Neural Network Toolbox User's Guide

Neural Network Toolbox User's Guide
Neural Network Toolbox
For Use with MATLAB
®
Howard Demuth
Mark Beale
Computation
Visualization
Programming
User’s Guide
Version 3.0
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Neural Network Toolbox User’s Guide
 COPYRIGHT 1992 - 1997 by The MathWorks, Inc. All Rights Reserved.
The software described in this document is furnished under a license agreement. The software may be used
or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc.
U.S. GOVERNMENT: If Licensee is acquiring the Programs on behalf of any unit or agency of the U.S.
Government, the following shall apply: (a) For units of the Department of Defense: the Government shall
have only the rights specified in the license under which the commercial computer software or commercial
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and accompanying documentation, the rights of the Government regarding its use, reproduction, and disclosure are as set forth in Clause 52.227-19 (c)(2) of the FAR.
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and Target Language Compiler are trademarks of The MathWorks, Inc.
Other product or brand names are trademarks or registered trademarks of their respective holders.
Printing History: June 1992
April 1993
January 1997
July 1997
First printing
Second printing
Third printing
Fourth Printing
Contents
Introduction
1
Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2
Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4
Basic Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4
Help and Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4
Whats New in 3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5
Other New Algorithms and Functions . . . . . . . . . . . . . . . . . . . . . 1-5
Modular Network Representation . . . . . . . . . . . . . . . . . . . . . . . . 1-5
Simulink Simulation Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5
General Toolbox Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5
Reduced Memory Levenberg-Marquardt Algorithm . . . . . . . 1-6
Other New Networks, Algorithms and Improvements . . . . . 1-7
Resilient Backpropagation (Rprop) . . . . . . . . . . . . . . . . . . . . . . . 1-7
Conjugate Gradient Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7
Quasi-Newton Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7
BFGS Quasi Newton Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 1-7
A One Step Secant Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7
Speed Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8
Improving Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9
Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9
Early Stopping With Validation . . . . . . . . . . . . . . . . . . . . . . . . 1-9
Pre and Post Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9
Scale Minimum and Maximum . . . . . . . . . . . . . . . . . . . . . . . 1-10
Scale Mean and Standard Deviation . . . . . . . . . . . . . . . . . . . 1-10
Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . 1-10
Post-training Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10
New Training Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10
Probabilistic Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11
Generalized Regression Networks . . . . . . . . . . . . . . . . . . . . . . . 1-11
i
Modular Network Representation . . . . . . . . . . . . . . . . . . . . . 1-12
Better Simulink Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-13
General Toolbox Improvements . . . . . . . . . . . . . . . . . . . . . . . 1-14
Simpler and More Extensible Toolbox . . . . . . . . . . . . . . . . . . . 1-14
Custom Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15
Neural Network Applications . . . . . . . . . . . . . . . . . . . . . . . . .
Aerospace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Automotive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Banking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Defense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Entertainment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Financial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Medical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Oil and Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Speech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Telecommunications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-16
1-16
1-16
1-16
1-16
1-16
1-17
1-17
1-17
1-17
1-17
1-17
1-17
1-18
1-18
1-18
1-18
1-18
Neural Network Design Book . . . . . . . . . . . . . . . . . . . . . . . . . 1-19
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-20
Neuron Model and Network Architectures
2
ii
Contents
Basic Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mathematical Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mathematical and Code Equivalents . . . . . . . . . . . . . . . . . . .
2-2
2-2
2-2
2-2
Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simple Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Neuron With Vector Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-4
2-4
2-5
2-7
Network Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Layer of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inputs and Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multiple Layers of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-10
2-10
2-11
2-13
Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation With Concurrent Inputs in a Static Network . . . .
Simulation With Sequential Inputs in a Dynamic Network . .
Simulation With Concurrent Inputs in a Dynamic Network .
2-15
2-15
2-16
2-18
Training Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Incremental Training (of Adaptive and Other Networks) . . . .
Incremental Training with Static Networks . . . . . . . . . . . .
Incremental Training With Dynamic Networks . . . . . . . . .
Batch Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Batch Training With Static Networks . . . . . . . . . . . . . . . . .
Batch Training With Dynamic Networks . . . . . . . . . . . . . . .
2-20
2-20
2-20
2-22
2-22
2-22
2-24
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figures and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simple Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hard Limit Transfer Function . . . . . . . . . . . . . . . . . . . . . . .
Purelin Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . .
Log Sigmoid Transfer Function . . . . . . . . . . . . . . . . . . . . . .
Neuron With Vector Input . . . . . . . . . . . . . . . . . . . . . . . . . .
Net Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Single Neuron Using Abbreviated Notation . . . . . . . . . . . .
Icons for Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . .
Layer of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Three Layers of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Layer of Neurons, Abbreviated Notation . . . . . . . . . . . . . . .
Layer of Neurons Showing Indices . . . . . . . . . . . . . . . . . . . .
Three Layers, Abbreviated Notation . . . . . . . . . . . . . . . . . .
Linear Neuron With Two Element Vector Input. . . . . . . . .
Dynamic Network With One Delay. . . . . . . . . . . . . . . . . . . .
2-26
2-28
2-28
2-28
2-28
2-29
2-29
2-29
2-30
2-30
2-30
2-31
2-31
2-32
2-32
2-33
2-34
2-34
iii
Perceptrons
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2
Important Perceptron Functions . . . . . . . . . . . . . . . . . . . . . . . . . 3-3
Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4
Perceptron Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6
Creating a Perceptron (NEWP) . . . . . . . . . . . . . . . . . . . . . . . . . 3-7
Simulation (SIM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8
Initialization (INIT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
Learning Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11
Perceptron Learning Rule (LEARNP) . . . . . . . . . . . . . . . . . . 3-12
Adaptive Training (ADAPT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15
Limitations and Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21
Outliers and the Normalized Perceptron Rule . . . . . . . . . . . . . 3-21
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figures and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perceptron Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perceptron Transfer Function, hardlim . . . . . . . . . . . . . .
Decision Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perceptron Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Perceptron Learning Rule . . . . . . . . . . . . . . . . . . . . . . .
One Perceptron Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-23
3-23
3-23
3-24
3-24
3-25
3-25
3-26
3-26
Adaptive Linear Filters
4
iv
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2
Important Linear Network Functions . . . . . . . . . . . . . . . . . . . . 4-3
Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4
Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5
Single ADALINE (NEWLIN) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6
Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9
Linear System Design (NEWLIND) . . . . . . . . . . . . . . . . . . . . . 4-10
LMS Algorithm (LEARNWH) . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11
Linear Classification (TRAIN) . . . . . . . . . . . . . . . . . . . . . . . . . 4-13
Adaptive Filtering (ADAPT) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tapped Delay Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adaptive Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adaptive Filter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prediction Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Noise Cancellation Example . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multiple Neuron Adaptive Filters . . . . . . . . . . . . . . . . . . . . . . .
4-16
4-16
4-17
4-18
4-20
4-21
4-23
Limitations and Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Overdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Underdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linearly Dependent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
Too Large a Learning Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-25
4-25
4-25
4-25
4-26
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figures and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Purelin Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . .
MADALINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ADALINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Decision Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LMS (Widrow-Hoff) Algorithm . . . . . . . . . . . . . . . . . . . . . . .
Tapped Delay Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adaptive Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adaptive Filter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-27
4-28
4-28
4-28
4-29
4-29
4-30
4-30
4-30
4-31
4-32
4-32
v
Prediction Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Noise Cancellation Example . . . . . . . . . . . . . . . . . . . . . . . . .
Multiple Neuron Adaptive Filter . . . . . . . . . . . . . . . . . . . . .
Abbreviated Form of Adaptive Filter . . . . . . . . . . . . . . . . . .
Specific Small Adaptive Filter . . . . . . . . . . . . . . . . . . . . . . .
New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-33
4-34
4-35
4-35
4-36
4-36
Backpropagation
5
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2
Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Neuron Model (TANSIG, LOGSIG, PURELIN) . . . . . . . . . . .
Feedforward Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation (SIM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Backpropagation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . .
Faster Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Variable Learning Rate (TRAINGDA, TRAINGDX) . . . . . . . .
Resilient Backpropagation (TRAINRP) . . . . . . . . . . . . . . . . . .
Conjugate Gradient Algorithms . . . . . . . . . . . . . . . . . . . . . . . .
Fletcher-Reeves Update (TRAINCGF) . . . . . . . . . . . . . . . . .
Polak-Ribiére Update (TRAINCGP) . . . . . . . . . . . . . . . . . .
Powell-Beale Restarts (TRAINCGB) . . . . . . . . . . . . . . . . . .
Scaled Conjugate Gradient (TRAINSCG) . . . . . . . . . . . . . .
Line Search Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Golden Section Search (SRCHGOL) . . . . . . . . . . . . . . . . . . .
Brent’s Search (SRCHBRE) . . . . . . . . . . . . . . . . . . . . . . . . .
Hybrid Bisection-Cubic Search (SRCHHYB) . . . . . . . . . . .
Charalambous’ Search (SRCHCHA) . . . . . . . . . . . . . . . . . . .
Backtracking (SRCHBAC) . . . . . . . . . . . . . . . . . . . . . . . . . .
Quasi-Newton Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BFGS Algorithm (TRAINBFG) . . . . . . . . . . . . . . . . . . . . . . .
One Step Secant Algorithm (TRAINOSS) . . . . . . . . . . . . . .
Levenberg-Marquardt (TRAINLM) . . . . . . . . . . . . . . . . . . . . .
Reduced Memory Levenberg-Marquardt (TRAINLM) . . . . . . .
vi
Contents
5-3
5-3
5-3
5-5
5-8
5-8
5-9
5-16
5-16
5-18
5-20
5-20
5-22
5-24
5-25
5-26
5-26
5-27
5-27
5-28
5-28
5-29
5-29
5-30
5-31
5-33
Speed and Memory Comparison . . . . . . . . . . . . . . . . . . . . . . . 5-35
Improving Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modified Performance Function . . . . . . . . . . . . . . . . . . . . . .
Automated Regularization (TRAINBR) . . . . . . . . . . . . . . . .
Early Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-37
5-38
5-38
5-39
5-41
Preprocessing and Postprocessing . . . . . . . . . . . . . . . . . . . . .
Min and Max (PREMNMX, POSTMNMX, TRAMNMX) . . . . .
Mean and Stand. Dev. (PRESTD, POSTSTD, TRASTD) . . . . .
Principal Component Analysis (PREPCA, TRAPCA) . . . . . . .
Post-training Analysis (POSTREG) . . . . . . . . . . . . . . . . . . . . .
5-44
5-44
5-45
5-46
5-47
Sample Training Session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-49
Limitations and Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-54
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-56
Radial Basis Networks
6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2
Important Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . 6-2
Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exact Design (NEWRBE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
More Efficient Design (NEWRB) . . . . . . . . . . . . . . . . . . . . . . . . .
Demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-3
6-3
6-4
6-5
6-7
6-8
Generalized Regression Networks . . . . . . . . . . . . . . . . . . . . . . 6-9
Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9
Design (NEWGRNN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11
vii
Probabilistic Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . 6-12
Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12
Design (NEWPNN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radial Basis Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radbas Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radial Basis Network Architecture . . . . . . . . . . . . . . . . . . .
Generalized Regression Neural Network Architecture . . . .
Probabilistic Neural Network Architecture . . . . . . . . . . . . .
New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-15
6-16
6-16
6-16
6-17
6-17
6-18
6-19
Self-Organizing Networks
7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2
Important Self-Organizing Functions . . . . . . . . . . . . . . . . . . . . . 7-2
Competitive Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creating a Competitive Neural Network (NEWC) . . . . . . . . . .
Kohonen Learning Rule (LEARNK) . . . . . . . . . . . . . . . . . . . . . .
Bias Learning Rule (LEARNCON) . . . . . . . . . . . . . . . . . . . . . . .
Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Graphical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Self-Organizing Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Topologies (GRIDTOP, HEXTOP, RANDTOP) . . . . . . . . . . . .
Distance Functions (DIST, LINKDIST,
MANDIST, BOXDIST) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creating a Self Organizing MAP Neural Network
(NEWSOM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Training (LEARNSOM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phase 1: Ordering Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phase 2: Tuning Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii Contents
7-3
7-3
7-4
7-5
7-5
7-6
7-8
7-10
7-12
7-16
7-19
7-20
7-22
7-22
7-22
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-25
One-Dimensional Self-Organizing Map . . . . . . . . . . . . . . . . 7-25
Two-Dimensional Self-Organizing Map . . . . . . . . . . . . . . . . 7-27
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Competitive Network Architecture . . . . . . . . . . . . . . . . . . . .
Self Organizing Feature Map Architecture . . . . . . . . . . . . .
New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-32
7-32
7-32
7-33
7-33
Learning Vector Quantization
8
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2
Important LVQ Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2
Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3
Creating an LVQ Network (NEWLVQ) . . . . . . . . . . . . . . . . . . . 8-5
LVQ Learning Rule(LEARNLV2) . . . . . . . . . . . . . . . . . . . . . . . 8-9
Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LVQ Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-14
8-14
8-14
8-14
ix
9
Recurrent Networks
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2
Important Recurrent Network Functions . . . . . . . . . . . . . . . . . . 9-2
Elman Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creating an Elman Network (NEWELM) . . . . . . . . . . . . . . . . .
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Training an Elman Network . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-3
9-3
9-4
9-4
9-6
Hopfield Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9
Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9
Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9
Design(NEWHOP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-13
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Elman Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hopfield Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-16
9-17
9-17
9-17
9-18
Applications
10
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2
Application Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2
Applin1: Linear Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
Contents
10-3
10-3
10-4
10-4
10-6
Applin2: Adaptive Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 10-7
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-7
Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8
Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8
Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8
Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-10
Applin3: Linear System Identification . . . . . . . . . . . . . . . .
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-11
10-11
10-12
10-12
10-14
Applin4: Adaptive System Identification . . . . . . . . . . . . . .
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-15
10-15
10-16
10-17
10-17
10-17
Appelm1: Amplitude Detection . . . . . . . . . . . . . . . . . . . . . . .
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Improving Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-19
10-19
10-20
10-20
10-21
10-22
10-23
Appcs1: Nonlinear System Identification . . . . . . . . . . . . . .
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-24
10-25
10-26
10-26
10-26
10-29
xi
Appcs2: Model Reference Control . . . . . . . . . . . . . . . . . . . . .
Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Neural Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-30
10-30
10-31
10-32
10-32
10-33
10-36
10-37
Appcr1: Character Recognition . . . . . . . . . . . . . . . . . . . . . . .
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Training Without Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Training With Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Training Without Noise Again . . . . . . . . . . . . . . . . . . . . . .
System Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-38
10-38
10-39
10-40
10-40
10-40
10-41
10-41
10-42
10-42
10-43
Advanced Topics
11
xii
Contents
Custom Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2
Custom Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4
Network Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5
Architecture Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5
Number of Outputs and Targets . . . . . . . . . . . . . . . . . . . . . . 11-7
Subobject Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8
Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8
Network Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-12
Weight and Bias Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-13
Network Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-14
Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-14
Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-14
Additional Toolbox Functions . . . . . . . . . . . . . . . . . . . . . . . .
Initialization Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
randnc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
randnr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
satlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
softmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
tribas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Learning Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnhd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11-17
11-17
11-17
11-17
11-17
11-17
11-17
11-17
11-18
11-18
11-18
11-18
11-18
Custom Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Net Input Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Initialization Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network Initialization Functions . . . . . . . . . . . . . . . . . . . .
Layer Initialization Functions . . . . . . . . . . . . . . . . . . . . . .
Weight and Bias Initialization Functions . . . . . . . . . . . . .
Learning Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Training Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adapt Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Performance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Weight and Bias Learning Functions . . . . . . . . . . . . . . . . .
Self-Organizing Map Functions . . . . . . . . . . . . . . . . . . . . . . .
Topology Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11-19
11-20
11-20
11-22
11-24
11-26
11-26
11-26
11-27
11-29
11-29
11-32
11-34
11-36
11-39
11-39
11-40
xiii
Network Object Reference
12
xiv Contents
Network Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2
Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2
numInputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2
numLayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2
biasConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-3
inputConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-4
layerConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-4
outputConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-4
targetConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5
numOutputs (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5
numTargets (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5
numInputDelays (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . 12-5
numLayerDelays (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . 12-6
Subobject Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-6
inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-7
layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-7
outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-7
targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-8
biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-8
inputWeights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-9
layerWeights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-9
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-10
adaptFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-10
initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-10
performFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-11
trainFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-11
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-13
adaptParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-13
initParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-13
performParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-13
trainParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-14
Weight and Bias Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-14
IW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-14
LW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-15
b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-16
Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-16
userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-16
Subobject Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
distanceFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
distances (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
netInputFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
positions (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
topologyFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
transferFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Input Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
weightFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12-17
12-17
12-17
12-17
12-18
12-18
12-18
12-19
12-19
12-20
12-20
12-21
12-22
12-22
12-23
12-24
12-25
12-25
12-25
12-25
12-25
12-25
12-26
12-26
12-26
12-27
12-27
12-28
12-28
12-28
12-28
12-29
12-29
12-30
12-31
12-31
12-31
12-32
xv
LayerWeights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
learnParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
weightFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12-32
12-32
12-32
12-33
12-33
12-35
12-35
12-35
12-36
Reference
13
Functions Listed by Network Type . . . . . . . . . . . . . . . . . . . . 13-2
Functions by Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-3
Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-13
Transfer Function Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-14
Reference Page Headings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-18
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-19
Glossary
xvi Contents
Notation
A
Mathematical Notation for Equations and Figures . . . . . . . B-2
Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Weight Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Scalar Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Column Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Row Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Bias Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Scalar Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Layer Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
Input Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3
Layer Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3
Figure and Equation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . B-3
Mathematics and Code Equivalents . . . . . . . . . . . . . . . . . . . . . B-4
Bibliography
B
Demonstrations and Applications
C
Tables of Demonstrations and Application . . . . . . . . . . . . . . D-2
Chapter 2 Neuron Model & Network Architectures . . . . . . . . . D-2
Chapter 3 Perceptrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2
Chapter 4 Adaptive Linear Filters . . . . . . . . . . . . . . . . . . . . . . . D-3
Chapter 5 Backpropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-3
Chapter 6 Radial Basis Networks . . . . . . . . . . . . . . . . . . . . . . . . D-4
Chapter 7 Self-Organizing Networks . . . . . . . . . . . . . . . . . . . . . D-4
Chapter 8 Learning Vector Quantization . . . . . . . . . . . . . . . . . . D-4
Chapter 9 Recurrent Networks . . . . . . . . . . . . . . . . . . . . . . . . . . D-5
Chapter 10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-5
xvii
Simulink
D
Block Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2
Transfer Function Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-3
Net Input Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-3
Weight Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-4
Block Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-5
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-5
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-7
Changing Input Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-7
Discrete Sample Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-7
Index
xviii Contents
1
Introduction
Neural Networks . . . . . . . . . . . . . . . . . . 1-2
Getting Started . . . . . . . . . . . . . . . . . . 1-4
Basic Chapters . . . . . . . . . . . . . . . . . . . 1-4
Help and Installation . . . . . . . . . . . . . . . . . 1-4
Whats New in 3.0 . . . . . . . .
Other New Algorithms and Functions
Modular Network Representation . .
Simulink Simulation Support . . .
General Toolbox Improvements . . .
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1-5
1-5
1-5
1-5
1-5
Reduced Memory Levenberg-Marquardt Algorithm . . 1-6
Other New Networks, Algorithms and Improvements
Resilient Backpropagation (Rprop) . . . . . . . . . .
Conjugate Gradient Algorithms . . . . . . . . . . . .
Quasi-Newton Algorithms . . . . . . . . . . . . . .
Speed Comparison . . . . . . . . . . . . . . . . .
Improving Generalization . . . . . . . . . . . . . .
Pre and Post Processing . . . . . . . . . . . . . . .
New Training Options . . . . . . . . . . . . . . .
Probabilistic Neural Networks . . . . . . . . . . . .
Generalized Regression Networks . . . . . . . . . . .
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1-7
1-7
1-7
1-7
1-8
1-9
1-9
1-10
1-11
1-11
Modular Network Representation . . . . . . . . . . 1-12
Better Simulink Support
. . . . . . . . . . . . . . 1-13
General Toolbox Improvements . . . . . . . . . . . 1-14
Simpler and More Extensible Toolbox . . . . . . . . . . 1-14
Custom Functions . . . . . . . . . . . . . . . . . . 1-15
Neural Network Applications . . . . . . . . . . . . 1-16
Neural Network Design Book . . . . . . . . . . . . 1-19
Acknowledgments
. . . . . . . . . . . . . . . . . 1-20
1
Introduction
Neural Networks
Neural networks are composed of simple elements operating in parallel. These
elements are inspired by biological nervous systems. As in nature, the network
function is determined largely by the connections between elements. We can
train a neural network to perform a particular function by adjusting the values
of the connections (weights) between elements.
Commonly neural networks are adjusted, or trained, so that a particular input
leads to a specific target output. Such a situation is shown below. There, the
network is adjusted, based on a comparison of the output and the target, until
the network output matches the target. Typically many such input/target pairs
are used, in this supervised learning, to train a network.
Target
Input
Neural Network
including connections
(called weights)
between neurons
Compare
Output
Adjust
weights
Neural networks have been trained to perform complex functions in various
fields of application including pattern recognition, identification, classification,
speech, vision and control systems. A list of applications is given later in this
chapter.
Today neural networks can be trained to solve problems that are difficult for
conventional computers or human beings. Throughout the toolbox emphasis is
placed on neural network paradigms that build up to or are themselves used in
engineering, financial and other practical applications.
The supervised training methods are commonly used, but other networks can
be obtained from unsupervised training techniques or from direct design
methods. Unsupervised networks can be used, for instance, to identify groups
of data. Certain kinds of linear networks and Hopfield networks are designed
1-2
Neural Networks
directly. In summary, there are a variety of kinds of design and learning
techniques that enrich the choices that a user can make.
The field of neural networks has a history of some five decades but has found
solid application only in the past fifteen years, and the field is still developing
rapidly. Thus, it is distinctly different from the fields of control systems or
optimization where the terminology, basic mathematics, and design
procedures have been firmly established and applied for many years. We do not
view the Neural Network Toolbox as simply a summary of established
procedures that are known to work well. Rather, we hope that it will be a useful
tool for industry, education and research, a tool that will help users find what
works and what doesn’t, and a tool that will help develop and extend the field
of neural networks. Because the field and the material are so new, this toolbox
will explain the procedures, tell how to apply them, and illustrate their
successes and failures with examples. We believe that an understanding of the
paradigms and their application is essential to the satisfactory and successful
use of this toolbox, and that without such understanding user complaints and
inquiries would bury us. So please be patient if we include a lot of explanatory
material. We hope that such material will be helpful to you.
This chapter includes a few comments on getting started with the Neural
Network Toolbox. It also describes the new graphical user interface, and new
algorithms and architectures; and it explains the increased flexibility of the
Toolbox due to its use of modular network object representation.
Finally this chapter gives a list of some practical neural network applications
and describes a new text, Neural Network Design. This book presents the
theory of neural networks as well as their design and application, and makes
considerable use of MATLAB® and the Neural Network Toolbox.
1-3
1
Introduction
Getting Started
Basic Chapters
Chapter 2 contains basic material about network architectures and notation
specific to this toolbox. Chapter 3 includes the first reference to basic functions
such as init and adapt. Chapter 4 describes the use of the functions designd
and train, and discusses delays. Chapter 2, 3 and 4 should be read before going
to later chapters
Help and Installation
The Neural Network Toolbox is contained in a directory called nnet. Type help
nnet for a listing of help topics.
A number of demonstrations are included in the Toolbox. Each example states
a problem, shows the network used to solve the problem and presents the final
results. Lists of the neural network demonstration and application scripts that
are discussed in this guide can be found by typing help nndemos
Instructions for installing the Neural Network Toolbox are found in one of two
MATLAB documents, the Installation Guide for MS-Windows and Macintosh or
the Installation Guide for UNIX.
1-4
Whats New in 3.0
Whats New in 3.0
A few of the new features and improvements introduced with this version of the
Neural Network Toolbox are listed below.
Reduced Memory Levenberg-Marquardt Algorithm
The Neural Network Toolbox version 2.0 introduced the Levenberg-Marquardt
(LM) algorithm which is faster than other algorithms by a factor of from 10 to
100. Now version 3.0 introduces the Reduced Memory Levenberg-Marquardt
algorithm, which allows for a time/memory trade off. This means that the LM
algorithm can now be used in much larger problems, with perhaps only a slight
increase in running time.
Other New Algorithms and Functions
Conjugate gradient and R-Prop algorithms have been added, as have
Probabilistic, and Generalized Regression Networks. Automatic
regularization, new training options and a method for early stopping of
training have also been included here for the first time. New training options,
including training on variations of mean square error for better generalization,
training against a validation set, and training until the gradient of the error
reaches a minimum are now available. Finally, various pre and post processing
function have been included.
Modular Network Representation
The modular representation in the Toolbox version 3.0 allows a great deal of
flexibility for the design of one’s own custom networks. Virtually any
combination of neurons and layers, with delays if required, can be trained in
any one of Other New Networks, Algorithms and Improvements.
Simulink® Simulation Support
You can now generate network simulation blocks for use with Simulink.
General Toolbox Improvements
This toolbox is simpler but more powerful than ever. It has fewer functions but
each of them, including INIT (initialization), SIM (simulation), TRAIN,
(training) and ADAPT (adaptive learning) can be applied to a broad variety of
networks.
1-5
1
Introduction
Reduced Memory Levenberg-Marquardt Algorithm
A low-memory-use Levenberg-Marquardt algorithm has been developed by
Professor Martin Hagan of Oklahoma State University for the Neural Network
Toolbox. This algorithm achieves nearly the same speed of the original very
fast Levenberg-Marquardt algorithm, but uses less memory required by the
original. (See Neural Network Toolbox Version 2.0 and Hagan, M.T., and M.
Menhaj, “Training Feedforward Networks with the Marquardt Algorithm,”
IEEE Transactions on Neural Networks, vol. 5, no. 6, 1994.)
There is a drawback to using memory reduction. A significant computational
overhead is associated with computing the Jacobian in submatrices. If you
have enough memory available, then it is better to set mem_reduc to 1 and to
compute the full Jacobian. If you have a large training set, and you are running
out of memory, then you should set mem_reduc to 2, and try again. If you still
run out of memory, continue to increase mem_reduc.
Even if you use memory reduction, the Levenberg-Marquardt algorithm will
always compute the approximate Hessian matrix, which has dimensions n × n .
If your network is very large, then you may run out of memory. If this is the
case, then you will want to try trainoss, trainrp, or one of the conjugate
gradient algorithms.
1-6
Other New Networks, Algorithms and Improvements
Other New Networks, Algorithms and Improvements
Resilient Backpropagation (Rprop)
The resilient backpropagation (Rprop) training algorithm eliminates the
harmful effect of having a small slope at the extreme ends of sigmoid squashing
transfer functions. Only the sign of the derivative of the transfer function is
used to determine the direction of the weight update; the magnitude of the
derivative has no effect on the weight update. Rprop is generally much faster
than the standard steepest descent algorithm. It also has the nice property that
it requires only a modest increase in memory requirements.
Conjugate Gradient Algorithms
Various forms of a conjugate gradient backprop algorithm have been added. In
the conjugate gradient algorithms a search is performed along conjugate
directions, which produces generally faster convergence than steepest descent
directions. This is a well know, highly efficient algorithm that gives good
results on a broad spectrum of problems.
Quasi-Newton Algorithms
Quasi-Newton (or secant) methods are based on Newton’s method but don’t
require calculation of second derivatives. They update an approximate Hessian
matrix at each iteration of the algorithm. The update is computed as a function
of the gradient. Two quasi-newton algorithms are included in the Neural
Network Toolbox.
BFGS Quasi Newton Algorithm
This algorithm requires more computation in each iteration and more storage
than the conjugate gradient methods, although it generally converges in fewer
iterations. For very large networks it may be better to use Rprop or one of the
conjugate gradient algorithms. For smaller networks, however, trainbfg can
be an efficient training function.
A One Step Secant Algorithm
This algorithm requires less storage and computation per epoch than the BFGS
algorithm. It requires slightly more storage and computation per epoch than
the conjugate gradient algorithms. It can be considered a compromise between
full quasi-Newton algorithms and conjugate gradient algorithms.
1-7
1
Introduction
Speed Comparison
The following table gives some example convergence times for the various
algorithms on one particular regression problem. In this problem a 1-10-1
network was trained on a data set with 41 input/output pairs until a mean
square error performance of 0.01 was obtained. Twenty different test runs were
made for each training algorithm on a Macintosh Powerbook 1400 to obtain the
average numbers shown in the table. These numbers should be used with
caution, since the performances shown here may not be typical for these
algorithms on other types of problems. (You may notice that there is not a clear
relationship between the number of floating point operations and the time
required to reach convergence. This is because some of the algorithms can take
advantage of efficient built-in MATLAB functions. This is especially true for the
Levenberg-Marquardt algorithm.)
Function
Technique
Time
Epochs
Mflops
traingdx
Variable Learning Rate
57.71
980
2.50
trainrp
Rprop
12.95
185
0.56
trainscg
Scaled Conj. Grad.
16.06
106
0.70
traincgf
Fletcher-Powell CG
16.40
81
0.99
traincgp
Polak-Ribiére CG
19.16
89
0.75
traincgb
Powell-Beale CG
15.03
74
0.59
trainoss
One-Step-Secant
18.46
101
0.75
trainbfg
BFGS quasi-Newton
10.86
44
1.02
trainlm
Levenberg-Marquardt
1.87
6
0.46
For most situations, we recommend that you try the Levenberg-Marquardt
algorithm first. If this algorithm requires too much memory, then try the BFGS
algorithm trainbfg, or one of the conjugate gradient methods. The Rprop
algorithm trainrp is also very fast, and has relatively small memory
requirements.
Radial basis networks can be designed very quickly, typically in less time than
it takes the Levenberg-Marquardt algorithm to be trained. You might also
1-8
Other New Networks, Algorithms and Improvements
consider them. However, they have the disadvantage that, once designed, the
computation associated with their use may be greater than that for
conventional feedforward networks.
Improving Generalization
One of the problems that occurs during neural network training is called
overfitting. The error on the training set is driven to a very small value, but
when new data is presented to the network the error is large. The network has
memorized the training examples, but it has not learned to generalize to new
situations. Two solutions to the overfitting problem are presented here.
Regularization
Regularization involves modifying the performance function, which is normally
chosen to be the sum of squares of the network errors on the training set. We
have included two routines which will automatically set the optimal
performance function to achieve the best generalization.
Regularization helps take the mystery out of how to pick the number of
neurons in a network and consistently leads to good networks that are not
overtrained.
Early Stopping With Validation
Early stopping is a technique based on dividing the data into three subsets. The
first subset is the training set used for computing the gradient and updating
the network weights and biases. The second subset is the validation set. The
error on the validation set is monitored during the training process. The
validation error will normally decrease during the initial phase of training, as
does the training set error. However, when the network begins to overfit the
data, the error on the validation set will typically begin to rise. When the
validation error increases for a specified number of iterations, the training is
stopped, and the weights and biases at the minimum of the validation error are
returned.
Pre and Post Processing
Neural network training can be made more efficient if certain preprocessing
steps are performed on the network inputs and targets. Thus, we have included
the following functions.
1-9
1
Introduction
Scale Minimum and Maximum
The function premnmx can be used to scale inputs and targets so that they fall
in the range [-1,1].
Scale Mean and Standard Deviation
The function prestd normalizes the mean and standard deviation of the
training set.
Principal Component Analysis
The principle components analysis program prepca can be used to reduce the
dimensions of the input vectors.
Post-training Analysis
We have included a post training function postreg that performs a regression
analysis between the network response and the corresponding targets.
New Training Options
In this toolbox we can not only minimize mean squared error as before, but we
can also:
• Minimize with variations of mean squared error for better generalization.
Such training simplifies the problem of picking the number of hidden
neurons and produces good networks that are not overtrained.
• Train with validation to achieve appropriately early stopping. Here the
training result is checked against a validation set of input output data to
make sure that overtraining has not occurred.
• Stop training when the error gradient reaches a minimum. This avoids
wasting computation time when further training is having little effect.
• The low memory use Levenberg Marquardt algorithm has been incorporated
into both new and old algorithms.
1-10
Other New Networks, Algorithms and Improvements
Probabilistic Neural Networks
Probabilistic neural networks can be used for classification problems. Their
design is straightforward and does not depend on training. These networks
generalize well.
Generalized Regression Networks
A generalized regression neural network (GRNN) is often used for function
approximation. Given a sufficient number of hidden neurons, GRNNs can
approximate a continuous function to an arbitrary accuracy.
1-11
1
Introduction
Modular Network Representation
The modular representation in the Toolbox version 3.0 allows a great deal of
flexibility, including the following options:
• Networks can have any number of sets of inputs, layers.
• Any input or layer can be connected to any layer with a weight.
• Each layer can have a bias or not.
• Each layer can be a network output or not.
• Weights can have tapped delays.
• Weights can be partially connected
• Each layer can have a target or not
1-12
Better Simulink Support
Better Simulink Support
The Neural Network Toolbox Version 3.0 can now generate network simulation
blocks for use with Simulink. The Neural Network Toolbox version 2.0
provided transfer function blocks but didn't help import entire networks. We
can do that now.
1-13
1
Introduction
General Toolbox Improvements
Simpler and More Extensible Toolbox
The new Toolbox has fewer functions, but each of them does more than the old
ones. For instance, the following functions can be applied to a broad variety of
networks.
• init – initialization
• sim – simulation
• train – training
• adapt – adaptive learning
Now the Neural Network Toolbox Version 3.0 is more extensible in the
following ways:
• Network properties can be altered.
• Custom properties can be added to a network object.
1-14
General Toolbox Improvements
Custom Functions
The toolbox allows you to create and use many kinds of functions, giving you a
great deal of control over the algorithms used to initialize, simulate, and train,
your networks. The following sections indicate the kinds of functions you can
create:
• Simulation functions
- transfer
- net input
- weight
• Initialization functions
- network initialization
- layer initialization
- weight and bias initialization
• Learning functions
- network training
- network adapt
- network performance
- weight and bias learning
• Self-organizing map functions
- topology
- distance
1-15
1
Introduction
Neural Network Applications
The 1988 DARPA Neural Network Study [DARP88] lists various neural
network application, s beginning in about 1984 with the adaptive channel
equalizer. This device, which is an outstanding commercial success, is a singleneuron network used in long distance telephone systems to stabilize voice
signals. The DARPA report goes on to list other commercial applications,
including a small word recognizer, a process monitor, a sonar classifier, and a
risk analysis system.
Neural networks have been applied in many other fields since the DARPA
report was written. A list of some applications mentioned in the literature
follows:
Aerospace
• High performance aircraft autopilot, flight path simulation, aircraft control
systems, autopilot enhancements, aircraft component simulation, aircraft
component fault detection
Automotive
• Automobile automatic guidance system, warranty activity analysis
Banking
• Check and other document reading, credit application evaluation
Defense
• Weapon steering, target tracking, object discrimination, facial recognition,
new kinds of sensors, sonar, radar and image signal processing including
data compression, feature extraction and noise suppression, signal/image
identification
Electronics
• Code sequence prediction, integrated circuit chip layout, process control,
chip failure analysis, machine vision, voice synthesis, nonlinear modeling
1-16
Neural Network Applications
Entertainment
• Animation, special effects, market forecasting
Financial
• Real estate appraisal, loan advisor, mortgage screening, corporate bond
rating, credit line use analysis, portfolio trading program, corporate
financial analysis, currency price prediction
Insurance
• Policy application evaluation, product optimization
Manufacturing
• Manufacturing process control, product design and analysis, process and
machine diagnosis, real-time particle identification, visual quality
inspection systems, beer testing, welding quality analysis, paper quality
prediction, computer chip quality analysis, analysis of grinding operations,
chemical product design analysis, machine maintenance analysis, project
bidding, planning and management, dynamic modeling of chemical process
system
Medical
• Breast cancer cell analysis, EEG and ECG analysis, prosthesis design,
optimization of transplant times, hospital expense reduction, hospital
quality improvement, emergency room test advisement
Oil and Gas
• Exploration
Robotics
• Trajectory control, forklift robot, manipulator controllers, vision systems
1-17
1
Introduction
Speech
• Speech recognition, speech compression, vowel classification, text to speech
synthesis
Securities
• Market analysis, automatic bond rating, stock trading advisory systems
Telecommunications
• Image and data compression, automated information services, real-time
translation of spoken language, customer payment processing systems
Transportation
• Truck brake diagnosis systems, vehicle scheduling, routing systems
Summary
The list of additional neural network applications, the money that has been
invested in neural network software and hardware, and the depth and breadth
of interest in these devices have been growing rapidly. It is hoped that this
toolbox will be useful for neural network educational and design purposes
within a broad field of neural network applications.
A variety of neural network applications are described in Chapter 10.
1-18
Neural Network Design Book
Neural Network Design Book
Professor Martin Hagan of Oklahoma State University, and Neural Network
Toolbox authors Howard Demuth and Mark Beale have written a textbook,
Neural Network Design, published by PWS Publishing Company in 1996 (ISBN
0-534-94332-2). The book presents the theory of neural networks as well as
their design and application, and makes considerable use of MATLAB and the
Neural Network Toolbox. Demonstration programs from the book are used in
various chapters of this Guide. The book has a instructor’s manual containing
problem solutions (ISBN 0-534-95049-3), and overheads for class use. (The
overheads, in hard copy form, come one to a page for instructor use and three
to a page for student use.) For information about obtaining this text, please
contact International Thomson Publishing Customer Service, phone
1-800-347-7707.
1-19
1
Introduction
Acknowledgments
The authors would like to thank:
Martin Hagan, Oklahoma State University for providing the original
Levenberg-Marquardt algorithm in the Neural Network Toolbox version 2.0
and various algorithms found here in version 3.0, including the new reduced
memory use version of the Levenberg-Marquardt algorithm, the coujugate
gradient algorithm, RPROP, and generalized regression method. Martin also
wrote Chapter 5 of this Toolbox, Backpropagation, which contains descriptions
of new algorithms and suggestions for pre and post processing of data.
Joe Hicklin, of The MathWorks for getting Howard into neural network
research years ago at the University of Idaho, for encouraging Howard to write
the toolbox, for providing crucial help in getting the first toolbox version 1.0 out
the door, and for continuing to be a good friend.
Liz Callanan of The MathWorks for getting us off the such a good start with
the Neural Network Toolbox version 1.0.
Jim Tung of The MathWorks for his consistent long term support for this
project.
Jeff Faneuff and Roy Lurie of The MathWorks for their vigilant reviews of
the developing material in this version of the toolbox.
Matthew Simoneau of The MathWorks for his help with demos, test suite
routines, and getting user feedback.
Kathy Ford of The MathWorks for her careful help with the documentation.
Jane Price of The MathWorks for her help in getting constructive comments
from users.
Margaret Jenks of Moscow Idaho, for her patient, persistent and effective
editing.
Teri for running the show while Mark worked on this toolbox and finally,
PWS Publishing, for their permission to include various problems,
demonstrations and other material from Neural Network Design, Jan. 1996.
1-20
2
Neuron Model and
Network Architectures
Basic Chapters . . . . . . . . . . . . . . . . . . . 2-2
Notation . . . . . . . . . . . . . . . . . . . . . . 2-2
Neuron Model . . . .
Simple Neuron . . . .
Transfer Functions . . .
Neuron With Vector Input
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2-4
2-4
2-5
2-7
Network Architectures . . . . . . . . . . . . . . . 2-10
A Layer of Neurons . . . . . . . . . . . . . . . . . . 2-10
Multiple Layers of Neurons . . . . . . . . . . . . . . 2-13
Data Structures . . . . . . . . . . . . . . . . .
Simulation With Concurrent Inputs in a Static Network .
Simulation With Sequential Inputs in a Dynamic Network
Simulation With Concurrent Inputs in a Dynamic Network
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2-15
2-15
2-16
2-18
Training Styles . . . . . . . . . . . . . . . . . . . 2-20
Incremental Training (of Adaptive and other Networks) . . . 2-20
Batch Training . . . . . . . . . . . . . . . . . . . 2-22
Summary . . . . . . . . . . . . . . . . . . . . . 2-26
Figures and Equations . . . . . . . . . . . . . . . . 2-28
2
Neuron Model and Network Architectures
Basic Chapters
The Neural Network Toolbox is written so that if you read Chapter 2, 3 and 4
you can proceed to a later chapter, read it and use its functions without
difficulty. To make this possible, Chapter 2 presents the fundamentals of the
neuron model, the architectures of neural networks. It also will discuss
notation used in the architectures. All of this is basic material. It is to your
advantage to understand this Chapter 2 material thoroughly.
The neuron model and the architecture of a neural network describe how a
network transforms its input into an output. This transformation can be
