
cdf(x, loc=0, scale=1)
sf(x, loc=0, scale=1)
ppf(q, loc=0, scale=1)
isf(q, loc=0, scale=1)
stats(loc=0, scale=1, moments=’mv’)
entropy(loc=0, scale=1)
fit(data, loc=0, scale=1)
Random variates.
Probability density function.
Cumulative density function.
Survival function (1-cdf — sometimes more accurate).
Percent point function (inverse of cdf — percentiles).
Inverse survival function (inverse of sf).
Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
(Differential) entropy of the RV.
Parameter estimates for generic data.
alpha(momtype=1, a=None, b=None, xa=-10.0, xb=10.0, xtol=1e-14, badvalue=None, name=None, longname=None, shapes=None, extradoc=None)
A alpha continuous random variable.
Continuous random variables are defined from a standard form and may require some shape parameters to
complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as
given below:
Parameters
x : array-like
quantiles
q : array-like
lower or upper tail probability
a : array-like
shape parameters
loc : array-like, optional
location parameter (default=0)
scale : array-like, optional
scale parameter (default=1)
size : int or tuple of ints, optional
shape of random variates (default computed from input arguments )
moments : str, optional
composed of letters [’mvsk’] specifying which moments to compute where ‘m’
= mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)
Alternatively, the object may be called (as a function) to fix the shape, :
location, and scale parameters returning a “frozen” continuous RV object: :
rv = alpha(a, loc=0, scale=1) :
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• Frozen RV object with the same methods but holding the given shape, location, and scale
fixed.
Notes
Alpha distribution
alpha.pdf(x,a) = 1/(x**2*Phi(a)*sqrt(2*pi)) * exp(-1/2 * (a-1/x)**2) where Phi(alpha) is the normal CDF, x >
0, and a > 0.
Examples
>>>
>>>
>>>
>>>
import matplotlib.pyplot as plt
numargs = alpha.numargs
[ a ] = [0.9,] * numargs
rv = alpha(a)
Display frozen pdf
>>> x = np.linspace(0, np.minimum(rv.dist.b, 3))
>>> h = plt.plot(x, rv.pdf(x))
Check accuracy of cdf and ppf
>>> prb = alpha.cdf(x, a)
>>> h = plt.semilogy(np.abs(x - alpha.ppf(prb, a)) + 1e-20)
Random number generation
>>> R = alpha.rvs(a, size=100)
Methods
rvs(a, loc=0, scale=1, size=1)
pdf(x, a, loc=0, scale=1)
cdf(x, a, loc=0, scale=1)
sf(x, a, loc=0, scale=1)
ppf(q, a, loc=0, scale=1)
isf(q, a, loc=0, scale=1)
stats(a, loc=0, scale=1, moments=’mv’)
entropy(a, loc=0, scale=1)
fit(data, a, loc=0, scale=1)
Random variates.
Probability density function.
Cumulative density function.
Survival function (1-cdf — sometimes more accurate).
Percent point function (inverse of cdf — percentiles).
Inverse survival function (inverse of sf).
Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
(Differential) entropy of the RV.
Parameter estimates for generic data.
anglit(momtype=1, a=None, b=None, xa=-10.0, xb=10.0, xtol=1e-14, badvalue=None, name=None, longname=None, shapes=None, extradoc=None)
A anglit continuous random variable.
Continuous random variables are defined from a standard form and may require some shape parameters to
complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as
given below:
Parameters
x : array-like
quantiles
q : array-like
lower or upper tail probability
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loc : array-like, optional
location parameter (default=0)
scale : array-like, optional
scale parameter (default=1)
size : int or tuple of ints, optional
shape of random variates (default computed from input arguments )
moments : str, optional
composed of letters [’mvsk’] specifying which moments to compute where ‘m’
= mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)
Alternatively, the object may be called (as a function) to fix the shape, :
location, and scale parameters returning a “frozen” continuous RV object: :
rv = anglit(loc=0, scale=1) :
• Frozen RV object with the same methods but holding the given shape, location, and scale
fixed.
Notes
Anglit distribution
anglit.pdf(x) = sin(2*x+pi/2) = cos(2*x) for -pi/4 <= x <= pi/4
Examples
>>>
>>>
>>>
>>>
import matplotlib.pyplot as plt
numargs = anglit.numargs
[ ] = [0.9,] * numargs
rv = anglit()
Display frozen pdf
>>> x = np.li