viewed as a computation. The model and the architecture each place
limitations on what a particular neural network can compute. The way a
network computes its output must be understood before training methods for
the network can be explained.
Notation
Mathematical Notation
The basic notation used here for equations and figures is given below.
• Scalars-small italic letters.....a,b,c
• Vectors - small bold non-italic letters.....a,b,c
• Matrices - capital BOLD non-italic letters.....A,B,C
• Vector means a column of numbers.
Mathematical and Code Equivalents
The transition from mathematical to code notation or vice versa can be made
with the aid of a few rules. They are listed here for future reference.
2-2
Basic Chapters
To change from Mathematical notation to MATLAB notation the user needs to:
• Change superscripts to cell array indices
1
For example, p → p { 1 }
• Change subscripts to parentheses indices
1
For example, p 2 → p ( 2 ) , and p 2 → p { 1 } ( 2 )
• Change parentheses indices to a second cell array index
1
For example, p ( k – 1 ) → p { 1, k – 1 }
• Change mathematics operators to MATLAB operators and toolbox functions
For example, ab → a*b
See Appendix B for additional information on notation in this Toolbox.
2-3
2
Neuron Model and Network Architectures
Neuron Model
Simple Neuron
A neuron with a single scalar input and no bias is shown on the left below.
Input
p
- Title
- bias
Neuron
without
w
AA
AA
n
a-=Exp
f (wp- )
f
a
- Title
Neuron
with- bias
Input
p
w
AA
AA
AAAA
n
f
a
b
1
- b)
a =- fExp
(wp +
The scalar input p is transmitted through a connection that multiplies its
strength by the scalar weight w, to form the product wp, again a scalar. Here
the weighted input wp is the only argument of the transfer function f, which
produces the scalar output a. The neuron on the right has a scalar bias, b. You
may view the bias as simply being added to the product wp as shown by the
summing junction or as shifting the function f to the left by an amount b. The
bias is much like a weight, except that it has a constant input of 1. The transfer
function net input n, again a scalar, is the sum of the weighted input wp and
the bias b. This sum is the argument of the transfer function f. (Chapter 6
discusses a different way to form the net input n.) Here f is a transfer function,
typically a step function or a sigmoid function, that takes the argument n and
produces the output a. Examples of various transfer functions are given in the
next section. Note that w and b are both adjustable scalar parameters of the
neuron. The central idea of neural networks is that such parameters can be
adjusted so that the network exhibits some desired or interesting behavior.
Thus, we can train the network to do a particular job by adjusting the weight
or bias parameters, or perhaps the network itself will adjust these parameters
to achieve some desired end.
All of the neurons in this toolbox have provision for a bias, and a bias is used
in many of our examples and will be assumed in most of this toolbox. However,
you may omit a bias in a neuron if you wish.
2-4
Neuron Model
As noted above, the bias b is an adjustable (scalar) parameter of the neuron. It
is not an input. However, the constant 1 that drives the bias is an input and
must be treated as such when considering the linear dependence of input
vectors in Chapter 4.
Transfer Functions
Many transfer functions have been included in this toolbox. A complete list of
them can be found in “Transfer Function Graphs” in Chapter 13. Three of the
most commonly used functions are shown below.
a
+1
0
-1
n
AA
a = hardlim(n)
Hard Limit Transfer Function
The hard limit transfer function shown above limits the output of the neuron
to either 0, if the net input argument n is less than 0, or 1, if n is greater than
or equal to 0. We will use this function in Chapter 3 “Perceptrons” to create
neurons that make classification decisions.
The Toolbox has a function, hardlim, to realize the mathematical hard limit
transfer function shown above. Your might try the code shown below.
n = -5:0.1:5;
plot(n,hardlim(n),'c+:');
It produces a plot of the function hardlim over the range -5 to +5.
All of the mathematical transfer functions in the toolbox can be realized with
a function having the same name.
2-5
2
Neuron Model and Network Architectures
The linear transfer function is shown below.
a
AA
AA
+1
n
0
-1
a = purelin(n)
Linear Transfer Function
Neurons of this type are used as linear approximators in “Adaptive Linear
Filters” in Chapter 4.
The sigmoid transfer function shown below takes the input, which may have
any value between plus and minus infinity, and squashes the output into the
range 0 to 1.
a
+1
n
0
-1
AA
AA
a = logsig(n)
Log-Sigmoid Transfer Function
This transfer function is commonly used in backpropagation networks, in part
because it is differentiable.
The symbol in the square to the right of each transfer function graph shown
above represents the associated transfer function. These icons will replace the
general f in the boxes of network diagrams to show the particular transfer
function that is being used.
For a complete listing of transfer functions and their icons, see the “Transfer
Function Graphs” in Chapter 13. You can also specify your own transfer
functions. You are not limited to the transfer functions listed in Chapter 13.
You can experiment with a simple neuron and various transfer functions by
running the demonstration program nnd2n1.
2-6
Neuron Model
Neuron With Vector Input
A neuron with a single R-element input vector is shown below. Here the
individual element inputs
p 1 , p 2 ,... p R
are multiplied by weights
w 1, 1 , w 1, 2 ,...w 1, R
and the weighted values are fed to the summing junction. Their sum is simply
Wp, the dot product of the (single row) matrix W and the vector p.
Input Neuron w Vector Input
AA A
A
p1
p2
p3
w1,1
pR
w1, R
n
f
Where...
a
R = # Elements
in input vector
b
1
a = f(Wp +b)
The neuron has a bias b, which is summed with the weighted inputs to form
the net input n. This sum, n, is the argument of the transfer function f.
n = w 1, 1 p 1 + w 1, 2 p 2 + ... + w 1, R p R + b
This expression can, of course, be written in MATLAB code as:
n = W*p + b
However, the user will seldom be writing code at this low level, for such code is
already built into functions to define and simulate entire networks.
The figure of a single neuron shown above contains a lot of detail. When we
consider networks with many neurons and perhaps layers of many neurons,
there is so much detail that the main thoughts tend to be lost. Thus, the
authors have devised an abbreviated notation for an individual neuron. This
2-7
2
Neuron Model and Network Architectures
notation, which will be used later in circuits of multiple neurons, is illustrated
in the diagram shown below.
Input
AA
AA
AA
AA
p
Rx1
W
1xR
1
R
b
1x1
Neuron
AA
AA
AA
AA
a
n
1x1
1x1
f
Where...
R = # of elements
in input vector
1
a = f(Wp +b)
Here the input vector p is represented by the solid dark vertical bar at the left.
The dimensions of p are shown below the symbol p in the figure as Rx1. (Note
that we will use a capital letter, such as R in the previous sentence, when
referring to the size of a vector.) Thus, p is a vector of R input elements. These
inputs post multiply the single row, R column matrix W. As before, a constant
1 enters the neuron as an input and is multiplied by a scalar bias b. The net
input to the transfer function f is n, the sum of the bias b and the product Wp.
This sum is passed to the transfer function f to get the neuron’s output a, which
in this case is a scalar. Note that if we had more than one neuron, the network
output would be a vector.
A layer of a network is defined in the figure shown above. A layer includes the
combination of the weights, the multiplication and summing operation (here
realized as a vector product Wp), the bias b, and the transfer function f. The
array of inputs, vector p, will not be included in or called a layer.
Each time this abbreviated network notation is used, the size of the matrices
will be shown just below their matrix variable names. We hope that this
notation will allow you to understand the architectures and follow the matrix
mathematics associated with them.
2-8
Neuron Model
As discussed previously, when a specific transfer function is to be used in a
figure, the symbol for that transfer function will replace the f shown above.
Here are some examples.
A AA AA
AA AA
AA
AA AA
hardlim
purelin
logsig
You can experiment with a 2 element neuron by running the demonstration
program nnd2n2.
2-9
2
Neuron Model and Network Architectures
Network Architectures
Two or more of the neurons shown above may be combined in a layer, and a
particular network might contain one or more such layers. First consider a
single layer of neurons.
A Layer of Neurons
A one layer network with R input elements and S neurons is shown below.
Input
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AAAA
Layer of Neurons
n1
w1, 1
p1
p2
p3
pR
f
a1
b1
1
n2
f
a2
b2
1
nS
wS, R
f
aS
Where...
R = # of elements
in input vector
S = # Neurons
in Layer
bS
1
a= f (Wp + b)
In this network, each element of the input vector p is connected to each neuron
input through the weight matrix W. The ith neuron has a summer that gathers
its weighted inputs and bias to form its own scalar output n(i). The various n(i)
taken together form an S-element net input vector n. Finally, the neuron layer
outputs form a column vector a. We show the expression for a at the bottom of
the figure.
Note that it is common for the number of inputs to a layer to be different from
the number of neurons (i.e. R ≠ S). A layer is not constrained to have the
number of its inputs equal to the number of its neurons.
You can create a single (composite) layer of neurons having different transfer
functions simply by putting two of the networks shown above in parallel. Both
2-10
Network Architectures
networks would have the same inputs, and each network would create some of
the outputs.
The input vector elements enter the network through the weight matrix W.
w 1, 1 w 1, 2 … w 1, R
W =
w 2, 1 w 2, 2 … w 2, R
w S, 1 w S, 2 … w S, R
Note that the row indices on the elements of matrix W indicate the destination
neuron of the weight and the column indices indicate which source is the input
for that weight. Thus, the indices in w 1, 2 say that the strength of the signal
from the second input element to the first (and only) neuron is w 1, 2 .
The S neuron R input one layer network also can be drawn in abbreviated
notation.
Layer of Neurons
Input
p
AA
AA
AA
AA
AA AA
Rx1
a
W
n
SxR
Sx1
1
R
Sx1
R = # of elements
in input vector
f
b
Sx1
Where...
S = # Neurons
in Layer
S
a= f (Wp + b)
Here p is an R length input vector, W is an SxR matrix, and a and b are S
length vectors. As defined previously, the neuron layer includes the weight
matrix, the multiplication operations, the bias vector b, the summer, and the
transfer function boxes.
Inputs and Layers
We are about to discuss networks having multiple layers so we will need to
extend our notation to talk about such networks. Specifically, we need to make
a distinction between weight matrices that are connected to inputs and weight
2-11
2
Neuron Model and Network Architectures
matrices that are connected between layers. We also need to identify the source
and destination for the weight matrices.
We will call weight matrices connected to inputs, input weights, and we will call
weight matrices coming from layer outputs layer weights. Further, we will use
superscripts to identify the source (second index) and the destination (first
index) for the various weights and other elements of the network. To illustrate,
we have re-drawn the one layer multiple input network shown above in
abbreviated form below.
Input
p
R x1
AA
AA
AA
AA
IW1,1
S1xR
1
R
AA
AA
AA
AA
Layer 1
b1
S1x1
n1
S1 x1
Where...
a1
S1x 1
f1
S1
R = # of elements
in input
S1
= # Neurons in
layer 1
a1 = f1(IW1,1p +b1)
As you can see, we have labeled the weight matrix connected to the input vector
p as an Input Weight matrix (IW1,1) having a source 1 (second index) and a
destination 1 (first index). Also, elements of layer one, such as its bias, net
input and output have a superscript 1 to say that they are associated with the
first layer.
In the next section we will use Layer Weight (LW) matrices as well as Input
Weight (IW) matrices.
You might recall from the notation section at the beginning of this chapter that
conversion of the layer weight matrix from math to code for a particular
network called net is:
IW
1, 1
→ net.IW { 1, 1 }
Thus, we could write the code to obtain the net input to the transfer function as:
n{1} = net.IW{1,1}*p + net.b{1};
2-12
Network Architectures
Multiple Layers of Neurons
A network can have several layers. Each layer has a weight matrix W, a bias
vector b, and an output vector a. To distinguish between the weight matrices,
output vectors, etc., for each of these layers in our figures, we will append the
number of the layer as a superscript to the variable of interest. You can see the
use of this layer notation in the three layer network shown below, and in the
equations below the figure.
iw1,1
1, 1
p
AA
AA
A
1
n1
1
b1
1
1
p
2
p
n1
2
b1
3
2
p
1
1
R
AA
AA
A
First Layer
Input
iw1,1
S, R
n
1 1
b1 1
1
S
S
a1
f1
a1 = f1 (IW1,1p +b1)
n2
lw2,1
1
f1
f1
AA
AA
AA
AA
AA
AA
AAAA
AAAA
Second Layer
1
1,1
a2
1
f2
lw3,2
2
n2
2
f2
a2
2
a
S
lw2,1
2
n
a
2 2
S
1
S ,S
b2
2
S
1
a2 = f2 (LW2,1a1 +b2)
f3
a3
2
b3
2
1
2 2
f2
n3
2
2
1
1
f3
1
1
b2
1 1
a3
1
b3
1
a1
n3
1,1
b2
1
AA
AA
AA
AA
AA
AA
AAAA
AAAA
Third Layer
S
n3
lw3,2
S
3
a3
3
f3
3
S
2
b3
S ,S
3
S
1
a3 =f3 (LW3,2 a2 + b3)
a3 =f3 (LW3,2 f2 (LW2,1f1 (IW1,1p +b1)+ b2)+ b3)
The network shown above has R1 inputs, S1 neurons in the first layer, S2
neurons in the second layer, etc. It is common for different layers to have
different numbers of neurons. A constant input 1 is fed to the biases for each
neuron.
Note that the outputs of each intermediate layer are the inputs to the following
layer. Thus layer 2 can be analyzed as a one layer network with S1 inputs, S2
neurons, and an S1xS2 weight matrix W2. The input to layer 2 is a1, the output
is a2. Now that we have identified all the vectors and matrices of layer 2 we can
treat it as a single layer network on its own. This approach can be taken with
any layer of the network.
2-13
2
Neuron Model and Network Architectures
The layers of a multilayer network play different roles. A layer that produces
the network output is called an output layer. All other layers are called hidden
layers. The three layer network shown above has one output layer (layer 3) and
two hidden layers (layer 1 and layer 2). Some authors refer to the inputs as a
fourth layer. We will not use that designation.
The same three layer network discussed previously also can be drawn using
our abbreviated notation.
Input
AA
AA
AA
p
Rx1
IW1,1
S1xR
1
R
b
1
S1x1
AA
AA
AA
AA
AAAA
First Layer
Second Layer
a1
n1
S1x1
S1x1
S2xS1
f1
1
S1
a1 = f1 (IW1,1p +b1)
LW2,1
b
2
S2x1
A
AAA
AA
A
AA
AAA
AAAA AA
Third Layer
a3 = y
a2
n2
S2x1
S2x1
LW3,2
S 3x S 2
f2
n3
S3 x1
b
1
S2
3
S3x1
a2 = f2 (LW2,1 a1 +b2)
a3 =f3 (LW3,2 f2 (LW2,1f1 (IW1,1p +b1)+ b2)+ b3
S3 x1
f3
S3
a3 =f3 (LW3,2a2 +b3)
=
y
Multiple layer networks are quite powerful. For instance, a network of two
layers, where the first layer is sigmoid and the second layer is linear, can be
trained to approximate any function (with a finite number of discontinuities)
arbitrarily well. This kind of two-layer network is used extensively in Chapter
5, “Backpropagation.”
Note that we have labeled the output of the a3 layer as y. We will use this
notation to specify the output of such networks.
2-14
Data Structures
Data Structures
This section will discuss how the format of input data structures effects the
simulation of networks. We will begin with static networks and then move to
dynamic networks.
We will be concerned about two basic types of input vectors: those that occur
concurrently (at the same time, or in no particular time sequence) and those
that occur sequentially in time. For sequential vectors, the order in which the
vectors appear is important. For concurrent vectors, the order is not important,
and if we had a number of networks running in parallel we could present one
input vector to each of the networks.
Simulation With Concurrent Inputs in a Static
Network
The simplest situation for simulating a network occurs when the network to be
simulated is static (has no feedback or delays). In this case we do not have to
be concerned about whether or not the input vectors occur in a particular time
sequence, so we can treat the inputs as concurrent. In addition, to make the
problem even simpler, we will begin by assuming that the network has only one
input vector. We will use the following network as an example.
Inputs
Linear Neuron
AA A
A
p1
w1,1
p2
w1,2
n
a
b
1
a = purelin (Wp + b)
To set up this feedforward network we can use the following command.
net = newlin([-1 1;-1 1],1);
For simplicity we will assign the weight matrix and bias to be
W = 1 2 ,b = 0 .
2-15
2
Neuron Model and Network Architectures
The commands for these assignments are
net.IW{1,1} = [1 2];
net.b{1} = 0;
Suppose that the network simulation data set consists of Q = 4 concurrent
vectors:
p1 = 1 , p 2 = 2 , p 3 = 2 , p 4 = 3
2
1
3
1
Concurrent vectors are presented to the network as a single matrix:
P = [1 2 2 3; 2 1 3 1];
We can now simulate the network:
A = sim(net,P)
A =
5
4
8
5
A single matrix of concurrent vectors is presented to the network and the
network produces a single matrix of concurrent vectors as output. The result
would be the same if there were four networks operating in parallel and each
network received one of the input vectors and produced one of the outputs. The
ordering of the input vectors is not important as they do not interact with each
other.
Simulation With Sequential Inputs in a Dynamic
Network
When a network contains delays, the input to the network would normally be
a sequence of input vectors which occur in a certain time order. To illustrate
this case we will use a simple network which contains one delay.
2-16
Data Structures
Inputs
p(t)
Linear Neuron
AA
AA
AA
AAAAAA
w1,1
n(t)
D
a(t)
w1,2
a(t) = w1,1 p(t) + w1,2 p(t - 1)
The following commands will create this network:
net = newlin([-1 1],1,[0 1]);
net.biasConnect = 0;
Assign the weight matrix to be
W = 1 2 .
The command is
net.IW{1,1} = [1 2];
Suppose that the input sequence is
p(1) = 1 , p(2) = 2 , p(3) = 3 , p(4) = 4
Sequential inputs are presented to the network as elements of a cell array:
P = {1 2 3 4};
We can now simulate the network:
A = sim(net,P)
A =
[1]
[4]
[7]
[10]
We input a cell array containing a sequence of inputs, and the network
produced a cell array containing a sequence of outputs. Note that the order of
the inputs is important when they are presented as a sequence. In this case the
current output is obtained by multiplying the current input by 1 and the
2-17
2
Neuron Model and Network Architectures
preceding input by 2 and summing the result. If we were to change the order of
the inputs it would change the numbers we would obtain in the output.
Simulation With Concurrent Inputs in a Dynamic
Network
If we were to apply the same inputs from the previous example as a set of
concurrent inputs instead of a sequence of inputs we would obtain a completely
different response. (Although it is not clear why we would want to do this with
a dynamic network.) It would be as if each input were applied concurrently to
a separate parallel network. For the previous example, if we use a concurrent
set of inputs we have
p1 = 1 , p 2 = 2 , p 3 = 3 , p 4 = 4 ,
which can be created with the following code:
P = [1 2 3 4];
When we simulate with concurrent inputs we obtain
A = sim(net,P)
A =
1
2
3
4
The result is the same as if we had concurrently applied each one of the inputs
to a separate network and computed one output. Note that since we did not
assign any initial conditions to the network delays they were assumed to be
zero. For this case the output will simply be 1 times the input, since the weight
which multiplies the current input is 1.
In certain special cases we might want to simulate the network response to
several different sequences at the same time. In this case we would want to
present the network with a concurrent set of sequences. For example, let’s say
we wanted to present the following two sequences to the network:
p1 ( 1 ) = 1 , p1 ( 2 ) = 2 , p1 ( 3 ) = 3 , p1 ( 4 ) = 4 ,
p2 ( 1 ) = 4 , p2 ( 2 ) = 3 , p2 ( 3 ) = 2 , p2 ( 4 ) = 1 .
The input P should be a cell array, where each element of the array contains
the two elements of the two sequences which occur at the same time:
P = {[1 4] [2 3] [3 2] [4 1]};
2-18
Data Structures
We can now simulate the network:
A = sim(net,P);
The resulting network output would be
A = {[ 1 4] [4 11] [7 8] [10 5]}
As you can see, the first column of each matrix makes up the output sequence
produced by the first input sequence, which was the one we used in an earlier
example. The second column of each matrix makes up the output sequence
produced by the second input sequence. There is no interaction between the
two concurrent sequences. It is as if they were each applied to separate
networks running in parallel.
The following diagram shows the general format for the input P to the sim
function when we have Q concurrent sequences of TS time steps. It covers all
cases where there is a single input vector. Each element of the cell array is a
matrix of concurrent vectors which correspond to the same point in time for
each sequence. If there are multiple input vectors there will be multiple rows
of matrices in the cell array.
Qth Sequence
{ [ p 1 ( 1 ), p 2 ( 1 ), …, p Q ( 1 ) ], [ p 1 ( 2 ), p 2 ( 2˙), …, p Q ( 2 ) ], …, [ p 1 ( TS ), p 2 ( TS ), …, p Q ( TS ) ] }
First Sequence
In this section we have applied sequential and concurrent inputs to dynamic
networks. In the previous section we applied concurrent inputs to static
networks. It is also possible to apply sequential inputs to static networks. It
will not change the simulated response of the network, but it can affect the way
in which the network is trained. This will become clear in the next section.
2-19
2
Neuron Model and Network Architectures
Training Styles
In this section we will describe two different styles of training. In incremental
training the weights and biases of the network are updated each time an input
is presented to the network. In batch training the weights and biases are only
updated after all of the inputs have been presented.
Incremental Training (of Adaptive and Other
Networks)
Incremental training can be applied to both static and dynamic networks,
although it is more commonly used with dynamic networks, such as adaptive
filters. In this section we will demonstrate how incremental training can be
performed on both static and dynamic networks.
Incremental Training with Static Networks
Consider again the static network we used for our first example. We want to
train it incrementally, so that the weights and biases will be updated after each
input is presented. In this case we use the function adapt, and we present the
inputs and targets as sequences.
Suppose we want to train the network to create the linear function
t = 2 p1 + p2 .
Then for the previous inputs we used,
p1 = 1 , p 2 = 2 , p 3 = 2 , p 4 = 3 ,
2
1
3
1
the targets would be
t1 = 4 , t 2 = 5 , t 3 = 7 , t 4 = 7 .
We will first set up the network with zero initial weights and biases. We will
also set the learning rate to zero initially, in order to show the effect of the
incremental training.
net = newlin([-1 1;-1 1],1,0,0);
net.IW{1,1} = [0 0];
net.b{1} = 0;
2-20
Training Styles
For incremental training we want to present the inputs and targets as
sequences:
P = {[1;2] [2;1] [2;3] [3;1]};
T = {4 5 7 7};
Recall from the earlier discussion that for a static network the simulation of the
network will produce the same outputs whether the inputs are presented as a
matrix of concurrent vectors or as a cell array of sequential vectors. This is not
true when training the network, however. When using the adapt function, if
the inputs are presented as a cell array of sequential vectors, then the weights
will be updated as each input is presented (incremental mode). As we will see
in the next section, if the inputs are presented as a matrix of concurrent
vectors, then the weights will be updated only after all inputs have been
presented (batch mode).
We are now ready to train the network incrementally.
[net,a,e,pf] = adapt(net,P,T);
The network outputs will remain zero, since the learning rate is zero, and the
weights are not updated. The errors will be equal to the targets:
a = [0]
e = [4]
[0]
[5]
[0]
[7]
[0]
[7]
If we now set the learning rate to 0.1 we can see how the network is adjusted
as each input is presented:
net.inputWeights{1,1}.learnParam.lr=0.1;
net.biases{1,1}.learnParam.lr=0.1;
[net,a,e,pf] = adapt(net,P,T);
a = [0]
[2]
[6.0]
[5.8]
e = [4]
[3]
[1.0]
[1.2]
The first output is the same as it was with zero learning rate, since no update
is made until the first input is presented. The second output is different, since
the weights have been updated. The weights continue to be modified as each
error is computed. If the network is capable and the learning rate is set
correctly, the error will eventually be driven to zero.
2-21
2
Neuron Model and Network Architectures
Incremental Training With Dynamic Networks
We can also train dynamic networks incrementally. In fact, this would be the
most common situation. Let’s take the linear network with one delay at the
input that we used in a previous example. We will initialize the weights to zero
and set the learning rate to 0.1.
net = newlin([-1 1],1,[0 1],0.1);
net.IW{1,1} = [0 0];
net.biasConnect = 0;
To train this network incrementally we will present the inputs and targets as
elements of cell arrays.
Pi = {1};
P = {2 3 4};
T = {3 5 7};
Here we are attempting to train the network to sum the current and previous
inputs to create the current output. This is the same input sequence we used
in the previous example of using sim, except that we are assigning the first
term in the sequence as the initial condition for the delay. We are now ready to
sequentially train the network using adapt.
[net,a,e,pf] = adapt(net,P,T,Pi);
a = [0] [2.4] [ 7.98]
e = [3] [2.6] [-1.98]
The first output is zero, since the weights have not yet been updated. The
weights change at each subsequent time step.
Batch Training
Batch training, in which weights and biases are only updated after all of the
inputs and targets have been presented, can be applied to both static and
dynamic networks. We will discuss both types of networks in this section.
Batch Training With Static Networks
Batch training can be done using either adapt or train, although train is
generally the best option, since it typically has access to more efficient training
algorithms. Incremental training can only be done with adapt; train can only
perform batch training.
2-22
Training Styles
Let’s begin with the static network we used in previous examples. The learning
rate will be set to 0.1.
net = newlin([-1 1;-1 1],1,0,0.1);
net.IW{1,1} = [0 0];
net.b{1} = 0;
For batch training of a static network with adapt, the input vectors must be
placed in one matrix of concurrent vectors.
P = [1 2 2 3; 2 1 3 1];
T = [4 5 7 7];
When we call adapt it will invoke adaptwb, which is the default adaptation
function for the linear network, and learnwh is the default learning function
for the weights and biases. Therefore, Widrow-Hoff learning will be used.
[net,a,e,pf] = adapt(net,P,T);
a = 0 0 0 0
e = 4 5 7 7
Note that the outputs of the network are all zero, because the weights are not
updated until all of the training set has been presented. If we display the
weights we find:
»net.IW{1,1}
ans = 4.9000
»net.b{1}
ans =
2.3000
4.1000
This is different that the result we had after one pass of adapt with
incremental updating.
Now let’s perform the same batch training using train. Since the Widrow-Hoff
rule can be used in incremental or batch mode, it can be invoked by adapt or
train. There are several algorithms which can only be used in batch mode (e.g.,
Levenberg-Marquardt), and so these algorithms can only be invoked by train.
The network will be set up in the same way.
net = newlin([-1 1;-1 1],1,0,0.1);
net.IW{1,1} = [0 0];
net.b{1} = 0;
2-23
2
Neuron Model and Network Architectures
For this case the input vectors can either be placed in a matrix of concurrent
vectors or in a cell array of sequential vectors. Within train any cell array of
sequential vectors would be converted to a matrix of concurrent vectors. This
is because the network is static, and because train always operates in the
batch mode. Concurrent mode operation is generally used whenever possible,
because it has a more efficient MATLAB implementation.
P = [1 2 2 3; 2 1 3 1];
T = [4 5 7 7];
Now we are ready to train the network. We will train it for only one epoch, since
we used only one pass of adapt. The default training function for the linear
network is trainwb, and the default learning function for the weights and
biases is learnwh, so we should get the same results that we obtained using
adapt in the previous example, where the default adaptation function was
adaptwb.
net.inputWeights{1,1}.learnParam.lr = 0.1;
net.biases{1}.learnParam.lr = 0.1;
net.trainParam.epochs = 1;
net = train(net,P,T);
If we display the weights after one epoch of training we find:
»net.IW{1,1}
ans = 4.9000
»net.b{1}
ans =
2.3000
4.1000
This is the same result we had with the batch mode training in adapt. With
static networks the adapt function can implement incremental or batch
training depending on the format of the input data. If the data is presented as
a matrix of concurrent vectors batch training will occur. If the data is presented
as a sequence, incremental training will occur. This is not true for train, which
always performs batch training, regardless of the format of the input.
Batch Training With Dynamic Networks
Training static networks is relatively straightforward. If we use train the
network will be trained in the batch mode and the inputs will be converted to
concurrent vectors (columns of a matrix), even if they are originally passed as
a sequence (elements of a cell array). If we use adapt, the format of the input
2-24
Training Styles
will determine the method of training. If the inputs are passed as a sequence,
then the network will be trained in incremental mode. If the inputs are passed
as concurrent vectors, then batch mode training will be used.
With dynamic networks batch mode training would typically be done with
train only, especially if only one training sequence exists. To illustrate this,
let’s consider again the linear network with a delay. We will use a learning rate
of 0.02 for the training. (When using a gradient descent algorithm, we will
typically use a smaller learning rate for batch mode training than incremental
training, because all of the individual gradients are summed together before
determining the step change to the weights.)
net = newlin([-1 1],1,[0 1],0.02);
net.IW{1,1}=[0 0];
net.biasConnect=0;
net.trainParam.epochs = 1;
Pi = {1};
P = {2 3 4};
T = {3 5 6};
We want to train the network with the same sequence we used for the
incremental training earlier, but this time we want to update the weights only
after all of the inputs have been applied (batch mode). The network will be
simulated in sequential mode because the input is a sequence, but the weights
will be updated in batch mode.
net=train(net,P,T,Pi);
The weights after one epoch of training are
»net.IW{1,1}
ans = 0.9000
0.6200
These are different weights than we would obtain using incremental training,
where the weights would have been updated three times during one pass
through the training set. For batch training the weights are only updated once
in each epoch.
2-25
2
Neuron Model and Network Architectures
Summary
The inputs to a neuron include its bias and the sum of its weighted inputs
(using the inner product). The output of a neuron depends on the neuron’s
inputs and on its transfer function. There are many useful transfer functions.
A single neuron cannot do very much. However, several neurons can be
combined into a layer or multiple layers that have great power. Hopefully this
toolbox makes it easy to create and understand such large networks.
The architecture of a network consists of a description of how many layers a
network has, the number of neurons in each layer, each layer’s transfer
function, and how the layers are connected to each other. The best architecture
to use depends on the type of problem to be represented by the network.
A network effects a computation by mapping input values to output values. The
particular mapping problem to be performed fixes the number of inputs as well
as the number of outputs for the network.
Aside from the number of neurons in a network’s output layer, the number of
neurons in each layer is up to the designer. Except for purely linear networks,
the more neurons in a hidden layer the more powerful the network.
If a linear mapping needs to be represented linear neurons should be used.
However, linear networks cannot perform any nonlinear computation. Use of a
nonlinear transfer function makes a network capable of storing nonlinear
relationships between input and output.
A very simple problem may be represented by a single layer of neurons.
However, single layer networks cannot solve certain problems. Multiple
feed-forward layers give a network greater freedom. For example, any
reasonable function can be represented with a two layer network: a sigmoid
layer feeding a linear output layer.
Networks with biases can represent relationships between inputs and outputs
more easily than networks without biases. (For example, a neuron without a
bias will always have a net input to the transfer function of zero when all of its
inputs are zero. However, a neuron with a bias can learn to have any net
transfer function input under the same conditions by learning an appropriate
value for the bias.)
Feed-forward networks cannot perform temporal computation. More complex
networks with internal feedback paths are required for temporal behavior.
2-26
Summary
If several input vectors are to be presented to a network, they may be presented
sequentially or concurrently. Batching of concurrent inputs is computationally
more efficient and may be what is desired in any case. The matrix notation
used in MATLAB makes batching simple.
2-27
2
Neuron Model and Network Architectures
Figures and Equations
Simple Neuron
- Title
- bias
Neuron
without
Input
AA
AA
w
p
n
f
a
a-=Exp
f (wp- )
Hard Limit Transfer Function
a
+1
0
n
-1
AA
a = hardlim(n)
Hard Limit Transfer Function
Purelin Transfer Function
a
+1
n
0
-1
a = purelin(n)
AA
AA
Linear Transfer Function
2-28
- Title
Neuron
with- bias
Input
p
w
AA
AA
AAAA
n
f
b
1
- b)
a =- fExp
(wp +
a
Summary
Log Sigmoid Transfer Function
a
+1
n
0
-1
AA
AA
a = logsig(n)
Log-Sigmoid Transfer Function
Neuron With Vector Input
Input Neuron w Vector Input
AA A
A
p1
p2
p3
w1,1
pR
w1, R
n
f
Where...
a
R = # Elements
in input vector
b
1
a = f(Wp +b)
Net Input
n = w 1, 1 p 1 + w 1, 2 p 2 + ... + w 1, R p R + b
2-29
2
Neuron Model and Network Architectures
Single Neuron Using Abbreviated Notation
AA
AA
AA
AA
AA AA
AA
A AA
Input
Neuron
p
Rx1
a
W
f
1x1
b
R
1x1
n
1xR
1
AA
AA
AA
AA
AA
AA
AA
Where...
R = # of elements
in input vector
1
1x1
a = f(Wp +b)
Icons for Transfer Functions
hardlim
purelin
logsig
Layer of Neurons
Layer of Neurons
Input
p
AA
AA
AA
Rx1
n
SxR
1
R
Sx1
b
Sx1
a= f (Wp + b)
2-30
AA
AA
AA
a
W
Sx1
Where...
R = # of elements
in input vector
f
S
S = # Neurons
in Layer
Summary
Three Layers of Neurons
Input
AA
AA
AAAA
AA
AA
AA
AA
AA
AA
AAAA
Layer of Neurons
n1
w1, 1
p1
p2
p3
pR
f
a1
b1
1
n2
f
a2
b2
1
nS
wS, R
f
aS
Where...
R = # of elements
in input vector
S = # Neurons
in Layer
bS
1
a= f (Wp + b)
Weight Matrix
w 1, 1 w 1, 2 … w 1, R
W =
w 2, 1 w 2, 2 … w 2, R
w S, 1 w S, 2 … w S, R
2-31
2
Neuron Model and Network Architectures
Layer of Neurons, Abbreviated Notation
Layer of Neurons
Input
p
AA
AA
AA
Rx1
n
SxR
1
Sx1
b
Sx1
R
AA
AA
AA
a
W
Sx1
Where...
R = # of elements
in input vector
f
S = # Neurons
in Layer
S
a= f (Wp + b)
Layer of Neurons Showing Indices
Input
p
R x1
AA
AA
AA
AA
IW1,1
S1xR
1
R
AA
AA
AA
AA
Layer 1
b1
S1x1
n1
S1 x1
a1 = f1(IW1,1p +b1)
2-32
Where...
a1
S1x 1
f1
S1
R = # of elements
in input
S1
= # Neurons in
layer 1
Summary
Three Layers of Neurons
Input
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AAAA
First Layer
n1
iw1,1
1, 1
p
1
1
p
2
p
3
p
b1
1
R
f1
n1
S, R
1
S
1
a1 = f1 (IW1,1p +b1)
1
2
f2
a2
lw2,1
S
b3
2
2
b2
1
a2 = f2 (LW2,1a1 +b2)
n3
3
S
3
1
a3
2
2
1
lw3,2
2
S
f3
2
S
1
S ,S
n3
b3
a3
f3
1
1
a2 2
f2
1
1,1
2
n2 2
AA
A
AA
A
AA
A
AA
A
AA
A
AAA
n3
lw3,2
b2
S
b1 1
n2
1
a1 1
f1
a2
f2
1
2
1
S
1
Third Layer
b2
2
iw1,1
n2
1,1
a1
f1
2
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AAAA
lw2,1
1
1
n1
b1
1
a1
1
Second Layer
a3
3
S
f3
2
S ,S
b3
3
S
1
a3 =f3 (LW3,2 a2 + b3)
a3 =f3 (LW3,2 f2 (LW2,1f1 (IW1,1p +b1)+ b2)+ b3)
Three Layers, Abbreviated Notation
AAA
AA
AA
AAA
AA
AA
AAA AAAA
First Layer
Input
p
Rx1
a1
IW1,1
S1xR
n1
S1x1
1
R
S1x1
1
S1
a1 = f1 (IW1,1p +b1)
LW2,1
S2xS1
f1
b1
S1x1
AA
AA
AA
AA
AA
AA
AAAA AA
Second Layer
b2
S2x1
Third Layer
a3 = y
a2
n2
S2x1
S2x1
LW3,2
S 3x S 2
f2
n3
S3 x1
b3
1
S2
S3x1
a2 = f2 (LW2,1 a1 +b2)
a3 =f3 (LW3,2 f2 (LW2,1f1 (IW1,1p +b1)+ b2)+ b3
S3 x1
f3
S3
a3 =f3 (LW3,2a2 +b3)
=
y
2-33
2
Neuron Model and Network Architectures
Linear Neuron With Two Element Vector Input.
Inputs
Linear Neuron
AA A
A
p1
w1,1
p2
w1,2
n
a
b
1
a = purelin (Wp + b)
Dynamic Network With One Delay.
Inputs
p(t)
Linear Neuron
AA
AA
AA
AAAAAA
w1,1
n(t)
D
a(t)
w1,2
a(t) = w1,1 p(t) + w1,2 p(t - 1)
2-34
3
Perceptrons
Introduction . . . . . . . . . . . . . . . . . . . . 3-2
Important Perceptron Functions. . . . . . . . . . . . . 3-3
Neuron Model . . . . . . . . . . . . . . . . . . . 3-4
Perceptron Architecture
. . . . . . . . . . . . . . 3-6
Creating a Perceptron (NEWP) . . . . . . . . . . . 3-7
Simulation (SIM) . . . . . . . . . . . . . . . . . . . 3-8
Initialization (INIT) . . . . . . . . . . . . . . . . . 3-9
Learning Rules . . . . . . . . . . . . . . . . . . . 3-11
Perceptron Learning Rule (LEARNP)
. . . . . . . . 3-12
Adaptive Training (ADAPT) . . . . . . . . . . . . . 3-15
Limitations and Cautions . . . . . . . . . . . . . . 3-21
Outliers and the Normalized Perceptron Rule . . . . . . . 3-21
Summary . . . . . . . . . . . . . . . . . . . . . 3-23
Figures and Equations . . . . . . . . . . . . . . . . 3-23
New Functions . . . . . . . . . . . . . . . . . . . 3-26
3
Perceptrons
Introduction
This chapter has a number of objectives. First we want to introduce you to
learning rules, methods of deriving the next changes that might be made in a
network, and training, a procedure whereby a network is actually adjusted to
do a particular job. Along the way we will discuss a toolbox function to create a
simple perceptron network, and we will also cover functions to initialize and
simulate such networks. We will use the perceptron as a vehicle for tying these
concepts together.
Rosenblatt [Rose61] created many variations of the perceptron. One of the
simplest was a single layer network whose weights and biases could be trained
to produce a correct target vector when presented with the corresponding input
vector. The training technique used is called the perceptron learning rule. The
perceptron generated great interest due to its ability to generalize from its
training vectors and learn from initially randomly distributed connections.
Perceptrons are especially suited for simple problems in pattern classification.
They are fast and reliable networks for the problems they can solve. In
addition, an understanding of the operations of the perceptron provides a good
basis for understanding more complex networks.
In this chapter we will define what we mean by a learning rule, explain the
perceptron network and its learning rule, and tell you how to initialize and
simulate perceptron networks.
The discussion of perceptron in this chapter is necessarily brief. You may wish
to read a more thorough discussion such as that given in Chapter 4 “Perceptron
Learning Rule,” of [HDB1996]. This Chapter discusses the use of multiple
layers of perceptrons to solve more difficult problems beyond the capability of
one layer.
You also may want to refer to the original book on the perceptron, Rosenblatt,
F., Principles of Neurodynamics, Washington D.C.: Spartan Press, 1961.
[Rose61].
3-2
Introduction
Important Perceptron Functions
Entering help percept at the MATLAB command line displays all the
functions which are related to perceptrons.
Perceptron networks can be created with the function newp. These networks
can be initialized and simulated with the init and sim. The following material
describes how perceptrons work and introduces these functions.
3-3
3
Perceptrons
Neuron Model
A perceptron neuron, which uses the hard limit transfer function hardlim, is
shown below.
Input
Perceptron Neuron
AA
AA
AAAA
p1
p2
p3
w1,1
pR
w1, R
n
f
a
Where...
R = # Elements
in input vector
b
1
a = hardlim
- Exp -(Wp + b)
Each external input is weighted with an appropriate weight w1j, and the sum
of the weighted inputs is sent to the hard limit transfer function, which also
has an input of 1 transmitted to it through the bias. The hard limit transfer
function, which returns a 0 or a 1, is shown below.
a
+1
0
-1
n
A
a = hardlim(n)
Hard Limit Transfer Function
The perceptron neuron produces a 1 if the net input into the transfer function
is equal to or greater than 0, otherwise it produces a 0.
The hard limit transfer function gives a perceptron the ability to classify input
vectors by dividing the input space into two regions. Specifically, outputs will
be 0 if the net input n is less than 0, or 1 if the net input n is 0 or greater. The
input space of a 2-input hard limit neuron with the weights
w 1, 1 = – 1, w 1, 2 = 1 and a bias b = 1 , is shown below.
3-4
Neuron Model
L
p
W
2
Wp+b > 0
a=1
+1
Wp+b = 0
a=0
-b/w
1,1
p
1
-1
+1
Wp+b < 0
a=0
-b/w
-1
1,2
Where... w
= -1
w
= +1
1,1
1,2
and
b = +1
Two classification regions are formed by the decision boundary line L at
Wp + b = 0 . This line is perpendicular to the weight matrix W and shifted
according to the bias b. Input vectors above and to the left of the line L will
result in a net input greater than 0, and therefore cause the hard limit neuron
to output a 1. Input vectors below and to the right of the line L cause the neuron
to output 0. The dividing line can be oriented and moved anywhere to classify
the input space as desired by picking the weight and bias values.
Hard limit neurons without a bias will always have a classification line going
through the origin. Adding a bias allows the neuron to solve problems where
the two sets of input vectors are not located on different sides of the origin. The
bias allows the decision boundary to be shifted away from the origin as shown
in the plot above.
You may wish to run the demonstration program nnd4db. With it you can move
a decision boundary around, pick new inputs to classify, and see how the
repeated application of the learning rule yields a network that does classify the
input vectors properly.
3-5
3
Perceptrons
Perceptron Architecture
The perceptron network consists of a single layer of S perceptron neurons
connected to R inputs through a set of weights wi,j as shown below in two
forms. As before, the network indices i and j indicate that wi,j is the strength
of the connection from the jth input to the ith neuron.
Input #1
AAAA
AA
AAA
AAA
Input # 1
Perceptron Layer
iw1,1
1, 1
p
1
n11
a11
p
R x1
b1
1
p
2
1
iw1,1S1,R
n1S 1
n1
S1x1
a
1
1
2
b12
3
pR
n
1
AA
AA
AA
AA
AA AA
a1
IW1,1
S1xR
1
p2
Layer 1
R
f1
b1
S1x1
S1
a1 = hardlim(IW1,1p1 +b1)
a1S1
b1 1
S 1x 1
Where...
S
1
a1 = hardlim (IW1,1p1 + b1)
R = # of elements in Input
S1 = # of Neurons in layer 1
The perceptron learning rule that we will describe shortly is capable of training
only a single layer. Thus, here we will consider only one layer networks. This
restriction places limitations on the computation a perceptron can perform.
The types of problems that perceptrons are capable of solving are discussed
later in this chapter in the “Limitations and Cautions” section.
3-6
Creating a Perceptron (NEWP)
Creating a Perceptron (NEWP)
A perceptron can be created with the function newp.
net = newp(PR, S)
where the input arguments are:
PR is an Rx2 matrix of minimum and maximum values for R input elements.
S is the Number of neurons.
Commonly the hardlim function is used in perceptrons, so it is the default.
The code below creates a peceptron network with a single one-element input
vector and one neuron. The range for the single element of the single input
vector is [0 2].
net = newp([0 2],1);
We can see what network has been created by executing the following code:
inputweights = net.inputweights{1,1}
which yields:
inputweights =
delays:
initFcn:
learn:
learnFcn:
learnParam:
size:
userdata:
weightFcn:
0
'initzero'
1
'learnp'
[]
[1 1]
[1x1 struct]
'dotprod'
Note that the default learning function is learnp, which will be discussed later
in this chapter. The net input to the hardlim transfer function is dotprod,
which generates the product of the input vector and weight matrix and adds
the bias to compute the net input.
Also note that the default initialization function, initzero, is used to set the
initial values of the weights to zero.
Similarly,
biases = net.biases{1}
3-7
3
Perceptrons
gives
biases =
initFcn:
learn:
learnFcn:
learnParam:
size:
userdata:
'initzero'
1
'learnp'
[]
1
[1x1 struct].
We can see that the default initialization for the bias is also 0.
Simulation (SIM)
To show how sim works we will examine a simple problem.
Suppose we take a perceptron with a single two element input vector, like that
discussed in the decision boundary figure. We define the network with:
net = newp([-2 2;-2 +2],1);
As noted above, this will give us zero weights and biases, so if we want a
particular set other than zeros, we will have to create them. We can set the two
weights and the one bias to -1, 1 and 1 as they were in the decision boundary
figure with the following two lines of code.
net.IW{1,1}= [-1 1];
net.b{1} = [1];
To make sure that these parameters were set correctly, we will check them
with:
net.IW{1,1}
ans =
-1
1
net.b{1}
ans =
1
3-8
Creating a Perceptron (NEWP)
Now let us see if the network responds to two signals, one on each side of the
perceptron boundary.
p1 = [1;1];
a1 = sim(net,p1)
a1 =
1
and for
p2 = [1;-1]
a2 = sim(net,p2)
a2 =
0
Sure enough, the perceptron has classified the two inputs correctly.
Note that we could have presented the two inputs in a sequence and gotten the
outputs in a sequence as well.
p3 = {[1;1] [1;-1]};
a3 = sim(net,p3)
a3 =
[1]
[0]
Initialization (INIT)
You can use the function init to reset the network weights and biases to their
original values. Suppose, for instance that you start with the network:
net = newp([-2 2;-2 +2],1);
Now check its weights with
wts = net.IW{1,1}
which gives, as expected,
wts =
0
0
In the same way, you can verify that the bias is 0 with
bias = net.b{1}
3-9
3
Perceptrons
which gives
bias =
0.
Now set the weights to the values 3 and 4 and the bias to the value 5 with
net.IW{1,1} = [3,4];
net.b{1} = 5;
Recheck the weights and bias as shown above to verify that the change has
been made. Sure enough,
wts =
3
4
bias =
5.
Now use init to reset the weights and bias to their original values.
net = init(net);
We can check as shown above to verify that:
wts =
0
bias =
0.
0
We can change the as way that a perceptron is initialized with init. For
instance, suppose that we define the network input weights and bias initFcns
as rands and then apply init as shown below.
net.inputweights{1,1}.initFcn = 'rands';
net.biases{1}.initFcn = 'rands';
net = init(net);
Now check on the weights and bias.
wts =
0.2309
biases =
-0.1106
0.5839
We can see that the weights and bias have been given random numbers.
3-10
Learning Rules
Learning Rules
We will define a learning rule as a procedure for modifying the weights and
biases of a network. (This procedure may also be referred to as a training
algorithm.) The learning rule is applied to train the network to perform some
particular task. Learning rules in this toolbox fall into two broad categories:
supervised learning and unsupervised learning.
In supervised learning, the learning rule is provided with a set of examples (the
training set) of proper network behavior:
{p1, t 1} , {p2, t 2} , …, {pQ, tQ}
where p q is an input to the network, and t q is the corresponding correct
(target) output. As the inputs are applied to the network, the network outputs
are compared to the targets. The learning rule is then used to adjust the
weights and biases of the network in order to move the network outputs closer
to the targets. The perceptron learning rule falls in this supervised learning
category.
In unsupervised learning, the weights and biases are modified in response to
network inputs only. There are no target outputs available. Most of these
algorithms perform clustering operations. They categorize the input patterns
into a finite number of classes. This is especially useful in such applications as
vector quantization.
As noted, the perceptron discussed in this chapter is trained with supervised
learning. Hopefully, a network that produces the right output for a particular
input will be obtained.
3-11
3
Perceptrons
Perceptron Learning Rule (LEARNP)
Perceptrons are trained on examples of desired behavior. The desired behavior
can be summarized by a set of input, output pairs.
p 1 t 1 ,p 2 t 1 ,..., p Q t Q
where p is an input to the network and t is the corresponding correct (target)
output. The objective is to reduce the error, e which is the difference t – a
between the neuron response a, and the target vector t. The perceptron
learning rule learnp calculates desired changes to the perceptron’s weights
and biases given an input vector p, and the associated error e. The target
vector t must contain values of either 0 or 1, as perceptrons (with hardlim
transfer functions) can only output such values.
Each time learnp is executed, the perceptron will have a better chance of
producing the correct outputs. The perceptron rule has been proven to converge
on a solution in a finite number of iterations if a solution exists.
If a bias is not used, learnp works to find a solution by altering only the weight
vector w to point toward input vectors to be classified as 1, and away from
vectors to be classified as 0. This results in a decision boundary that is
perpendicular to w and which properly classifies the input vectors.
There are three conditions that can occur for a single neuron once an input
vector p is presented and the network’s response a is calculated:
CASE 1. If an input vector is presented and the output of the neuron is correct
(a = t, and e = t – a = 0), then the weight vector w is not altered.
CASE 2. If the neuron output is 0 and should have been 1 (a = 0 and t = 1, and
e = t – a = 1) the input vector p is added to the weight vector w. This makes
the weight vector point closer to the input vector, increasing the chance that
the input vector will be classified as a 1 in the future.
CASE 3. If the neuron output is 1 and should have been 0 (a = 0 and t = 1, and
e = t – a = –1) the input vector p is subtracted from the weight vector w. This
makes the weight vector point farther away from the input vector, increasing
the chance that the input vector will be classified as a 0 in the future.
The perceptron learning rule can be written more succinctly in terms of the
error e = t – a, and the change to be made to the weight vector ∆w:
3-12
Perceptron Learning Rule (LEARNP)
CASE 1. If e = 0, then make a change ∆w equal to 0.
CASE 2. If e = 1, then make a change ∆w equal to pT.
CASE 3. If e = –1, then make a change ∆w equal to –pT.
All three cases can then be written with a single expression:
∆w = ( t – a )p T = ep T
We can get the expression for changes in a neuron’s bias by noting that the bias
is simply a weight which always has an input of 1:
∆b = ( t – a ) ( 1 ) = e
For the case of a layer of neurons we have:
∆W = ( t – a ) ( p ) T = e ( p ) T and
∆b = ( t – a ) = E
The Perceptron Learning Rule can be summarized as follows:
W
b
new
new
= W
= b
old
old
+ ep
T
and
+e
where e = t – a .
Now let us try a simple example. We will start with a single neuron having a
input vector with just two elements.
net = newp([-2 2;-2 +2],1);
To simplify matters we will set the bias equal to 0 and the weights to 1 and -0.8.
net.biases{1}.value = [0];
w = [1 -0.8];
net.IW{1,1}.value = w;
The input target pair is given by:
p = [1; 2];
t = [1];
3-13
3
Perceptrons
We can compute the output and error with
a = sim(net,p)
a =
0
e = t-a
e =
1
and finally use the function learnp to find the change in the weights.
dw = learnp(w,p,[],[],[],[],e,[],[],[])
dw =
1
2.
The new weights, then, are obtained as
w = w + dw
w =
2.0000
1.2000
The process of finding new weights (and biases) can be repeated until there are
no errors. Note that the perceptron learning rule is guaranteed to converge in
a finite number of steps for all problems that can be solved by a perceptron.
These include all classification problems that are “linearly separable.” The
objects to be classified in such cases can be separated by a single line.
You might want to try demo nnd4pr. It allows you to pick new input vectors and
apply the learning rule to classify them.
3-14
Adaptive Training (ADAPT)
Adaptive Training (ADAPT)
If sim and learnp are used repeatedly to present inputs to a perceptron, and to
change the perceptron weights and biases according to the error, the
perceptron will eventually find weight and bias values which solve the
problem, given that the perceptron can solve it. Each traverse through all of the
training input and target vectors is called a pass.
The function adapt carries out such a loop of calculation. In each pass the
function adapt will proceed through the specified sequence of inputs,
calculating the output, error and network adjustment for each input vector in
the sequence as the inputs are presented.
Note that adapt does not guarantee that the resulting network does its job.
The new values of W and b must be checked by computing the network output
for each input vector to see if all targets are reached. If a network does not
perform successfully it can be trained further by again calling adapt with the
new weights and biases for more training passes, or the problem can be
analyzed to see if it is a suitable problem for the perceptron. Problems which
are not solvable by the perceptron network are discussed in the “Limitations
and Cautions” section.
To illustrate the adaptation procedure, we will work through a simple problem.
Consider a one neuron perceptron with a single vector input having two
elements.
Perceptron Neuron
Input
p
1
p
2
AAAA
AAAA
w
1,1
w
1, 2
n
f
a
b
1
a = hardlim
- Exp -(Wp + b)
This network, and the problem we are about to consider are simple enough that
you can follow through what is done with hand calculations if you wish. The
problem discussed below follows that found in [HDB1996].
3-15
3
Perceptrons
Let us suppose we have the following classification problem and would like to
solve it with our single vector input, 2 element perceptron network.

2 , t = 0   p = 1 , t = 1   p = –2 , t = 0   p = –1 , t = 1 
 p1 =
  2
  3
  4

1
2
3
4
2
–2
2
1

 
 
 

Use the initial weights and bias. We denote the variables at each step of this
calculation by using a number in parentheses after the variable. Thus, above,
we have the initial values, W(0) and b(0).
W(0) = 0 0
b(0) = 0
We start by calculating the perceptron’s output a for the first input vector p1,
using the initial weights and bias.
a = hardlim ( W ( 0 )p 1 + b ( 0 ) )


= hardlim  0 0 2 + 0 = hardlim ( 0 ) = 1


2
The output a does not equal the target value t1, so we use the perceptron rule
to find the incremental changes to the weights and biases based on the error.
e = t1 – a = 0 – 1 = –1
T
∆W = ep 1 = ( – 1 ) 2 2 = – 2 – 2
∆b = e = ( – 1 ) = – 1
You can calculate the new weights and bias using the Perceptron update rules
shown previously.
W
new
b
3-16
= W
new
old
= b
+ ep
old
T
= 0 0 + –2 –2 = –2 –2 = W ( 1 )
+ e = 0 + ( –1 ) = –1 = b ( 1 )
Adaptive Training (ADAPT)
Now present the next input vector, p2. The output is calculated below.
a = hardlim ( W ( 1 )p 2 + b ( 1 ) )


= hardlim  – 2 – 2 – 2 – 1 = hardlim ( 1 ) = 1


–2
On this occasion, the target is 1, so the error is zero. Thus there are no changes
in weights or bias so W ( 2 ) = W ( 1 ) = – 2 – 2 and p ( 2 ) = p ( 1 ) = – 1
We can continue in this fashion, presenting p3 next, calculating an output and
the error, and making changes in the weights and bias, etc. After making one
pass through all of the four inputs, you will get the values: W ( 4 ) = – 3 – 1
and b ( 4 ) = 0 . To determine if we have obtained a satisfactory solution, we
must make one pass through all input vectors to see if they all produce the
desired target values. This is not true for the 4th input, but the algorithm does
converge on the 6th presentation of an input. The final values are:
W ( 6 ) = – 2 – 3 and b ( 6 ) = 1 .
This concludes our hand calculation. Now, how can we do this using the adapt
function?
The following code defines a perceptron like that shown in the previous figure,
with initial weights and bias values of 0.
net = newp([-2 2;-2 +2],1);
Let us first consider the application of a single input. We will define the first
input vector and target as sequences (cell arrays in curly brackets).
p = {[2; 2]};
t = {0}
Now set passes to 1, so that adapt will go through the input vectors (only one
here) just one time.
net.adaptParam.passes = 1;
[net,a,e] = adapt(net,p,t);
3-17
3
Perceptrons
The output and error that are returned are:
a =
[1]
e =
[-1]
The new weights and bias are:
twts = net.IW{1,1}
twts =
-2
-2
tbiase = net.b{1}
tbiase =
-1
Thus, the initial weights and bias are 0, and after training on just the first
vector they have the values [-2 -2] and -1, just as we hand calculated.
We now apply the second input vector p 2 . The output is 1, as it will be until the
weights and bias are changed, but now the target is 1, the error will be 0 and
the change will be zero. We could proceed in this way, starting from the
previous result and applying a new input vector time after time. But we can do
this job automatically with adapt.
Now let’s apply adapt for one pass through the sequence of all four input
vectors. Start with the network definition.
net = newp([-2 2;-2 +2],1);
net.trainParam.passes = 1;
The input vectors and targets are:
p = {[2;2] [1;-2] [-2;2] [-1;1]}
t = {0 1 0 1}.
Now train the network with:
[net,a,e] = adapt(net,p,t);
3-18
Adaptive Training (ADAPT)
The output and error that are returned are:
a =
[1]
[1]
[0]
[0]
e =
[-1]
[0]
[0]
[1]
Note that these outputs and errors are the values obtained when each input
vector is applied to the network as it existed at the time.
The new weights and bias are:
twts =
-3
tbias =
0
-1
Finally simulate the trained network for each of the inputs.
a1 = sim(net,p)
a1 =
[0]
[0]
[1]
[1]
The outputs do not yet equal the targets, so we need to train the network for
more than one pass. This time let us run the problem again for two passes. We
get the weights
twts =
-2
tbiase =
1
-3
and the simulated output and errors for the various inputs is:
a1 =
[0]
[1]
[0]
[1]
The second pass does the job. The network has converged and produces the
correct outputs for the four input vectors. To check we can find the error for
each of the inputs.
error = {a1{1}-t{1} a1{2}-t{2} a1{3}-t{3} a1{4}-t{4}}
error =
[0]
[0]
[0]
[0]
Sure enough, all of the inputs are correctly classified.
3-19
3
Perceptrons
Note that adapt uses the perceptron learning rule in its pure form, in that
corrections to the weights and bias are made after each presentation of an
input vector. Thus, adapt will converge in a finite number of steps unless the
problem presented can not be solved with a simple perceptron.
The function adapt can be used in various ways by other networks as well. You
might type help adapt to read more about this basic function.
You may wish to try various demonstration programs. For instance, demop1
illustrates classification and training of a simple perceptron.
3-20
Limitations and Cautions
Limitations and Cautions
Perceptron networks should be trained with adapt, which presents the input
vectors to the network one at a time and makes corrections to the network
based on the results of each presentation. Use of adapt in this way guarantees
that any linearly separable problem will be solved in a finite number of
training presentations. Perceptrons can also be trained with the function
train, which is presented in the next chapter. When train is used for
perceptrons, it presents the inputs to the network in batches, and makes
corrections to the network based on the sum of all the individual corrections.
Unfortunately, there is no proof that such a training algorithm converges for
perceptrons. On that account the use of train for perceptrons is not
recommended.
Perceptron networks have several limitations. First, the output values of a
perceptron can take on only one of two values (0 or 1) due to the hard limit
transfer function. Second, perceptrons can only classify linearly separable sets
of vectors. If a straight line or a plane can be drawn to separate the input
vectors into their correct categories, the input vectors are linearly separable. If
the vectors are not linearly separable, learning will never reach a point where
all vectors are classified properly. Note, however, that it has been proven that
if the vectors are linearly separable, perceptrons trained adaptively will always
find a solution in finite time. You might want to try demop6. It shows the
difficulty of trying to classify input vectors that are not linearly separable.
It is only fair, however, to point out that networks with more than one
perceptron can be used to solve more difficult problems. For instance, suppose
that you have a set of four vectors that you would like to classify into distinct
groups, and that in fact, two lines can be drawn to separate them. A two neuron
network can be found such that its two decision boundaries classify the inputs
into four categories. For additional discussion about perceptrons and to
examine more complex perceptron problems, see [HDB1996].
Outliers and the Normalized Perceptron Rule
Long training times can be caused by the presence of an outlier input vector
whose length is much larger or smaller than the other input vectors. Applying
the perceptron learning rule involves adding and subtracting input vectors
from the current weights and biases in response to error. Thus, an input vector
with large elements can lead to changes in the weights and biases that take a
3-21
3
Perceptrons
long time for a much smaller input vector to overcome. You might wish to try
demop4 to see how an outlier effects the training.
By changing the perceptron learning rule slightly, training times can be made
insensitive to extremely large or small outlier input vectors.
Here is the original rule for updating weights:
∆w = ( t – a )p T = ep T
As shown above, the larger an input vector p, the larger its effect on the weight
vector w. Thus, if an input vector is much larger than other input vectors, the
smaller input vectors must be presented many times to have an effect.
The solution is to normalize the rule so that effect of each input vector on the
weights is of the same magnitude:
pT
pT
∆w = ( t – a ) -------- = e -------p
p
The normalized perceptron rule is implemented with the function learnpn
which is called exactly like learnpn. The normalized perceptron rule function
learnpn takes slightly more time to execute, but reduces number of epochs
considerably if there are outlier input vectors. You might try demop5 to see how
this normalized training rule works.
3-22
Summary
Summary
Perceptrons are useful as classifiers. They can classify linearly separable input
vectors very well. Convergence is guaranteed in a finite number of steps
providing the perceptron can solve the problem.
The design of a perceptron network is constrained completely by the problem
to be solved. Perceptrons have a single layer of hard limit neurons. The number
of network inputs and the number of neurons in the layer are constrained by
the number of inputs and outputs required by the problem.
Training time is sensitive to outliers, but outlier input vectors do not stop the
network from finding a solution.
Single-layer perceptrons can solve problems only when data is linearly
separable. This is seldom the case. One solution to this difficulty is to use a
preprocessing method that results in linearly separable vectors. Or you might
use multiple perceptrons in multiple layers. Alternatively, you can use other
kinds of networks such as linear networks or backpropagation networks, which
can classify nonlinearly separable input vectors.
Figures and Equations
Perceptron Neuron
Input
Perceptron Neuron
AAAA
AAAA
p1
p2
p3
w1,1
pR
w1, R
n
f
a
Where...
R = # Elements
in input vector
b
1
a = hardlim
- Exp -(Wp + b)
3-23
3
Perceptrons
Perceptron Transfer Function, hardlim
a
+1
n
0
-1
AA
a = hardlim(n)
Hard Limit Transfer Function
Decision Boundary
L
p
W
2
Wp+b > 0
a=1
+1
Wp+b = 0
a=0
-b/w
1,1
p
1
-1
+1
Wp+b < 0
a=0
-b/w
-1
1,2
Where... w
= -1
w
= +1
1,1
1,2
3-24
and
b = +1
Summary
Perceptron Architecture
Input #1
iw1,11, 1
p
1
b1
n11
1
1
p2
p3
n12
b12
1
p
R
AA
AA
AA
A
A
A
A
A
A
Input # 1
Perceptron Layer
iw1,1S1,R
n1
S
1
b1S1
1
a11
p
AA
AA
AA
R x1
Layer 1
IW1,1
S1xR
a12
1
R
AA
AA
AA
a1
n1
S1x1
b1
S1x1
S 1x 1
f
1
S1
a1 = hardlim(IW1,1p1 +b1)
a1 1
S
Where...
R = # of elements in Input
S1 = # of Neurons in layer 1
a1 = hardlim (IW1,1p1 + b1)
The Perceptron Learning Rule
W
b
new
new
= W
= b
old
old
+ ep
T
+e
where
e = t–a
3-25
3
Perceptrons
One Perceptron Neuron
Perceptron Neuron
Input
p
1
p
AA
AA
AAAA
w
1,1
w
2
1, 2
n
f
a
b
1
a = hardlim
- Exp -(Wp + b)
New Functions
This chapter introduces the following new functions:
3-26
Function
Description
hardlim
A hard limit transfer function
initzero
Zero weight/bias initialization function
dotprod
Dot product weight function
newp
Creates a new perceptron network.
sim
Simulates a neural network.
init
Initializes a neural network
learnp
Perceptron learning function
adapt
Trains a network using a sequence of inputs
learnpn
Normalized perceptron learning function
4
Adaptive Linear Filters
Introduction . . . . . . . . . . . . . . . . . . . . 4-2
Important Linear Network Functions . . . . . . . . . . 4-3
Neuron Model . . . . . . . . . . . . . . . . . . . 4-4
Network Architecture . . . . . . . . . . . . . . . 4-5
Single ADALINE (NEWLIN) . . . . . . . . . . . . . . 4-6
Mean Square Error . . . . . . . . . . . . . . . . . 4-9
. . . . . . . . . 4-10
Linear System Design (NEWLIND)
LMS Algorithm (LEARNWH)
. . . . . . . . . . . . 4-11
Linear Classification (TRAIN) . . . . . . . . . . . . 4-13
Adaptive Filtering (ADAPT) .
Tapped Delay Line . . . . . .
Adaptive Filter . . . . . . .
Adaptive Filter Example . . .
Prediction Example . . . . . .
Noise Cancellation Example . .
Multiple Neuron Adaptive Filters
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4-16
4-16
4-17
4-18
4-20
4-21
4-23
Limitations and Cautions
Overdetermined Systems .
Underdetermined Systems .
Linearly Dependent Vectors
Too Large a Learning Rate .
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4-25
4-25
4-25
4-25
4-26
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Summary . . . . . . . . . . . . . . . . . . . . . 4-27
Figures and Equations . . . . . . . . . . . . . . . . 4-28
New Functions . . . . . . . . . . . . . . . . . . . 4-36
4
Adaptive Linear Filters
Introduction
The ADALINE (Adaptive Linear Neuron networks) networks discussed in this
chapter are similar to the perceptron, but their transfer function is linear
rather than hard-limiting. This allows their outputs to take on any value,
whereas the perceptron output is limited to either 0 or 1. Both the ADALINE
and the perceptron can only solve linearly separable problems. However, here
we will make use of a so called LMS (Least Mean Squares) learning rule which
is much more powerful that the perceptron learning rule. The LMS or
Widrow-Hoff learning rule minimizes the mean square error and thus moves
the decision boundaries as far as it can from the training patterns.
First we will design a linear network that, when presented with a set of given
input vectors, produces outputs of corresponding target vectors. For each input
vector we can calculate the network’s output vector. The difference between an
output vector and its target vector is the error. We would like to find values for
the network weights and biases such that the sum of the squares of the errors
is minimized or below a specific value. This problem is manageable because
linear systems have a single error minimum. In most cases we can calculate a
linear network directly, such that its error is a minimum for the given input
vectors and targets vectors. In other cases numerical problems prohibit direct
calculation. Fortunately, we can always train the network to have a minimum
error by using the Widrow-Hoff learning rule.
Later we will design an adaptive linear system that responds to changes in its
environment as it is operating. Linear networks which are adjusted at each
time step based on new input and target vectors can find weights and biases
which minimize the network’s sum-squared error for recent input and target
vectors. Networks of this sort are often used in error cancellation, signal
processing and control systems.
The pioneering work in this field was done by Widrow and Hoff, who gave the
name ADALINE to adaptive linear elements. The basic reference on this
subject is: Widrow B. and S. D. Sterns, Adaptive Signal Processing, New York:
Prentice-Hall 1985.
4-2
Introduction
Important Linear Network Functions
This chapter introduces the newlin, a function that creates a linear layer, the
function newlind that designs a linear layer, the function learnwh, the
Widrow-Hoff weight/bias learning rule, the function train, that trains a neural
network and finally, the function adapt, that changes the weights and biases
of a network incrementally during training.
You might type help linnet to see a list of linear network functions,
demonstrations, and applications.
4-3
4
Adaptive Linear Filters
Neuron Model
A linear neuron with R inputs is shown below.
Linear Neuron w
Vector Input
Input
AA A
A
p1
p2
p
w1,1
pR
w1, R
n
3
f
Where...
a
R = # Elements
in input vector
b
1
a = purelin (Wp + b)
This network has the same basic structure as the perceptron. The only
difference is that the linear neuron uses a linear transfer function which we
will give the name purelin.
a
+1
n
0
-1
a = purelin(n)
AA
AA
Linear Transfer Function
The linear transfer function calculates the neuron’s output by simply returning
the value passed to it.
a = purelin ( n ) = purelin ( Wp + b ) = Wp + b
This neuron can be trained to learn an affine function of its inputs, or to find a
linear approximation to a nonlinear function. A linear network cannot, of
course, be made to perform a nonlinear computation.
4-4
Network Architecture
Network Architecture
The linear network shown below has one layer of S neurons connected to R
inputs through a matrix of weights W.
Input
Layer of Linear
Neurons
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AAAA
n1
w1, 1
p1
p
2
p
3
Input
a1
p
Rx1
b
1
2
a
1
2
R
b
2
nS
n
bS
a= purelin (Wp + b)
b
S
Sx1
a= purelin (Wp + b)
aS
wS, R
1
Sx1
Sx1
1
pR
a
W
SxR
1
n
AA
AA
AA
AA
AA AA
Layer of Linear Neurons
Where...
R = # of elements
in input vector
S = # Neurons
in Layer
This network is sometimes called a MADALINE for Many ADALINES. Note
that the figure on the right defines an S-length output vector a.
The Widrow-Hoff rule can only train single-layer linear networks. This is not
much of a disadvantage, however, as single-layer linear networks are just as
capable as multi-layer linear networks. For every multi-layer linear network,
there is an equivalent single-layer linear network.
4-5
4
Adaptive Linear Filters
Single ADALINE (NEWLIN)
Consider a single ADALINE with two inputs. The diagram for this network is
shown below.
Input
Simple ADALINE
AA
AA
AAAA
p1
w1,1
p2
w1,2
n
a
b
1
a = purelin(Wp+b)
The weight matrix W in this case has only one row. The network output is:
a = purelin ( n ) = purelin ( Wp + b ) = Wp + b
or
a = w 1, 1 p 1 + w 1, 2 p 2 + b
Like the perceptron, the ADALINE has a decision boundary which is
determined by the input vectors for which the net input n is zero. For n = 0
the equation Wp + b = 0 specifies such a decision boundary as shown below
(adapted with thanks from [HDB96]).
p
2
a<0
a>0
-b/w
1,2
W
Wp+b=0
p
-b/w
1
1,1
Input vectors in the upper right gray area will lead to an output greater than
0. Input vectors in the lower left white area will lead to an output less than 0.
Thus, the ADALINE can be used to classify objects into two categories.
4-6
Network Architecture
However, it can classify in this way only if the objects are linearly separable.
Thus, the ADALINE has the same limitation as the perceptron.
We can create a network like that shown above with the command:
net = newlin( [-1 1; -1 1],1);
The first matrix of arguments specify the range of the two scalar inputs. The
last argument, 1, says that the network has a single output.
The network weights and biases are set to zero by default. You can see the
current values with the commands:
W = net.IW{1,1}
W =
0
0
and
b= net.b{1}
b =
0
However, you can give the weights any value that you wish, such as 2 and 3
respectively, with:
net.IW{1,1} = [2 3];
W = net.IW{1,1}
W =
2
3
The bias can be set and checked in the same way.
net.b{1} =[-4];
b = net.b{1}
b =
-4
You can simulate the ADALINE for a particular input vector. Let us try
p = [5;6];
4-7
4
Adaptive Linear Filters
Now you can find the network output with the function sim.
a = sim(net,p)
a =
24
To summarize, you can create an ADALINE network with newlin, adjust its
elements as you wish and simulate it with sim. You can find more about newlin
by typing help newlin.
4-8
Mean Square Error
Mean Square Error
Like the perceptron learning rule, the least mean square error (LMS)
algorithm is an example of supervised training, in which the learning rule is
provided with a set of examples of desired network behavior:
{p 1, t 1} , { p 2, t 2} , …, {p Q, tQ}
Here p q is an input to the network, and t q is the corresponding target output.
As each input is applied to the network, the network output is compared to the
target. The error is calculated as the difference between the target output and
the network output. We want to minimize the average of the sum of these
errors.
1
mse = ---Q
Q
∑
k=1
1
e ( k ) = ---Q
2
Q
∑ (t(k) – a(k))
2
k=1
The LMS algorithm adjusts the weights and biases of the ADALINE so as to
minimize this mean square error.
Fortunately, the mean square error performance index for the ADALINE
network is a quadratic function. Thus, the performance index will either have
one global minimum, a weak minimum or no minimum, depending on the
characteristics of the input vectors. Specifically, the characteristics of the
input vectors determine whether or not a unique solution exists.
You can find more about this topic in Ch. 10 of [HDB96].
4-9
4
Adaptive Linear Filters
Linear System Design (NEWLIND)
Unlike most other network architectures, linear networks can be designed
directly if all input/target vector pairs are known. Specific network values for
weights and biases can be obtained to minimize the mean square error by using
the function newlind.
Suppose that the inputs and targets are:
P = [1 2 3];
T= [2.0 4.1 5.9];
Now you can design a network.
net = newlind(P,T);
You can simulate the network behavior to check that the design was done
properly.
Y = sim(net,P)
Y =
2.0500
4.0000
5.9500
Note that the network outputs are quite close to the desired targets.
You might try demolin1. It shows error surfaces for a particular problem,
illustrates the design and plots the designed solution.
Next we will discuss the LMS algorithm. We can use it to train a network to
minimize the mean square error.
4-10
LMS Algorithm (LEARNWH)
LMS Algorithm (LEARNWH)
The LMS algorithm or Widrow-Hoff learning algorithm, is based on an
approximate steepest descent procedure. Here again, linear networks are
trained on examples of correct behavior.
Widrow and Hoff had the insight that they could estimate the mean square
error by using the squared error at each iteration. If we take the partial
derivative of the squared error with respect to the weights and biases at the kth
iteration we have:
2
∂e ( k )
∂e ( k )
----------------- = 2e ( k ) --------------∂w 1, j
∂w 1, j
for j = 1, 2, …, R and
2
∂e ( k )
∂e ( k )
----------------- = 2e ( k ) -------------∂b
∂b
Next look at the partial derivative with respect to the error.
∂[ t ( k ) – a ( k ) ]
∂
∂e ( k )
--------------- = ------------------------------------ =
[ t ( k ) – ( Wp ( k ) + b ) ] or
∂w1, j
∂w 1, j
∂ w1, j
∂
∂e ( k )
--------------- =
∂w 1, j
∂ w 1,
j

t(k) – 

R

∑ w1, i pi ( k ) + b
i=1
Here pi(k) is the ith element of the input vector at the kth iteration.
Similarly,
∂e ( k )
--------------- = – p j ( k )
∂w 1, j
This can be simplified to:
∂e ( k )
--------------- = – p j ( k ) and
∂w 1, j
∂e ( k )
-------------- = – 1
∂b
4-11
4
Adaptive Linear Filters
Finally, the change to the weight matrix and the bias will be:
2αe ( k )p ( k ) and 2αe ( k ) . These two equations form the basis of the
Widrow-Hoff (LMS) learning algorithm.
These results can be extended to the case of multiple neurons, and written in
matrix form as:
T
W ( k + 1 ) = W ( k ) + 2αe ( k )p ( k )
b ( k + 1 ) = b ( k ) + 2αe ( k ) .
Here the error e and the bias b are vectors and α is a learning rate. If α is
large, learning occurs quickly, but if it is too large it may lead to instability and
errors may even increase. To ensure stable learning, the learning rate must be
less than the reciprocal of the largest eigenvector of the correlation matrix
p T p of the input vectors.
You might want to read some of Chapter 10 of [HDB96] for more information
about the LMS algorithm and its convergence.
Fortunately we have a toolbox function learnwh that does all of the calculation
for us. It calculates the change in weights as
dw = lr*e*p'
and the bias change as
db = lr*e.
The constant 2 shown a few lines above has been absorbed into the code
learning rate lr. The function maxlinlr calculates this maximum stable
learning rate lr as 0.999 * P'*P.
Type help learnwh and help maxlinlr for more details about these two
functions.
4-12
Linear Classification (TRAIN)
Linear Classification (TRAIN)
Linear networks can be trained to perform linear classification with the
function train. This function applies each vector of a set of input vectors and
calculates the network weight and bias increments due to each of the inputs
according to learnp. Then the network is adjusted with the sum of all these
corrections. We will call each pass through the input vectors an epoch. This
contrasts with adapt, which adjusts weights for each input vector as it is
presented.
Finally, train applies the inputs to the new network, calculates the outputs,
compares them to the associated targets, and calculates a mean square error.
If the error goal is met, or if the maximum number of epochs is reached, the
training is stopped and train returns the new network and a training record.
Otherwise train goes through another epoch. Fortunately, the LMS algorithm
converges when this procedure is executed.
To illustrate this procedure, we will work through a simple problem. Consider
the ADALINE network introduced earlier in this chapter.
Input
Simple ADALINE
AA
AA
AAAA
p1
w1,1
p2
w1,2
n
a
b
1
a = purelin(Wp+b)
Next suppose we have the classification problem presented in Chapter 3 on
perceptrons.

2 , t = 0   p = 1 , t = 1   p = –2 , t = 0   p = –1 , t = 1 
 p1 =
  2
  3
  4

1
2
3
4
2
–2
2
1

 
 
 

Here we have four input vectors, and we would like a network that produces
the output corresponding to each input vector when that vector is presented.
4-13
4
Adaptive Linear Filters
We will use train to get the weights and biases for a network that produces
the correct targets for each input vector. The initial weights and bias for the
new network will be 0 by default. We will set the error goal to 0.1 rather than
accept its default of 0.
P = [2 1 -2 -1;2 -2 2 1];
t = [0 1 0 1];
net = newlin( [-2 2; -2 2],1);
net.trainParam.goal= 0.1;
[net, tr] = train(net,P,t);
The problem runs, producing the following training record.
TRAINWB,
TRAINWB,
TRAINWB,
TRAINWB,
TRAINWB,
Epoch 0/100, MSE 0.5/0.1.
Epoch 25/100, MSE 0.181122/0.1.
Epoch 50/100, MSE 0.111233/0.1.
Epoch 64/100, MSE 0.0999066/0.1.
Performance goal met.
Thus, the performance goal is met in 64 epochs. The new weights and bias are:
weights = net.iw{1,1}
weights =
-0.0615
-0.2194
bias = net.b(1)
bias =
[0.5899]
We can simulate the new network as shown below.
A = sim(net, p)
A =
0.0282
0.9672
0.2741
0.4320,
We also can calculate the error.
err = t - sim(net,P)
err =
-0.0282
0.0328
-0.2741
0.5680
Note that the targets are not realized exactly. The problem would have run
longer in an attempt to get perfect results had we chosen a smaller error goal,
but in this problem it is not possible to obtain a goal of 0. The network is limited
4-14
Linear Classification (TRAIN)
in its capability. See demolin4 in “Limitations and Cautions” at the end of this
chapter for an example of this limitation.
This demonstration program demolin2 shows the training of a linear neuron,
and plots the weight trajectory and error during training
You also might try running the demonstration program nnd10lc. It addresses
a classic and historically interesting problem, shows how a network can be
trained to classify various patterns, and how the trained network responds
when noisy patterns are presented.
4-15
4
Adaptive Linear Filters
Adaptive Filtering (ADAPT)
The ADALINE network, much like the perceptron, can only solve linearly
separable problems. Nevertheless, the ADALINE has been and is today one of
the most widely used neural networks found in practical applications. Adaptive
filtering is one of its major application areas.
Tapped Delay Line
We need a new component, the tapped delay line, to make full use of the
ADALINE network. Such a delay line is shown below. There the input signal
enters from the left, and passes through N-1 delays.The output of the tapped
delay line (TDL) is an N-dimensional vector, made up of the input signal at the
current time, the previous input signal, etc.
TDL
AA
AA
AA
AA
D
pd (k)
2
D
pdN (k)
N
4-16
pd1(k)
Adaptive Filtering (ADAPT)
Adaptive Filter
We can combine a tapped delay line with an ADALINE network to create the
adaptive filter shown below.
Linear Layer
TDL
p(k)
AA
AA
AA
AA
pd1(k)
D
p(k - 1)
D
w1,1
pd (k)
2
pd (k)
N
AAAA
n(k)
w1,2
a(k)
SxR
b
1
w1, N
N
The output of the filter is given by
R
a ( k ) = purelin ( Wp + b ) =
∑ w1, i a ( k – i + 1 ) + b
i=1
The network shown above is referred to in the digital signal processing field as
a finite impulse response (FIR) filter [WiSt85]. Let us take a look at the code
that we will use to generate and simulate such an adaptive network.
4-17
4
Adaptive Linear Filters
Adaptive Filter Example
First we will define a new linear network using newlin.
Input
p1(t) = p(t)
AA AAAA
AA AAAA
D
p2(t) = p(t - 1)
Linear Digital Filter
D
w1,1
w1,2
n(t)
a(t)
b
w1,3 1
p3(t) = p(t - 2)
a = purelin
- Exp(Wp
- + b)
Assume that the input values have a range from 0 to 10. We can now define our
single output network.
net = newlin([0,10],1);
We can specify the delays in the tapped delay line with
net.inputWeights{1,1}.delays = [0 1 2];
This says that the delay line is connected to the network weight matrix through
delays of 0, 1 and 2 time units. (You can specify as many delays as you wish,
and can omit some values if you like. They must be in ascending order.)
We can give the various weights and the bias values with:
net.IW{1,1} = [7 8 9];
net.b{1} = [0];
Finally we will define the initial values of the outputs of the delays as:
pi ={1 2}
Note that these are ordered from left to right to correspond to the delays taken
from top to bottom in the figure. This concludes the setup of the network. Now
how about the input?
4-18
Adaptive Filtering (ADAPT)
We will assume that the input scalars arrive in a sequence, first the value 3,
then the value 4, next the value 5 and finally the value 6. We can indicate this
sequence by defining the values as elements of a cell array. (Note the curly
brackets.)
p = {3 4 5 6}
Now we have a network and a sequence of inputs. We can simulate the network
to see what its output is as a function of time.
[a,pf] = sim(net,p,pi);
This yields an output sequence
a =
[46]
[70]
[94]
[118]
and final values for the delay outputs of
pf =
[5]
[6].
The example is sufficiently simple that you can check it by hand to make sure
that you understand the inputs, initial values of the delays, etc.
The network that we have defined can be trained with the function adapt to
produce a particular output sequence. Suppose, for instance, we would like the
network to produce the sequence of values 10, 20, 30, and 40.
T = {10 20 30 40}
We can train our defined network to do this, starting from the initial delay
conditions that we used above. We will specify ten passes through the input
sequence with:
net.adaptParam.passes = 10;
Then we can do the training with:
[net,y,E pf,af] = adapt(net,p,T,pi);
4-19
4
Adaptive Linear Filters
This code returns final weights, bias and output sequence shown below.
wts = net.IW{1,1}
wts =
0.5059
3.1053
5.7046
bias = net.b{1}
bias =
-1.5993
y =
[11.8558]
[20.7735]
[29.6679]
[39.0036]
Presumably if we had run for additional passes the output sequence would
have been even closer to the desired values of 10, 20, 30 and 40.
Thus, adaptive networks can be specified, simulated and finally trained with
adapt. However, the outstanding value of adaptive networks lies in their use
to perform a particular function, such as or prediction or noise cancellation.
Prediction Example
Suppose that we would like to use an adaptive filter to predict the next value
of a stationary random process, p(t). We will use the network shown below to
do this.
Input
p1(t) = p(t)
AA
AA
AA
A AAAA
A
D
p2(t) = p(t - 1)
Linear Digital Filter
D
p3(t) = p(t - 2)
w1,2
n(t)
+
a(t)
e(t)
-
b
w1,3 1
Target = p(t)
Adjust weights
a = purelin (Wp + b)
Predictive Filter:
a(t) is approximation to p(t)
The signal to be predicted, p(t), enters from the left into a tapped delay line.
The previous two values of p(t) are available as outputs from the tapped delay
4-20
Adaptive Filtering (ADAPT)
line. The network uses adapt to change the weights on each time step so as to
minimize the error e(t) on the far right. If this error is zero, then the network
output a(t) is exactly equal to p(t), and the network has done its prediction
properly.
A detailed analysis of this network is not appropriate here, but we can state the
main points. Given the autocorrelation function of the stationary random
process p(t), the error surface, the maximum learning rate, and the optimum
values of the weights can be calculated. Commonly, of course, one does not have
detailed information about the random process, so these calculations cannot be
performed. But this lack does not matter to the network. The network, once
initialized and operating, adapts at each time step to minimize the error and
in a relatively short time is able to predict the input p(t).
Chapter 10 of [HDB96] presents this problem, goes through the analysis, and
shows the weight trajectory during training. The network finds the optimum
weights on its own without any difficulty whatsoever.
You also might want to try demonstration program nnd10nc to see an adaptive
noise cancellation program example in action. This demonstration allows you
to pick a learning rate and momentum, (see Chapter 5), and shows the learning
trajectory, and the original and cancellation signals verses time.
Noise Cancellation Example
Consider a pilot in an airplane. When the pilot speaks into as microphone, the
engine noise in the cockpit is added to the voice signal, and the resultant signal
heard by passengers would be of low quality. We would like to obtain a signal
which contains the pilot’s voice but not the engine noise. We can do this with
an adaptive filter if we can obtain a sample of the engine noise and apply it as
the input to the adaptive filter.
4-21
4
Adaptive Linear Filters
Pilot’s
Voice
Pilot’s Voice
Contaminated with
Engine Noise
v
m
Restored Signal
e
+
-
Contaminating
Noise
c
"Error"
Filtered Noise to Cancel
Contamination
Noise Path
Filter
n
Engine Noise
Adaptive
Filter
a
Adaptive Filter Adjusts to Minimize Error.
This removes the engine noise from contaminated
signal, leaving the pilot’s voice as the “error.”
Here we will adaptively train the neural linear network to predict the
combined pilot/engine signal m from an engine signal n. Notice that the engine
signal n does not tell the adaptive network anything about the pilot’s voice
signal contained in m. However, the engine signal n. does give the network
information it can use to predict the engine’s contribution to the pilot/engine
signal m. The network will do its best to adaptively output m. However, in this
case, the network can only predict the engine interference noise in the pilot/
engine signal m. The network error e will be equal to m, the pilot/engine signal,
minus the predicted contaminating engine noise signal. Thus e contains only
the pilot’s voice! Our linear adaptive network adaptively learns to cancel the
engine noise. Note, in closing, that such adaptive noise canceling generally
does a better job than a classical filter because the noise here is subtracted from
rather than filtered out of the signal m.
4-22
Adaptive Filtering (ADAPT)
Multiple Neuron Adaptive Filters
We may want to use more than one neuron in an adaptive system, so we need
some additional notation. A tapped delay line can be used with an S linear
neurons as shown below.
TDL
p(k)
Linear Layer
AA
AA
AA
AA
D
pd1(k)
w1,1
A
A
A
A
A
n1(k)
a1(k)
n2(k)
a2(k)
nS (k)
aS (k)
b1
pd2(k)
1
p(k - 1)
D
A
A
A
A
A
b2
1
pdN (k)
wS, N
bS
1
N
Alternatively, we can show this same network in abbreviated form.
AAAAA AA
AAAAA
AAA
AA
AAA AA
Linear Layer of S Neurons
p(k)
Qx1
a(k)
pd(k)
TDL
(Q*N) x 1
W
n(k)
S x1
S x (Q*N)
N
S x1
1
b
S x1
S
4-23
4
Adaptive Linear Filters
If we want to show more of the detail of the tapped delay line and there are not
too many delays we can use the following notation.
Abreviated Notation
A
AA
A
AA
AA
A
AA
pd(k)
p(k)
TDL
1x1
3x1
0
1
2
2
W
3x2
1
b
3x1
AA
AA
AA
a(k)
n(k)
3x1
3x1
Linear Layer
Here we have a tapped delay line that sends the current signal, the previous
signal and the signal delayed before that to the weight matrix. We could have
a longer list, and some delay values could be omitted if desired. The only
requirement is that the delays are shown in increasing order as they go from
top to bottom.
4-24
Limitations and Cautions
Limitations and Cautions
ADALINEs may only learn linear relationships between input and output
vectors. Thus ADALINEs cannot find solutions to some problems. However,
even if a perfect solution does not exist, the ADALINE will minimize the sum
of squared errors if the learning rate lr is sufficiently small. The network will
find as close a solution as is possible given the linear nature of the network’s
architecture. This property holds because the error surface of a linear network
is a multi-dimensional parabola. Since parabolas have only one minimum, a
gradient descent algorithm (such as the LMS rule) must produce a solution at
that minimum.
ADALINES have other various limitations. Some of them are discussed below.
Overdetermined Systems
Linear networks have a number of limitations. For instance, the system may
be overdetermined. Suppose that we have a network to be trained with four
1-element input vectors and four targets. A perfect solution to wp + b = t for
each of the inputs may not exist, for there are four constraining equations and
only one weight and one bias to adjust. However, the LMS rule will still
minimize the error. You might try demolin4 to see how this is done.
Underdetermined Systems
Consider a single linear neuron with one input. This time, in demolin5, we will
train it on only one 1-element input vector and its 1-element target vector:
P = [+1.0];
T = [+0.5];
Note that while there is only one constraint arising from the single input/target
pair, there are two variables, the weight and the bias. Having more variables
than constraints results in an underdetermined problem with an infinite
number of solutions. You might wish to try demoin5 to explore this topic.
Linearly Dependent Vectors
Normally it is a straightforward job to determine whether or not a linear
network can solve a problem. Commonly, if a linear network has at least as
many degrees of freedom (S*R+S = number of weights and biases) as
constraints (Q = pairs of input/target vectors), then the network can solve the
4-25
4
Adaptive Linear Filters
problem. This is true except when the input vectors are linearly dependent and
they are applied to a network without biases. In this case, as shown with
demonstration script demolin6, the network cannot solve the problem with
zero error. You might want to try demolin6.
Too Large a Learning Rate
A linear network can always be trained with the Widrow-Hoff rule to find the
minimum error solution for its weights and biases, as long as the learning rate
is small enough. Demonstration script demolin7 shows what happens when a
neuron with one input and a bias is trained with a learning rate larger than
that recommended by maxlinlr. The network is trained with two different
learning rates to show the results of using too large a learning rate.
4-26
Summary
Summary
Single-layer linear networks can perform linear function approximation or
pattern association.
Single-layer linear networks can be designed directly or trained with the
Widrow-Hoff rule to find a minimum error solution. In addition, linear
networks can be trained adaptively allowing the network to track changes in
its environment.
The design of a single-layer linear network is constrained completely by the
problem to be solved. The number of network inputs and the number of
neurons in the layer are determined by the number of inputs and outputs
required by the problem.
Multiple layers in a linear network do not result in a more powerful network
so the single layer is not a limitation. However, linear networks can solve only
linear problems.
Nonlinear relationships between inputs and targets cannot be represented
exactly by a linear network. The networks discussed in this chapter make a
linear approximation with the minimum sum-squared error.
If the relationship between inputs and targets is linear or a linear
approximation is desired then linear networks are made for the job. Otherwise,
backpropagation may be a good alternative.
Adaptive linear filters have many practical applications such as noise
cancellation and prediction in control and communication systems.
4-27
4
Adaptive Linear Filters
Figures and Equations
Linear Neuron
Linear Neuron w
Vector Input
Input
AA A
A
p1
p
p23
w1,1
pR
w1, R
n
f
Where...
a
b
1
a = purelin (Wp + b)
Purelin Transfer Function
a
+1
n
0
-1
a = purelin(n)
AA
AA
Linear Transfer Function
4-28
R = # Elements
in input vector
Summary
MADALINE
Input
Layer of Linear
Neurons
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AAAA
n1
w1, 1
p1
Input
a1
p
Rx1
b
1
p
2
p
1
2
R
b
3
2
nS
aS
wS, R
1
n
b
S
Sx1
a= purelin (Wp + b)
1
pR
Sx1
Sx1
a
2
a
W
SxR
1
n
AA
AA
AA
AA
AA AA
Layer of Linear Neurons
Where...
bS
R = # of elements
in input vector
S = # Neurons
in Layer
a= purelin (Wp + b)
ADALINE
Input
Simple ADALINE
AAAA
AAAA
p1
w1,1
p2
w1,2
n
a
b
1
a = purelin(Wp+b)
4-29
4
Adaptive Linear Filters
Decision Boundary
p
2
a<0
a>0
-b/w
1,2
W
Wp+b=0
p
1
-b/w
1,1
Mean Square Error
1
mse = ---Q
Q
∑
k=1
1
2
e ( k ) = ---Q
Q
∑ (t(k) – a(k))
k=1
LMS (Widrow-Hoff) Algorithm
T
W ( k + 1 ) = W ( k ) + 2αe ( k )p ( k )
b ( k + 1 ) = b ( k ) + 2αe ( k ) .
4-30
2
Summary
Tapped Delay Line
TDL
AA
AA
AA
AA
pd1(k)
D
pd2(k)
D
pdN (k)
N
4-31
4
Adaptive Linear Filters
Adaptive Filter
Linear Layer
TDL
p(k)
AA
AA
AA
AA
pd1(k)
D
p(k - 1)
D
w1,1
AAAA
pd (k)
2
n(k)
SxR
w1,2
b
1
pd (k)
N
w1, N
N
Adaptive Filter Example
Input
p1(t) = p(t)
AA
AA
AA
AA AAAA
D
p2(t) = p(t - 1)
Linear Digital Filter
D
w1,1
w1,2
n(t)
a(t)
b
w1,3 1
p3(t) = p(t - 2)
a = purelin
- Exp(Wp
- + b)
4-32
a(k)
Summary
Prediction Example
Input
p1(t) = p(t)
AA AAAA
AA AAAA
D
p2(t) = p(t - 1)
Linear Digital Filter
D
p3(t) = p(t - 2)
w1,2
n(t)
+
a(t)
e(t)
-
b
w1,3 1
Target = p(t)
Adjust weights
a = purelin (Wp + b)
Predictive Filter:
a(t) is approximation to p(t)
4-33
4
Adaptive Linear Filters
Noise Cancellation Example
Pilot’s
Voice
Pilot’s Voice
Contaminated with
Engine Noise
v
m
Restored Signal
e
+
-
Contaminating
Noise
c
Filtered Noise to Cancel
Contamination
Noise Path
Filter
n
Engine Noise
Adaptive
Filter
a
Adaptive Filter Adjusts to Minimize Error.
This removes the engine noise from contaminated
signal, leaving the pilot’s voice as the “error.”
4-34
"Error"
Summary
Multiple Neuron Adaptive Filter
TDL
p(k)
Linear Layer
AA
AA
AA
AA
D
pd1(k)
w1,1
A
A
A
A
A
n1(k)
a1(k)
n2(k)
a2(k)
nS (k)
aS (k)
b1
pd2(k)
1
p(k - 1)
D
A
A
A
A
A
b2
1
pdN (k)
wS, N
bS
1
N
Abbreviated Form of Adaptive Filter
AA
AAA
AAAAA AA
AAA
AA
AAA AA
Linear Layer of S Neurons
p(k)
Qx1
a(k)
pd(k)
TDL
(Q*N) x 1
W
n(k)
S x1
S x (Q*N)
N
S x1
1
b
S x1
S
4-35
4
Adaptive Linear Filters
Specific Small Adaptive Filter
Abreviated Notation
AA
A
AA
A
AA
A
AA
AA
AA
AA
pd(k)
p(k)
TDL
1x1
3x1
0
1
2
a(k)
W
n(k)
3x2
1
3x1
b
3x1
2
3x1
Linear Layer
New Functions
This chapter introduces the following new functions:
4-36
Function
Description
newlin
Creates a linear layer.
newlind
Design a linear layer.
learnwh
Widrow-Hoff weight/bias learning rule.
purelin
A hard limit transfer function.
train
Trains a neural network.
5
Backpropagation
Overview
. . . . . . . . . . . . . . . . . . . . . 5-2
Fundamentals
Architecture . .
Simulation (SIM)
Training . . .
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5-3
5-3
5-8
5-8
Faster Training . . . . . . . . . . . . . .
Variable Learning Rate (TRAINGDA, TRAINGDX) .
Resilient Backpropagation (TRAINRP) . . . . . .
Conjugate Gradient Algorithms . . . . . . . . .
Line Search Routines . . . . . . . . . . . . .
Quasi-Newton Algorithms . . . . . . . . . . .
Levenberg-Marquardt (TRAINLM) . . . . . . .
Reduced Memory Levenberg-Marquardt (TRAINLM)
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. 5-16
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. 5-20
. 5-26
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. 5-31
. 5-33
Speed and Memory Comparison . . . . . . . . . . . 5-35
Improving Generalization . . . . . . . . . . . . . . 5-37
Regularization . . . . . . . . . . . . . . . . . . . . 5-38
Early Stopping . . . . . . . . . . . . . . . . . . . . 5-41
Preprocessing and Postprocessing . . . . . . .
Min and Max (PREMNMX, POSTMNMX TRAMNMX)
Mean and Stand. Dev. (PRESTD, POSTSTD, TRASTD)
Principal Component Analysis (PREPCA, TRAPCA) .
Post-training Analysis (POSTREG) . . . . . . . .
Sample Training Session
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. 5-44
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Limitations and Cautions . . . . . . . . . . . . . . 5-54
Summary
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5
Backpropagation
Overview
Backpropagation was created by generalizing the Widrow-Hoff learning rule to
multiple-layer networks and nonlinear differentiable transfer functions. Input
vectors and the corresponding output vectors are used to train a network until
it can approximate a function, associate input vectors with specific output
vectors, or classify input vectors in an appropriate way as defined by you.
Networks with biases, a sigmoid layer, and a linear output layer are capable of
approximating any function with a finite number of discontinuities.
Standard backpropagation is a gradient descent algorithm, as is the
Widrow-Hoff learning rule. The term backpropagation refers to the manner in
which the gradient is computed for nonlinear multilayer networks. There are
a number of variations on the basic algorithm which are based on other
standard optimization techniques, such as conjugate gradient and Newton
methods. The Neural Network Toolbox implements a number of these
variations. This chapter will explain how to use each of these routines and will
discuss the advantages and disadvantages of each.
Properly-trained backpropagation networks tend to give reasonable answers
when presented with inputs that they have never seen. Typically, a new input
will lead to an output similar to the correct output for input vectors used in
training that are similar to the new input being presented. This generalization
property makes it possible to train a network on a representative set of input/
target pairs and get good results without training the network on all possible
input/output pairs. There are two features of the Neural Network Toolbox
which are designed to improve network generalization - regularization and
early stopping. These features and their use will be discussed later in this
chapter.
This chapter will also discuss preprocessing and postprocessing techniques
which can improve the efficiency of network training.
Before beginning this chapter you may want to read a basic reference on
backpropagation, such as D.E Rumelhart, G.E. Hinton, R.J. Williams,
“Learning internal representations by error propagation,”, D. Rumelhart and
J. McClelland, editors. Parallel Data Processing, Vol.1, Chapter 8, the M.I.T.
Press, Cambridge, MA 1986 pp. 318-362. This subject is also covered in detail
in Chapters 11 and 12 of M.T. Hagan, H.B. Demuth, M.H. Beale, Neural
Network Design, PWS Publishing Company, Boston, MA 1996.
5-2
Fundamentals
Fundamentals
Architecture
In this section we want to present the architecture of the network which is most
commonly used with the backpropagation algorithm - the multilayer
feedforward network. The routines in the Neural Network Toolbox can be used
to train more general networks, some of these will be briefly discussed in later
chapters.
Neuron Model (TANSIG, LOGSIG, PURELIN)
An elementary neuron with R inputs is shown below. Each input is weighted
with an appropriate w. The sum of the weighted inputs and the bias forms the
input to the transfer function f. Neurons may use any differentiable transfer
function f to generate their output.
Input
General Neuron
AA A
A
p1
p2
p3
w1,1
pR
w1, R
n
f
Where...
a
b
R = # Elements
in input vector
1
a =- fExp
(Wp-+ b)
Multilayer networks often use the log-sigmoid transfer function logsig.
a
+1
n
0
-1
AA
AA
a = logsig(n)
Log-Sigmoid Transfer Function
5-3
5
Backpropagation
The function logsig generates outputs between 0 and 1 as the neuron’s net
input goes from negative to positive infinity.
Alternatively, multilayer networks may use the tan-sigmoid transfer function
tansig.
a
+1
0
n
-1
a = tansig(n)
Tan-Sigmoid Transfer Function
Occasionally, the linear transfer function purelin is used in backpropagation
networks.
a
+1
n
0
-1
a = purelin(n)
AA
AA
Linear Transfer Function
If the last layer of a multilayer network has sigmoid neurons, then the outputs
of the network are limited to a small range. If linear output neurons are used
the network outputs can take on any value.
In backpropagation it is important to be able to calculate the derivatives of any
transfer functions used. Each of the transfer functions above, tansig, logsig,
and purelin, have a corresponding derivative function: dtansig, dlogsig and
dpurelin. To get the name of a transfer function’s associated derivative
function, call the transfer function with the string 'deriv'.
tansig('deriv')
ans = dtansig
5-4
Fundamentals
The three transfer functions described here are the most commonly used
transfer functions for backpropagation, but other differentiable transfer
functions can be created and used with backpropagation if desired. See
Chapter 11, “Advanced Topics” for more information.
Feedforward Network
A single-layer network of S logsig neurons having R inputs is shown below in
full detail on the left and with a layer diagram on the right.
Input
p1
p2
p3
pR
w1, 1
Layer of Neurons
AAAA
AA
AA
AA
AA
AA
AA
AAAA
n1
Input
a1
p
Rx1
b1
1
n2
1
wS, R
nS
bS
1
AA
AA
AA
W
n
SxR
a2
b2
Layer of Neurons
1
R
Sx1
b
Sx1
A
A
A
a
Sx1
S
a= f (Wp + b)
aS
Where...
R = # of elements in input vector
S = # Neurons in Layer
a= f (Wp + b)
Feedforward networks often have one or more hidden layers of sigmoid
neurons followed by an output layer of linear neurons. Multiple layers of
neurons with nonlinear transfer functions allow the network to learn nonlinear
and linear relationships between input and output vectors. The linear output
layer lets the network produce values outside the range –1 to +1.
On the other hand, if it is desirable to constrain the outputs of a network (such
as between 0 and 1) then the output layer should use a sigmoid transfer
function (such as logsig).
As noted in Chapter 2, for multiple-layer networks we use the number of the
layers to determine the superscript on the weight matrices. The appropriate
notation is used in the two-layer tansig/purelin network shown next.
5-5
5
Backpropagation
Input
Hidden Layer
Output Layer
AAAA AA
AAA
AAA
AA
AA
AA
AAA AAAA AA
p1
2 x1
IW1,1
4x2
a1
4x1
n1
LW2,1
3 x4
4 x1
1
2
n2
3 x1
1
b
1
4 x1
a2
4
a1 = tansig (IW1,1p1 +b1)
3 x1
f2
b
2
3 x1
3
a2 =purelin (LW2,1a1 +b2)
This network can be used as a general function approximator. It can
approximate any function with a finite number of discontinuities, arbitrarily
well, given sufficient neurons in the hidden layer.
Creating a Network (NEWFF). The first step in training a feedforward network is to
create the network object. The function newff creates a trainable feedforward
network. It requires four inputs and returns the network object. The first input
is an R by 2 matrix of minimum and maximum values for each of the R
elements of the input vector. The second input is an array containing the sizes
of each layer. The third input is a cell array containing the names of the
transfer functions to be used in each layer. The final input contains the name
of the training function to be used.
For example, the following command will create a two-layer network. There
will be one input vector with two elements, three neurons in the first layer and
one neuron in the second (output) layer. The transfer function in the first layer
will be tan-sigmoid, and the output layer transfer function will be linear. The
values for the first element of the input vector will range between -1 and 2, the
values of the second element of the input vector will range between 0 and 5, and
the training function will be traingd (which will be described in a later
section).
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traingd');
This command creates the network object and also initializes the weights and
biases of the network; therefore the network is ready for training. There are
times when you may wish to re-initialize the weights, or to perform a custom
initialization. The next section explains the details of the initialization process.
5-6
Fundamentals
Initializing Weights (INIT, INITNW, RANDS). Before training a feedforward network,
the weights and biases must be initialized. The initial weights and biases are
created with the command init. This function takes a network object as input
and returns a network object with all weights and biases initialized. Here is
how a network is initialized:
net = init(net);
The specific technique which is used to initialize a given network will depend
on how the network parameters net.initFcn and net.layers{i}.initFcn are
set. The parameter net.initFcn is used to determine the overall initialization
function for the network. The default initialization function for the feedforward
network is initlay, which allows each layer to use its own initialization
function. With this setting for net.initFcn, the parameters
net.layers{i}.initFcn are used to determine the initialization method for
each layer.
For feedforward networks there are two different layer initialization methods
which are normally used: initwb and initnw. The initwb function causes the
initialization to revert to the individual initialization parameters for each
weight matrix (net.inputWeights{i,j}.initFcn) and bias. For the
feedforward networks the weight initialization is usually set to rands, which
sets weights to random values between -1 and 1. It is normally used when the
layer transfer function is linear.
The function initnw is normally used for layers of feedforward networks where
the transfer function is sigmoid. It is based on the technique of Nguyen and
Widrow [NgWi90] and generates initial weight and bias values for a layer so
that the active regions of the layer's neurons will be distributed roughly evenly
over the input space. It has several advantages over purely random weights
and biases: (1) few neurons are wasted (since the active regions of all the
neurons are in the input space), (2) training works faster (since each area of the
input space has active neuron regions).
The initialization function init is called by newff, therefore the network is
automatically initialized with the default parameters when it is created, and
init does not have to be called separately. However, the user may want to
re-initialize the weights and biases, or to use a specific method of initialization.
For example, in the network that we just created, using newff, the default
initialization for the first layer would be initnw. If we wanted to re-initialize
5-7
5
Backpropagation
the weights and biases in the first layer using the rands function, we would
issue the following commands:
net.layers{1}.initFcn = 'initwb';
net.inputWeights{1,1}.initFcn = 'rands';
net.biases{1,1}.initFcn = 'rands';
net.biases{2,1}.initFcn = 'rands';
net = init(net);
Simulation (SIM)
The function sim simulates a network. sim takes the network input p, and the
network object net, and returns the network outputs a. Here is how simuff can
be used to simulate the network we created above for a single input vector:
p = [1;2];
a = sim(net,p)
a =
-0.1011
(If you try these commands, your output may be different, depending on the
state of your random number generator when the network was initialized.)
Below, sim is called to calculate the outputs for a concurrent set of three input
vectors.
p = [1 3 2;2 4 1];
a=sim(net,p)
a =
-0.1011
-0.2308
0.4955
Training
Once the network weights and biases have been initialized, the network is
ready for training. The network can be trained for function approximation
(nonlinear regression), pattern association, or pattern classification. The
training process requires a set of examples of proper network behavior network inputs p and target outputs t. During training the weights and biases
of the network are iteratively adjusted to minimize the network performance
function net.performFcn. The default performance function for feedforward
networks is mean square error mse - the average squared error between the
network outputs a and the target outputs t.
5-8
Fundamentals
The remainder of this chapter will describe several different training
algorithms for feedforward networks. All of these algorithms use the gradient
of the performance function to determine how to adjust the weights to
minimize performance. The gradient is determined using a technique called
backpropagation, which involves performing computations backwards through
the network. The backpropagation computation is derived using the chain rule
of calculus and is described in Chapter 11 of [HDB96]. The basic
backpropagation training algorithm, in which the weights are moved in the
direction of the negative gradient, is described in the next section. Later
sections will describe more complex algorithms that increase the speed of
convergence.
Backpropagation Algorithm
There are many variations of the backpropagation algorithm, several of which
will be discussed in this chapter. The simplest implementation of
backpropagation learning updates the network weights and biases in the
direction in which the performance function decreases most rapidly - the
negative of the gradient. One iteration of this algorithm can be written
xk + 1 = xk – αk g k ,
where x k is a vector of current weights and biases, g k is the current gradient,
and α k is the learning rate.
There are two different ways in which this gradient descent algorithm can be
implemented: incremental mode and batch mode. In the incremental mode, the
gradient is computed and the weights are updated after each input is applied
to the network. In the batch mode all of the inputs are applied to the network
before the weights are updated. The next section will describe the incremental
training, and the following section will describe batch training.
Incremental Training(ADAPT)
The function adapt is used to train networks in the incremental mode. This
function takes the network object and the inputs and the targets from the
training set, and returns the trained network object and the outputs and errors
of the network for the final weights and biases.
There are several network parameters which must be set in order guide the
incremental training. The first is net.adaptFcn, which determines which
incremental mode training function is to be used. The default for feedforward
5-9
5
Backpropagation
networks is adaptwb, which allows each weight and bias to assign its own
function. These individual learning functions for the weights and biases are set
by the parameters net.biases{i,j}.learnFcn,
net.inputWeights{i,j}.learnFcn, and net.layerWeights{i,j}.learnFcn.
Gradient Descent (LEARDGD). For the basic steepest (gradient) descent algorithm,
the weights and biases are moved in the direction of the negative gradient of
the performance function. For this algorithm, the individual learning function
parameters for the weights and biases are set to 'learngd'. The following
commands illustrate how these parameters are set for the feedforward network
we created earlier.
net.biases{1,1}.learnFcn = 'learngd';
net.biases{2,1}.learnFcn = 'learngd';
net.layerWeights{2,1}.learnFcn = 'learngd';
net.inputWeights{1,1}.learnFcn = 'learngd';
The function learngd has one learning parameter associated with it - the
learning rate lr. The changes to the weights and biases of the network are
obtained by multiplying the learning rate times the negative of the gradient.
The larger the learning rate, the bigger the step. If the learning rate is made
too large the algorithm will become unstable. If the learning rate is set too
small, the algorithm will take a long time to converge. See page 12-8 of
[HDB96] for a discussion of the choice of learning rate.
The learning rate parameter is set to the default value for each weight and bias
when the learnFcn is set to learngd, as in the code above, although you can
change its value if you desire. The following command demonstrates how you
can set the learning rate to 0.2 for the layer weights. The learning rate can be
set separately for each weight and bias.
net.layerWeights{2,1}.learnParam.lr= 0.2;
The final parameter to be set for sequential training is
net.adaptParam.passes, which determines the number of passes through the
training set during training. Here we set the number of passes to 200.
net.adaptParam.passes = 200;
5-10
Fundamentals
We are now almost ready to train the network. It remains to set up the training
set. Here is a simple set of inputs and targets which we will use to illustrate
the training procedure:
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
If we want the learning algorithm to update the weights after each input
pattern is presented, we need to convert the matrices of inputs and targets into
cell arrays, with a cell for each input vector and target:
p = num2cell(p,1);
t = num2cell(t,1);
We are now ready to perform the incremental training using the adapt
function:
[net,a,e]=adapt(net,p,t);
After the training is complete we can simulate the network to test the quality
of the training.
a = sim(net,p)
a =
[-0.9995]
[-1.0000]
[1.0001]
[1.0000]
Gradient Descent With Momentum (LEARDGDM). In addition to learngd, there is
another incremental learning algorithm for feedforward networks that often
provides faster convergence - learngdm, steepest descent with momentum.
Momentum allows a network to respond not only to the local gradient, but also
to recent trends in the error surface. Acting like a low pass filter, momentum
allows the network to ignore small features in the error surface. Without
momentum a network may get stuck in a shallow local minimum. With
momentum a network can slide through such a minimum. See page 12-9 of
[HDB96] for a discussion of momentum.
Momentum can be added to backpropagation learning by making weight
changes equal to the sum of a fraction of the last weight change and the new
change suggested by the backpropagation rule. The magnitude of the effect
that the last weight change is allowed to have is mediated by a momentum
constant, mc, which can be any number between 0 and 1. When the momentum
constant is 0 a weight change is based solely on the gradient. When the
5-11
5
Backpropagation
momentum constant is 1 the new weight change is set to equal the last weight
change and the gradient is simply ignored.
The learngdm function is invoked using the same steps shown above for the
learngd function, except that both the mc and lr learning parameters can be
set. Different parameter values can be used for each weight and bias, since
each weight and bias has its own learning parameters.
The following commands will cause the previously created network to be
incrementally trained using learngdm with the default learning parameters.
net.biases{1,1}.learnFcn = 'learngdm';
net.biases{2,1}.learnFcn = 'learngdm';
net.layerWeights{2,1}.learnFcn = 'learngdm';
net.inputWeights{1,1}.learnFcn = 'learngdm';
[net,a,e]=adapt(net,p,t);
Batch Training (TRAIN). The alternative to incremental training is batch training,
which is invoked using the function train. In batch mode the weights and
biases of the network are updated only after the entire training set has been
applied to the network. The gradients calculated at each training example are
added together to determine the change in the weights and biases. For a
discussion of batch training with the backpropagation algorithm see page 12-7
of [HDB96].
Batch Gradient Descent (TRAINGD). The batching equivalent of the incremental
function learngd is traingd, which implements the batching form of the
standard steepest descent training function. The weights and biases are
updated in the direction of the negative gradient of the performance function.
If you wish to train a network using batch steepest descent, you should set the
network trainFcn to traingd and then call the function train. Unlike the
learning functions in the previous section, which were assigned separately to
each weight matrix and bias vector in the network, there is only one training
function associated with a given network.
There are seven training parameters associated with traingd: epochs, show,
goal, time, min_grad, max_fail, and lr. The learning rate lr has the same
meaning here as it did for learngd - it is multiplied times the negative of the
gradient to determine the changes to the weights and biases. The training
status will be displayed every show iterations of the algorithm. The other
parameters determine when the training is stopped. The training will stop if
the number of iterations exceeds epochs, if the performance function drops
5-12
Fundamentals
below goal, if the magnitude of the gradient is less than mingrad, or if the
training time is longer than time seconds. We will discuss max_fail, which is
associated with the early stopping technique, in the section on improving
generalization.
The following code will recreate our earlier network, and then train it using
batch steepest descent. (Note that for batch training all of the inputs in the
training set are placed in one matrix.)
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traingd');
net.trainParam.show = 50;
net.trainParam.lr = 0.05;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINGD, Epoch 0/300, MSE 1.59423/1e-05, Gradient 2.76799/
1e-10
TRAINGD, Epoch 50/300, MSE 0.00236382/1e-05, Gradient
0.0495292/1e-10
TRAINGD, Epoch 100/300, MSE 0.000435947/1e-05, Gradient
0.0161202/1e-10
TRAINGD, Epoch 150/300, MSE 8.68462e-05/1e-05, Gradient
0.00769588/1e-10
TRAINGD, Epoch 200/300, MSE 1.45042e-05/1e-05, Gradient
0.00325667/1e-10
TRAINGD, Epoch 211/300, MSE 9.64816e-06/1e-05, Gradient
0.00266775/1e-10
TRAINGD, Performance goal met.
a = sim(net,p)
a =
-1.0010
-0.9989
1.0018
0.9985
Try the Neural Network Design Demonstration nnd12sd1[HDB96] for an
illustration of the performance of the batch gradient descent algorithm.
Batch Gradient Descent With Momentum (TRAINGDM). The batch form of gradient
descent with momentum is invoked using the training function traingdm. This
algorithm is equivalent to learngdm, with two exceptions. First, the gradient is
computed by summing the gradients calculated at each training example, and
5-13
5
Backpropagation
the weights and biases are only updated after all training examples have been
presented. Second, if the new performance function on a given iteration
exceeds the performance function on a previous iteration by more than a
predefined ratio max_perf_inc (typically 1.04), the new weights and biases are
discarded, and the momentum coefficient mc is set to zero.
In the following code we recreate our previous network and retrain it using
gradient descent with momentum. The training parameters for traingdm are
the same as those for traingd, with the addition of the momentum factor mc
and the maximum performance increase max_perf_inc. (The training
parameters are reset to the default values whenever net.trainFcn is set to
traingdm.)
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traingdm');
net.trainParam.show = 50;
net.trainParam.lr = 0.05;
net.trainParam.mc = 0.9;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINGDM, Epoch 0/300, MSE 3.6913/1e-05, Gradient 4.54729/
1e-10
TRAINGDM, Epoch 50/300, MSE 0.00532188/1e-05, Gradient
0.213222/1e-10
TRAINGDM, Epoch 100/300, MSE 6.34868e-05/1e-05, Gradient
0.0409749/1e-10
TRAINGDM, Epoch 114/300, MSE 9.06235e-06/1e-05, Gradient
0.00908756/1e-10
TRAINGDM, Performance goal met.
a = sim(net,p)
a =
-1.0026
-1.0044
0.9969
0.9992
5-14
Fundamentals
Note that since we re-initialized the weights and biases before training, we
obtain a different mean square error than we did using traingd. If we were to
re-initialize and train again using traingdm, we would get yet a different mean
square error. The random choice of initial weights and biases will affect the
performance of the algorithm. If you wish to compare the performance of
different algorithms, you should test each using several different sets of initial
weights and biases.
Try the Neural Network Design Demonstration nnd12mo [HDB96] for an
illustration of the performance of the batch momentum algorithm.
5-15
5
Backpropagation
Faster Training
The previous section presented two backpropagation training algorithms:
gradient descent and gradient descent with momentum. These two methods
are often too slow for practical problems. In this section we will discuss several
high performance algorithms which can converge from ten to one hundred
times faster than the algorithms discussed previously. All of the algorithms in
this section operate in the batch mode and are invoked using train.
These faster algorithms fall into two main categories. The first category uses
heuristic techniques, which were developed from an analysis of the
performance of the standard steepest descent algorithm. One heuristic
modification is the momentum technique, which was presented in the previous
section. This section will discuss two more heuristic techniques: variable
learning rate backpropagation, traingda, and resilient backpropagation
trainrp.
The second category of fast algorithms uses standard numerical optimization
techniques. (See Chapter 9 of [HDB96] for a review of basic numerical
optimization.) Later in this section we will present three types of numerical
optimization techniques for neural network training: conjugate gradient
(traincgf, traincgp, traincgb, trainscg), quasi-Newton (trainbfg,
trainoss), and Levenberg-Marquardt (trainlm).
Variable Learning Rate (TRAINGDA, TRAINGDX)
With standard steepest descent, the learning rate is held constant throughout
training. The performance of the algorithm is very sensitive to the proper
setting of the learning rate. If the learning rate is set too high, the algorithm
may oscillate and become unstable. If the learning rate is too small, the
algorithm will take too long to converge. It is not practical to determine the
optimal setting for the learning rate before training, and, in fact, the optimal
learning rate changes during the training process, as the algorithm moves
across the performance surface.
The performance of the steepest descent algorithm can be improved if we allow
the learning rate to change during the training process. An adaptive learning
rate will attempt to keep the learning step size as large as possible while
keeping learning stable. The learning rate is made responsive to the
complexity of the local error surface.
5-16
Faster Training
An adaptive learning rate requires some changes in the training procedure
used by traingd. First, the initial network output and error are calculated. At
each epoch new weights and biases are calculated using the current learning
rate. New outputs and errors are then calculated.
As with momentum, if the new error exceeds the old error by more than a
predefined ratio max_perf_inc (typically 1.04), the new weights and biases are
discarded. In addition, the learning rate is decreased (typically by multiplying
by lr_dec = 0.7). Otherwise the new weights, etc., are kept. If the new error is
less than the old error, the learning rate is increased (typically by multiplying
by lr_inc = 1.05).
This procedure increases the learning rate, but only to the extent that the
network can learn without large error increases. Thus a near optimal learning
rate is obtained for the local terrain. When a larger learning rate could result
in stable learning, the learning rate is increased. When the learning rate is too
high to guarantee a decrease in error, it gets decreased until stable learning
resumes.
Try the Neural Network Design Demonstration nnd12vl [HDB96] for an
illustration of the performance of the variable learning rate algorithm.
5-17
5
Backpropagation
Backpropagation training with an adaptive learning rate is implemented with
the function traingda which is called just like traingd except for the
additional training parameters max_perf_inc, lr_dec, and lr_inc. Here is
how it is called to train our previous two-layer network:
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traingda');
net.trainParam.show = 50;
net.trainParam.lr = 0.05;
net.trainParam.lr_inc = 1.05;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINGDA, Epoch 0/300, MSE 1.71149/1e-05, Gradient 2.6397/
1e-06
TRAINGDA, Epoch 44/300, MSE 7.47952e-06/1e-05, Gradient
0.00251265/1e-06
TRAINGDA, Performance goal met.
a = sim(net,p)
a =
-1.0036
-0.9960
1.0008
0.9991
The function traingdx combines adaptive learning rate with momentum
training. It is invoked in the same way as traingda, except that it has the
momentum coefficient mc as an additional training parameter.
Resilient Backpropagation (TRAINRP)
Multilayer networks typically use sigmoid transfer functions in the hidden
layers. These functions are often called squashing functions, since they
compress an infinite input range into a finite output range. Sigmoid functions
are characterized by the fact that their slope must approach zero as the input
gets large. This causes a problem when using steepest descent to train a
multilayer network with sigmoid functions, since the gradient can have a very
small magnitude, and therefore cause small changes in the weights and biases,
even though the weights and biases are far from their optimal values.
The purpose of the resilient backpropagation (Rprop) training algorithm is to
eliminate these harmful effects of the magnitudes of the partial derivatives.
Only the sign of the derivative is used to determine the direction of the weight
5-18
Faster Training
update; the magnitude of the derivative has no effect on the weight update. The
size of the weight change is determined by a separate update value. The update
value for each weight and bias is increased by a factor delt_inc whenever the
derivative of the performance function with respect to that weight has the
same sign for two successive iterations. The update value is decreased by a
factor delt_dec whenever the derivative with respect that weight changes sign
from the previous iteration. If the derivative is zero, then the update value
remains the same. Whenever the weights are oscillating the weight change will
be reduced. If the weight continues to change in the same direction for several
iterations, then the magnitude of the weight change will be increased. A
complete description of the Rprop algorithm is given in [ReBr93].
In the following code we recreate our previous network and train it using the
Rprop algorithm. The training parameters for trainrp are epochs, show, goal,
time, min_grad, max_fail, delt_inc, delt_dec, delta0, deltamax. We have
previously discussed the first eight parameters. The last two are the initial step
size and the maximum step size, respectively. The performance of Rprop is not
very sensitive to the settings of the training parameters. For the example below
we leave most of the training parameters at the default values. We do reduce
show below our previous value, because Rprop generally converges much faster
than the previous algorithms.
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainrp');
net.trainParam.show = 10;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINRP, Epoch 0/300, MSE 0.469151/1e-05, Gradient 1.4258/
1e-06
TRAINRP, Epoch 10/300, MSE 0.000789506/1e-05, Gradient
0.0554529/1e-06
TRAINRP, Epoch 20/300, MSE 7.13065e-06/1e-05, Gradient
0.00346986/1e-06
TRAINRP, Performance goal met.
a = sim(net,p)
a =
-1.0026
-0.9963
0.9978
1.0017
5-19
5
Backpropagation
Rprop is generally much faster than the standard steepest descent algorithm.
It also has the nice property that it requires only a modest increase in memory
requirements. We do need to store the update values for each weight and bias,
which is equivalent to storage of the gradient.
Conjugate Gradient Algorithms
The basic backpropagation algorithm adjusts the weights in the steepest
descent direction (negative of the gradient). This is the direction in which the
performance function is decreasing most rapidly. It turns out that, although
the function decreases most rapidly along the negative of the gradient, this
does not necessarily produce the fastest convergence. In the conjugate gradient
algorithms a search is performed along conjugate directions, which produces
generally faster convergence than steepest descent directions. In this section
we will present four different variations of conjugate gradient algorithms.
See page 12-14 of [HDB96] for a discussion of conjugate gradient algorithms
and their application to neural networks.
In most of the training algorithms that we have discussed up to this point, a
learning rate is used to determine the length of the weight update (step size).
In most of the conjugate gradient algorithms the step size is adjusted at each
iteration. A search is made along the conjugate gradient direction to determine
the step size which will minimize the performance function along that line.
There are five different search functions that are included in the toolbox, and
these will be discussed at the end of this section. Any of these search functions
can be used interchangeably with a variety of the training functions described
in the remainder of this chapter. Some search functions are best suited to
certain training functions, although the optimum choice can vary according to
the specific application. An appropriate default search function is assigned to
each training function, but this can be modified by the user.
Fletcher-Reeves Update (TRAINCGF)
All of the conjugate gradient algorithms start out by searching in the steepest
descent direction (negative of the gradient) on the first iteration.
p0 = –g0
5-20
Faster Training
A line search is then performed to determine the optimal distance to move
along the current search direction:
xk + 1 = xk + αk pk
Then the next search direction is determined so that it is conjugate to previous
search directions. The general procedure for determining the new search
direction is to combine the new steepest descent direction with the previous
search direction:
pk = – gk + βk pk – 1
The various versions of conjugate gradient are distinguished by the manner in
which the constant β k is computed. For the Fletcher-Reeves update the
procedure is
T
gk gk
β k = --------------------------- .
T
gk – 1 gk – 1
This is the ratio of the norm squared of the current gradient to the norm
squared of the previous gradient.
See [FlRe64] or [HDB96] for a discussion of the Fletcher-Reeves conjugate
gradient algorithm.
In the following code we re-initialize our previous network and retrain it using
the Fletcher-Reeves version of the conjugate gradient algorithm. The training
parameters for traincgf are epochs, show, goal, time, min_grad, max_fail,
srchFcn, scal_tol, alpha, beta, delta, gama, low_lim, up_lim, maxstep,
minstep, bmax. We have previously discussed the first six parameters. The
parameter srchFcn is the name of the line search function. It can be any of the
functions described later in this section (or a user-supplied function). The
remaining parameters are associated with specific line search routines and are
described later in this section. The default line search routine srchcha is used
in this example. traincgf generally converges in fewer iterations than
trainrp (although there is more computation required in each iteration).
5-21
5
Backpropagation
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traincgf');
net.trainParam.show = 5;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINCGF-srchcha, Epoch 0/300, MSE 2.15911/1e-05, Gradient
3.17681/1e-06
TRAINCGF-srchcha, Epoch 5/300, MSE 0.111081/1e-05, Gradient
0.602109/1e-06
TRAINCGF-srchcha, Epoch 10/300, MSE 0.0095015/1e-05, Gradient
0.197436/1e-06
TRAINCGF-srchcha, Epoch 15/300, MSE 0.000508668/1e-05,
Gradient 0.0439273/1e-06
TRAINCGF-srchcha, Epoch 17/300, MSE 1.33611e-06/1e-05,
Gradient 0.00562836/1e-06
TRAINCGF, Performance goal met.
a = sim(net,p)
a =
-1.0001
-1.0023
0.9999
1.0002
The conjugate gradient algorithms are usually much faster than variable
learning rate backpropagation, and are sometimes faster than trainrp,
although the results will vary from one problem to another. The conjugate
gradient algorithms require only a little more storage than the simpler
algorithms, so they are often a good choice for networks with a large number of
weights.
Try the Neural Network Design Demonstration nnd12cg [HDB96] for an
illustration of the performance of a conjugate gradient algorithm.
Polak-Ribiére Update (TRAINCGP)
Another version of the conjugate gradient algorithm was proposed by Polak
and Ribiére. As with the Fletcher-Reeves algorithm, the search direction at
each iteration is determined by
pk = – gk + βk pk – 1 .
5-22
Faster Training
For the Polak-Ribiére update, the constant β k is computed by
T
∆g k – 1 g k
β k = --------------------------- .
T
gk – 1 gk – 1
This is the inner product of the previous change in the gradient with the
current gradient divided by the norm squared of the previous gradient. See
[FlRe64] or [HDB96] for a discussion of the Polak-Ribiére conjugate gradient
algorithm.
In the following code we recreate our previous network and train it using the
Polak-Ribiére version of the conjugate gradient algorithm. The training
parameters for traincgp are the same as those for traincgf. The default line
search routine srchcha is used in this example. The parameters show and
epoch are set to the same values as they were for traincgf.
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traincgp');
net.trainParam.show = 5;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINCGP-srchcha, Epoch 0/300, MSE 1.21966/1e-05, Gradient
1.77008/1e-06
TRAINCGP-srchcha, Epoch 5/300, MSE 0.227447/1e-05, Gradient
0.86507/1e-06
TRAINCGP-srchcha, Epoch 10/300, MSE 0.000237395/1e-05,
Gradient 0.0174276/1e-06
TRAINCGP-srchcha, Epoch 15/300, MSE 9.28243e-05/1e-05,
Gradient 0.00485746/1e-06
TRAINCGP-srchcha, Epoch 20/300, MSE 1.46146e-05/1e-05,
Gradient 0.000912838/1e-06
TRAINCGP-srchcha, Epoch 25/300, MSE 1.05893e-05/1e-05,
Gradient 0.00238173/1e-06
TRAINCGP-srchcha, Epoch 26/300, MSE 9.10561e-06/1e-05,
Gradient 0.00197441/1e-06
TRAINCGP, Performance goal met.
a = sim(net,p)
a =
-0.9967
-1.0018
0.9958
1.0022
5-23
5
Backpropagation
The traincgp routine has performance similar to traincgf. It is difficult to
predict which algorithm will perform best on a given problem. The storage
requirements for Polak-Ribiére (four vectors) are slightly larger than for
Fletcher-Reeves (three vectors).
Powell-Beale Restarts (TRAINCGB)
For all conjugate gradient algorithms, the search direction will be periodically
reset to the negative of the gradient. The standard reset point occurs when the
number of iterations is equal to the number of network parameters (weights
and biases), but there are other reset methods which can improve the efficiency
of training. One such reset method was proposed by Powell [Powe77], based on
an earlier version proposed by Beale [Beal72]. For this technique we will
restart if there is very little orthogonality left between the current gradient and
the previous gradient. This is tested with the following inequality:
T
g k – 1 g k ≥ 0.2 g k
2
.
If this condition is satisfied, the search direction is reset to the negative of the
gradient.
In the following code we recreate our previous network and train it using the
Powell-Beale version of the conjugate gradient algorithm. The training
parameters for traincgb are the same as those for traincgf. The default line
search routine srchcha is used in this example. The parameters show and
epoch are set to the same values as they were for traincgf.
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traincgb');
net.trainParam.show = 5;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINCGB-srchcha, Epoch 0/300, MSE 2.5245/1e-05, Gradient
3.66882/1e-06
TRAINCGB-srchcha, Epoch 5/300, MSE 4.86255e-07/1e-05, Gradient
0.00145878/1e-06
TRAINCGB, Performance goal met.
a = sim(net,p)
a =
-0.9997
-0.9998
1.0000
1.0014
5-24
Faster Training
The traincgb routine has performance which is somewhat better than
traincgp for some problems, although performance on any given problem is
difficult to predict. The storage requirements for the Powell-Beale algorithm
(six vectors) are slightly larger than for Polak-Ribiére (four vectors).
Scaled Conjugate Gradient (TRAINSCG)
Each of the conjugate gradient algorithms which we have discussed so far
require a line search at each iteration. This line search is computationally
expensive, since it requires that the network response to all training inputs be
computed several times for each search. The scaled conjugate gradient
algorithm (SCG), developed by Moller [Moll93], was designed to avoid the time
consuming line search.This algorithm is too complex to explain in a few lines,
but the basic idea is to combine the model-trust region approach, which is used
in the Levenberg-Marquardt algorithm described later, with the conjugate
gradient approach. See {Moll93] for a detailed explanation of the algorithm.
In the following code we re-initialize our previous network and retrain it using
the scaled conjugate gradient algorithm. The training parameters for trainscg
are epochs, show, goal, time, min_grad, max_fail, sigma, lambda. We have
previously discussed the first six parameters. The parameter sigma determines
the change in the weight for the second derivative approximation. The
parameter lambda regulates the indefiniteness of the Hessian. The parameters
show and epoch are set to 10 and 300, respectively.
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainscg');
net.trainParam.show = 10;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINSCG, Epoch 0/300, MSE 4.17697/1e-05, Gradient 5.32455/
1e-06
TRAINSCG, Epoch 10/300, MSE 2.09505e-05/1e-05, Gradient
0.00673703/1e-06
TRAINSCG, Epoch 11/300, MSE 9.38923e-06/1e-05, Gradient
0.0049926/1e-06
TRAINSCG, Performance goal met.
a = sim(net,p)
a =
-1.0057
-1.0008
1.0019
1.0005
5-25
5
Backpropagation
The trainscg routine may require more iterations to converge than the other
conjugate gradient algorithms, but the number of computations in each
iteration is significantly reduced because no line search is performed. The
storage requirements for the scaled conjugate gradient algorithm are about the
same as those of Fletcher-Reeves.
Line Search Routines
Several of the conjugate gradient and quasi-Newton algorithms require that a
line search be performed. In this section we describe five different line searches
which can be used. In order to use any of these search routines you simply set
the training parameter srchFcn equal to the name of the desired search
function, as has been described in previous sections. It is often difficult to
predict which of these routines will provide the best results for any given
problem, but we have set the default search function to an appropriate initial
choice for each training function, so you may never need to modify this
parameter.
Golden Section Search (SRCHGOL)
The golden section search srchgol is a linear search which does not require the
calculation of the slope. This routine begins by locating an interval in which the
minimum of the performance occurs. This is accomplished by evaluating the
performance at a sequence of points, starting at a distance of delta and
doubling in distance each step, along the search direction. When the
performance increases between two successive iterations, a minimum has been
bracketed. The next step is to reduce the size of the interval containing the
minimum. Two new points are located within the initial interval. The values of
the performance at these two points determines a section of the interval which
can be discarded, and a new interior point is placed within the new interval.
This procedure is continued until the interval of uncertainty is reduced to a
width of tol, which is equal to delta/scale_tol.
See [HDB96], starting on page 12-16, for a complete description of the golden
section search. Try the Neural Network Design Demonstration nnd12sd1
[HDB96] for an illustration of the performance of the golden section search in
combination with a conjugate gradient algorithm.
5-26
Faster Training
Brent’s Search (SRCHBRE)
Brent’s search is a linear search which is a hybrid combination of the golden
section search and a quadratic interpolation. Function comparison methods,
like the golden section search, have a first-order rate of convergence, while
polynomial interpolation methods have an asymptotic rate which is faster than
superlinear. On the other hand, the rate of convergence for the golden section
search starts when the algorithm is initialized, whereas the asymptotic
behavior for the polynomial interpolation methods may take many iterations
to become apparent. Brent’s search attempts to combine the best features of
both approaches.
For Brent’s search we begin with the same interval of uncertainty that we used
with the golden section search, but some additional points are computed. A
quadratic function is then fitted to these points and the minimum of the
quadratic function is computed. If this minimum is within the appropriate
interval of uncertainty, it is used in the next stage of the search and a new
quadratic approximation is performed. If the minimum falls outside the known
interval of uncertainty, then a step of the golden section search is performed.
See [Bren73] for a complete description of this algorithm. This algorithm has
the advantage that it does not require computation of the derivative. The
derivative computation requires a backpropagation through the network,
which involves more computation than a forward pass. However, the algorithm
may require more performance evaluations than algorithms which use
derivative information.
Hybrid Bisection-Cubic Search (SRCHHYB)
Like Brent’s search, srchhyb is a hybrid algorithm. It is a combination of
bisection and cubic interpolation. For the bisection algorithm, one point is
located in the interval of uncertainty and the performance and its derivative
are computed. Based on this information, half of the interval of uncertainty is
discarded. In the hybrid algorithm, a cubic interpolation of the function is
obtained by using the value of the performance and its derivative at the two
end points. If the minimum of the cubic interpolation falls within the known
interval of uncertainty, then it is used to reduce the interval of uncertainty.
Otherwise, a step of the bisection algorithm is used.
See [Scal85] for a complete description of the hybrid bisection-cubic search.
This algorithm does require derivative information, so it performs more
computations at each step of the algorithm than the golden section search or
Brent’s algorithm.
5-27
5
Backpropagation
Charalambous’ Search (SRCHCHA)
The method of Charalambous srchcha was designed to be used in combination
with a conjugate gradient algorithm for neural network training. Like the
previous two methods, it is a hybrid search. It uses a cubic interpolation,
together with a type of sectioning.
See [Char92] for a description of Charalambous’ search. We have used this
routine as the default search for most of the conjugate gradient algorithms,
since it appears to produce excellent results for many different problems. It
does require the computation of the derivatives (backpropagation) in addition
to the computation of performance, but it overcomes this limitation by locating
the minimum with fewer steps. This is not true for all problems, and you may
want to experiment with other line searches.
Backtracking (SRCHBAC)
The backtracking search routine srchbac is best suited to use with the
quasi-Newton optimization algorithms. It begins with a step multiplier of 1 and
then backtracks until an acceptable reduction in the performance is obtained.
On the first step it uses the value of performance at the current point and at a
step multiplier of 1, and also the derivative of performance at the current point,
to obtain a quadratic approximation to the performance function along the
search direction. The minimum of the quadratic approximation becomes a
tentative optimum point (under certain conditions) and the performance at this
point is tested. If the performance is not sufficiently reduced, a cubic
interpolation is obtained and the minimum of the cubic interpolation becomes
the new tentative optimum point. This process is continued until a sufficient
reduction in the performance is obtained.
The backtracking algorithm is described in [DeSc83]. It was used as the default
line search for the quasi-Newton algorithms, although it may not be the best
technique for all problems.
5-28
Faster Training
Quasi-Newton Algorithms
BFGS Algorithm (TRAINBFG)
Newton’s method is an alternative to the conjugate gradient methods for fast
optimization. The basic step of Newton’s method is
–1
xk + 1 = xk – A k g k ,
where A k is the Hessian matrix (second derivatives) of the performance index
at the current values of the weights and biases. Newton’s method often
converges faster than conjugate gradient methods. Unfortunately, it is complex
and expensive to compute the Hessian matrix for feedforward neural networks.
There is a class of algorithms that are based on Newton’s method but which
don’t require calculation of second derivatives. These are called quasi-Newton
(or secant) methods. They update an approximate Hessian matrix at each
iteration of the algorithm. The update is computed as a function of the
gradient. The quasi-Newton method which has been most successful in
published studies is the Broyden, Fletcher, Goldfarb, and Shanno (BFGS)
update. This algorithm has been implemented in the trainbfg routine.
In the following code we re-initialize our previous network and retrain it using
the BFGS quasi-Newton algorithm. The training parameters for trainbfg are
the same as those for traincgf. The default line search routine srchbac is used
in this example. The parameters show and epoch are set to 5 and 300,
respectively.
5-29
5
Backpropagation
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainbfg');
net.trainParam.show = 5;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINBFG-srchbac, Epoch 0/300, MSE 0.492231/1e-05, Gradient
2.16307/1e-06
TRAINBFG-srchbac, Epoch 5/300, MSE 0.000744953/1e-05, Gradient
0.0196826/1e-06
TRAINBFG-srchbac, Epoch 8/300, MSE 7.69867e-06/1e-05, Gradient
0.00497404/1e-06
TRAINBFG, Performance goal met.
a = sim(net,p)
a =
-0.9995
-1.0004
1.0008
0.9945
The BFGS algorithm is described in [DeSc83]. This algorithm requires more
computation in each iteration and more storage than the conjugate gradient
methods, although it generally converges in fewer iterations. The approximate
2
2
Hessian must be stored, and its dimension is n × n , where n is equal to the
number of weights and biases in the network. For very large networks it may
be better to use Rprop or one of the conjugate gradient algorithms. For smaller
networks, however, trainbfg can be an efficient training function.
One Step Secant Algorithm (TRAINOSS)
Since the BFGS algorithm requires more storage and computation in each
iteration than the conjugate gradient algorithms, there is need for a secant
approximation with smaller storage and computation requirements. The one
step secant (OSS) method is an attempt to bridge the gap between the
conjugate gradient algorithms and the quasi-Newton (secant) algorithms. This
algorithm does not store the complete Hessian matrix; it assumes that at each
iteration the previous Hessian was the identity matrix. This has the additional
advantage that the new search direction can be calculated without computing
a matrix inverse.
In the following code we re-initialize our previous network and retrain it using
the one step secant algorithm. The training parameters for trainoss are the
same as those for traincgf. The default line search routine srchbac is used in
5-30
Faster Training
this example. The parameters show and epoch are set to 10 and 300,
respectively.
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainoss');
net.trainParam.show = 5;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINOSS-srchbac, Epoch 0/300, MSE 0.665136/1e-05, Gradient
1.61966/1e-06
TRAINOSS-srchbac, Epoch 5/300, MSE 0.000321921/1e-05, Gradient
0.0261425/1e-06
TRAINOSS-srchbac, Epoch 7/300, MSE 7.85697e-06/1e-05, Gradient
0.00527342/1e-06
TRAINOSS, Performance goal met.
a = sim(net,p)
a =
-1.0035
-0.9958
1.0014
0.9997
The one step secant method is described in [Batt92]. This algorithm requires
less storage and computation per epoch than the BFGS algorithm. It requires
slightly more storage and computation per epoch than the conjugate gradient
algorithms. It can be considered a compromise between full quasi-Newton
algorithms and conjugate gradient algorithms.
Levenberg-Marquardt (TRAINLM)
Like the quasi-Newton methods, the Levenberg-Marquardt algorithm was
designed to approach second-order training speed without having to compute
the Hessian matrix. When the performance function has the form of a sum of
squares (as is typical in training feedforward networks), then the Hessian
matrix can be approximated as
T
H = J J,
and the gradient can be computed as
T
g = J e
5-31
5
Backpropagation
where J is the Jacobian matrix, which contains first derivatives of the network
errors with respect to the weights and biases, and e is a vector of network
errors. The Jacobian matrix can be computed through a standard
backpropagation technique (see [HaMe94]) that is much less complex than
computing the Hessian matrix.
The Levenberg-Marquardt algorithm uses this approximation to the Hessian
matrix in the following Newton-like update:
T
–1 T
x k + 1 = x k – [ J J + µI ] J e .
When the scalar µ is zero, this is just Newton’s method, using the approximate
Hessian matrix. When µ is large, this becomes gradient descent with a small
step size. Newton’s method is faster and more accurate near an error
minimum, so the aim is to shift towards Newton’s method as quickly as
possible. Thus, µ is decreased after each successful step (reduction in
performance function) and is increased only when a tentative step would
increase the performance function. In this way, the performance function will
always be reduced at each iteration of the algorithm.
In the following code we re-initialize our previous network and retrain it using
the Levenberg-Marquardt algorithm. The training parameters for trainlm are
epochs, show, goal, time, min_grad, max_fail, mu, mu_dec, mu_inc, mu_max,
mem_reduc. We have discussed the first six parameters earlier. The parameter
mu is the initial value for µ. This value is multiplied by mu_dec whenever the
performance function is reduced by a step. It is multiplied by mu_inc whenever
a step would increase the performance function. If mu becomes larger than
mu_max, the algorithm is stopped. The parameter mem_reduc is used to control
5-32
Faster Training
the amount of memory used by the algorithm. It will be discussed in the next
section. The parameters show and epoch are set to 5 and 300, respectively.
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainlm');
net.trainParam.show = 5;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
TRAINLM, Epoch 0/300, MSE 2.7808/1e-05, Gradient 7.77931/1e-10
TRAINLM, Epoch 4/300, MSE 3.67935e-08/1e-05, Gradient
0.000808272/1e-10
TRAINLM, Performance goal met.
a = sim(net,p)
a =
-1.0000
-1.0000
1.0000
0.9996
The original description of the Levenberg-Marquardt algorithm is given in
[Marq63]. The application of Levenberg-Marquardt to neural network training
is described in [HaMe94] and starting on page 12-19 of [HDB96]. This
algorithm appears to be the fastest method for training moderate-sized
feedforward neural networks (up to several hundred weights). It also has a
very efficient MATLAB implementation, since the solution of the matrix
equation is a built-in function, so its attributes become even more pronounced
in a MATLAB setting.
Try the Neural Network Design Demonstration nnd12m [HDB96] for an
illustration of the performance of the batch gradient descent algorithm.
Reduced Memory Levenberg-Marquardt (TRAINLM)
The main drawback of the Levenberg-Marquardt algorithm is that it requires
the storage of some matrices which can be quite large for certain problems. The
size of the Jacobian matrix is Q × n , where Q is the number of training sets and
n is the number of weights and biases in the network. It turns out that this
matrix does not have to be computed and stored as a whole. For example, if we
were to divide the Jacobian into two equal submatrices we could compute the
approximate Hessian matrix as follows:
5-33
5
Backpropagation
T
T
H = J J =
J1
T
J2
T
T
J1 J2 = J1 J1 + J2 J2 .
Therefore the full Jacobian does not have to exist at one time. The approximate
Hessian can be computed by summing a series of subterms. Once one subterm
has been computed, the corresponding submatrix of the Jacobian can be
cleared.
When using the training function trainlm, the parameter mem_reduc is used to
determine how many rows of the Jacobian are to be computed in each
submatrix. If mem_reduc is set to 1, then the full Jacobian is computed, and no
memory reduction is achieved. If mem_reduc is set to 2, then only half of the
Jacobian will be computed at one time. This saves half of the memory used by
the calculation of the full Jacobian.
There is a drawback to using memory reduction. A significant computational
overhead is associated with computing the Jacobian in submatrices. If you
have enough memory available, then it is better to set mem_reduc to 1 and to
compute the full Jacobian. If you have a large training set, and you are running
out of memory, then you should set mem_reduc to 2, and try again. If you still
run out of memory, continue to increase mem_reduc.
Even if you use memory reduction, the Levenberg-Marquardt algorithm will
always compute the approximate Hessian matrix, which has dimensions n × n .
If your network is very large, then you may run out of memory. If this is the
case, then you will want to try trainoss, trainrp, or one of the conjugate
gradient algorithms.
5-34
Speed and Memory Comparison
Speed and Memory Comparison
It is very difficult to know which training algorithm will be the fastest for a
given problem. It will depend on many factors, including the complexity of the
problem, the number of data points in the training set, the number of weights
and biases in the network, and the error goal. In general, on networks which
contain up to a few hundred weights the Levenberg-Marquardt algorithm will
have the fastest convergence. This advantage is especially noticeable if very
accurate training is required. The quasi-Newton methods are often the next
fastest algorithms on networks of moderate size. The BFGS algorithm does
require storage of the approximate Hessian matrix, but is generally faster than
the conjugate gradient algorithms. Of the conjugate gradient algorithms, the
Powell-Beale procedure requires the most storage, but usually has the fastest
convergence. Rprop and the scaled conjugate gradient algorithm do not require
a line search and have small storage requirements. They are reasonably fast,
and are very useful for large problems. The variable learning rate algorithm is
usually much slower than the other methods, and has about the same storage
requirements as Rprop, but it can still be useful for some problems. There are
certain situations in which it is better to converge more slowly. For example,
when using early stopping you may have inconsistent results if you use an
algorithm which converges too quickly. You may overshoot the point at which
the error on the validation set is minimized.
For most situations, we recommend that you try the Levenberg-Marquardt
algorithm first. If this algorithm requires too much memory, then try the BFGS
algorithm, or one of the conjugate gradient methods. The Rprop algorithm is
also very fast, and has relatively small memory requirements.
The following table gives some example convergence times for the various
algorithms on one particular regression problem. In this problem a 1-10-1
network was trained on a data set with 41 input/output pairs until a mean
square error performance of 0.01 was obtained. Twenty different test runs were
made for each training algorithm on a Macintosh Powerbook 1400 to obtain the
average numbers shown in the table. These numbers should be used with
caution, since the performances shown here may not be typical for these
algorithms on other types of problems.
5-35
5
Backpropagation
You may notice that there is not a clear relationship between the number of
floating point operations and the time required to reach convergence. This is
because some of the algorithms can take advantage of efficient built-in
MATLAB functions. This is especially true for the Levenberg-Marquardt
algorithm.
Function
5-36
Technique
Time
Epochs
Mflops
traingdx
Variable Learning Rate
57.71
980
2.50
trainrp
Rprop
12.95
185
0.56
trainscg
Scaled Conj. Grad.
16.06
106
0.70
traincgf
Fletcher-Powell CG
16.40
81
0.99
traincgp
Polak-Ribiére CG
19.16
89
0.75
traincgb
Powell-Beale CG
15.03
74
0.59
trainoss
One-Step-Secant
18.46
101
0.75
trainbfg
BFGS quasi-Newton
10.86
44
1.02
trainlm
Levenberg-Marquardt
1.87
6
0.46
Improving Generalization
Improving Generalization
One of the problems that occurs during neural network training is called
overfitting. The error on the training set is driven to a very small value, but
when new data is presented to the network the error is large. The network has
memorized the training examples, but it has not learned to generalize to new
situations.
The following figure shows the response of a 1-20-1 neural network which has
been trained to approximate a noisy sine function. The underlying sine
function is shown by the dotted line, the noisy measurements are given by the
‘+’ symbols, and the neural network response is given by the solid line. Clearly
this network has overfit the data and will not generalize well.
Function Approximation
1.5
1
Output
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1
One method for improving network generalization is to use a network which is
just large enough to provide an adequate fit. The larger a network you use, the
more complex the functions that the network can create. If we use a small
enough network, it will not have enough power to overfit the data. Run the
Neural Network Design Demonstration nnd11gn [HDB96] to investigate how
reducing the size of a network can prevent overfitting.
The problem is that it is difficult to know beforehand how large a network
should be for a specific application. There are two other methods for improving
5-37
5
Backpropagation
generalization which are implemented in the Neural Network Toolbox:
regularization and early stopping. The next few subsections will describe these
two techniques, and the routines to implement them.
Regularization
The first method for improving generalization is called regularization. This
involves modifying the performance function, which is normally chosen to be
the sum of squares of the network errors on the training set. The next
subsection will explain how the performance function can be modified, and the
following subsection will describe two routines which will automatically set the
optimal performance function to achieve the best generalization.
Modified Performance Function
The typical performance function that is used for training feedforward neural
networks is the mean sum of squares of the network errors:
1
F = mse = ----N
N
∑
i=1
1
( e i ) = ----N
N
∑ ( ti – ai )
2
2
.
i=1
It is possible to improve generalization if we modify the performance function
by adding a term that consists of the mean of the sum of squares of the network
weights and biases:
msereg = γmse + ( 1 – γ )msw ,
where γ is the performance ratio, and
1
msw = --n
n
∑ wj .
2
j=1
Using this performance function will cause the network to have smaller
weights and biases, and this will force the network response to be smoother and
less likely to overfit.
In the following code we re-initialize our previous network and retrain it using
the BFGS algorithm with the regularized performance function. Here we set
5-38
Improving Generalization
the performance ratio to 0.5, which gives equal weight to the mean square
errors and the mean square weights.
net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainbfg');
net.performFcn = 'msereg';
net.performParam.ratio = 0.5;
net.trainParam.show = 5;
net.trainParam.epochs = 300;
net.trainParam.goal = 1e-5;
p = [-1 -1 2 2;0 5 0 5];
t = [-1 -1 1 1];
net=train(net,p,t);
The problem with regularization is that it is difficult to determine the optimum
value for the performance ratio parameter. If we make this parameter too
large, we may get overfitting. If the ratio is too small, the network will not
adequately fit the training data. In the next section we will describe two
routines which automatically set the regularization parameters.
Automated Regularization (TRAINBR)
It is desirable to determine the optimal regularization parameters in an
automated fashion. One approach to this process is the Bayesian framework of
David MacKay [MacK92]. In this framework the weights and biases of the
network are assumed to be random variables with specified distributions. The
regularization parameters are related to the unknown variances associated
with these distributions. We can then estimate these parameters using
statistical techniques.
A detailed discussion of Bayesian regularization is beyond the scope of this
users guide. A detailed discussion of the use of Bayesian regularization, in
combination with Levenberg-Marquardt training, can be found in [FoHa97].
5-39
5
Backpropagation
Bayesian regularization has been implemented in the function trainbr. The
following code shows how we can train a 1-20-1 network using this function to
approximate the noisy sine wave shown earlier in this section.
net=newff([-1 1],[20,1],{'tansig','purelin'},'trainbr');
net.trainParam.show = 10;
net.trainParam.epochs = 50;
randn('seed',192736547);
p = [-1:.05:1];
t = sin(2*pi*p)+0.1*randn(size(p));
net = init(net);
net=train(net,p,t);
TRAINBR, Epoch 0/50, SSE 107.962/0, Gradient 9.51e+01/1.00e-10,
#Par 6.10e+01/61
TRAINBR, Epoch 10/50, SSE 20.5/0, Gradient 1.88e-01/1.00e-10,
#Par 1.82e+01/61
TRAINBR, Epoch 20/50, SSE 20.5/0, Gradient 4.07e-02/1.00e-10,
#Par 1.65e+01/61
TRAINBR, Epoch 30/50, SSE 20.5/0, Gradient 5.57e-01/1.00e-10,
#Par 1.55e+01/61
TRAINBR, Epoch 40/50, SSE 20.5/0, Gradient 2.76e-01/1.00e-10,
#Par 1.48e+01/61
TRAINBR, Epoch 50/50, SSE 20.5/0, Gradient 4.05e-01/1.00e-10,
#Par 1.42e+01/61
One feature of this algorithm is that it provides a measure of how many
network parameters (weights and biases) are being effectively used by the
network. In this case, the final trained network uses approximately 14
parameters (indicated by #Par in the printout), out of the 61 total weights and
biases in the 1-20-1 network. This effective number of parameters should
remain the same, no matter how large the total number of parameters in the
network becomes.
The following figure shows the response of the network. In contrast to the
previous figure, in which a 1-20-1 network overfit the data, here we see that
the network response is very close to the underlying sine function (dotted line),
and, therefore, the network will generalize well to new inputs. We could have
tried an even larger network, but the network response would never overfit the
data. This eliminates the guesswork required in determining the optimum
network size.
5-40
Improving Generalization
Function Approximation
1.5
1
Output
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0
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0.8
1
Early Stopping
Another method for improving generalization is called early stopping. In this
technique the available data is divided into three subsets. The first subset is
the training set which is used for computing the gradient and updating the
network weights and biases. The second subset is the validation set. The error
on the validation set is monitored during the training process. The validation
error will normally decrease during the initial phase of training, as does the
training set error. However, when the network begins to overfit the data, the
error on the validation set will typically begin to rise. When the validation error
increases for a specified number of iterations, the training is stopped, and the
weights and biases at the minimum of the validation error are returned.
The test set error is not used during the training, but it is used to compare
different models. It is also useful to plot the test set error during the training
process. If the error in the test set reaches a minimum at a significantly
different iteration number than the validation set error, this may indicate a
poor division of the data set.
Early stopping can be used with any of the training functions which were
described earlier in this chapter. You simply need to pass the validation data
to the training function. The following sequence of commands demonstrates
how to use the early stopping function.
5-41
5
Backpropagation
First we will create a simple test problem. For our training set we will generate
a noisy sine wave with input points ranging from -1 to 1 at steps of 0.05.
p = [-1:0.05:1];
t = sin(2*pi*p)+0.1*randn(size(p));
Next we will generate the validation set. The inputs will range from -1 to 1, as
in the test set, but we will offset them slightly. To make the problem more
realistic, we also add a different noise sequence to the underlying sine wave.
Notice that the validation set is contained in a structure which contains both
the inputs and the targets.
v.P = [-0.975:.05:0.975];
v.T = sin(2*pi*v.P)+0.1*randn(size(v.P));
We will now create a 1-20-1 network, as in our previous example with
regularization, and train it. (Notice that the validation structure is passed to
train after the initial input and layer conditions, which are null vectors in this
case since the network contains no delays.) For this example we will use the
training function traingdx, although early stopping can be used with any of
the other training functions we have discussed in this chapter.
net=newff([-1 1],[20,1],{'tansig','purelin'},'traingdx');
net.trainParam.show = 25;
net.trainParam.epochs = 300;
net = init(net);
[net,tr]=train(net,p,t,[],[],v);
TRAINGDX, Epoch 0/300, MSE 9.39342/0, Gradient 17.7789/1e-06
TRAINGDX, Epoch 25/300, MSE 0.312465/0, Gradient 0.873551/1e-06
TRAINGDX, Epoch 50/300, MSE 0.102526/0, Gradient 0.206456/1e-06
TRAINGDX, Epoch 75/300, MSE 0.0459503/0, Gradient 0.0954717/1e-06
TRAINGDX, Epoch 100/300, MSE 0.015725/0, Gradient 0.0299898/1e-06
TRAINGDX, Epoch 125/300, MSE 0.00628898/0, Gradient 0.042467/
1e-06
TRAINGDX, Epoch 131/300, MSE 0.00650734/0, Gradient 0.133314/
1e-06
TRAINGDX, Validation stop.
5-42
Improving Generalization
In the following figure we have a graph of the network response. We can see
that the network did not overfit the data, as in the earlier example, although
the response is not extremely smooth, as when using regularization. This is
characteristic of early stopping.
Function Approximation
1.5
1
Output
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5-43
5
Backpropagation
Preprocessing and Postprocessing
Neural network training can be made more efficient if certain preprocessing
steps are performed on the network inputs and targets. In this section we
describe several preprocessing routines which can be used.
Min and Max (PREMNMX, POSTMNMX, TRAMNMX)
Before training, it is often useful to scale the inputs and targets so that they
always fall within a specified range. The function premnmx can be used to scale
inputs and targets so that they fall in the range [-1,1]. The following code
illustrates the use of this function.
[pn,minp,maxp,tn,mint,maxt] = premnmx(p,t);
net=train(net,pn,tn);
The original network inputs and targets are given in the matrices p and t. The
normalized inputs and targets, pn and tn, that are returned will all fall in the
interval [-1,1]. The vectors minp and maxp contain the minimum and maximum
values of the original inputs, and the vectors mint and maxt contain the
minimum and maximum values of the original targets. After the network has
been trained, these vectors should be used to transform any future inputs
which are applied to the network. They effectively become a part of the
network, just like the network weights and biases.
If premnmx is used to scale both the inputs and targets, then the output of the
network will be trained to produce outputs in the range [-1,1]. If you want to
convert these outputs back into the same units which were used for the original
targets, then you should use the routine postmnmx. In the following code we
simulate the network which was trained in the previous code and then convert
the network output back into the original units.
an = sim(net,pn);
a = postmnmx(an,mint,maxt);
The network output an will correspond to the normalized targets tn. The
un-normalized network output a is in the same units as the original targets t.
If premnmx is used to preprocess the training set data, then whenever the
trained network is used with new inputs they should be preprocessed with the
minimum and maximums which were computed for the training set. This can
5-44
Preprocessing and Postprocessing
be accomplished with the routine tramnmx. In the following code we apply a new
set of inputs to the network we have already trained.
pnewn = tramnmx(pnew,minp,maxp);
anewn = sim(net,pnewn);
anew = postmnmx(anewn,mint,maxt);
Mean and Stand. Dev. (PRESTD, POSTSTD, TRASTD)
Another approach for scaling network inputs and targets is to normalize the
mean and standard deviation of the training set. This procedure is
implemented in the function prestd. It normalizes the inputs and targets so
that they will have zero mean and unity standard deviation. The following code
illustrates the use of prestd.
[pn,meanp,stdp,tn,meant,stdt] = prestd(p,t);
The original network inputs and targets are given in the matrices p and t. The
normalized inputs and targets, pn and tn, that are returned will have zero
means and unity standard deviation. The vectors meanp and stdp contain the
mean and standard deviations of the original inputs, and the vectors meant and
stdt contain the means and standard deviations of the original targets. After
the network has been trained, these vectors should be used to transform any
future inputs which are applied to the network. They effectively become a part
of the network, just like the network weights and biases.
If prestd is used to scale both the inputs and targets, then the output of the
network will be trained to produce outputs with zero mean and unity standard
deviation. If you want to convert these outputs back into the same units which
were used for the original targets, then you should use the routine poststd. In
the following code we simulate the network which was trained in the previous
code and then convert the network output back into the original units.
an = sim(net,pn);
a = poststd(an,meant,stdt);
The network output an will correspond to the normalized targets tn. The
un-normalized network output a is in the same units as the original targets t.
If prestd is used to preprocess the training set data, then whenever the trained
network is used with new inputs they should be preprocessed with the means
and standard deviations which were computed for the training set. This can be
5-45
5
Backpropagation
accomplished with the routine trastd. In the following code we apply a new set
of inputs to the network we have already trained.
pnewn = trastd(pnew,meanp,stdp);
anewn = sim(net,pnewn);
anew = poststd(anewn,meant,stdt);
Principal Component Analysis (PREPCA, TRAPCA)
In some situations the dimension of the input vector is large, but the
components of the vectors are highly correlated (redundant). It is useful in this
situation to reduce the dimension of the input vectors. An effective procedure
for performing this operation is principal component analysis. This technique
has three effects: it orthogonalizes the components of the input vectors (so that
they are uncorrelated with each other); it orders the resulting orthogonal
components (principal components) so that those with the largest variation
come first; and it eliminates those components which contribute the least to the
variation in the data set. The following code illustrates the use of prepca,
which performs a principal component analysis.
[pn,meanp,stdp] = prestd(p);
[ptrans,transMat] = prepca(pn,0.02);
Note that we first normalize the input vectors, using prestd, so that they have
zero mean and unity variance. This is a standard procedure when using
principal components. In this example, the second argument passed to prepca
is 0.02. This means that prepca will eliminate those principal components
which contribute less than 2% to the total variation in the data set. The matrix
ptrans contains the transformed input vectors. The matrix transMat contains
the principal component transformation matrix. After the network has been
trained, this matrix should be used to transform any future inputs which are
applied to the network. It effectively becomes a part of the network, just like
the network weights and biases. If you multiply the normalized input vectors
pn by the transformation matrix transMat, you will obtain the transformed
input vectors ptrans.
If prepca is used to preprocess the training set data, then whenever the trained
network is used with new inputs they should be preprocessed with the
transformation matrix which was computed for the training set.
5-46
Preprocessing and Postprocessing
This can be accomplished with the routine trapca. In the following code we
apply a new set of inputs to a network we have already trained.
pnewn = trastd(pnew,meanp,stdp);
pnewtrans = trapca(pnewn,transMat);
a = sim(net,pnewtrans);
Post-training Analysis (POSTREG)
The performance of a trained network can be measured to some extent by the
errors on the training, validation and test sets, but it is often useful to
investigate the network response in more detail. One option is to perform a
regression analysis between the network response and the corresponding
targets. The routine postreg is designed to perform this analysis.
The following commands illustrate how we can perform a regression analysis
on the network which we previously trained in the early stopping section.
a = sim(net,p);
[m,b,r] = postreg(a,t)
m =
0.9874
b =
-0.0067
r =
0.9935
Here we pass the network output and the corresponding targets to postreg. It
returns three parameters. The first two, m and b, correspond to the slope and
the y-intercept of the best linear regression relating targets to network
outputs. If we had a perfect fit (outputs exactly equal to targets), the slope
would be 1, and the y-intercept would be 0. In this example we can see that the
numbers are very close. The third variable returned by postreg is the
correlation coefficient (R-value) between the outputs and targets. It is a
measure of how well the variation in the output is explained by the targets. If
this number is equal to 1, then there is perfect correlation between targets and
outputs. In our example here the number is very close to 1, which indicates a
good fit.
The following figure illustrates the graphical output provided by postreg. The
network outputs are plotted versus the targets as open circles. The best linear
fit is indicated by a dashed line. The perfect fit (output equal to targets) is
5-47
5
Backpropagation
indicated by the solid line. In this example it is difficult to distinguish the best
linear fit line from the perfect fit line, because the fit is so good.
Best Linear Fit: A = (0.987) T + (-0.00667)
1.5
Data Points
A=T
Best Linear Fit
1
A
0.5
0
-0.5
R = 0.994
-1
-1.5
-1.5
5-48
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0
T
0.5
1
1.5
Sample Training Session
Sample Training Session
We have covered a number of different concepts in this chapter. At this point it
might be useful to put some of these ideas together with an example of how a
typical training session might go.
For this example we are going to use data from a medical application [PuLu92].
We want to design an instrument which can determine serum cholesterol levels
from measurements of spectral content of a blood sample. We have a total of
264 patients for which we have measurements of 21 wavelengths of the
spectrum. For the same patients we also have measurements of hdl, ldl and
vldl cholesterol levels, based on serum separation. The first step is to load the
data into the workspace and perform a principal component analysis.
load choles_all
[pn,meanp,stdp,tn,meant,stdt] = prestd(p,t);
[ptrans,transMat] = prepca(pn,0.001);
Here we have conservatively retained those principal components which
account for 99.9% of the variation in the data set. Let’s check the size of the
transformed data.
[R,Q] = size(ptrans)
R =
4
Q =
264
There was apparently significant redundancy in the data set, since the
principal component analysis has reduced the size of the input vectors from 21
to 4.
The next step is to divide the data up into training, validation and test subsets.
We will take one fourth of the data for the validation set, one fourth for the test
set and one half for the training set. We will pick the sets as equally spaced
points throughout the original data.
iitst = 2:4:Q;
iival = 4:4:Q;
iitr = [1:4:Q 3:4:Q];
v.P = ptrans(:,iival); v.T = tn(:,iival);
t.P = ptrans(:,iitst); t.V = tn(:,iitst);
ptr = ptrans(:,iitr); ttr = tn(:,iitr);
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5
Backpropagation
We are now ready to create a network and train it. For this example we will try
a two-layer network, with tan-sigmoid transfer function in the hidden layer
and a linear transfer function in the output layer. This is a useful structure for
function approximation (or regression) problems. As an initial guess, we will
use five neurons in the hidden layer. The network should have three output
neurons since there are three targets. We will use the Levenberg-Marquardt
algorithm for training.
net = newff(minmax(ptr),[5 3],{'tansig' 'purelin'},'trainlm');
[net,tr]=train(net,ptr,ttr,[],[],v,t);
TRAINLM, Epoch 0/100, MSE 3.11023/0, Gradient 804.959/1e-10
TRAINLM, Epoch 15/100, MSE 0.330295/0, Gradient 104.219/1e-10
TRAINLM, Validation stop.
The training stopped after 15 iterations because the validation error increased.
It is a useful diagnostic tool to plot the training, validation and test errors to
check the progress of training. We can do that with the following commands.
plot(tr.epoch,tr.perf,tr.epoch,tr.vperf,tr.epoch,tr.tperf)
legend('Training','Validation','Test',-1);
ylabel('Squared Error'); xlabel('Epoch')
The result is shown in the following figure. The result here is reasonable, since
the test set error and the validation set error have similar characteristics, and
it doesn’t appear that any significant overfitting has occurred.
5-50
Sample Training Session
3.5
3
Training
Validation
Test
Squared Error
2.5
2
1.5
1
0.5
0
0
5
10
15
Epoch
The next step is to perform some analysis of the network response. We will put
the entire data set through the network (training, validation and test) and will
perform a linear regression between the network outputs and the
corresponding targets. First we will need to un-normalize the network outputs.
an = sim(net,ptrans);
a = poststd(an,meant,stdt);
for i=1:3
figure(i)
[m(i),b(i),r(i)] = postreg(a(i,:),t(i,:));
end
In this case we have three outputs, so we perform three regressions. The
results are shown in the following figures.
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5
Backpropagation
Best Linear Fit: A = (0.764) T + (14)
400
350
Data Points
A=T
Best Linear Fit
300
250
A
200
150
100
R = 0.886
50
0
-50
0
50
100
150
200
T
250
300
350
400
Best Linear Fit: A = (0.753) T + (31.7)
350
300
Data Points
A=T
Best Linear Fit
250
A
200
150
100
R = 0.862
50
0
0
50
100
150
200
T
5-52
250
300
350
Sample Training Session
Best Linear Fit: A = (0.346) T + (28.3)
120
Data Points
A=T
Best Linear Fit
100
A
80
60
40
R = 0.563
20
0
0
20
40
60
T
80
100
120
The first two outputs seem to track the targets reasonably well (this is a
difficult problem), and the R-values are almost 0.9. The third output (vldl
levels) is not well modeled. We probably need to work more on that problem.
We might go on to try other network architectures (more hidden layer
neurons), or to try Bayesian regularization instead of early stopping for our
training technique. Of course there is also the possibility that vldl levels cannot
be accurately computed based on the given spectral components.
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Backpropagation
Limitations and Cautions
The gradient descent algorithm is generally very slow, because it requires
small learning rates for stable learning. The momentum variation is usually
faster than simple gradient descent, since it allows higher learning rates while
maintaining stability, but it is still too slow for many practical applications.
These two methods would normally be used only when incremental training is
desired. You would normally use Levenberg-Marquardt training if you have
enough memory available. If memory is a problem, then there are a variety of
other fast algorithms available.
Multi-layered networks are capable of performing just about any linear or
nonlinear computation, and can approximate any reasonable function
arbitrarily well. Such networks overcome the problems associated with the
perceptron and linear networks. However, while the network being trained
may be capable theoretically of performing correctly, backpropagation and its
variations may not always find a solution. See page 12-8 of [HDB96] for a
discussion of convergence to local minimum points.
Picking the learning rate for a nonlinear network is a challenge. As with linear
networks, a learning rate that is too large leads to unstable learning.
Conversely, a learning rate that is too small results in incredibly long training
times. Unlike linear networks, there is no easy way of picking a good learning
rate for nonlinear multilayer networks. See page 12-8 of [HDB96] for examples
of choosing the learning rate. With the faster training algorithms, the default
parameter values will normally perform adequately.
The error surface of a nonlinear network is more complex than the error
surface of a linear network. To understand this complexity see the figures on
pages 12-5 to 12-7 of [HDB96], which show three different error surfaces for a
multilayer network. The problem is that nonlinear transfer functions in
multilayer networks introduce many local minima in the error surface. As
gradient descent is performed on the error surface it is possible for the network
solution to become trapped in one of these local minima. This may happen
depending on the initial starting conditions. Settling in a local minimum may
be good or bad depending on how close the local minimum is to the global
minimum and how low an error is required. In any case, be cautioned that
although a multilayer backpropagation network with enough neurons can
implement just about any function, backpropagation will not always find the
correct weights for the optimum solution. You may wish to re-initialize the
network and retrain several times to guarantee that you have the best solution.
5-54
Limitations and Cautions
Networks are also sensitive to the number of neurons in their hidden layers.
Too few neurons can lead to underfitting. Too many neurons can contribute to
overfitting, in which all training points are well fit, but the fitting curve takes
wild oscillations between these points. Ways of dealing with various of these
issues have been discussed in the section on improving generalization. This
topic is also discussed starting on page 11-21 of [HDB96].
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5
Backpropagation
Summary
Backpropagation can train multilayer feed-forward networks with
differentiable transfer functions to perform function approximation, pattern
association, and pattern classification. (Other types of networks can be trained
as well, although the multilayer network is most commonly used.) The term
backpropagation refers to the process by which derivatives of network error,
with respect to network weights and biases, can be computed. This process can
be used with a number of different optimization strategies.
The architecture of a multilayer network is not completely constrained by the
problem to be solved. The number of inputs to the network is constrained by
the problem, and the number of neurons in the output layer is constrained by
the number of outputs required by the problem. However, the number of layers
between network inputs and the output layer and the sizes of the layers are up
to the designer.
The two-layer sigmoid/linear network can represent any functional
relationship between inputs and outputs if the sigmoid layer has enough
neurons.
There are several different backpropagation training algorithms. They have a
variety of different computation and storage requirements, and no one
algorithm is best suited to all locations. The following list summarizes the
training algorithms included in the toolbox.
5-56
Function
Description
traingd
Basic gradient descent. Slow response, can be used in
incremental mode training.
traingdm
Gradient descent with momentum. Generally faster than
traingd. Can be used in incremental mode training.
traingdx
Adaptive learning rate. Faster training than traingd, but
can only be used in batch mode training.
trainrp
Resilient backpropagation. Simple batch mode training
algorithm with fast convergence and minimal storage
requirements.
Summary
Function
Description
traincgf
Fletcher-Reeves conjugate gradient algorithm. Has
smallest storage requirements of the conjugate gradient
algorithms.
traincgp
Polak-Ribiére conjugate gradient algorithm. Slightly larger
storage requirements than traincgf. Faster convergence
on some problems.
traincgb
Powell-Beale conjugate gradient algorithm. Slightly larger
storage requirements than traincgp. Generally faster
convergence.
trainscg
Scaled conjugate gradient algorithm. The only conjugate
gradient algorithm that requires no line search.
trainbfg
BFGS quasi-Newton method. Requires storage of
approximate Hessian matrix and has more computation in
each iteration than conjugate gradient algorithms, but
usually converges in fewer iterations.
trainoss
One step secant method. Compromise between conjugate
gradient methods and quasi-Newton methods.
trainlm
Levenberg-Marquardt algorithm. Fastest training
algorithm for networks of moderate size. Has memory
reduction feature for use when the training set is large.
trainbr
Bayesian regularization. Modification of the
Levenberg-Marquardt training algorithm to produce
networks which generalize well. Reduces the difficulty of
determining the optimum network architecture.
One problem which can occur when training neural networks is that the
network can overfit on the training set and not generalize well to new data
outside the training set. This can be prevented by training with trainbr, but
it can also be prevented by using early stopping with any of the other training
routines. This requires that the user pass a validation set to the training
algorithm, in addition to the standard training set.
5-57
5
Backpropagation
In order to produce the most efficient training, it is often helpful to preprocess
the data before training. It is also helpful to analyze the network response after
training is complete. The toolbox contains a number of routines for pre- and
post-processing. They are summarized in the following table.
5-58
Function
Description
premnmx
Normalize data to fall in the range [-1,1].
postmnmx
Inverse of premnmx. Used to convert data back to standard
units.
tramnmx
Normalize data using previously computed minimums and
maximums. Used to preprocess new inputs to networks
which have been trained with data normalized with premnmx.
prestd
Normalize data to have zero mean and unity standard
deviation.
poststd
Inverse of prestd. Used to convert data back to standard
units.
trastd
Normalize data using previously computed means and
standard deviations. Used to preprocess new inputs to
networks which have been trained with data normalized
with prestd.
prepca
Principal component analysis. Reduces dimension of input
vector and un-correlates components of input vectors.
trapca
Preprocess data using previously computed principal
component transformation matrix. Used to preprocess new
inputs to networks which have been trained with data
transformed with prepca.
postreg
Linear regression between network outputs and targets.
Used to determine the adequacy of network fit.
6
Radial Basis Networks
Introduction . . . . . . . . . . . . . . . . . . . . 6-2
Important Radial Basis Functions . . . . . . . . . . . . 6-2
Radial Basis Functions . . .
Neuron Model . . . . . . . .
Network Architecture . . . . .
Exact Design (NEWRBE) . . .
More Efficient Design (NEWRB)
Demonstrations . . . . . . .
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6-3
6-3
6-4
6-5
6-7
6-8
Generalized Regression Networks . . . . . . . . . . 6-9
Network Architecture . . . . . . . . . . . . . . . . . 6-9
Design (NEWGRNN) . . . . . . . . . . . . . . . . . 6-11
Probabilistic Neural Networks . . . . . . . . . . . 6-12
Network Architecture . . . . . . . . . . . . . . . . . 6-12
Design (NEWPNN) . . . . . . . . . . . . . . . . . . 6-13
Summary . . . . . . . . . . . . . . . . . . . . . 6-16
Figures . . . . . . . . . . . . . . . . . . . . . . . 6-17
New Functions . . . . . . . . . . . . . . . . . . . 6-19
6
Radial Basis Networks
Introduction
Radial basis networks may require more neurons than standard feed-forward
backpropagation networks, but often they can be designed in a fraction of the
time it takes to train standard feed-forward networks. They work best when
many training vectors are available.
You may want to consult the following paper on this subject:
Chen, S., C.F.N. Cowan, and P. M. Grant, “Orthogonal Least Squares Learning
Algorithm for Radial Basis Function Networks,” IEEE Transactions on Neural
Networks, vol. 2, no. 2, March 1991, pp. 302-309.
This chapter discusses two variants of radial basis networks, Generalized
Regression networks (GRNN) and Probabilistic neural networks (PNN). You
may wish to read about them in P.D. Wasserman, Advanced Methods in Neural
Computing, New York: Van Nostrand Reinhold, 1993 on pp. 155-61, and pp.
35-55 respectively.
Important Radial Basis Functions
Radial basis networks can be designed with either newrbe or newrb. GRNN and
PNN can be designed with newgrnn and newpnn respectively.
Type help radbasis to see a listing of all functions and demonstrations related
to radial basis networks.
6-2
Radial Basis Functions
Radial Basis Functions
Neuron Model
Here is a radial basis network with R inputs.
Input
Radial Basis Neuron
1,1
p
p1
p2
3
... w1, R
AA
A
AA A
w
+
_
n
|| dist ||
a
b
p
R
1
a = radbas( || w-p || b)
Notice that the expression for the net input of a radbas neuron is different from
that of neurons in previous chapters. Here the net input to the radbas transfer
function is the vector distance between its weight vector w and the input vector
p, multiplied by the bias b. (The dist box in this figure accepts the input
vector p and the single row input weight matrix, and produces the dot product
of the two.)
The transfer function for a radial basis neuron is:
radbas ( n ) = e
–n
2
Here is a plot of the radbas transfer function.
a
1.0
0.5
0.0
AA
AA
n
-0.833
+0.833
a = radbas(n)
Radial Basis Function
6-3
6
Radial Basis Networks
The radial basis function has a maximum of 1 when its input is 0. As the
distance between w and p decreases, the output increases. Thus a radial basis
neuron acts as a detector which produces 1 whenever the input p is identical to
its weight vector p.
The bias b allows the sensitivity of the radbas neuron to be adjusted. For
example, if a neuron had a bias of 0.1 it would output 0.5 for any input vector
p at vector distance of 8.326 (0.8326/b) from its weight vector w.
Network Architecture
Radial basis networks consist of two layers: a hidden radial basis layer of S1
neurons and an output linear layer of S2 neurons.
AAA
AAA
AAA
AA
AA
AA
AA
AA
AA
AAA
AAA AAAA AA
Radial Basis Layer
Input
S1xR
p
Rx1
1
R
Linear Layer
IW1,1
|| dist ||
a1
S1x1
.*
S1x1
n1
S2xS1
a2
S2x1
n2
S2x1
S1x1
b1
S1x1
LW2,1
1
S1
a 1 = radbas ( || IW1,1 - p || b 1)
i
i
i
b2
S2x1
Where...
R = # elements
in input
vector
S2
a2 = purelin (LW2,1 a1 +b2)
S1 = # Neurons
in layer 1
S2 = # Neurons
in layer 2
ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1
The dist box in this figure accepts the input vector p and the input weight
matrix IW1,1, and produces a vector having S1 elements. The elements are the
distances between the input vector and vectors iIW1,1 formed from the rows of
the input weight matrix.
The bias vector b1 and the output of dist are combined with the MATLAB
operation .* , which does element-by-element multiplication.
The output of the first layer for a feed forward network net can be obtained with
the following code:
a{1} = radbas(netprod(dist(net.IW{1,1},p),net.b{1}))
6-4
Radial Basis Functions
Fortunately, you won’t have to write such lines of code. All of the details of
designing this network are built into design functions newrbe and newrb, and
their outputs can be obtained with sim.
We can understand how this network behaves by following an input vector p
through the network to the output a2. If we present an input vector to such a
network, each neuron in the radial basis layer will output a value according to
how close the input vector is to each neuron’s weight vector.
Thus, radial basis neurons with weight vectors quite different from the input
vector p will have outputs near zero. These small outputs will have only a
negligible effect on the linear output neurons.
In contrast, a radial basis neuron with a weight vector close to the input vector
p will produce a value near 1. If a neuron has an output of 1 its output weights
in the second layer pass their values to the linear neurons in the second layer.
In fact, if only one radial basis neuron had an output of 1, and all others had
outputs of 0’s (or very close to 0), the output of the linear layer would be the
active neuron’s output weights. This would, however, be an extreme case.
Typically several neurons are always firing, to varying degrees.
Now let us look in detail at how the first layer operates. Each neuron's
weighted input is the distance between the input vector and its weight vector,
calculated with dist. Each neuron's net input is the element by element
product of its weighted input with its bias, calculated with netprod. Each
neurons' output is its net input passed through radbas. If a neuron's weight
vector is equal to the input vector (transposed) its weighted input will be 0, its
net input will be 0, and its output will be 1. If a neuron's weight vector is a
distance of spread from the input vector, its weighted input will be spread, its
net input will be sqrt(-log(.5)) (or 0.8326), therefore its output will be 0.5.
Exact Design (NEWRBE)
Radial basis networks can be designed with the function newrbe. This function
can produce a network with zero error on training vectors. It is called in the
following way
net = newrbe(P,T,SPREAD)
The function newrbe takes matrices of input vectors P and target vectors T, and
a spread constant SPREAD for the radial basis layer, and returns a network with
weights and biases such that the outputs are exactly T when the inputs are P.
6-5
6
Radial Basis Networks
This function newrbe creates as many radbas neurons as there are input
vectors in P, and sets the first layer weights to P'. Thus, we have a layer of
radbas neurons in which each neuron acts as a detector for a different input
vector. If there are Q input vectors, then there will be Q neurons.
Each bias in the first layer is set to 0.8326/SPREAD. This gives radial basis
functions that cross 0.5 at weighted inputs of +/- SPREAD. This determines the
width of an area in the input space to which each neuron responds. If SPREAD
is 4, then each radbas neuron will respond with 0.5 or more to any input vectors
within a vector distance of 4 from their weight vector. As we shall see, SPREAD
should be large enough that neurons respond strongly to overlapping regions
of the input space.
The second layer weights IW 2,1 (or in code, IW{2,1}) and biases b2 (or in code,
b{2}) are found by simulating the first layer outputs a1 (A{1}), and then solving
the following linear expression:
[W{2,1} b{2}] * [A{1}; ones] = T
We know the inputs to the second layer (A{1}) and the target (T), and the layer
is linear. We can use the following code to calculate the weights and biases of
the second layer to minimize the sum-squared error.
Wb = T/[P; ones(1,Q)]
Here Wb contains both weights and biases, with the biases in the last column.
The sum-squared error will always be 0, as explained below.
We have a problem with C constraints (input/target pairs) and each neuron has
C +1 variables (the C weights from the C radbas neurons, and a bias). A linear
problem with C constraints and more than C variables has an infinite number
of zero error solutions!
Thus newrbe creates a network with zero error on training vectors. The only
condition we have to meet is to make sure that SPREAD is large enough so that
the active input regions of the radbas neurons overlap enough so that several
radbas neurons always have fairly large outputs at any given moment. This
makes the network function smoother and results in better generalization for
new input vectors occurring between input vectors used in the design.
(However, SPREAD should not be so large that each neuron is effectively
responding in the same, large, area of the input space.)
6-6
Radial Basis Functions
The drawback to newrbe is that it produces a network with as many hidden
neurons as there are input vectors. For this reason, newrbe does not return an
acceptable solution when many input vectors are needed to properly define a
network, as is typically the case.
More Efficient Design (NEWRB)
The function newrb iteratively creates a radial basis network one neuron at a
time. Neurons are added to the network until the sum-squared error falls
beneath an error goal or a maximum number of neurons has been reached. The
call for this function is:
net = newrb(P,T,GOAL,SPREAD)
The function newrb takes matrices of input and target vectors, P and T, and
design parameters GOAL and, SPREAD, and returns the desired network.
The design method of newrb is similar to that of newrbe. The difference is that
newrb creates neurons one at a time. At each iteration the input vector which
will result in lowering the network error the most, is used to create a radbas
neuron. The error of the new network is checked, and if low enough newrb is
finished. Otherwise the next neuron is added. This procedure is repeated until
the error goal is met, or the maximum number of neurons is reached.
As with newrbe, it is important that the spread parameter be large enough that
the radbas neurons respond to overlapping regions of the input space, but not
so large that all the neurons respond in essentially the same manner.
Why not always use a radial basis network instead of a standard feed-forward
network? Radial basis networks, even when designed efficiently with newrbe,
tend to have many times more neurons than a comparable feed-forward
network with tansig or logsig neurons in the hidden layer.
This is because sigmoid neurons can have outputs over a large region of the
input space, while radbas neurons only respond to relatively small regions of
the input space. The result is that the larger the input space (in terms of
number of inputs, and the ranges those inputs vary over) the more radbas
neurons required.
On the other hand, designing a radial basis network often takes much less time
than training a sigmoid/linear network, and can sometimes result in fewer
neurons being used, as can be seen in the next demonstration.
6-7
6
Radial Basis Networks
Demonstrations
The demonstration script demorb1 shows how a radial basis network is used to
fit a function. Here the problem is solved with only five neurons.
Demonstration scripts demorb3 and demorb4 examine how the spread constant
affects the design process for radial basis networks.
In demorb3, a radial basis network is designed to solve the same problem as in
demorb1. However, this time the spread constant used is 0.01. Thus, each
radial basis neuron returns 0.5 or lower, for any input vectors with a distance
of 0.01 or more from its weight vector.
Because the training inputs occur at intervals of 0.1, no two radial basis
neurons have a strong output for any given input.
In demorb3, it was demonstrated that having too small a spread constant can
result in a solution which does not generalize from the input/target vectors
used in the design. This demonstration, demorb4, shows the opposite problem.
If the spread constant is large enough, the radial basis neurons will output
large values (near 1.0) for all the inputs used to design the network.
If all the radial basis neurons always output 1, any information presented to
the network becomes lost. No matter what the input, the second layer outputs
ones. The function newrb will attempt to find a network, but will not be able to
do so because to numerical problems that arise in this situation.
The moral of the story is, choose a spread constant larger than the distance
between adjacent input vectors, so as to get good generalization, but smaller
than the distance across the whole input space.
For this problem that would mean picking a spread constant greater than 0.1,
the interval between inputs, and less than 2, the distance between the left most
and right most inputs.
6-8
Generalized Regression Networks
Generalized Regression Networks
A generalized regression neural network (GRNN) is often used for function
approximation. As discussed below, it has a radial basis layer and a special
linear layer.
Network Architecture
The architecture for the GRNN is shown below. It is similar to the radial basis
network, but has a slightly different second layer.
AAA
AAA
AA
AAA
AA
AAA
A
AA
AA
A
AAA
AAA
AAA AA
A
Input
Radial Basis Layer
Q xR
Special Linear Layer
Rx1
1
R
R = # elements
in input vector
IW1,1
Q xQ
p
|| dist ||
LW2,1
Q x1
.*
n
a
1
n
1
Q x1
nprod
Q x1
2
Q x1
Where...
a2
Q
= # Neurons
in layer 1
Q
= # Neurons
in layer 2
Q
= # of input/
target pairs
Q x1
b1
Q
Q x1
ai1 = radbas ( || iIW1,1 - p || bi1)
Q
a2 = purelin ( n2)
ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1
Here the nprod box shown above (code function normprod) produces S2
elements in vector n2. Each element is the dot product of a row of LW2,1 and
the input vector a1, all normalized by the sum of the elements of a1. For
instance, suppose that:
LW{1,2}= [1 -2;3 4;5 6];
a{1} = [7; -8;
Then
aout = normprod(LW{1,2},a{1})
aout =
-23
11
13
6-9
6
Radial Basis Networks
The first layer is just like that for newrbe networks. It has as many neurons as
there are input/ target vectors in P. Specifically, the first layer weights are set
to P'. The bias b1 is set to a column vector of 0.8326/SPREAD. The user chooses
SPREAD, the distance an input vector must be from a neuron’s weight vector to
be 0.5.
Again, the first layer operates just like the newbe radial basis layer described
previously. Each neuron's weighted input is the distance between the input
vector and its weight vector, calculated with dist. Each neuron's net input is
the product of its weighted input with its bias, calculated with netprod. Each
neurons' output is its net input passed through radbas. If a neuron's weight
vector is equal to the input vector (transposed), its weighted input will be 0, its
net input will be 0, and its output will be 1. If a neuron's weight vector is a
distance of spread from the input vector, its weighted input will be spread, and
its net input will be sqrt(-log(.5)) (or 0.8326). Therefore its output will be 0.5.
The second layer also has as many neurons as input/target vectors, but here
LW{2,1} is set to T.
Suppose we have an input vector p close to pi, one of the input vectors among
the input vector/target pairs used in designing layer one weights. This input p
produces a layer 1 ai output close to 1. This leads to a layer 2 output close to ti,
one of the targets used forming layer 2 weights.
A larger spread leads to a large area around the input vector where layer 1
neurons will respond with significant outputs.Therefore if spread is small the
radial basis function is very steep so that the neuron with the weight vector
closest to the input will have a much larger output than other neurons. The
network will tend to respond with the target vector associated with the nearest
design input vector.
As spread gets larger the radial basis function's slope gets smoother and
several neuron's may respond to an input vector. The network then acts like it
is taking a weighted average between target vectors whose design input
vectors are closest to the new input vector. As spread gets larger more and
more neurons contribute to the average with the result that the network
function becomes smoother.
6-10
Generalized Regression Networks
Design (NEWGRNN)
You can use the function newgrnn to create a GRNN. For instance, suppose that
three input and three target vectors are defined as:
P = [4 5 6];
T = [1.5 3.6 6.7];
We can now obtain a GRNN with:
net = newgrnn(P,T);
and simulate it with:
P = 4.5;
v = sim(net,P)
You might want to try demogrn1. It shows how to approximate a function with
a GRNN.
6-11
6
Radial Basis Networks
Probabilistic Neural Networks
Probabilistic neural networks can be used for classification problems. When an
input is presented, the first layer computes distances from the input vector to
the training input vectors, and produces a vector whose elements indicate how
close the input is to a training input. The second layer sums these contributions
for each class of inputs to produce as its net output a vector of probabilities.
Finally, a compete transfer function on the output of the second layer picks the
maximum of these probabilities, and produces a one for that class and a 0 for
the other classes. The architecture for this system is shown below.
Network Architecture
AAA
AAA
AAA
AAA
AAA
Input
Radial Basis Layer
Q xR
p
Rx1
1
R
IW1,1
|| dist ||
b
1
Q x1
Q x1
.*
Competitive Layer
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
a2
n1
a1
Q x1
ai1 = radbas ( || iIW1,1 - p || bi1)
Q x1
K x1
n2
LW2,1
K x1
Where...
C
R = # elements
in input vector
KxQ
Q
Q
a2 = compet ( LW2,1 a1)
ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1
Q = # input/target pairs
= # of neurons in layer 1
K = # of classes of input data = # of neurons in layer 2
It is assumed that there are Q input vector/target vector pairs. Each target
vector has K elements. One of these element is one and the rest is zero. Thus,
each input vector is associated with one of K classes.
The first layer input weights, IW1,1 (net.IW{1,1}) are set to the transpose of
the matrix formed from the Q training pairs, P'. When an input is presented
the ||dist|| box produces a vector whose elements indicate how close the
input is to the vectors of the training set. These elements are multiplied,
element by element, by the bias and sent the radbas transfer function. An input
vector close to a training vector will be represented by a number close to one in
the output vector a1. If an input is close to several training vectors of a single
class, it will be represented by several elements of a1 that are close to one.
6-12
Probabilistic Neural Networks
The second layer weights, LW1,2 (net.LW{2,1}), are set to the matrix T of
target vectors. Each vector has a one only in the row associated with that
particular class of input, and zeros elsewhere. (A function ind2vec is used to
create the proper vectors.) The multiplication Ta1 sums the elements of a1 due
to each of the K input classes. Finally, the second layer transfer function,
compete, produces a one corresponding to the largest element of n2, and zeros
elsewhere. Thus, the network has classified the input vector into a specific one
of K classes because that class had the maximum probability of being correct.
Design (NEWPNN)
You can use the function newpnn to create a PNN. For instance, suppose that
seven input vectors and their corresponding targets are:
P = [0 0;1 1;0 3;1 4;3 1;4 1;4 3]'
which yields
P =
0
1
0
0
1
3
Tc = [1 1 2 2 3 3 3];
1
4
3
1
4
1
4
3
2
3
3
3
Which yields
Tc =
1
1
2
We need a target matrix with ones in the right place. We can get it with the
function ind2vec. It gives a matrix with zeros except at the correct spots. So
execute
T = ind2vec(Tc)
which gives
T =
(1,1)
(1,2)
(2,3)
(2,4)
(3,5)
(3,6)
(3,7)
1
1
1
1
1
1
1
6-13
6
Radial Basis Networks
Now we can create a network and simulate it, using the input P to make sure
that it does produce the correct classifications. We will use the function
vec2ind to convert the output Y into a row Yc to make the classifications clear.
net = newpnn(P,T);
Y = sim(net,P)
Yc = vec2ind(Y)
Finally we get
Yc =
1
1
2
2
3
3
3
We might try classifying vectors other than those that were used to design the
network. We will try to classify the vectors shown below in P2.
P2 = [1 4;0 1;5 2]'
P2 =
1
4
0
1
5
2
Can you guess how these vectors will be classified? If we run the simulation
and plot the vectors as we did before we get:
Yc =
2
1
3
These results look good, for these test vectors were quite close to members of
classes 2, 1 and 3 respectively. The network has managed generalize its
operation to properly classify vectors other than those used to design the
network.
You might want to try demopnn1. It shows how to design a PNN, and how the
network can successfully classify a vector not used in the design.
6-14
Summary
Summary
Radial basis networks can be designed very quickly in two different ways.
The first design method, newrbe, finds an exact solution. The function newrbe
creates radial basis networks with as many radial basis neurons as there are
input vectors in the training data.
The second method, newrb, finds the smallest network that can solve the
problem within a given error goal. Typically, far fewer neurons are required by
newrb than are returned newrbe. However, because the number of radial basis
neurons is proportional to the size of the input space, and the complexity of the
problem, radial basis networks can still be larger than backpropagation
networks.
A generalized regression neural network (GRNN) is often used for function
approximation. It has been shown that, given a sufficient number of hidden
neurons, GRNNs can approximate a continuous function to an arbitrary
accuracy.
Probabilistic neural networks can be used for classification problems. Their
design is straightforward and does not depend on training. A PNN is
guaranteed to converge to a Bayesian classifier providing it is given enough
training data. These networks generalize well.
The GRNN and PNN have many advantages, but they both suffer from one
major disadvantage. They are slower to operate because they use more
computation than other kinds of networks to do their function approximation
or classification.
6-15
6
Radial Basis Networks
Figures
Radial Basis Neuron
Input
Radial Basis Neuron
1,1
p
p1
p2
3
... w1, R
AA
AA
w
+
_
p
R
n
|| dist ||
b
1
A
A
a = radbas( || w-p || b)
Radbas Transfer Function
a
1.0
0.5
0.0
AA
AA
n
-0.833
+0.833
a = radbas(n)
Radial Basis Function
6-16
a
Summary
Radial Basis Network Architecture
AAA
AAA
AAA
AA
AA
AAA
AA
AA
AAA AAAA
Radial Basis Layer
Input
IW1,1
S1xR
p
Rx1
.*
S1x1
n1
a2
LW2,1
b1
1
S1
a 1 = radbas ( || IW1,1 - p || b 1)
i
i
i
S2x1
n2
S2xS1
S1 = # Neurons
in layer 1
S2x1
S1x1
S1x1
Where...
R = # elements
in input
vector
AA
AA
AA
a1
S1x1
|| dist ||
1
R
Linear Layer
b2
S2x1
S2
S2 = # Neurons
in layer 2
a2 = purelin (LW2,1 a1 +b2)
ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1
Generalized Regression Neural Network Architecture
AAA
AA
AAA
AAA AAAAA
AA A
AAA
AA
AAA
A
AAA AA
A
Input
Radial Basis Layer
Q xR
Special Linear Layer
Rx1
1
R
R = # elements
in input vector
IW1,1
Q xQ
p
|| dist ||
LW2,1
Q x1
.*
n1
a1
Q x1
Q x1
nprod
n2
Q x1
b1
Q x1
Q
ai1 = radbas ( || iIW1,1 - p || bi1)
Q
Where...
a2
Q
= # Neurons
in layer 1
Q
= # Neurons
in layer 2
Q x1
Q = # of input/
target pairs
a2 = purelin ( n2)
ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1
6-17
6
Radial Basis Networks
Probabilistic Neural Network Architecture
AAA
AAA
AAA
AA
AA
A
AAA
AA
AA
A
AAA AA
A
Input
Radial Basis Layer
Q xR
p
Rx1
1
R
Competitive Layer
IW1,1
|| dist ||
a2
Q x1
.*
n1
a1
Q x1
Q x1
b1
Q x1
ai1 = radbas ( || iIW1,1 - p || bi1)
LW2,1
n2
K x1
Where...
K x1
C
R = # elements
in input vector
KxQ
Q
Q
a2 = compet ( LW2,1 a1)
ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1
Q = # input/target pairs
= # of neurons in layer 1
K = # of classes of input data = # of neurons in layer 2
6-18
Summary
New Functions
This chapter introduces the following new functions:
Function
Description
newrb
Design a radial basis network.
newrbe
Design an exact radial basis network.
newgrnn
Design a generalized regression neural network.
newpnn
Design a probabilistic neural network.
dist
Euclidean distance weight function.
negdist
Negative euclidean distance weight function
dotprod
Dot product weight function.
normprod
Normalized dot product weight function.
netprod
Product net input function.
compet
Competitive transfer function.
radbas
Radial basis transfer function.
ind2vec
Convert indices to vectors.
vec2ind
Convert vectors to indices.
6-19
6
Radial Basis Networks
6-20
7
Self-Organizing Networks
Introduction . . . . . . . . . . . . . . . . . . . . 7-2
Important Self-Organizing Functions . . . . . . . . . . 7-2
Competitive Learning . . . . . . . . . .
Architecture . . . . . . . . . . . . . . . .
Creating a Competitive Neural Network (NEWC)
Kohonen Learning Rule (LEARNK) . . . . . .
Bias Learning Rule (LEARNCON) . . . . . .
Training . . . . . . . . . . . . . . . . .
Graphical Example . . . . . . . . . . . . .
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7-3
7-3
7-4
7-5
7-5
7-6
7-8
Self-Organizing Maps . . . . . . . . . . . . . .
Topologies (GRIDTOP, HEXTOP, RANDTOP) . . . . .
Distance Functions(DIST, LINKD., MAND. BOXD.) . .
Architecture . . . . . . . . . . . . . . . . . . .
Creating a Self Organizing MAP Neural Net. (NEWSOM)
Training (LEARNSOM) . . . . . . . . . . . . . .
Examples . . . . . . . . . . . . . . . . . . . .
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7-10
7-12
7-16
7-19
7-20
7-22
7-25
Summary and Conclusions . . . . . . . . . . . . . 7-32
Figures . . . . . . . . . . . . . . . . . . . . . . . 7-32
New Functions . . . . . . . . . . . . . . . . . . . 7-33
7
Self-Organizing Networks
Introduction
Self-organizing in networks is one of the most fascinating topics in the neural
network field. Such networks can learn to detect regularities and correlations
in their input and adapt their future responses to that input accordingly. The
neurons of competitive networks learn to recognize groups of similar input
vectors. Self-organizing maps learn to recognize groups of similar input vectors
in such a way that neurons physically close together in the neuron layer
respond to similar input vectors. A basic reference is:
Kohonen, T. Self-Organization and Associative Memory, 2nd Edition, Berlin:
Springer-Verlag, 1987.
Important Self-Organizing Functions
Competitive layers and self organizing maps can be created with newc and
newsom respectively.
A listing of all self-organizing functions and demonstrations can be found by
typing help selforg.
7-2
Competitive Learning
Competitive Learning
The neurons in a competitive layer distribute themselves to recognize
frequently presented input vectors.
Architecture
The architecture for a competitive network is shown below.
AA
AA
AAA
AAA
AA
AA
Input
Competitive Layer
S1xR
IW1,1
p
Rx1
1
R
|| ndist ||
b1
S1x1
AA
AA
AA
a1
S1x1
n1
S1x1
S1x1
C
S1
The dist box in this figure accepts the input vector p and the input weight
matrix IW1,1, and produces a vector having S1 elements. The elements are the
negative of the distances between the input vector and vectors iIW1,1 formed
from the rows of the input weight matrix.
The net input n1 of a competitive layer is computed by finding the negative
distance between input vector p and the weight vectors and adding the biases
b. If all biases are zero, the maximum net input a neuron can have is 0. This
occurs when the input vector p equals that neuron’s weight vector.
The competitive transfer function accepts a net input vector for a layer and
returns neuron outputs of 0 for all neurons except for the winner, the neuron
associated with the most positive element of net input n1. The winner’s output
is 1. If all biases are 0 then the neuron whose weight vector is closest to the
input vector has the least negative net input, and therefore wins the
competition to output a 1.
Reasons for using biases with competitive layers are introduced in a later
section on training.
7-3
7
Self-Organizing Networks
Creating a Competitive Neural Network (NEWC)
A competitive neural network can be created with the function newc. We will
show how this works with a simple example.
Suppose that we wish to divide the following four two element vectors into two
classes.
p = [.1 .8
p =
0.1000
0.2000
.1 .9; .2 .9 .1 .8]
0.8000
0.9000
0.1000
0.1000
0.9000
0.8000
Thus, we have two vectors near the origin and two vectors near (1,1).
First create a two neuron layer is created with two input elements ranging
from 0 to 1. The first argument gives the range of the two input vectors and the
second argument says that there are to be 2 neurons.
net = newc([0 1; 0 1],2);
The weights will be initialized to the center of the input ranges with the
function midpoint. We can check to see these initial values as follows:
wts = net.IW{1,1}
wts =
0.5000
0.5000
0.5000
0.5000
These weights are indeed the values at the midpoint of the range (0 to 1) of the
inputs, as we would expect when using midpoint for initialization.
The biases are computed by initcon, which gives
biases =
5.4366
5.4366
Now we have a network, but we need to train it to do the classification job.
Recall that each neuron competes to respond to an input vector p. If the biases
are all 0, the neuron whose weight vector is closest to p gets the highest net
input and therefore wins the competition and outputs 1. All other neurons
output 0. We would like to adjust the winning neuron so as to move it closer to
the input. A learning rule to do this is discussed in the next section.
7-4
Competitive Learning
Kohonen Learning Rule (LEARNK)
The weights of the winning neuron (a row of the input weight matrix) are
adjusted with the Kohonen learning rule. Supposing that the ith neuron wins,
the ith row of the input weight matrix are adjusted as shown below.
iIW
1, 1
( q ) = iIW
1, 1
( q – 1 ) + α ( p ( q ) – iIW
1, 1
(q – 1))
The Kohonen rule allows the weights of a neuron to learn an input vector, and
because of this it is useful in recognition applications.
Thus, the neuron whose weight vector was closest to the input vector is
updated to be even closer. The result is that the winning neuron is more likely
to win the competition the next time a similar vector is presented, and less
likely to win when a very different input vector is presented. As more and more
inputs are presented, each neuron in the layer closest to a group of input
vectors soon adjusts its weight vector toward those input vectors. Eventually,
if there are enough neurons, every cluster of similar input vectors will have a
neuron which outputs 1 when a vector in the cluster is presented, while
outputting a 0 at all other times. Thus the competitive network learns to
categorize the input vectors it sees.
The function learnk is used to perform the Kohonen learning rule in this
toolbox.
Bias Learning Rule (LEARNCON)
One of the limitations of competitive networks is that some neurons may not
always get allocated. In other words, some neuron weight vectors may start out
far from any input vectors and never win the competition, no matter how long
the training is continued. The result is that their weights do not get to learn
and they never win. These unfortunate neurons, referred to as dead neurons,
never perform a useful function.
To stop this from happening, biases are used to give neurons that only win the
competition rarely (if ever) an advantage over neurons which win often. A
positive bias, added to the negative distance, makes a distant neuron more
likely to win.
7-5
7
Self-Organizing Networks
To do this job a running average of neuron outputs is kept. It is equivalent to
the percentages of times each output is 1. This average is used to update the
biases with the learning function learncon so that the biases of frequently
active neurons will get smaller, and biases of infrequently active neurons will
get larger.
The learning rates for learncon are typically set an order of magnitude or more
smaller than for learnk. Doing this helps make sure that the running average
is accurate.
The result is that biases of neurons which haven’t responded very frequently
will increase vs. biases of neurons that have responded frequently. As the
biases of infrequently active neurons increase the input space to which that
neuron responds increases. As that input space increases the infrequently
active neuron responds and moves toward more input vectors. Eventually the
neuron will respond to an equal number of vectors as other neurons.
This has two good effects. First, if a neuron never wins a competition because
its weights are far from any of the input vectors, its bias will eventually get
large enough so that it will be able to win. When this happens it will move
toward some group of input vectors. Once the neuron’s weights have moved
into a group of input vectors and the neuron is winning consistently its bias will
decrease to 0. Thus the problem of dead neurons is resolved.
The second advantage of biases is that they force each neuron to classify
roughly the same percentage of input vectors. Thus, if a region of the input
space is associated with a larger number of input vectors than another region,
the more densely filled region will attract more neurons and be classified into
smaller subsections.
Training
Now train the network for 500 epochs. Either train or adapt can be used.
net.trainParam.epochs = 500
net = train(net,p);
Note that train for competitive networks uses the training function trainwb1.
You can verify this by executing the following code after creating the network.
net.trainFcn
7-6
Competitive Learning
This code produces
ans =
trainwb1
Thus, during each epoch, a single vector is chosen randomly and presented to
the network and weight and bias values are updated accordingly.
Next supply the original vectors as input to the network, simulate the network
and finally convert its output vectors to class indices.
a = sim(net,p)
ac = vec2ind(a)
This yields
ac =
1
2
1
2
We see that the network has been trained to classify the input vectors into two
groups, those near the origin, class 1, and those near (1,1), class 2.
It might be interesting to look at the final weights and biases. They are
wts =
0.8208
0.1348
biases =
5.3699
5.5049
0.8263
0.1787
(You may get different answers if you run this problem, as a random seed is
used to pick the order of the vectors presented to the network for training.)
Note that the first vector (formed from the first row of the weight matrix) is
near the input vectors close to (1,1), while the vector formed from the second
row of the weight matrix is close to the input vectors near the origin. Thus, the
network has been trained, just by exposing it to the inputs, to classify them.
During training each neuron in the layer closest to a group of input vectors
adjusts its weight vector toward those input vectors. Eventually, if there are
enough neurons, every cluster of similar input vectors will have a neuron which
outputs 1 when a vector in the cluster is presented, while outputting a 0 at all
other times. Thus the competitive network learns to categorize the input.
7-7
7
Self-Organizing Networks
Graphical Example
Competitive layers can be understood better when their weight vectors and
input vectors are shown graphically. The diagram below shows forty-eight
two-element input vectors represented as with ‘+’ markers.
Input Vectors
1
0.8
0.6
0.4
0.2
0
-0.5
0
0.5
1
The input vectors above appear to fall into clusters. A competitive network of
eight neurons will be used to classify the vectors into such clusters.
7-8
Competitive Learning
The following plot shows the weights after training.
Competitive Learning: 500 epochs
1
0.8
0.6
0.4
0.2
0
-0.5
0
0.5
1
Note that seven of the weight vectors found clusters of input vectors to classify.
The eighth neuron’s weight vector is still in the middle of the input vectors.
Continued training of the layer would eventually cause this last neuron to
move toward some input vectors.
You might try democ1 to see a dynamic example of competitive learning.
7-9
7
Self-Organizing Networks
Self-Organizing Maps
Self-organizing feature maps (SOFM) learn to classify input vectors according
to how they are grouped in the input space. They differ from competitive layers
in that neighboring neurons in the self-organizing map learn to recognize
neighboring sections of the input space. Thus self-organizing maps learn both
the distribution (as do competitive layers) and topology of the input vectors
they are trained on.
The neurons in the layer of an SOFM are arranged originally in physical
positions according to a topology function. The functions gridtop, hextop or
randtop can arrange the neurons in a grid, hexagonal, or random topology.
Distances between neurons are calculated from their positions with a distance
function. There are four distance functions, dist, boxdist, linkdist and
mandist. Link distance is the most common. These topology and distance
functions are described in detail later in this section.
Here a self-organizing feature map network identifies a winning neuron i∗
using the same procedure as employed by a competitive layer. However,
instead of updating only the winning neuron, all neurons within a certain
neighborhood N i∗ ( d ) of the winning neuron are updated using the Kohonen
rule. Specifically, we will adjust all such neurons i ∈ N i∗ ( d ) as follows
iw ( q )
= iw ( q – 1 ) + α ( p ( q ) – iw ( q – 1 ) ) or
iw ( q )
= ( 1 – α ) iw ( q – 1 ) + αp ( q )
Here the neighborhood N i∗ ( d ) contains the indices for all of the neurons that
lie within a radius d of the winning neuron i∗ :
N i ( d ) = { j, d ij ≤ d }
Thus, when a vector p is presented, the weights of the winning neuron and its
close neighbors will move towards p . Consequently, after many presentations,
neighboring neurons will have learned vectors similar to each other.
7-10
Self-Organizing Maps
To illustrate the concept of neighborhoods, consider the figure given below. The
left diagram shows a two-dimensional neighborhood of radius d = 1 around
neuron 13 . The right diagram shows a neighborhood of radius d = 2 .
1
2
3
4
5
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
11
12
13
14
15
11
12
13
14
15
16
17
18
19
20
16
17
18
19
20
21
22
23
24
25
21
22
23
24
25
N (1)
13
N (2)
13
These neighborhoods could be written as:
N 13 ( 1 ) = { 8, 12, 13, 14, 18 } and
N 13 ( 2 ) = { 3, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 23 }
Note that the neurons in an SOFM do not have to be arranged in a
two-dimensional pattern. You use a one-dimensional arrangement, or even
three or more dimensions. For a one-dimensional SOFM, a neuron will only
have two neighbors within a radius of 1 (or a single neighbor if the neuron is at
the end of the line).You can also define distance in different ways, by using
rectangular and hexagonal arrangements of neurons and neighborhoods for
instance. The performance of the network is not sensitive to the exact shape of
the neighborhoods.
7-11
7
Self-Organizing Networks
Topologies (GRIDTOP, HEXTOP, RANDTOP)
You can specify different topologies for the original neuron locations with the
functions gridtop, hextop or randtop.
The gridtop topology starts with neurons in a rectangular grid similar to that
shown in the previous figure. For example, suppose that you want a 2 by 3
array of six neurons You can get this with:
pos = gridtop(2,3)
pos =
0
1
0
0
0
1
1
1
0
2
1
2
Here neuron 1 has the position (0,0), neuron 2 has the position (1,0), neuron 3
had the position (0,1), etc.
2
5
6
1
3
4
0
1
2
0
1
gridtop(2,3)
Note that had we asked for a gridtop with the arguments reversed we would
have gotten a slightly different arrangement.
pos = gridtop(3,2)
pos =
0
1
2
0
0
0
7-12
0
1
1
1
2
1
Self-Organizing Maps
An 8x10 set of neurons in a gridtop topology can be created and plotted with
the code shown below
pos = gridtop(8,10);
plotsom(pos)
to give the following graph.
Neuron Positions
9
8
7
position(2,i)
6
5
4
3
2
1
0
0
2
4
position(1,i)
6
8
As shown, the neurons in the gridtop topology do indeed lie on a grid.
The hextop function creates a similar set of neurons but they are in a
hexagonal pattern. A 2 by 3 pattern of hextop neurons is generated as follows:
pos = hextop(2,3)
pos =
0
1.0000
0
0
0.5000
0.8660
1.5000
0.8660
0
1.7321
1.0000
1.7321
7-13
7
Self-Organizing Networks
Note that hextop is the default pattern for SOFM networks generated with
newsom.
An 8x10 set of neurons in a hextop topology can be created and plotted with the
code shown below
pos = hextop(8,10);
plotsom(pos)
to give the following graph.
Neuron Positions
7
6
position(2,i)
5
4
3
2
1
0
0
1
2
3
4
position(1,i)
5
6
7
Note the positions of the neurons in a hexagonal arrangement.
7-14
8
Self-Organizing Maps
Finally, the randtop function creates neurons in an N dimensional random
pattern. The following code generates a random pattern of neurons.
pos = randtop(2,3)
pos =
0
0.7787
0
0.1925
0.4390
0.6476
1.0657
0.9106
0.1470
1.6490
0.9070
1.4027
An 8x10 set of neurons in a randtop topology can be created and plotted with
the code shown below
pos = randtop(8,10);
plotsom(pos)
to give the following graph.
Neuron Positions
6
5
position(2,i)
4
3
2
1
0
0
1
2
3
position(1,i)
4
5
6
You might take a look at the help for these topology functions. They contain a
number of examples.
7-15
7
Self-Organizing Networks
Distance Functions (DIST, LINKDIST, MANDIST,
BOXDIST)
In this toolbox there are four distinct ways to calculate distances from a
particular neuron to its neighbors. Each calculation method is implemented
with a special function.
The dist function has been discussed before. It calculates the Euclidean
distance from a home neuron to any other neuron. Suppose we have three
neurons:
pos2 = [ 0 1 2; 0 1 2]
pos2 =
0
1
2
0
1
2
We will find the distance from each neuron to the other with:
D2 = dist(pos2)
D2 =
0
1.4142
1.4142
0
2.8284
1.4142
2.8284
1.4142
0
Thus, the distance from neuron 1 to itself is 0, the distance from neuron 1 to
neuron 2 is 1.414, etc. These are indeed the Euclidean distances as we know
them.
The graph below shows a home neuron in a two-dimensional (gridtop) layer of
neurons. The home neuron has neighborhoods of increasing diameter
surrounding it. A neighborhood of diameter 1 includes the home neuron and its
7-16
Self-Organizing Maps
immediate neighbors. The neighborhood of diameter 2 includes the diameter 1
neurons and their immediate neighbors.
Columns
Home Neuron
Rows
2-Dimensional
Layer of Neurons
Neighborhood 1
Neighborhood 2
Neighborhood 3
As for the dist function, all the neighborhoods for an S neuron layer map are
represented by an SxS matrix of distances. The particular distances shown
above, 1 in the immediate neighborhood, 2 in neighborhood 2, etc., are
generated by the function boxdist. Suppose that we have 6 neurons in a
gridtop configuration.
pos = gridtop(2,3)
pos =
0
1
0
0
0
1
1
1
0
2
1
2
1
1
1
0
1
1
2
2
1
1
0
1
2
2
1
1
1
0
Then the box distances are:
d = boxdist(pos)
d =
0
1
1
0
1
1
1
1
2
2
2
2
1
1
0
1
1
1
The distance from neuron 1 to 2, 3 and 4 is just 1, for they are in the immediate
neighborhood. The distance from neuron 1 to both 5 and 6 is 2. The distance
from both 3 and 4 to all other neurons is just 1.
7-17
7
Self-Organizing Networks
The link distance from one neuron is just the number of links, or steps, that
must be taken to get to the neuron under consideration. Thus if we calculate
the distances from the same set of neurons with linkdist we get:
dlink =
0
1
1
2
2
3
1
0
2
1
3
2
1
2
0
1
1
2
2
1
1
0
2
1
2
3
1
2
0
1
3
2
2
1
1
0
The Manhattan distance between two vectors x and y is calculated as:
D = sum(abs(x-y))
Thus if we have
W1 = [ 1 2; 3 4; 5 6]
W1 =
1
2
3
4
5
6
and
P1= [1;1]
P1 =
1
1
then we get for the distances
Z1 = mandist(W1,P1)
Z1 =
1
5
9
The distances calculated with mandist do indeed follow the mathematical
expression given above.
7-18
Self-Organizing Maps
Architecture
The architecture for this SOFM is shown below.
Input
AA
AAA
AAA
IW1,1
S1xR
p
R x1
R
A
A
A
A
Self Organizing Map Layer
|| ndist ||
n1
S1x1
C
a1
S1x1
S1
ni1 = - || iIW1,1 - p ||
a1 = compet (n1)
This architecture is like that of a competitive network, except no bias is used
here. The competitive transfer function produces a 1 for output element a1i
corresponding to i∗ ,the winning neuron. All other output elements in a1 are 0.
Now, however, as described above, neurons close to the winning neuron are
updated along with the winning neuron. As described previously, one can chose
from various topologies of neurons. Similarly, one can choose from various
distance expressions to calculate neurons that are close to the winning neuron.
7-19
7
Self-Organizing Networks
Creating a Self Organizing MAP Neural Network
(NEWSOM)
You can create a new SOFM network with the function newsom. This function
defines variables used in two phases of learning:
• Ordering phase learning rate
• Ordering phase steps
• Tuning phase learning rate
• Tuning phase neighborhood distance
These values are used for training and adapting.
Consider the following example.
Suppose that we wish to create a network having input vectors with two
elements that fall in the range 0 to 2 and 0 to 1 respectively. Further suppose
that we want to have six neurons in a hexagonal 2 by 3 network. The code to
obtain this network is:
net = newsom([0 2; 0 1] , [2 3]);
Suppose also that the vectors to train on are:
P = [.1 .3 1.2 1.1 1.8 1.7 .1 .3 1.2 1.1 1.8 1.7;...
0.2 0.1 0.3 0.1 0.3 0.2 1.8 1.8 1.9 1.9 1.7 1.8]
We can plot all of this with
plot(P(1,:),P(2,:),'.g','markersize',20)
hold on
plotsom(net.iw{1,1},net.layers{1}.distances)
hold off
7-20
Self-Organizing Maps
to give
Weight Vectors
2
1.8
1.6
1.4
W(i,2)
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
W(i,1)
The various training vectors are seen as fuzzy gray spots around the perimeter
of this figure. The initialization for newsom is midpoint. Thus, the initial
network neurons are all concentrated at the black spot at (1, 0.5).
When simulating a network, the negative distances between each neuron's
weight vector and the input vector are calculated (negdist) to get the weighted
inputs. The weighted inputs are also the net inputs (netsum). The net inputs
compete (compete) so that only the neuron with the most positive net input will
output a 1.
7-21
7
Self-Organizing Networks
Training (LEARNSOM)
Learning in a self organizing feature map occurs for one vector at a time
independent of whether the network is trained directly (trainwb1) or whether
it is trained adaptively (adaptwb). In either case, learnsom is the
self-organizing map weight learning function.
First the network identifies the winning neuron. Then the weights of the
winning neuron, and the other neurons in its neighborhood, are moved closer
to the input vector at each learning step using the self-organizing map learning
function learnsom. The winning neuron's weights are altered proportional to
the learning rate. The weights of neurons in its neighborhood are altered
proportional to half the learning rate. The learning rate and the neighborhood
distance used to determine which neurons are in the winning neuron's
neighborhood are altered during training through two phases.
Phase 1: Ordering Phase
This phase lasts for the given number of steps. The neighborhood distance
starts as the maximum distance between two neurons, and decreases to the
tuning neighborhood distance. The learning rate starts at the ordering phase
learning rate and decreases until it reaches the tuning phase learning rate. As
the neighborhood distance and learning rate decrease over this phase the
neuron's of the network will typically order themselves in the input space with
the same topology which they are ordered physically.
Phase 2: Tuning Phase
This phase lasts for the rest of training or adaption. The neighborhood distance
stays at the tuning neighborhood distance (should include only close neighbors,
i.e. typically 1.0). The learning rate continues to decrease from the tuning
phase learning rate, but very slowly. The small neighborhood and slowly
decreasing learning rate fine tune the network, while keeping the ordering
learned in the previous phase stable. The number of epochs for the tuning part
of training (or time steps for adaption) should be much larger than the number
of steps in the ordering phase, because the tuning phase usually takes much
longer.
Now let us take a look at some of the specific values commonly used in these
networks.
7-22
Self-Organizing Maps
Learning occurs according to learnsom's learning parameter, shown here with
its default value.
LP.order_lr
0.9
Ordering phase learning rate.
LP.order_steps
1000
Ordering phase steps.
LP.tune_lr
0.02
Tuning phase learning rate.
LP.tune_nd
1
Tuning phase neighborhood distance.
learnsom calculates the weight change dW for a given neuron from the neuron's
input P, activation A2, and learning rate LR:
dw =
lr*a2*(p'-w)
where the activation A2 is found from the layer output A and neuron distances
D and the current neighborhood size ND:
a2(i,q) = 1,
if a(i,q) = 1
= 0.5, if a(j,q) = 1 and D(i,j) <= nd
= 0,
otherwise
The learning rate LR and neighborhood size NS are altered through two phases:
an ordering phase and a tuning phase.
The ordering phase lasts as many steps as LP.order_steps. During this phase
LR is adjusted from LP.order_lr down to LP.tune_lr, and ND is adjusted from
the maximum neuron distance down to 1. It is during this phase that neuron
weights are expected to order themselves in the input space consistent with the
associated neuron positions.
During the tuning phase LR decreases slowly from LP.tune_lr and ND is always
set to LP.tune_nd. During this phase the weights are expected to spread out
relatively evenly over the input space while retaining their topological order
found during the ordering phase.
Thus, the neuron’s weight vectors initially take large steps all together toward
the area of input space where input vectors are occurring. Then as the
neighborhood size decreases to 1, the map tends to order itself topologically
over the presented input vectors. Once the neighborhood size is 1, the network
should be fairly well ordered and the learning rate is slowly decreased over a
longer period to give the neurons time to spread out evenly across the input
vectors.
7-23
7
Self-Organizing Networks
As with competitive layers, the neurons of a self-organizing map will order
themselves with approximately equal distances between them if input vectors
appear with even probability throughout a section of the input space. Also, if
input vectors occur with varying frequency throughout the input space, the
feature map layer will tend to allocate neurons to an area in proportion to the
frequency of input vectors there.
Thus, feature maps, while learning to categorize their input, also learn both
the topology and distribution of their input.
We can train the network for 1000 epochs with
net.trainParam.epochs = 1000;
net = train(net,P);
This training produces the following plot.
Weight Vectors
1.8
1.6
1.4
W(i,2)
1.2
1
0.8
0.6
0.4
0.2
0
7-24
0.5
1
W(i,1)
1.5
2
Self-Organizing Maps
We can see that the neurons have started to move toward the various training
groups. Additional training will be required to get the neurons closer to the
various groups.
As noted previously, self-organizing maps differ from conventional competitive
learning in terms of which neurons get their weights updated. Instead of
updating only the winner, feature maps update the weights of the winner and
its neighbors. The result is that neighboring neurons tend to have similar
weight vectors and to be responsive to similar input vectors.
Examples
Two examples are described briefly below. You might try the demonstration
scripts demosm1 and demosm2 to see similar examples.
One-Dimensional Self-Organizing Map
Consider 100 two-element unit input vectors spread evenly between 0˚ and 90˚.
angles = 0:0.5∗pi/99:0.5∗pi;
Here is a plot of the data
P = [sin(angles); cos(angles)];
1
0.8
0.6
0.4
0.2
0
0
0.5
1
7-25
7
Self-Organizing Networks
We will define a a self-organizing map as a one-dimensional layer of 10
neurons. This map is to be trained on these input vectors shown above.
Originally these neurons will be at the center of the figure.
1.5
W(i,2)
1
0.5
0
-0.5
-1
0
1
2
W(i,1)
Of course, since all the weight vectors start in the middle of the input vector
space, all you see now is a single circle.
As training starts the weight vectors move together toward the input vectors.
They also become ordered as the neighborhood size decreases. Finally the layer
has adjusted its weights so that each neuron responds strongly to a region of
7-26
Self-Organizing Maps
the input space occupied by input vectors. The placement of neighboring
neuron weight vectors also reflects the topology of the input vectors.
1
W(i,2)
0.8
0.6
0.4
0.2
0
0
0.5
W(i,1)
1
Note that self-organizing maps are trained with input vectors in a random
order, so starting with the same initial vectors does not guarantee identical
training results.
Two-Dimensional Self-Organizing Map
This example shows how a two-dimensional self-organizing map can be
trained.
First some random input data is created with the following code.
P = rands(2,1000);
7-27
7
Self-Organizing Networks
Here is a plot of these 1000 input vectors.
1
0.5
0
-0.5
-1
-1
0
1
A two-dimensional map of 30 neurons is used to classify these input vectors.
The two-dimensional map is five neurons by six neurons, with distances
calculated according to the Manhattan distance neighborhood function
mandist.
The map is then trained for 5000 presentation cycles, with displays every 20
cycles.
7-28
Self-Organizing Maps
Here is what the self-organizing map looks like after 40 cycles.
1
W(i,2)
0.5
0
-0.5
-1
-0.5
0
0.5
1
W(i,1)
The weight vectors, shown with circles, are almost randomly placed. However,
even after only 40 presentation cycles, neighboring neurons, connected by
lines, have weight vectors close together.
Here is the map after 120 cycles.
1
W(i,2)
0.5
0
-0.5
-1
-1
0
W(i,1)
1
7-29
7
Self-Organizing Networks
After 120 cycles the map has begun to organize itself according to the topology
of the input space which constrains input vectors.
The following plot, after 500 cycles, shows the map is more evenly distributed
across the input space.
1
W(i,2)
0.5
0
-0.5
-1
-1
0
W(i,1)
1
Finally, after 5000 cycles, the map is rather evenly spread across the input
space. In addition, the neurons are very evenly spaced reflecting the even
distribution of input vectors in this problem.
1
W(i,2)
0.5
0
-0.5
-1
-1
7-30
0
W(i,1)
1
Self-Organizing Maps
Thus a two-dimensional self-organizing map has learned the topology of its
inputs’ space.
It is important to note that while a self-organizing map does not take long to
organize itself so that neighboring neurons recognize similar inputs, it can take
a long time for the map to finally arrange itself according to the distribution of
input vectors.
7-31
7
Self-Organizing Networks
Summary and Conclusions
A competitive network learns to categorize the input vectors presented to it. If
a neural network only needs to learn to categorize its input vectors, then a
competitive network will do. Competitive networks also learn the distribution
of inputs by dedicating more neurons to classifying parts of the input space
with higher densities of input.
A self-organizing map learns to categorize input vectors. It also learns the
distribution of input vectors. Feature maps allocate more neurons to recognize
parts of the input space where many input vectors occur and allocate fewer
neurons to parts of the input space where few input vectors occur.
Self-organizing maps also learn the topology of their input vectors. Neurons
next to each other in the network learn to respond to similar vectors. The layer
of neurons can be imagined to be a rubber net which is stretched over the
regions in the input space where input vectors occur.
Self-organizing maps allow neurons that are neighbors to the winning neuron
to output values. Thus the transition of output vectors is much smoother than
that obtained with competitive layers, where only one neuron has an output at
a time.
Figures
Competitive Network Architecture
AA
AA
AAA
AAA
AA
AA
Input
Competitive Layer
S1xR
IW1,1
p
Rx1
1
R
7-32
|| ndist ||
b1
S1x1
AA
AA
AA
a1
S1x1
n1
S1x1
S1x1
C
S1
Summary and Conclusions
Self Organizing Feature Map Architecture
Input
AA
AAA
AAA
IW1,1
S1xR
p
R x1
R
A
A
A
A
Self Organizing Map Layer
|| ndist ||
n1
S1x1
C
a1
S1x1
S1
ni1 = - || iIW1,1 - p ||
a1 = compet (n1)
New Functions
This chapter introduces the following new functions:
Function
Description
newc
Create a competitive layer.
newsom
Create a self-organizing map.
learncon
Conscience bias learning function.
boxdist
Distance between two position vectors.
dist
Euclidean distance weight function.
linkdist
Link distance function
mandist
Manhattan distance weight function.
gridtop
Gridtop layer topology function
hextop
Hexagonal layer topology function.
randtop
Random layer topology function.
7-33
7
Self-Organizing Networks
7-34
8
Learning Vector
Quantization
Introduction . . . . . . . . . . . . . . . . . . . . 8-2
Important LVQ Functions . . . . . . . . . . . . . . . 8-2
Network Architecture
. . . . . . . . . . . . . . . 8-3
Creating an LVQ Network (NEWLVQ)
. . . . . . . . 8-5
LVQ Learning Rule(LEARNLV2) . . . . . . . . . . . 8-9
Training . . . . . . . . . . . . . . . . . . . . . . 8-11
Summary . . . . . . . . . . . . . . . . . . . . . 8-14
Figures . . . . . . . . . . . . . . . . . . . . . . . 8-14
New Functions . . . . . . . . . . . . . . . . . . . 8-14
8
Learning Vector Quantization
Introduction
Learning vector quantization (LVQ) is a method for training competitive layers
in a supervised manner. A competitive layer will automatically learn to classify
input vectors. However, the classes that the competitive layer finds are
dependent only on the distance between input vectors. If two input vectors are
very similar, the competitive layer probably will put them into the same class.
There is no mechanism in a strictly competitive layer design to say whether or
not any two input vectors are in the same class or different classes.
LVQ networks, on the other hand, learn to classify input vectors into target
classes chosen by the user.
You might consult the following reference:
Kohonen, T. Self-Organization and Associative Memory, 2nd Edition, Berlin:
Springer-Verlag, 1987.
Important LVQ Functions
An LVQ network can be created with the function newlvq.
For a list of all LVQ functions and demonstrations type help lvq.
8-2
Network Architecture
Network Architecture
The LVQ network architecture is shown below.
A
AAAA AA
AA
AA
A
AA AA AA A
Input
Linear Layer
Competitive Layer
IW1,1
S1xR
p
R x1
|| ndist ||
n1
a1
S1x1
C
S1x1
LW2,1
n2
R
n 1 = - || IW1,1 - p ||
i
i
a2
S2x1
S2x1
S2xS1
S1
Where...
1
R = # elements in
input vector
S1= # competitive
neurons
S2= # linear neurons
S2
a2 = purelin(LW2,1 a1)
a = compet (n )
1
1
An LVQ network has a first competitive layer and a second linear layer. The
competitive layer learns to classify input vectors in much the same way as the
competitive layers of Chapter 7. The linear layer transforms the competitive
layer’s classes into target classifications defined by the user. We will refer to
the classes learned by the competitive layer as subclasses and the classes of the
linear layer as target classes.
Both the competitive and linear layers have one neuron per (sub or target)
class. Thus, the competitive layer can learn up to S1 subclasses. These, in turn
will be combined by the linear layer to form S2 target classes. (S1 will always
be larger than S2.)
For example, suppose neurons 1, 2, and 3 in the competitive layer all learn
subclasses of the input space which belong to the linear layer target class #2.
Then competitive neurons 1, 2, and 3, will have LW2,1 weights of 1.0 to neuron
n2 in the linear layer, and weights of 0 to all other linear neurons. Thus, the
linear neuron produces a 1 if any of the three competitive neurons (1,2, and 3)
win the competition and output a 1. This is how the subclasses of the
competitive layer are combined into target classes in the linear layer.
In short, a 1 in the ith row of a1 (the rest to the elements of a1 will be zero)
effectively picks the ith column of LW2,1 as the network output. Each such
column contains a single 1, corresponding to a specific class. Thus, subclass 1s
out of layer 1 get put into various classes, by the LW2,1a1 multiplication in
layer 2.
8-3
8
Learning Vector Quantization
We know ahead of time what fraction of the layer one neurons should be
classified into the various class outputs of layer 2, so we can specify the
elements of LW2,1 at the start. However, we will have to go through a training
procedure to get the first layer to produce the correct subclass output for each
vector of the training set. We will discuss this training shortly. First consider
how to create the original network.
8-4
Creating an LVQ Network (NEWLVQ)
Creating an LVQ Network (NEWLVQ)
An LVQ network can be created with the function newlvq.
net = newlvq(PR,S1,PC,LR,LF)
where
• PR is an Rx2 matrix of min and max values for R input elements,
• S1 is the number of first layer hidden neurons, and
• PC is an S2 element vector of typical class percentages.
The learning default is 0.01 and the default learning function is 'learnlv2'.
Suppose we have ten input vectors. We will create a network that assigns each
of these input vectors to one of four subclasses. Thus, we have four neurons in
the first competitive layer. These subclasses are then to be assigned to one of
two output classes by the two neurons in layer 2. The input vectors and targets
are specified by :
P = [-3 -2 -2 0 0 0 0 +2 + 2 +3; ...
0 +1 -1 +2 +1 -1 -2 +1 -1 0]
and
Tc = [1 1 1 2 2 2 2 1 1 1];
It may help to show the details of what we get from these two lines of code.
P =
-3
0
-2
1
-2
-1
0
2
0
1
0
-1
0
-2
2
1
2
-1
3
0
1
1
2
2
2
2
1
1
1
Tc =
1
8-5
8
Learning Vector Quantization
A plot of the input vectors follows.
3
p4
2
p2
1
p1
0
p10
p6
p3
-1
p8
p5
p9
-2
p7
-3
-5
0
5
Input Vectors
As you can see, there are four sub-classes of input vectors. We would like a
network that will classify p1, p2, p3, p8, p9, and p10 to produce an output of 1
and that will classify vectors p4, p5, p6 and p7 to produce an output of 2. Note
that this problem is nonlinearly separable, and so could not be solved by a
perceptron, but an LVQ network will have no difficulty.
Next we convert the Tc matrix to target vectors.
T = ind2vec(Tc)
This gives a sparse matrix T that can be displayed in full with:
targets = full(T)
which gives
targets =
1
0
1
0
1
0
0
1
0
1
0
1
0
1
1
0
1
0
1
0
This looks right. It says, for instance, that if we have the first column of P as
input, we should get the first column of targets as an output, and that output
says the input falls in class 1, which is correct. Now we are ready to call newlvq.
8-6
Creating an LVQ Network (NEWLVQ)
We will call newlvq with the proper arguments so that it will create a network
with four neurons in the first layer and two neurons in the second layer. The
first layer weights will be initialized to the center of the input ranges with the
function midpoint. The second layer weights will have 60% (6 of the 10 in Tc
above) of its columns with a 1 in the first row, corresponding to class 1, and 40%
of its columns will have a 1 in the second row, corresponding to class 2.
net = newlvq(range(P),4,[.6 .4]);
We can check to see the initial values of the first layer weight matrix:
net.IW{1,1}
ans =
0
0
0
0
0
0
0
0
These zero weights are indeed the values at the midpoint of the range (-3 to +3)
of the inputs, as we would expect when using midpoint for initialization.
We can look at the second layer weights with
net.LW{2,1}
ans =
1
1
0
0
0
1
0
1
This makes sense too. It says that if the competitive layer produces a 1 as the
first or second element, the input vector will be classified as class 1. Otherwise
it will be in class 2.
You may notice that the first two competitive neurons are connected to the first
linear neuron (with weights of 1), while the second two competitive neurons are
connected to the second linear neuron. All other weights between the
competitive neurons and linear neurons have values of 0. Thus, each of the two
target classes (the linear neurons) is in fact, the union of two subclasses (the
competitive neurons).
8-7
8
Learning Vector Quantization
We can simulate the network with sim. We will use the original P matrix as
input just to see what we get.
Y = sim(net,P);
Y = vec2ind(Yb4t)
Y =
1
1
1
1
1
1
1
1
1
1
The network classifies all inputs into class 1. This is not what we want of
course. We will have to train the network, adjusting the weights of layer 1 only,
before we can expect a good result. We will look at the training process shortly,
but first will discuss the LVQ learning rule.
8-8
LVQ Learning Rule(LEARNLV2)
LVQ Learning Rule(LEARNLV2)
LVQ learning in the competitive layer is based on a set of input/target pairs.
{ p 1, t 1 }, { p 2, t 2 }, …, { p Q, t Q }
Each target vector has a single 1. The rest of its elements are 0. The 1 tells the
proper classification of the associated input. For instance, consider the
following training pair.


0 

2

0 
 p1 = –1 , t1 =

1 

0

0 

Here we have input vectors of three elements, and each input vector is to be
assigned to one of four classes. The network is to be trained so that it will
classify the input vector shown above into the third of four classes.
To train the network an input vector p is presented, and the distance from p to
each row of the input weight matrix IW1,1 is computed with the function ndist.
The hidden neurons of layer 1 compete. Let us suppose that the ith element of
n1 is most positive, and neuron i* wins the competition. Then the competitive
transfer function produces a 1 as the i*th element of a1. All other elements of
a1 will be 0.
When a1 is multiplied by the layer 2 weights LW2,1, the single 1 in a1 selects
the class, k* associated with the input. Thus, the network has assigned the
input vector p to class k*.
Of course, this assignment may be a good one or a bad one. We will adjust the
i*th row of IW1,1 in such a way as to move this row closer to the input vector p
if the assignment is correct, and to move the row away from p if the assignment
is incorrect. So if p is classified correctly,
2
( a k∗ = t k∗ = 1 )
we will compute the new value of the i*th row of IW1,1 as:
1, 1
IW
i∗
(q) =
1, 1
IW
i∗
1, 1
( q – 1 ) + α ( p ( q ) – i∗IW
(q – 1)) .
8-9
8
Learning Vector Quantization
On the other hand, if p is classified incorrectly,
2
( a k∗ = 1 ≠ t k∗ = 0 ) ,
we will compute the new value of the i*th row of IW1,1 as:
IW
i∗
1, 1
(q) =
1, 1
IW
i∗
1, 1
( q – 1 ) – α ( p ( q ) – i∗IW
(q – 1))
Such corrections move the hidden neuron towards vectors that fall into the
class for which it forms a subclass, and away from vectors which fall into other
classes.
The default learning function that implements these changes in the layer 1
weights in LVQ networks is learnlv2. It will be applied during training.
8-10
Training
Training
Next we need to train the network to obtain first layer weights that will lead
to the correct classification of input vectors. We will do this with train as
shown below. First set the training epochs to 1000 and the learning rate to
0.05. Then use train.
net.trainParam.epochs = 1000;
net.trainParam.lr = 0.05;
net = train(net,P,T);
Now check on the first layer weights.
net.IW{1,1}
ans =
-2.2506
2.2432
-0.0023
0
-0.0948
-0.0623
1.3048
-1.2692
The following plot shows that these weights are fairly close to their respective
classification groups.
3
2
1
0
-1
-2
-3
-5
0
5
Weights (circles) after 1000 epochs
8-11
8
Learning Vector Quantization
To check to see that these weights do indeed lead to the correct classification,
take the matrix P as input and simulate the network. Then see what
classifications are produced by the network.
Y = sim(net,P)
Yc = vec2ind(Y)
This gives
Yc =
1
1
1
2
2
2
2
1
1
1
which is what we expected. As a last check, try an input close to a vector that
was used in training.
pchk1 = [0; 0.5];
Y = sim(net,pchk1);
Yc1 = vec2ind(Y)
Which gives
Yc1 =
2
This looks right, for pchk1 is close to other vectors classified as 2. Similarly,
pchk2 = [1; 0];
Y = sim(net,pchk2);
Yc2 = vec2ind(Y)
gives
Yc2 =
1
This looks right too, for pchk2 is close to other vectors classified as 1s.
You might try demolvq1 to see an LVQ network problem.
Note that you get about the same results by running adapt for 100 epochs
through the10 vectors. This will give us 1000 vector presentations just the
same as was done when using train. First set the epochs to 100.
net.adaptparam.passes = 100;
8-12
Training
Recall that the input vectors for adapt are to be in a cell array, so create such
an array from the original input signals with the following code.
Pseq = con2seq(P)
Tseq = con2seq(T)
Now apply adapt, and check on the results.
net = adapt(net,Pseq,Tseq);
wtsl1b = net.IW{1,1}
wtsl1b =
-2.2157
-0.0032
2.2221
-0.0027
-0.0027
1.2955
-0.0027
-1.2999
Now test the network with our original input vectors as we did before.
Y = sim(net,P);
Yc = vec2ind(Y);
Yc =
1
1
1
2
2
2
2
1
1
1
Here adapt and train give nearly the same results in about the same running
time. You may recall that when train is used for competitive networks, it
selects the input vectors in random order, and for that reason might give a
better result than adapt in some problems. In the problem studied here the
results given by train and adapt are about the same. Of course, one could
generate 1000 randomly ordered vectors from the 10 input vectors with the
following code.
TS = 1000;
ind = floor(rand(1,TS)*size(P,2))+1;
Pseq = con2seq(P(:,ind));
Tseq = con2seq(T(:,ind));
Finally then, we would need to make only one pass.
net.adaptparam.passes = 1;
net = adapt(net,Pseq,Tseq);
This code gives a network much like those from train and adapt that were
described previously.
8-13
8
Learning Vector Quantization
Summary
LVQ networks classify input vectors into target classes by using a competitive
layer to find subclasses of input vectors, and then combining them into the
target classes.
Unlike perceptrons, LVQ networks can classify any set of input vectors, not
just linearly separable sets of input vectors. The only requirement is that the
competitive layer must have enough neurons, and each class must be assigned
enough competitive neurons.
To insure that each class is assigned an appropriate amount of competitive
neurons, it is important that the target vectors used to initialize the LVQ
network have the same distributions of targets as the training data the
network will be trained on. If this is done, target classes with more vectors will
be the union of more subclasses.
Figures
LVQ Architecture
Input
AA
AAAA
AA
IW1,1
S1xR
p
R x1
R
A
A
A
A
|| ndist ||
n1
S1x1
C
a2
a1
S1x1
S1
ni1 = - || iIW1,1 - p ||
a1 = compet (n1)
AA
AA
AA
AAAA
AA
Linear Layer
Competitive Layer
LW2,1
S2x1
n2
S2x1
S2xS1
1
Where...
R = # elements in
input vector
S1= # competitive
neurons
S2= # linear neurons
S2
a2 = purelin(LW2,1 a1)
New Functions
This chapter introduces the following new functions:
8-14
Function
Description
newlvq
Create a learning vector quantization network.
learnlv2
LVQ2 weight learning function.
9‘
Recurrent Networks
Introduction . . . . . . . . . . . . . . . . . . . . 9-2
Important Recurrent Network Functions . . . . . . . . . 9-2
Elman Networks . . . . . . . . .
Architecture . . . . . . . . . . . .
Creating an Elman Network (NEWELM)
Training an Elman Network . . . . .
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9-3
9-3
9-4
9-6
Hopfield Network
Fundamentals . . .
Architecture . . . .
Design(NEWHOP) .
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9-9
9-9
9-9
9-11
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Summary . . . . . . . . . . . . . . . . . . . . . 9-16
Figures . . . . . . . . . . . . . . . . . . . . . . . 9-17
New Functions . . . . . . . . . . . . . . . . . . . 9-18
9
Recurrent Networks
Introduction
Recurrent networks is a topic of considerable interest. This chapter covers two
recurrent networks: Elman and Hopfield networks.
Elman networks are two-layer backpropagation networks, with the addition of
a feedback connection from the output of the hidden layer to its input. This
feedback path allows Elman networks to learn to recognize and generate
temporal patterns, as well as spatial patterns. The best paper on the Elman
network is:
Elman, J. L., “Finding structure in time,” Cognitive Science, vol. 14, 1990, pp.
179-211.
The Hopfield network is used to store one or more stable target vectors. These
stable vectors can be viewed as memories which the network recalls when
provided with similar vectors which act as a cue to the network memory. You
may want to pursue a basic paper in this field:
Li, J., A. N. Michel, and W. Porod, “Analysis and synthesis of a class of neural
networks: linear systems operating on a closed hypercube,” IEEE Transactions
on Circuits and Systems, vol. 36, no. 11, November 1989, pp. 1405-1422.
Important Recurrent Network Functions
Elman networks can be created with the function newelm.
Hopfield networks can be created with the function newhop.
Type help elman or help hopfield to see a list of function and demonstrations
related to either of these networks.
9-2
Elman Networks
Elman Networks
Architecture
The Elman network commonly is a two-layer network with feedback from the
first layer output to the first layer input. This recurrent connection allows the
Elman network to both detect and generate time-varying patterns. A two layer
Elman network is shown below.
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA AA
D
a (k-1)
1
LW1,1
p
R1 x 1
IW1,1
S 1 x R1
a1(k)
LW2,1
S1x1
n1
S2xS1
S1x1
1
R1
Input
b
1
S1x1
S1
Recurrent tansig layer
a1(k) = tansig (IW1,1p +LW1,1a1(k-1) + b1)
AA
AA
AA
AA
1
b
2
S1x1
n2
S2x1
A
A
A
A
a2(k)
S2x1
S2
Output purelin Layer
a2(k) = purelin (LW2,1a1(k) + b2)
The Elman network has tansig neurons in its hidden (recurrent) layer, and
purelin neurons in its output layer. This combination is special in that
two-layer networks with these transfer functions can approximate any
function (with a finite number of discontinuities) with arbitrary accuracy. The
only requirement is that the hidden layer must have enough neurons. More
hidden neurons are needed as the function being fit increases in complexity.
Note that the Elman network differs from conventional two-layer networks in
that the first layer has a recurrent connection. The delay in this connection
stores values from the previous time step, which can be used in the current
time step.
Thus, even if two Elman networks, with the same weights and biases, are given
identical inputs at a given time step, their outputs can be different due to
different feedback states.
9-3
9
Recurrent Networks
Because the network can store information for future reference it is able to
learn temporal patterns as well as spatial patterns. The Elman network can be
trained to respond to, and to generate, both kinds of patterns.
Creating an Elman Network (NEWELM)
An Elman network with two or more layers can be created with the function
newelm. The hidden layers commonly have tansig transfer functions, so that is
the default for newelm. As shown in the architecture diagram, purelin is
commonly the output layer transfer function.
The default backpropagation training function is trainbfg. One might use
trainlm, but it tends to proceed so rapidly that it does not necessarily do well
in the Elman network. The backprop weight/bias learning function default is
learngdm, and the default performance function is mse.
When the network is created, each layers weights and biases are initialized
with the Nguyen-Widrow layer initialization method implemented in the
function initnw.
Now consider an example. Suppose that we have a sequence of single element
input vectors in the range from 0 to 1. Suppose further that we want to have
five hidden layer tansig neurons and a single logsig output layer. The following
code creates the desired network.
net = newelm([0 1],[5 1],{'tansig','logsig'});
Simulation
Suppose that we would like to find the response of this network to an input
sequence of eight digits which are either 0 or 1.
P = round(rand(1,8))
P =
0
1
0
1
1
0
0
0
Recall that a sequence to be presented to a network is to be in cell array form.
We can convert P to this form with
Pseq = con2seq(P)
Pseq =
[0]
[1]
[0]
9-4
[1]
[1]
[0]
[0]
[0]
Elman Networks
Now we can find the output of the network with the function sim.
Y = sim(net,Pseq)
Y =
Columns 1 through 5
[1.9875e-04]
[0.1146]
[5.0677e-05]
[0.0017]
Columns 6 through 8
[0.0014]
[5.7241e-05]
[3.6413e-05]
[0.9544]
We will convert this back to concurrent form with
z = seq2con(Y);
and can finally display the output in concurrent form with
z{1,1}
ans =
Columns 1 through 7
0.0002
0.1146
0.0001
Column 8
0.0000
0.0017
0.9544
0.0014
0.0001
Thus, once the network is created and the input specified, one need only
call sim.
9-5
9
Recurrent Networks
Training an Elman Network
Elman networks can be trained with either of two functions, train or adapt.
When using the function train to train an Elman network the following occurs.
At each epoch:
1 The entire input sequence is presented to the network, and its outputs are
calculated and compared with the target sequence to generate an error
sequence.
2 For each time step the error is backpropagated to find gradients of errors for
each weight and bias. This gradient is actually an approximation since the
contributions of weights and biases to errors via the delayed recurrent
connection are ignored.
3 This gradient is then used to update the weights with the backprop training
function chosen by the user. The function traingdx is recommended.
When using the function adapt to train an Elman network the following occurs.
At each time step:
1 Input vectors are presented to the network, and it generates an error.
2 The error is backpropagated to find gradients of errors for each weight and
bias. This gradient is actually an approximation since the contributions of
weights and biases to the error, via the delayed recurrent connection are
ignored.
3 This approximate gradient is then used to update the weights with the
learning function chosen by the user. The function learngdm is
recommended.
Elman network's are not as reliable as some other kinds of networks because
both training and adaption happen using an approximation of the error
gradient.
For an Elman to have the best chance at learning a problem it needs more
hidden neurons in its hidden layer than are actually required for a solution by
another method, for while a solution may be available with fewer neurons, the
9-6
Elman Networks
Elman network is less able to find the most appropriate weights for hidden
neurons since the error gradient is approximated. Therefore having a fair
number of neurons to begin with makes it more likely that the hidden neurons
will start out dividing up the input space in useful ways.
The function train trains an Elman network to generate a sequence of target
vectors when it is presented with a given sequence of input vectors. The input
vectors and target vectors are passed to train as matrices P and T. Train takes
these vectors and the initial weights and biases of the network, trains the
network using backpropagation with momentum and an adaptive learning
rate, and returns new weights and biases.
Let us continue with the example of the previous section, and suppose that we
wish to train a network with an input P and targets T as defined below.
P = round(rand(1,8))
P =
1
0
1
1
1
0
1
1
T = [0 (P(1:end-1)+P(2:end) == 2)]
T =
0
0
0
1
1
0
0
1
and
Here T is defined to be 0 except when two ones occur in P, in which case T will
be 1.
As noted previously, our network will have five hidden neurons in the first
layer.
net = newelm([0 1],[5 1],{'tansig','logsig'});
We will use trainbfg as the training function and train for 100 epochs. After
training we will simulate the network with the input P and calculate the
difference between the target output and the simulated network output.
net = train(net,Pseq,Tseq);
Y = sim(net,Pseq);
z = seq2con(Y);
z{1,1};
diff1 = T - z{1,1}
9-7
9
Recurrent Networks
Note that the difference between the target and the simulated output of the
trained network is very small. Thus, the network have been trained to produce
the desired output sequence on presentation of the input vector P.
See Chapter 10 for an application of the Elman network to the detection of
wave amplitudes.
9-8
Hopfield Network
Hopfield Network
Fundamentals
The goal here is to design a network that stores a specific set of equilibrium
points such that, when an initial condition is provided, the network eventually
comes to rest at such a design point. The network is recursive in that the output
is fed back as the input once the network is in operation. Hopefully the network
output will settle on one of the original design points
The design method that we present is not perfect in that the designed network
may have undesired spurious equilibrium points in addition to the desired
ones. However, the number of these undesired points is made as small as
possible by the design method. Further, the domain of attraction of the
designed equilibrium points is as large as possible.
The design method is based on a system of first-order linear ordinary
differential equations that are defined on a closed hypercube of the state space.
The solutions exist on the boundary of the hypercube. These systems have the
basic structure of the Hopfield model but are easier to understand and design
than the Hopfield model.
The material in this section is based on the following paper: Jian-Hua Li,
Anthony N. Michel and Wolfgang Porod, “Analysis and synthesis of a class of
neural networks: linear systems operating on a closed hypercube,” IEEE
Trans. on Circuits and Systems vol 36, no. 11, pp. 1405-22, November 1989.
You may wish to read Chapter 18 of [HDB96] for further information on
Hopfield networks.
Architecture
The architecture of the network that we will use follows.
9-9
9
Recurrent Networks
AA
AA
AA
AA
AA
AA
AA
AA
AA AA
a1(k-1)
p
LW1,1
S 1 x R1
a (0)
1
R1 x 1
D
n1
a1(k)
S1x1
S1x1
1
b1
R1
S1x1
S1
Sym. Sat. Linear Layer
a (0) = p and then for k = 1, 2, ...
Initial
Conditions
1
a1(k) = satlins (LW1,1a1(k-1)) + b1)
As noted, the input p to this network merely supplies the initial conditions.
The Hopfield network uses the saturated linear transfer function satlins.
a
+1
n
-1
0 +1
-1
AA
AA
a = satlins(n)
Satlins Transfer Function
For inputs less than -1 satlins produces -1. For inputs in the range -1 to +1 it
simply returns the input value. For inputs greater than +1 it produces +1.
This network can be tested with one or more input vectors which are presented
as initial conditions to the network. After the initial conditions are given, the
network produces an output which is then fed back to become the input. This
process is repeated over and over until the output stabilizes. Hopefully, each
output vector eventually converges to one of the design equilibrium point
vectors that is closest to the input that provoked it.
9-10
Hopfield Network
Design(NEWHOP)
Li et. al. [LiMi89] have studied a system that has the basic structure of the
Hopfield network but is, in Li’s own words, “easier to analyze, synthesize and
implement than the Hopfield model.” The authors are enthusiastic about the
reference article, as it has many excellent points and is one of the most
readable in the field. However, the design is mathematically complex, and even
a short justification of it would burden this guide. Thus, we present the Li
design method, with thanks to Li et al., as a recipe that is found in the function
newhop.
Given a set of target equilibrium points represented as a matrix T of vectors,
newhop returns weights and biases for a recursive network. The network is
guaranteed to have stable equilibrium points at the target vectors, but it may
contain other spurious equilibrium points as well. The number of these
undesired points is made as small as possible by the design method.
Once the network has been designed, it can be tested with one or more input
vectors. Hopefully those input vectors close to target equilibrium points will
find their targets. As suggested by the network figure, an array of input vectors
may be presented at one time or in a batch. The network proceeds to give
output vectors which are fed back as inputs. These output vectors can be can
be compared to the target vectors to see how the solution is proceeding.
The ability to run batches of trial input vectors quickly allows you to check the
design in a relatively short time. First you might check to see that the target
equilibrium point vectors are indeed contained in the network. Then you could
try other input vectors to determine the domains of attraction of the target
equilibrium points and the locations of spurious equilibrium points if they are
present.
Consider the following design example. Suppose that we wish to design a
network with two stable points in a three dimensional space.
T = [-1 -1 1; 1 -1 1]'
T =
-1
1
-1
-1
1
1
We can execute the design with
net = newhop(T);
9-11
9
Recurrent Networks
Next we might check to make sure that the designed network is at these two
points. We can do this as follows. (Since Hopfield networks have no inputs, the
second argument to sim below is Q = 2 when using matrix notation).
Ai = T;
[Y,Pf,Af] = sim(net,2,[],Ai);
Y
This gives us
Y =
-1
-1
1
1
-1
1
Thus, the network has indeed been designed to be stable at its design points.
Next we might try another input condition that is not a design point, such as:
Ai = {[-0.9; -0.8; 0.7]}
which gives
Ai =
-0.9000
-0.8000
0.7000
.This point is reasonably close to the first design point, so one might anticipate
that the network would converge to that first point. To see if this happens, we
will run the following code. Note, incidentally, that we specified the original
point in cell array form. This allows us to run the network for more than one
step.
[Y,Pf,Af] = sim(net,{1 5},{},Ai);
Y{1}
We get
Y =
-1
-1
1
Thus, an original condition close to a design point did converge to that point.
9-12
Hopfield Network
This is, of course, our hope for all such inputs. Unfortunately, even the best
known Hopfield designs occasionally include undesired spurious stable points
that attract the solution.
Example
Consider a Hopfield network with just two neurons. Each neuron has a bias
and weights to accommodate two element input vectors weighted. We define
the target equilibrium points to be stored in the network as the two columns of
the matrix T.
T = [1 -1; -1 1]'
T =
1
-1
-1
1
Here is a plot of the Hopfield state space with the two stable points labeled
with ‘*’ markers.
Hopfield Network State Space
1
a(2
)
0.5
0
-0.5
-1
-1
0
a(1)
1
These target stable points are given to newhop to obtain weights and biases of
a Hopfield network.
net = newhop(T);
9-13
9
Recurrent Networks
The design returns a set of weights and a bias for each neuron. The results are
obtained from:
W= net.LW{1,1}
which gives
W =
0.6925
-0.4694
-0.4694
0.6925
and from
b = net.b{1,1}
which gives
b =
1.0e-16 *
0.6900
0.6900
Next the design is tested with the target vectors T to see if they are stored in
the network. The targets are used as inputs for the simulation function sim.
Ai = T;
[Y,Pf,Af] = sim(net,2,[],Ai);
Y =
1
-1
-1
1
As hoped, the new network outputs are the target vectors. The solution stays
at its initial conditions after a single update, and therefore will stay there for
any number of updates.
9-14
Hopfield Network
Now you might wonder how the network performs with various random input
vectors. Here is a plot showing the paths that the network took through its
state space, to arrive at a target point.
Hopfield Network State Space
1
a(2
)
0.5
0
-0.5
-1
-1
0
a(1)
1
This plot show the trajectories of the solution for various starting points. You
might try the demonstration demohop1 to see more of this kind of network
behavior.
Hopfield networks can be designed for an arbitrary number of dimensions. You
might try demohop3 to see a three dimensional design.
Unfortunately, Hopfield networks may have both unstable equilibrium points
and spurious stable points. You might try demonstration programs demohop2
and demohop4 to investigate these issues.
9-15
9
Recurrent Networks
Summary
Elman networks, by having an internal feedback loop, are capable of learning
to detect and generate temporal patterns. This makes Elman networks useful
in such areas as signal processing and prediction where time plays a dominant
role.
Because Elman networks are an extension of the two-layer sigmoid/linear
architecture, they inherit the ability to fit any input/output function with a
finite number of discontinuities. They are also able to fit temporal patterns, but
may need many neurons in the recurrent layer to fit a complex function.
Hopfield networks can act as error correction or vector categorization
networks. Input vectors are used as the initial conditions to the network, which
recurrently updates until it reaches a stable output vector.
Hopfield networks are interesting from a theoretical standpoint, but are
seldom used in practice. Even the best Hopfield designs may have spurious
stable points that lead to incorrect answers. More efficient and reliable error
correction techniques, such as backpropagation, are available.
9-16
Summary
Figures
Elman Network
AA
AAA
AA
AAA
AA
AA
AA
AA
AA
AA AA
D
a1(k-1)
LW1,1
p
R1 x 1
a1(k)
IW1,1
LW2,1
S1x1
n1
S 1 x R1
S2xS1
S1x1
1
b1
S1x1
R1
Input
S1
Recurrent tansig layer
AA
AA
AA
AA
1
b2
S1x1
AA
AA
AA
LW1,1
S 1 x R1
1
R1
Initial
Conditions
b
S2
a2(k) = purelin (LW2,1a1(k) + b2)
D
a1(k-1)
R1 x 1
S2x1
AA
AA
Hopfield Network
a1(0)
S2x1
a2(k)
Output purelin Layer
a1(k) = tansig (IW1,1p +LW1,1a1(k-1) + b1)
p
n2
A
A
A
A
1
S1x1
AA
AA
AA
a1(k)
S1x1
n1
S1x1
S1
Sym. Sat. Linear Layer
a1(0) = p and then for k = 1, 2, ...
a1(k) = satlins (LW1,1a1(k-1)) + b1)
9-17
9
Recurrent Networks
New Functions
This chapter introduces the following new functions:
9-18
Function
Description
newelm
Create an Elman backpropagation network.
newhop
Create a Hopfield recurrent network.
satlins
Symmetric saturating linear transfer function.
10
Applications
Introduction . . . . . . . . . . . . . . . . . . . . 10-2
Applin1: Linear Design . . . . . . . . . . . . . . . 10-3
Applin2: Adaptive Prediction . . . . . . . . . . . . 10-7
Applin3: Linear System Identification . . . . . . . 10-11
Applin4: Adaptive System Identification . . . . . . 10-15
Appelm1: Amplitude Detection
. . . . . . . . . . 10-19
Appcs1: Nonlinear System Identification . . . . . . 10-24
Appcs2: Model Reference Control
Appcr1: Character Recognition
. . . . . . . . . 10-30
. . . . . . . . . . 10-38
10
Applications
Introduction
Today, problems of economic importance that could not be approached
previously in any practical way can now be solved with neural networks. Some
of the recent neural network applications are discussed in this chapter. See
Chapter 1 for a list of many areas where neural networks already have been
applied.
The rest of this chapter describes applications that are practical and make
extensive use of the neural network functions described in this User’s Guide.
Application Scripts
The linear network applications are contained in scripts applin1 through
applin4.
The Elman network amplitude detection application is contained in the script
appelm1.
The control system applications are contained in scripts appcs1 and appcs2.
The character recognition application is in appcr1.
Type help nndemos to see a listing of all neural network demonstrations or
applications.
10-2
Applin1: Linear Design
Applin1: Linear Design
Problem Definition
Here is the definition of a signal T which lasts five seconds, and is defined at a
sampling rate of 40 samples per second.
time = 0:0.025:5;
T = sin(time*4*pi);
At any given time step, the network is given the last five values of the signal t,
and expected to give the next value. The inputs P are found by delaying the
signal T from one to five time steps.
P = zeros(5,Q);
P(1,2:Q) = T(1,1:(Q-1));
P(2,3:Q) = T(1,1:(Q-2));
P(3,4:Q) = T(1,1:(Q-3));
P(4,5:Q) = T(1,1:(Q-4));
P(5,6:Q) = T(1,1:(Q-5));
Here is a plot of the signal T.
Signal to be Predicted
1
0.8
0.6
Target Signal
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
10-3
10
Applications
Network Design
Because the relationship between past and future values of the signal is not
changing, the network can be designed directly from examples using newlind.
The problem as defined above has five inputs, the five delayed signal values,
and one output, the next signal value. Thus the network solution must consist
of a single neuron with five inputs.
Linear Neuron
Input
p1
p2
p3
p4
p5
AA A
A
w1,1
n
w1, 5
a
b
1
a = purelin (Wp +b)
Here newlind finds the weights and biases, for the neuron above, that
minimize the sum-squared error for this problem.
net = newlind(P,T);
The resulting network can now be tested.
Network Testing
To test the network, its output a is computed for the five delayed signals P and
compared with the actual signal T.
a = sim(net,P);
Here is a plot of a compared to T.
10-4
Applin1: Linear Design
Output and Target Signals
1
0.8
0.6
Output - Target +
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
The network’s output a and the actual signal t appear to match up perfectly.
Just to be sure, let us plot the error e = T – a.
Error Signal
0.35
0.3
0.25
Error
0.2
0.15
0.1
0.05
0
-0.05
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
The network did have some error for the first few time steps. This occurred
because the network did not actually have five delayed signal values available
until the fifth time step. However, after the fifth time step error was negligible.
The linear network did a good job. Run the script applin1 to see these plots.
10-5
10
Applications
Thoughts and Conclusions
While newlind is not able to return a zero error solution for nonlinear
problems, it does minimize the sum-squared error. In many cases the solution,
while not perfect, may model a nonlinear relationship well enough to meet the
application specifications. Giving the linear network many delayed signal
values gives it more information with which to find the lowest error linear fit
for a nonlinear problem.
Of course, if the problem is very nonlinear and/or the desired error is very low,
backpropagation or radial basis networks would be more appropriate.
10-6
Applin2: Adaptive Prediction
Applin2: Adaptive Prediction
In application script applin2, a linear network is trained incrementally with
adapt to predict a time series. Because the linear network is trained
incrementally, it can respond to changes in the relationship between past and
future values of the signal.
Problem Definition
The signal T to be predicted lasts six seconds with a sampling rate of 20
samples per second. However, after four seconds the signal’s frequency
suddenly doubles.
time1 = 0:0.05:4;
time2 = 4.05:0.024:6;
time = [time1 time2];
T = [sin(time1*4*pi) sin(time2*8*pi)];
Since we will be training the network incrementally, we will change t to a
sequence.
T = con2seq(T);
Here is a plot of this signal:
Signal to be Predicted
1
0.8
0.6
Target Signal
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
Time
4
5
6
10-7
10
Applications
The input to the network is the same signal which makes up the target:
P = T;
Network Initialization
The network has only one neuron, as only one output value of the signal T is
being generated at each time step. This neuron has five inputs, the five delayed
values of the signal T.
Linear Layer
AA
A
AA
A
AA
A
AA
pd(k)
p(k)
TDL
1x1
1
5x1
W
3
4
a(k)
n(k)
1x3
2
1
AA
AA
AA
1x1
1x1
b
1x1
5
The function newlin creates the network shown above. We will use a learning
rate of 0.1 for incremental training.
lr = 0.1;
delays = [1 2 3 4 5];
net = newlin(minmax(cat(2,P{:})),1,delays,lr);
[w,b] = initlin(P,t)
Network Training
The above neuron will be trained incrementally with adapt. Here is the code to
train the network on input/target signals P and T.
[net,a,e]=adapt(net,P,T);
Network Testing
Once the network has been adapted, we can plot its output signal and compare
it to the target signal.
10-8
Applin2: Adaptive Prediction
Output and Target Signals
1.5
1
Output --- Target - -
0.5
0
-0.5
-1
-1.5
0
1
2
3
Time
4
5
6
Initially, it takes the network a second and a half (30 samples) to track the
target signal. Then the predictions are accurate until the fourth second when
the target signal suddenly changes frequency. However, the adaptive network
learns to track the new signal in an even shorter interval as it has already
learned a behavior (a sine wave) similar to the new signal.
A plot of the error signal makes these effects easier to see.
Error Signal
1
0.8
0.6
0.4
Error
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
Time
4
5
6
10-9
10
Applications
Thoughts and Conclusions
The linear network was able to adapt very quickly to the change in the target
signal. The 30 samples required to learn the wave form are very impressive
when one considers that in a typical signal processing application a signal may
be sampled at 20 kHz. At such a sampling frequency 30 samples go by in 1.5
milliseconds.
10-10
Applin3: Linear System Identification
Applin3: Linear System Identification
Linear networks may be used to model real systems. If the real system is linear
or near linear then the linear network can act as a zero, or low, error model.
Problem Definition
Here is an input signal x which is given to a finite impulse response linear
system over a period of five seconds at 25 msec intervals.
time = 0:0.025:5;
X = sin(sin(time).*time*10);
Here is a plot of this input signal.
Input Signal to the System
1
0.8
0.6
0.4
Input Signal
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
Here are the measured outputs T of the system.
T = filter([1 0.5 –1.5],1,X);
Here is a plot of the systems output T.
10-11
10
Applications
Output Signal of the System
2
1.5
1
Output Signal
0.5
0
-0.5
-1
-1.5
-2
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
The network attempts to predict the system’s output T given the current and
previous two input signals P from X.
P = zeros(3,Q);
P(1,1:Q) = X(1,1:Q);
P(2,2:Q) = X(1,1:(Q-1));
P(3,3:Q) = X(1,1:(Q-2));
Network Design
The network required to solve this problem has only one neuron because the
system has only one output. That neuron has to have three inputs to receive
the current and two delayed values of the input signal.
The function newlind designs such a neuron for us.
net = newlind(P,T);
Network Testing
Once the weights and biases are obtained, the network can be tested. Here the
linear network computes its approximation a of the system output T given the
current and delayed values of the system input.
a = sim(net,P);
10-12
Applin3: Linear System Identification
The system output T and the network output a are compared in the following
plot.
Network and System Output Signals
2
1.5
Network Output - System Output +
1
0.5
0
-0.5
-1
-1.5
-2
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
The network appears to do a perfect job of modeling the system.
Here is a plot of the difference between network and system outputs.
-15
2
Error Signal
x 10
Error
1
0
-1
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
Sure enough, the difference is incredibly small (less than 1e–15).
10-13
10
Applications
Thoughts and Conclusions
As with linear predictors, linear system models can be used to model linear
systems with zero error, and nonlinear systems with minimum sum-squared
error.
Giving the linear network many delayed values from the system’s input signal
will allow newlind to minimize the error associated with modeling a nonlinear
system. However, if the system is highly nonlinear, a backpropagation or radial
basis network would be more appropriate.
10-14
Applin4: Adaptive System Identification
Applin4: Adaptive System Identification
Demonstration applin4 shows how to train a linear network to adaptively
model a linear system. By training the network incrementally, it can change its
behavior as the system it is modeling changes.
Problem Definition
Here is the input signal X to the system over a period of six seconds with a
sampling rate of 200 samples per second.
time1 = 0:0.005:4;
time2 = 4.005:0.005:6;
time = [time1 time2];
X = sin(sin(time*4).*time*8);
The input signal X to the system is plotted below.
Input Signal to System
1
0.8
0.6
0.4
Input Signal
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
Time
4
5
6
Here is the output signal T of the system. Note that the system acts differently
in the intervals before and after the fourth second.
steps1 = length(time1);
[T1,state] = filter([1 –0.5],1,X(1:steps1));
steps2 = length(time2);
T2 = filter([0.9 –0.6],1,x((1:steps2) + steps1),state);
T = [T1 T2];
10-15
10
Applications
Here is a plot of the system’s outputs. Note that due to the complexity of the
signal, it is not apparent that the system changed at four seconds.
Output Signal of System
0.8
0.6
0.4
Output Signal
0.2
0
-0.2
-0.4
-0.6
-0.8
0
1
2
3
Time
4
5
6
The input to the network P is the system input signal X. The network must
estimate the system’s output T from the last two values of the input signal. We
need to convert the input and output to sequences, so that the training can be
done incrementally.
T = con2seq(T);
P = con2seq(X);
Network Initialization
The network is created with newlin, which generates the weights and biases
for the two-input linear neuron required for this problem. There will be a
tapped delay line with two delays at the input of the network. The last two
values of the input signal will be used to predict the system’s output.
lr = 0.5;
delays = [0 1];
net = newlin(minmax(cat(2,P{:})),1,delays,lr);
10-16
Applin4: Adaptive System Identification
Network Training
These weights and biases can now be trained incrementally on the signal with
adapt using the learning rate of 0.5, which was set by newlin.
[net,a,e]=adapt(net,P,T);
Network Testing
To see how well the network did, we could plot the network’s estimate a of the
system’s output against the actual output T. It turns out that the signals are
complex enough that this plot is not of much help. (You can run applin4 to
see it.)
Instead we can take a look at the error between the network output and the
system output. This plot is easier to interpret.
Error Signal
0.25
0.2
0.15
0.1
Error
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0
1
2
3
Time
4
5
6
The error plot shows that the network took 2.5 seconds to track the model with
a very high accuracy. Then at four seconds, when the system changed abruptly,
another 0.2 seconds were required for network to properly model it again.
Thoughts and Conclusions
Besides being interesting, an adaptive linear model of a system can be used to
obtain a great deal of information. For example, the adaptive model can be
analyzed at any given time to determine characteristics of the actual system.
10-17
10
Applications
For example, the adaptive network might be monitored so as to give a warning
its constants were nearing values that would result in instability.
Another use for an adaptive linear model is suggested by its ability to find a
minimum sum-squared error linear estimate of a nonlinear system’s behavior.
An adaptive linear model will be highly accurate as long as the nonlinear
system stays near a given operating point. If the nonlinear system moves to a
different operating point, the adaptive linear network will change to model it
at the new point.
The sampling rate should be high to obtain the linear model of the nonlinear
system at its current operating point in the shortest amount of time. However,
there is a minimum amount of time that must occur for the network to see
enough of the system’s behavior to properly model it. To minimize this time, a
small amount of noise can be added to the input signals of the nonlinear
system. This allows the network to adapt faster as more of the operating points
dynamics will be expressed in a shorter amount of time. Of course, this noise
should be small enough so it does not affect the system’s usefulness.
10-18
Appelm1: Amplitude Detection
Appelm1: Amplitude Detection
Elman networks can be trained to recognize and produce both spatial and
temporal patterns. An example of a problem where temporal patterns are
recognized and classified with a spatial pattern is amplitude detection.
Amplitude detection requires that a wave form be presented to a network
through time, and that the network output the amplitude of the wave form.
This is not a difficult problem, but it demonstrates the Elman network design
process.
The following material describes code which is contained in the demonstration
script appelm1.
Problem Definition
The following code defines two sine wave forms, one with an amplitude of 1.0,
the other with an amplitude of 2.0:
p1 = sin(1:20);
p2 = sin(1:20)*2;
The target outputs for these wave forms will be their amplitudes.
t1 = ones(1,20);
t2 = ones(1,20)*2;
These wave forms can be combined into a sequence where each wave form
occurs twice. These longer wave forms will be used to train the Elman network.
p = [p1 p2 p1 p2];
t = [t1 t2 t1 t2];
We want the inputs and targets to be considered a sequence, so we need to
make the conversion from the matrix format.
Pseq = con2seq(p);
Tseq = con2seq(t);
10-19
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Applications
Network Initialization
This problem requires that the Elman network detect a single value (the
signal), and output a single value (the amplitude), at each time step. Therefore
the network must have one input element, and one output neuron.
R = 1;% 1 input eleement
S2 = 1;% 1 layer 2 output neuron
The recurrent layer can have any number of neurons. However, as the
complexity of the problem grows, more neurons are needed in the recurrent
layer for the network to do a good job.
This problem is fairly simple, so only 10 recurrent neurons will be used in the
first layer.
S1 = 10;% 10 recurrent neurons in the first layer
Now the function newelm can be used to create initial weight matrices and bias
vectors for a network with one input which can vary between –2 and +2. We
will use variable learning rate (traingdx) for this example.
net = newelm([-2 2],[S1 S2],{'tansig','purelin'},'traingdx');
Network Training
Now call train.
[net,tr] = train(net,Pseq,Tseq);
As this function finishes training at 500 epochs, it displays the following plot
of errors.
10-20
Appelm1: Amplitude Detection
Mean Squared Error of Elman Network
1
10
0
Mean Squared Error
10
-1
10
-2
10
0
50
100
150
Epoch
200
250
300
The final mean-squared error was about 1.8e-2. We can test the network to see
what this means.
Network Testing
To test the network, the original inputs are presented, and its outputs are
calculated with simuelm.
a = sim(net,Pseq);
Here is the plot:
10-21
10
Applications
Testing Amplitute Detection
2.2
2
Target - - Output ---
1.8
1.6
1.4
1.2
1
0.8
0
10
20
30
40
Time Step
50
60
70
80
The network does a good job. New wave amplitudes are detected with a few
samples. More neurons in the recurrent layer and longer training times would
result in even better performance.
The network has successfully learned to detect the amplitudes of incoming sine
waves.
Network Generalization
Of course, even if the network detects the amplitudes of the training wave
forms, it may not detect the amplitude of a sine wave with an amplitude it has
not seen before.
The following code defines a new wave form made up of two repetitions of a sine
wave with amplitude 1.6 and another with amplitude 1.2.
p3 = sin(1:20)*1.6;
t3 = ones(1,20)*1.6;
p4 = sin(1:20)*1.2;
t4 = ones(1,20)*1.2;
pg = [p3 p4 p3 p4];
tg = [t3 t4 t3 t4];
pgseq = con2seq(pg);
The input sequence pg and target sequence tg will be used to test the ability of
our network to generalize to new amplitudes.
10-22
Appelm1: Amplitude Detection
Once again the function sim is used to simulate the Elman network and the
results are plotted.
a = sim(net,pgseq);
Testing Generalization
2
1.9
1.8
Target - - Output ---
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0
10
20
30
40
Time Step
50
60
70
80
This time the network did not do as well. It seems to have a vague idea as to
what it should do, but is not very accurate!
Improved generalization could be obtained by training the network on more
amplitudes than just 1.0 and 2.0. The use of three or four different wave forms
with different amplitudes can result in a much better amplitude detector.
Improving Performance
Run appelm1 to see plots similar to those above. Then make a copy of this file
and try improving the network by adding more neurons to the recurrent layer,
using longer training times, and giving the network more examples in its
training data.
10-23
10
Applications
Appcs1: Nonlinear System Identification
Application script appcs1 demonstrates how to perform system identification
on the following inverted pendulum system:
0º
u(t)
Ø(t)
180º
Here the antenna arm angle Ø is controlled by applying a current to a dc motor
which is attached to the pendulum. This system may be represented by the
following equations:
Nonlinear System Model (NSM)
x2
d x
----- 1 =
dt x
9.81 sin x 1 – 2x 2 +u
2
where:
x1 = ∅
d∅
x 2 = -------dt
and u is the force applied to the pendulum by a motor. A positive force tends to
turn the pendulum clockwise. The 9.81sin x1 term is the force of gravity on the
pendulum and the –2x2 term is viscous friction acting against the velocity.
10-24
Appcs1: Nonlinear System Identification
The inverted pendulum system is summarized with the function pmodel which
takes the current time t, pendulum angle, pendulum velocity, and the current,
and returns the derivatives of angle, velocity, and force.
x = [angle; vel; force];
dx = pmodel(t,x)
Because the current is an input to the system, its derivative is always returned
as 0.
We can simulate the pendulum from 0 to 0.05 seconds using ode23.
[time,X] = ode23(‘pmodel’,[0 0.05],x)
This function returns a row vector of times, and the matrix X of state vectors
associated with those times.
The problem is, given only the behavior of this pendulum, create a network
model that behaves in an identical manner.
Problem Definition
Examples of pendulum behavior must be created so that the network can be
trained. The following lines of code define several different pendulum angles,
pendulum velocities, and forces.
deg2rad = pi/180;
angle = [–20:40:200]*deg2rad;
vel = [–90:36:90]*deg2rad;
force = –30:6:30;
By taking all possible combinations of these values, and also a set of steady
state conditions at various angles, we get a matrix Pm of 749 pendulum state/
input conditions, where the pendulum’s state is its angle and velocity, and its
input is the current.
angle2 = [-20:10:200]*deg2rad;
Pm = [combvec(angle,vel,force);
[angle2; zeros(2,length(angle2))]];
Next each of these 749 initial state/input conditions are applied to the
pendulum, and its next state, 0.05 seconds later is measured. The next state
consists of the pendulum angle and velocity after 0.05 seconds. The result is a
matrix Tm of the 749 2-element next states. (Actually, we will use the difference
10-25
10
Applications
between the next state and the current state as the target. The neural network
model will learn to predict the change in state over 0.05 seconds. This is done
because the state does not change by a large amount in this period of time, and
we can improve the performance of the model if we predict only the change in
state. If we need to know the actual state we simply need to add the change to
the previous state.)
Network Initialization
The network must transform the pendulum state and input into a next state
(or change in state). As there are two states (angle and velocity) and one input,
the network requires three inputs and one output.
The function initff is used to create a two-layer tansig/purelin network with
these number of inputs and outputs and eight hidden neurons.
S1 = 8;
[S2,Q] = size(Tm);
mnet = newff(minmax(Pm),[S1 S2],{'tansig' 'purelin'},'trainlm');
We use this network to model the nonlinear pendulum system.
Network Training
The Levenberg-Marquardt training function trainlm is used to obtain a
solution in the minimum amount of time. The network is trained for up to 500
epochs, displaying progress every epoch, and with a typical error of 0.0037
radians (0.25 degrees) for the 749 training vectors and the network’s two
outputs.
mnet.trainParam.goal = (0.0037^2);
mnet = train(mnet,Pm,Tm);
Network Testing
The network is tested by simulating its response for the following initial
conditions and comparing the results to the pendulum response.
angle = 5*deg2rad;
vel = 0*deg2rad;
force = 0;
x = [angle; vel; force];
10-26
Appcs1: Nonlinear System Identification
The following plots may be viewed by running appcs1b. This script loads a
previously trained model network.
Here is a plot of the pendulum’s response to those initial conditions.
Pendulum Response
250
Angle (deg)
200
150
100
50
0
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
250
Velocity (deg/sec)
200
150
100
50
0
-50
-100
As can be seen, the pendulum starts out with a slightly positive angle (leaning
slightly to the right), but quickly falls. After an oscillation it approaches an
angle of 180 degrees, pointing straight down.
The network is first simulated in an open loop fashion. This means that at each
time step the network computes its estimate of the pendulum’s next state. This
estimate then becomes the input to the network for the next time step. The
network never receives feedback from the real system. The open loop network
response is compared to the pendulum in the following plot.
10-27
10
Applications
Pendulum and Open Network Response
Angle (deg): P + N -
250
200
150
100
50
0
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
Velocity (deg/sec): P + N -
250
200
150
100
50
0
-50
-100
Note that errors build as the network uses its own estimates as inputs, but the
results are still very accurate.
To get a better idea of how accurate the model network is, we simulate it in a
closed loop manner. At each time step the network is given the actual
pendulum state and is required to estimate the pendulum’s next state. Thus,
the network’s estimates are not fed back into the network. Here is a plot of the
pendulum response and the network closed loop response.
Pendulum and Closed Network Response
Angle (deg): P + N -
250
200
150
100
50
0
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
Velocity (deg/sec): P + N -
250
200
150
100
50
0
-50
-100
10-28
Appcs1: Nonlinear System Identification
The network response (shown with solid line) is so accurate that it perfectly
overlaps the pendulum response (‘+’ markers).
Thoughts and Conclusions
This example demonstrates how to use a nonlinear network (a two-layer
tansig/purelin network) to identify a nonlinear system. Note that the model
was obtained in almost the same way as in applin3, where a linear model was
created. The only difference is that a linear network was being designed in
applin3, so the network could be designed instead of trained.
10-29
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Applications
Appcs2: Model Reference Control
The script appcs2 shows how a neural network can be trained to act as a model
reference controller, if a neural network model is available for the system. In
this script a neural controller is created for the inverted pendulum using the
neural model created in appcs1. For model reference control we want to control
a system so that its output follows the output of a reference model. In this
application we will train a neural network controller which will drive the
antenna arm system to follow a linear reference model.
Mathematical Model
Consider the antenna arm system for which a neural model was created in
appcs1:
0º
u(t)
Ø(t)
180º
Here the antenna arm angle Ø is to be controlled by applying a current to a dc
motor which is attached to the pendulum. We represent this system by the
following equations:
Nonlinear System Model (NSM)
x2
d x
----- 1 =
dt x
9.81 sin x 1 – 2x 2 +u
2
where:
x1 = ∅
d∅
x 2 = -------dt
10-30
Appcs2: Model Reference Control
and u is the force applied to the pendulum by a motor. A positive force turns
the pendulum clockwise. The 9.81sin x1 term is the force of gravity on the
pendulum and the –2x2 term is viscous friction acting against the velocity.
Suppose that we would like the closed loop system to respond with the
dynamics given by the Linear Reference Model (LRM).
x2
d x
----- 1 =
+ 0
dt x
– 9x 1 – 6x 2
9r
2
where r is the desired output angle.
Now let us train a neural network to help perform this model reference control.
Neural Design
A neural controller can be created for the case where the mathematical model
is not available. All that is required is a neural model of the original system
(which is created in appcs1) and a model of our desired system.
The desired linear reference model, described mathematically above, is
available in the function plinear. It takes the current time t, pendulum angle,
pendulum velocity, and the demand angle, and returns the derivatives of angle,
velocity, and demand.
x = [angle; vel; demand];
dx = pmodel(t,x)
Because the demand is an input to the system, its derivative is always returned as 0.
We can simulate the desired linear reference model from 0 to 0.05 seconds
using ode23.
[time,X] = ode23('plinear',0,0.05,x)
This function returns a row vector of times, and the matrix X of state vectors
associated with those times.
We would like to find a controller network which takes the current pendulum
angle, velocity, and the demand angle as inputs, and outputs a current which
can be applied to the pendulum. This current value should make the
pendulum’s next state (in 0.05 seconds) identical to that defined by the desired
linear reference model.
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Applications
Problem Definition
Before the controller can be trained, examples of initial states and desired next
states must be created.
The following lines of code define several different pendulum angles, pendulum
velocities, and demand angles.
deg2rad = pi/180;
angle = [–200:40:200]*deg2rad;
vel = [–90:36:90]*deg2rad;
demand = –180:40:180*deg2rad;
By taking all possible combinations of these values in addition to a set of steady
state initial conditions (velocity = 0, demand = angle) we get a matrix P of 381
pendulum state/input conditions, where the pendulum’s state is its angle and
velocity, and its input is the current.
angle2 = [-10:10:190]*deg2rad;
Pc = [combvec(anglevel,demand)
[angle2; angle2*0; angle2]];
Next each of these 381 initial state/input conditions are applied to the desired
linear model, and its next state, 0.05 seconds later is measured. The next state
consists of the desired pendulum angle and velocity after 0.05 seconds. The
result is a matrix Tc of the 381 two-element target next states. (As in the
identification problem appcs1, we will actually use the difference between the
next state and the current state as the target. The neural network model
learned to predict the change in state over 0.05 seconds. This was done because
the state does not change by a large amount in this period of time, and we can
improve the performance of the model if we predict only the change in state. If
we need to know the actual state we simply need to add the change to the
previous state.)
Network Initialization
The neural controller, like the neural pendulum model, is a
tansig/purelin network. Here newff creates such a network for the controller
which will have eight hidden neurons, and one output (the current to the
pendulum).
S1 = 8;
cnet = newff(minmax(Pc),[S1 1],{'tansig' 'purelin'});
10-32
Appcs2: Model Reference Control
Network Training
First, take a look at the following diagram of the entire neural controller/
pendulum system.
Next
State
State
Demand
Controller
Network
Force
Pendulum
We would like the pendulum to respond with target states Tc once 0.05 seconds
has passed since the pendulum was in the initial state Pc. The problem is that
the error between the actual pendulum behavior and the desired linear
behavior occurs on the outputs of the pendulum. How can these errors be used
to adjust the controller?
The trick is to replace the pendulum with its neural model, for purposes of
training the controller. Here is a diagram of the neural controller/model
system.
Linear
Reference
Model
Target
State
Demand
Controller
Network
Force
Model
Network
Next
State
Error
Now the error occurs at the output of the model network. The derivatives of this
error can be backpropagated through the model network to the control
network. The derivatives are then backpropagated through the controller and
used to adjust its weights and biases. (The neural model’s weights and biases
are not changed.) Thus the control network must learn how to control the
pendulum (represented temporarily by the model network) so that it behaves
like the linear reference model.
In the following code we will set up a combination network that includes both
the model network and the controller network. There will be two different
inputs to this total network, the state and the demand, and there will be four
10-33
10
Applications
layers. (The following code uses advanced features of the Neural Network
Toolbox. You may want to look through Chapter 11 and Chapter 12 before
reviewing the rest of this application.)
numInputs = 2;
numLayers = 4;
tnet = network(numInputs,numLayers);
We next need to set up the network connections.
tnet.biasConnect = [1 1 1 1]';
tnet.inputConnect = [1 0 1 0; 1 0 0 0]';
tnet.layerConnect = [0 0 0 0; 1 0 0 0; 0 1 0 0; 0 0 1 0];
tnet.outputConnect = [0 0 0 1];
tnet.targetConnect = [0 0 0 1];
Now we define the input and layer parameters. The first input corresponds to
the states, and the second input is the demand.
tnet.inputs{1}.range = minmax(Pc(1:2,:));
tnet.inputs{2}.range = minmax(Pc(3,:));
Next we define the size of each layer and the transfer functions.
tnet.layers{1}.size = S1;
tnet.layers{1}.transferFcn
tnet.layers{2}.size = 1;
tnet.layers{2}.transferFcn
tnet.layers{3}.size = 8;
tnet.layers{3}.transferFcn
tnet.layers{4}.size = 2;
tnet.layers{4}.transferFcn
= 'tansig';
= 'purelin';
= 'tansig';
= 'purelin';
We will use the quasi-Newton training function trainbfg to train the network
to minimize the mean square error.
tnet.performFcn = 'mse';
tnet.trainFcn = 'trainbfg';
10-34
Appcs2: Model Reference Control
We now set the initial weights and biases in the total network that correspond
to the controller network. These will be adjusted during the training process,
so the learn parameter is set to 1.
tnet.IW{1,1} = cnet.IW{1,1}(:,1:2);
tnet.inputWeights{1,1}.learn = 1;
tnet.IW{1,2} = cnet.IW{1,1}(:,3);
tnet.inputWeights{1,2}.learn = 1;
tnet.b{1} = cnet.b{1};
tnet.biases{1}.learn = 1;
tnet.b{2} = cnet.b{2};
tnet.biases{2}.learn = 1;
tnet.LW{2,1} = cnet.LW{2,1};
tnet.layerWeights{2,1}.learn = 1;
Finally, we set the weights and biases in the total network that correspond to
the model network. These will not be adjusted during the training process, so
the learn parameter is set to 0.
tnet.IW{3,1} = mnet.IW{1,1}(:,1:2);
tnet.inputWeights{3,1}.learn = 0;
tnet.LW{3,2} = mnet.IW{1,1}(:,3);
tnet.layerWeights{3,2}.learn = 0;
tnet.b{3} = mnet.b{1};
tnet.biases{3}.learn = 0;
tnet.LW{4,3} = mnet.LW{2,1};
tnet.layerWeights{4,3}.learn = 0;
tnet.b{4} = mnet.b{2};
tnet.biases{4}.learn = 0;
Here train is called to do this training for us. The controller will be trained for
up to 600 epochs, displaying progress every 5 epochs, to a typical error of 0.002
radians (0.11 degrees) for each of the 381 training vectors for each of the 2
model network outputs.
tnet.trainParam.show = 5;
tnet.trainParam.epochs = 600;
tnet.trainParam.goal = (0.002^2);
[tnet,tr] = train(tnet,{Pc(1:2,:);
Pc(3,:)},{Tc});
The network contains the weights and biases of both the controller and model
networks, but the model weights do not change. Only the controller weights are
10-35
10
Applications
updated. Now we want to place the new controller weights and biases back into
the controller network.
cnet.IW{1,1}(:,1:2) = tnet.IW{1,1};
cnet.IW{1,1}(:,3) = tnet.IW{1,2};
cnet.b{1} = tnet.b{1};
cnet.b{2} = tnet.b{2};
cnet.LW{2,1} = tnet.LW{2,1};
Network Testing
To test the control network, the neural controller/pendulum system is
simulated and its response compared to the linear reference model. Run
appcs2b to see the following plots.
Here are the results of simulating the linear reference model from an initial
angle of 10 degrees with a velocity of 0 degrees/second, and a constant demand
angle of 90 degrees.
Pendulum Response
100
Angle (deg)
80
60
40
20
0
0
0.5
1
1.5
2
Time
2.5
3
3.5
4
0
0.5
1
1.5
2
Time
2.5
3
3.5
4
Velocity (deg/sec)
100
80
60
40
20
0
The linear reference model quickly moves to 90 degrees, and then holds there.
10-36
Appcs2: Model Reference Control
Here are the results found by simulating the controlled pendulum:
Desired and Actual Response
Angle (deg): D + A -
100
80
60
40
20
0
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
Velocity (deg/sec): D+ A -
100
80
60
40
20
0
The network does a near perfect job of making the nonlinear pendulum system
(solid line) act like the linear reference model (‘+’ markers).
Thoughts and Conclusions
A neural network can be used to control a nonlinear system so that the system
output follows the response of a desired reference model. It is first necessary to
develop a neural network model of the nonlinear system, which is then used in
the training process for the neural network controller.
10-37
10
Applications
Appcr1: Character Recognition
It is often useful to have a machine perform pattern recognition. In particular,
machines which can read symbols are very cost effective. A machine that reads
banking checks can process many more checks than a human being in the same
time. This kind of application saves time and money, and eliminates the
requirement that a human perform such a repetitive task. The script appcr1
demonstrates how character recognition can be done with a backpropagation
network.
Problem Statement
A network is to be designed and trained to recognize the 26 letters of the
alphabet. An imaging system that digitizes each letter centered in the system’s
field of vision is available. The result is that each letter is represented as a 5 by
7 grid of boolean values.
For example, here is the letter A:
10-38
Appcr1: Character Recognition
However, the imaging system is not perfect and the letters may suffer from
noise:
Perfect classification of ideal input vectors is required, and reasonably accurate
classification of noisy vectors.
The twenty-six 35-element input vectors are defined in the function prprob as
a matrix of input vectors called alphabet. The target vectors are also defined
in this file with a variable called targets. Each target vector is a 26-element
vector with a 1 in the position of the letter it represents, and 0’s everywhere
else. For example, the letter A is to be represented by a 1 in the first element
(as A is the first letter of the alphabet), and 0’s in elements two through
twenty-six.
Neural Network
The network will receive the 35 boolean values as a 35-element input vector. It
will then be required to identify the letter by responding with a 26-element
output vector. The 26 elements of the output vector each represent a letter. To
operate correctly the network should respond with a 1 in the position of the
letter being presented to the network. All other values in the output vector
should be 0.
In addition, the network should be able to handle noise. In practice the network
will not receive a perfect boolean vector as input. Specifically, the network
should make as few mistakes as possible when classifying vectors with noise of
mean 0 and standard deviation of 0.2 or less.
10-39
10
Applications
Architecture
The neural network needs 35 inputs and 26 neurons in its output layer to
identify the letters. The network is a two-layer log-sigmoid/log-sigmoid
network. The log-sigmoid transfer function was picked because its output
range (0 to 1) is perfect for learning to output boolean values.
Input
Hidden Layer
AAA
AA
AA
AA
AAA
AA
AAA AAAA
p1
35 x 1
IW1,1
10 x 35
1
35
b1
10 x 1
a1
10 x 1
n1
LW2,1
26 x 10
10
x1
1
10
a1 = logsig (IW1,1p1 +b1)
b2
26 x 1
Output Layer
AA
AA
AA
a2
26 x 1
n2
26 x 1
26
a2 = logsig(LW2,1a1 +b2)
The hidden (first) layer has 10 neurons. This number was picked by guesswork
and experience. If the network has trouble learning, then neurons can be added
to this layer.
The network is trained to output a 1 in the correct position of the output vector
and to fill the rest of the output vector with 0’s. However, noisy input vectors
may result in the network not creating perfect 1’s and 0’s. After the network
has been trained the output will be passed through the competitive transfer
function compet. This makes sure that the output corresponding to the letter
most like the noisy input vector takes on a value of 1 and all others have a value
of 0. The result of this post-processing is the output that is actually used.
Initialization
The two layer network is created with newff.
S1 = 10; S2 = 26;
net = newff(minmax(P),[S1 S2],{'logsig' 'logsig'},'traingdx');
Training
To create a network that can handle noisy input vectors it is best to train the
network on both ideal and noisy vectors. To do this the network will first be
trained on ideal vectors until it has a low sum-squared error.
10-40
Appcr1: Character Recognition
Then the network will be trained on 10 sets of ideal and noisy vectors. The
network is trained on two copies of the noise-free alphabet at the same time as
it is trained on noisy vectors. The two copies of the noise-free alphabet are used
to maintain the network’s ability to classify ideal input vectors.
Unfortunately, after the training described above the network may have
learned to classify some difficult noisy vectors at the expense of properly
classifying a noise free vector. Therefore, the network will again be trained on
just ideal vectors. This ensures that the network will respond perfectly when
presented with an ideal letter.
All training is done using backpropagation with both adaptive learning rate
and momentum with the function trainbpx.
Training Without Noise
The network is initially trained without noise for a maximum of 5000 epochs
or until the network sum-squared error falls beneath 0.1.
P = alphabet;
T = targets;
net.performFcn = 'sse';
net.trainParam.goal = 0.1;
net.trainParam.show = 20;
net.trainParam.epochs = 5000;
net.trainParam.mc = 0.95;
[net,tr] = train(net,P,T);
Training With Noise
To obtain a network not sensitive to noise, we trained with two ideal copies and
two noisy copies of the vectors in alphabet. The target vectors consist of four
copies of the vectors in target. The noisy vectors have noise of mean 0.1 and
0.2 added to them. This forces the neuron to learn how to properly identify
noisy letters, while requiring that it can still respond well to ideal vectors.
10-41
10
Applications
To train with noise the maximum number of epochs is reduced to 300 and the
error goal is increased to 0.6, reflecting that higher error is expected due to
more vectors, including some with noise, are being presented.
netn = net;
netn.trainParam.goal = 0.6;
netn.trainParam.epochs = 300;
T = [targets targets targets targets];
for pass = 1:10
P = [alphabet, alphabet, ...
(alphabet + randn(R,Q)*0.1), ...
(alphabet + randn(R,Q)*0.2)];
[netn,tr] = train(netn,P,T);
end
Training Without Noise Again
Once the network has been trained with noise it makes sense to train it without
noise once more to ensure that ideal input vectors are always classified
correctly. Therefore the network is again trained with code identical to the
"Training Without Noise" section.
System Performance
The reliability of the neural network pattern recognition system is measured
by testing the network with hundreds of input vectors with varying quantities
of noise. The script file appcr1 tests the network at various noise levels and
then graphs the percentage of network errors vs. noise. Noise with mean of 0
and standard deviation from 0 to 0.5 are added to input vectors. At each noise
level 100 presentations of different noisy versions of each letter are made and
the network’s output is calculated. The output is then passed through the
competitive transfer function so that only one of the 26 outputs, representing
the letters of the alphabet, has a value of 1.
10-42
Appcr1: Character Recognition
The number of erroneous classifications are then added and percentages are
obtained:
Percentage of Recognition Errors
50
45
40
Network 1 - - Network 2 ---
35
30
25
20
15
10
5
0
0
0.05
0.1
0.15
0.2
0.25
Noise Level
0.3
0.35
0.4
0.45
0.5
The solid line on the graph shows the reliability for the network trained with
and without noise. The reliability of the same network when it had only been
trained without noise is shown with a dashed line. Thus, training the network
on noisy input vectors greatly reduced its errors when it had to classify noisy
vectors.
The network did not make any errors for vectors with noise of mean 0.00 or
0.05. When noise of mean 0.2 was added to the vectors both networks began to
make errors.
If a higher accuracy is needed the network could be trained for a longer time or
retrained with more neurons in its hidden layer. Also, the resolution of the
input vectors could be increased to say, a 10 by 14 grid. Finally, the network
could be trained on input vectors with greater amounts of noise if greater
reliability were needed for higher levels of noise.
10-43
10
Applications
To test the system a letter with noise can be created and presented to the
network.
noisyJ = alphabet(:,10)+randn(35,1) ∗ 0.2;
plotchar(noisyJ);
A2 = sim(net,noisyJ);
A2 = compet(A2);
answer = find(compet(A2) == 1);
plotchar(alphabet(:,answer));
Here is the noisy letter and the letter the network picked (correctly):
Summary
This problem demonstrates how a simple pattern recognition system can be
designed. Note that the training process did not consist of a single call to a
training function. Instead, the network was trained several times on various
input vectors.
In this case training a network on different sets of noisy vectors forced the
network to learn how to deal with noise, a common problem in the real world.
10-44
12
Network Object Reference
Network Properties
Architecture . . . . .
Subobject Structures .
Functions . . . . . .
Parameters . . . . .
Weight and Bias Values
Other . . . . . . .
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. 12-6
12-10
12-13
12-14
12-16
Subobject Properties
Inputs . . . . . . .
Layers . . . . . . .
Outputs . . . . . .
Targets . . . . . . .
Biases . . . . . . .
Input Weights . . . .
LayerWeights . . . .
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12-17
12-17
12-18
12-25
12-25
12-26
12-28
12-32
12
Network Object Reference
Network Properties
The properties define the basic features of a network. A later section,
“Subobject Properties” describes properties which define network details.
Architecture
These properties determine the number of network subobjects (which include
inputs, layers, outputs, targets, biases, and weights), and how they are
connected.
numInputs
This property defines the number of inputs a network receives.
net.numInputs
It can be set to 0 or a positive integer.
Clarification. The number of network inputs and the size of a network input are
not the same thing. The number of inputs defines how many sets of vectors the
network receives as input. The size of each input (i.e. the number of elements
in each input vector) is determined by the input size (net.inputs{i}.size).
Most networks have only one input, whose size is determined by the problem.
Side Effects. Any change to this property results in a change in the size of the
matrix defining connections to layers from inputs, (net.inputConnect) and the
size of the cell array of input subobjects (net.inputs).
numLayers
This property defines the number of layers a network has.
net.numLayers
It can be set to 0 or a positive integer.
12-2
Network Properties
Side Effects. Any change to this property changes the size of each of these
boolean matrices that define connections to and from layers,
net.biasConnect
net.inputConnect
net.layerConnect
net.outputConnect
net.targetConnect
and changes the size each cell array of subobject structures whose size depends
on the number of layers,
net.biases
net.inputWeights
net.layerWeights
net.outputs
net.targets
and also changes the size of each of the network’s adjustable parameters
properties.
net.IW
net.LW
net.b
biasConnect
This property defines which layers have biases.
net.biasConnect
It can be set to any N l × 1 matrix of boolean values, where N l is the number
of network layers (net.numLayers). The presence (or absence) of a bias to the
ith layer is indicated by a 1 (or 0) at:
net.biasConnect(i)
Side Effects. Any change to this property will alter the presence or absence of
structures in the cell array of biases (net.biases) and in the presence or
absence of vectors in the cell array of bias vectors (net.b).
12-3
12
Network Object Reference
inputConnect
This property defines which layers have weights coming from inputs.
net.inputConnect
It can be set to any N l × N i matrix of boolean values, where N l is the number
of network layers (net.numLayers), and N i is the number of network inputs
(net.numInputs). The presence (or absence) of a weight going to the ith layer
from the jth input is indicated by a 1 (or 0) at:
net.inputConnect(i,j)
Side Effects. Any change to this property will alter the presence or absence of
structures in the cell array of input weight subobjects (net.inputWeights) and
in the presence or absence of matrices in the cell array of input weight matrices
(net.IW).
layerConnect
This property defines which layers have weights coming from other layers.
net.layerConnect
It can be set to any N l × N l matrix of boolean values, where N l is the number
of network layers (net.numLayers). The presence (or absence) of a weight going
to the ith layer from the jth layer is indicated by a 1 (or 0) at:
net.layerConnect(i,j)
Side Effects. Any change to this property will alter the presence or absence of
structures in the cell array of layer weight subobjects (net.layerWeights) and
in the presence or absence of matrices in the cell array of layer weight matrices
(net.LW).
outputConnect
This property defines which layers generate network outputs.
net.outputConnect
It can be set to any 1 × N l matrix of boolean values, where N l is the number
of network layers (net.numLayers). The presence (or absence) of a network
output from the ith layer is indicated by a 1 (or 0) at:
net.outputConnect(i)
12-4
Network Properties
Side Effects. Any change to this property will alter the number of network
outputs (net.numOutputs) and the presence or absence of structures in the cell
array of output subobjects (net.outputs).
targetConnect
This property defines which layers have associated targets.
net.targetConnect
It can be set to any 1 × N l matrix of boolean values, where N l is the number
of network layers (net.numLayers). The presence (or absence) of a target
associated with the ith layer is indicated by a 1 (or 0) at:
net.targetConnect(i)
Side Effects. Any change to this property will alter the number of network
targets (net.numTargets) and the presence or absence of structures in the cell
array of target subobjects (net.targets).
numOutputs (read-only)
This property indicates how many outputs the network has.
net.numOutputs
It is always set to the number 1’s in the matrix of output connections.
numOutputs = sum(net.outputConnect)
numTargets (read-only)
This property indicates how many targets the network has.
net.numTargets
It is always set to the number of 1’s in the matrix of target connections.
numTargets = sum(net.targetConnect)
numInputDelays (read-only)
This property indicates the number of time steps of past inputs that must be
supplied to simulate the network.
net.numInputDelays
12-5
12
Network Object Reference
It is always set to the maximum delay value associated any of the network’s
input weights.
numInputDelays = 0;
for i=1:net.numLayers
for j=1:net.numInputs
if net.inputConnect(i,j)
numInputDelays = max( ...
[numInputDelays net.inputWeights{i,j}.delays]);
end
end
end
numLayerDelays (read-only)
This property indicates the number of time steps of past layer outputs that
must be supplied to simulate the network.
net.numLayerDelays
It is always set to the maximum delay value associated any of the network’s
layer weights.
numLayerDelays = 0;
for i=1:net.numLayers
for j=1:net.numLayers
if net.layerConnect(i,j)
numLayerDelays = max( ...
[numLayerDelays net.layerWeights{i,j}.delays]);
end
end
end
Subobject Structures
These properties consist of cell arrays of structures which define each of the
network’s inputs, layers, outputs, targets, biases, and weights.
The properties for each kind of subobject are described in the “Subobject
Properties” section which follows this “Network Properties” section.
12-6
Network Properties
inputs
This property holds structures of properties for each of the network’s inputs.
net.inputs
It is always an N i × 1 cell array of input structures, where N i is the number
of network inputs (net.numInputs).
The structure defining the properties of the ith network input is located at:
net.inputs{i}
Input Properties. See “Inputs” in the “Subobject Properties” section for
descriptions of input properties.
layers
This property holds structures of properties for each of the network’s layers.
net.layers
It is always an N l × 1 cell array of input structures, where N l is the number
of network layers (net.numLayers).
The structure defining the properties of the ith layer is located at:
net.layers{i}
Layer Properties. See “Layers” in the “Subobject Properties” section for
descriptions of layer properties.
outputs
This property holds structures of properties for each of the network’s outputs.
net.outputs
It is always an N l × 1 cell array, where N l is the number of network layers
(net.numLayers).
The structure defining the properties of the output from the ith layer (or a null
matrix []) is located at:
net.outputs{i}
12-7
12
Network Object Reference
if the corresponding output connection is 1 (or 0).
net.outputConnect(i)
Output Properties. See “Outputs” in the “Subobject Properties” section for
descriptions of output properties.
targets
This property holds structures of properties for each of the network’s targets.
net.targets
It is always an N l × 1 cell array, where N l is the number of network layers
(net.numLayers).
The structure defining the properties of the target associated with the ith layer
(or a null matrix []) is located at:
net.targets{i}
if the corresponding target connection is 1 (or 0).
net.targetConnect(i)
Target Properties. See “Targets” in the “Subobject Properties” section for
descriptions of target properties.
biases
This property holds structures of properties for each of the network’s biases.
net.biases
It is always an N l × 1 cell array, where N l is the number of network layers
(net.numLayers).
The structure defining the properties of the bias associated with the ith layer
(or a null matrix []) is located at:
net.biases{i}
if the corresponding bias connection is 1 (or 0).
net.biasConnect(i)
12-8
Network Properties
Bias Properties. See “Biases” in the “Subobject Properties” section for
descriptions of bias properties.
inputWeights
This property holds structures of properties for each of the network’s input
weights.
net.inputWeights
It is always an N l × N i cell array, where N l is the number of network layers
(net.numLayers) and N i is the number of network inputs (net.numInputs).
The structure defining the properties of the weight going to the ith layer from
the jth input (or a null matrix []) is located at:
net.inputWeights{i,j}
if the corresponding input connection is 1 (or 0).
net.inputConnect(i,j)
Input Weight Properties. See “Input Weights” in the “Subobject Properties” section
for descriptions of input weight properties.
layerWeights
This property holds structures of properties for each of the network’s layer
weights.
net.layerWeights
It is always an N l × N l cell array, where N l is the number of network layers
(net.numLayers).
The structure defining the properties of the weight going to the ith layer from
the jth layer (or a null matrix []) is located at:
net.layerWeights{i,j}
if the corresponding layer connection is 1 (or 0).
net.layerConnect(i,j)
Layer Weight Properties. See “LayerWeights” in the “Subobject Properties” section
for descriptions of layer weight properties.
12-9
12
Network Object Reference
Functions
These properties define the algorithms to use when a network is to adapt, is to
be initialized, is to have its performance measured, or is to be trained.
adaptFcn
This property defines the function to be used when the network adapts.
net.adaptFcn
It can be set to the name of any network adapt function, including this toolbox
function:
adaptwb
- By-weight-and-bias network adaption function.
The network adapt function is used to perform adaption whenever adapt is
called.
[net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai)
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom adapt functions.
Side Effects. Whenever this property is altered the network’s adaption
parameters (net.adaptParam) are set to contain the parameters and default
values of the new function.
initFcn
This property defines the function used to initialize the network’s weight
matrices and bias vectors.
net.initFcn
It can be set to the name of any network initialization function, including this
toolbox function.
initlay
- Layer-by-layer network initialization function.
The initialization function is used to initialize the network whenever init is
called.
net = init(net)
12-10
Network Properties
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom initialization functions.
Side Effects. Whenever this property is altered the network’s initialization
parameters (net.initParam) are set to contain the parameters and default
values of the new function.
performFcn
This property defines the function used to measure the network’s performance.
net.performFcn
It can be set to the name of any performance function, including these toolbox
functions:
Performance Functions
mae
Mean absolute error performance function.
mse
Mean squared error performance function.
msereg
Mean squared error w/reg performance function.
sse
Sum squared error performance function.
The performance function is used to calculate network performance during
training whenever train is called.
[net,tr] = train(NET,P,T,Pi,Ai)
Custom functions. See Chapter 11, “Advanced Topics” for information on creating
custom performance functions.
Side Effects. Whenever this property is altered the network’s performance
parameters (net.performParam) are set to contain the parameters and default
values of the new function.
trainFcn
This property defines the function used to train the network.
net.trainFcn
12-11
12
Network Object Reference
It can be set to the name of any training function, including these toolbox
functions:
Training Functions
trainbfg
BFGS quasi-Newton backpropagation.
trainbr
Bayesian regularization.
traincgb
Powell-Beale conjugate gradient backpropagation.
traincgf
Fletcher-Powell conjugate gradient backpropagation.
traincgp
Polak-Ribiere conjugate gradient backpropagation.
traingd
Gradient descent backpropagation.
traingda
Gradient descent w/adaptive lr backpropagation.
traingdm
Gradient descent w/momentum backpropagation.
traingdx
Gradient descent w/momentum & adaptive lr backprop.
trainlm
Levenberg-Marquardt backpropagation.
trainoss
One step secant backpropagation.
trainrp
Resilient backpropagation (Rprop)
trainscg
Scaled conjugate gradient backpropagation.
trainwb
By-weight-and-bias network training function.
trainwb1
By-weight-&-bias 1-vector-at-a-time training function.
The training function is used to train the network whenever train is called.
[net,tr] = train(NET,P,T,Pi,Ai)
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom training functions.
Side Effects. Whenever this property is altered the network’s training
parameters (net.trainParam) are set to contain the parameters and default
values of the new function.
12-12
Network Properties
Parameters
adaptParam
This property defines the parameters and values of the current adapt function.
net.adaptParam
The fields of this property depend on the current adapt function
(net.adaptFcn). Evaluate the above reference to see the fields of the current
adapt function.
Call help on the current adapt function to get a description of what each field
means.
help(net.adaptFcn)
initParam
This property defines the parameters and values of the current initialization
function.
net.initParam
The fields of this property depend on the current initialization function
(net.initFcn). Evaluate the above reference to see the fields of the current
initialization function.
Call help on the current initialization function to get a description of what each
field means.
help(net.initFcn)
performParam
This property defines the parameters and values of the current performance
function.
net.performParam
The fields of this property depend on the current performance function
(net.performFcn). Evaluate the above reference to see the fields of the current
performance function.
12-13
12
Network Object Reference
Call help on the current performance function to get a description of what each
field means.
help(net.performFcn)
trainParam
This property defines the parameters and values of the current training
function.
net.trainParam
The fields of this property depend on the current training function
(net.trainFcn). Evaluate the above reference to see the fields of the current
training function.
Call help on the current training function to get a description of what each field
means.
help(net.trainFcn)
Weight and Bias Values
These properties define the network’s adjustable parameters: its weight
matrices and bias vectors.
IW
This property defines the weight matrices of weights going to layers from
network inputs.
net.IW
It is always an N l × N i cell array, where N l is the number of network layers
(net.numLayers) and N i is the number of network inputs (net.numInputs).
The weight matrix for the weight going to the ith layer from the jth input (or a
null matrix []) is located at:
net.IW{i,j}
if the corresponding input connection is 1 (or 0).
net.inputConnect(i,j)
12-14
Network Properties
The weight matrix will have as many rows as the size of the layer it goes to
(net.layers{i}.size). It will have as many columns as the product of the
input size with the number of delays associated with the weight.
net.inputs{j}.size * length(net.inputWeights{i,j}.delays)
These dimensions can also be obtained from the input weight properties.
net.inputWeights{i,j}.size
LW
This property defines the weight matrices of weights going to layers from other
layers.
net.LW
It is always an N l × N l cell array, where N l is the number of network layers
(net.numLayers).
The weight matrix for the weight going to the ith layer from the jth layer (or a
null matrix []) is located at:
net.LW{i,j}
if the corresponding layer connection is 1 (or 0).
net.layerConnect(i,j)
The weight matrix will have as many rows as the size of the layer it goes to
(net.layers{i}.size). It will have as many columns as the product of the size
of the layer it comes from with the number of delays associated with the
weight.
net.layers{j}.size * length(net.layerWeights{i,j}.delays)
These dimensions can also be obtained from the layer weight properties.
net.layerWeights{i,j}.size
12-15
12
Network Object Reference
b
This property defines the bias vectors for each layer with a bias.
net.b
It is always an N l × 1 cell array, where N l is the number of network layers
(net.numLayers).
The bias vector for the ith layer (or a null matrix []) is located at:
net.b{i}
if the corresponding bias connection is 1 (or 0).
net.biasConnect(i)
The number of elements in the bias vector is always equal to the size of the
layer it is associated with (net.layers{i}.size).
This dimension can also be obtained from the bias properties.
net.biases{i}.size
Other
The only other property is a user data property.
userdata
This property provides a place for users to add custom information to a network
object.
net.userdata
Only one field is predefined. It contains a secret message to all Neural Network
Toolbox users.
net.userdata.note
Please keep this information confidential.
12-16
Subobject Properties
Subobject Properties
These properties define the details of a network’s inputs, layers, outputs,
targets, biases, and weights.
Inputs
These properties define the details of each ith network input.
net.inputs{i}
range
This property defines the ranges of each element of the ith network input.
net.inputs{i}.range
It can be set to any R i × 2 matrix, where R i is the number of elements in the
input (net.inputs{i}.size) and each element in column 1 is less than the
element next to it in column 2.
Each jth row defines the minimum and maximum values of the jth input
element, in that order:
net.inputs{i}(j,:)
Uses. Some initialization functions use input ranges to find appropriate initial
values for input weight matrices.
Side Effects. Whenever the number of rows in this property is altered, the
layers’s size (net.inputs{i}.size) will change to remain consistent. The size
of any weights coming from this input (net.inputWeights{:,i}.size) and the
dimensions of their weight matrices (net.IW{:,i}) will also change size.
size
This property defines the number of elements in the ith network input.
net.inputs{i}.size
It can be set to 0 or a positive integer.
Side Effects. Whenever this property is altered, the input’s ranges
(net.inputs{i}.ranges), any input weights (net.inputWeights{:,i}.size)
and their weight matrices (net.IW{:,i}) will change size to remain consistent.
12-17
12
Network Object Reference
userdata
This property provides a place for users to add custom information to the ith
network input.
net.inputs{i}.userdata
Only one field is predefined. It contains a secret message to all Neural Network
Toolbox users.
net.inputs{i}.userdata.note
Layers
These properties define the details of each ith network layer.
net.layers{i}
dimensions
This property defines the physical dimensions of the ith layer’s neurons. Being
able to arrange a layer’s neurons in a multi-dimensional manner is important
for self-organizing maps.
net.layers{i}.dimensions
It can be set to any row vector of 0 or positive integer elements, where the
product of all the elements will become the number of neuron’s in the layer
(net.layers{i}.size).
Uses. Layer dimensions are used to calculate the neuron positions within the
layer (net.layers{i}.positions) using the layer’s topology function
(net.layers{i}.topologyFcn).
Side Effects. Whenever this property is altered, the layers’s size
(net.layers{i}.size) will change to remain consistent. The layer’s neuron
positions (net.layers{i}.positions) and the distances between the neurons
(net.layers{i}.distances) will also be updated.
12-18
Subobject Properties
distanceFcn
This property defines the function used to calculate distances between neurons
in the ith layer (net.layers{i}.distances) from the neuron positions
(net.layers{i}.positions). Neuron distances are used by self-organizing
maps.
net.layers{i}.distanceFcn
It can be set to the name of any distance function, including these toolbox
functions:
Distance Functions
boxdist
Distance between two position vectors
dist
Euclidean distance weight function.
linkdist
Link distance function.
mandist
Manhattan distance weight function.
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom distance functions.
Side Effects. Whenever this property is altered, the distance between the layer’s
neurons (net.layers{i}.distances) will be updated.
distances (read-only)
This property defines the distances between neurons in the ith layer. These
distances are used by self-organizing maps.
net.layers{i}.distances
It is always set to the result of applying the layer’s distance function
(net.layers{i}.distanceFcn) to the positions of the layers neurons
(net.layers{i}.positions).
12-19
12
Network Object Reference
initFcn
This property defines the initialization function used to initialize the ith layer,
if the network initialization function (net.initFcn) is initlay.
net.layers{i}.initFcn
It can be set to the name of any layer initialization function, including these
toolbox functions:
Layer Initialization Functions
initnw
Nguyen-Widrow layer initialization function.
initwb
By-weight-and-bias layer initialization function
If the network initialization is set to initlay, then the function indicated by
this property will be used to initialize the layer’s weights and biases when init
is called.
net = init(net)
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom initialization functions.
netInputFcn
This property defines the net input function use to calculate the ith layer’s net
input, given the layer’s weighted inputs and bias.
net.layers{i}.netInputFcn
It can be set to the name of any net input function, including these toolbox
functions:
Net Input Functions
netprod
Product net input function.
netsum
Sum net input function.
The net input function is used to simulate the network when sim is called.
[Y,Pf,Af] = sim(net,P,Pi,Ai)
12-20
Subobject Properties
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom net input functions.
positions (read-only)
This property defines the positions of neurons in the ith layer. These positions
are used by self-organizing maps.
net.layers{i}.positions
It is always set to the result of applying the layer’s topology function
(net.layers{i}.topologyFcn) to the positions of the layer’s dimensions
(net.layers{i}.dimensions).
Plotting. Use plotsom to plot the positions of a layer’s neurons.
For instance, if the first layer neurons of a network are arranged with
dimensions (net.layers{1}.dimensions) of [4 5] and the topology function
(net.layers{1}.topologyFcn) is hextop, the neuron’s positions can be plotted
as shown below.
plotsom(net.layers{1}.positions)
Neuron Positions
3
position(2,i)
2.5
2
1.5
1
0.5
0
0
1
2
position(1,i)
3
12-21
12
Network Object Reference
size
This property defines the number of neurons in the ith layer.
net.layers{i}.size
It can be set to 0 or a positive integer.
Side Effects. Whenever this property is altered, the sizes of any input weights
going to the layer (net.inputWeights{i,:}.size) and any layer weights going
to the layer (net.layerWeights{i,:}.size) or coming from the layer
(net.inputWeights{i,:}.size), and the layer’s bias (net.biases{i}.size)
will change.
The dimensions of the corresponding weight matrices (net.IW{i,:},
net.LW{i,:}, net.LW{:,i}) and biases (net.b{i}) will also change.
Changing this property also changes the size of the layer’s output
(net.outputs{i}.size) and target (net.targets{i}.size) if they exist.
Finally, when this property is altered the dimensions of the layer’s neurons
(net.layers{i}.dimension) are set to the same value. (This results in a
one-dimensional arrangement of neurons. If another arrangement is required
set the dimensions property directly instead of using size).
topologyFcn
This property defines the function used to calculate the ith layer’s neuron
positions (net.layers{i}.positions) from the layer’s dimensions
(net.layers{i}.dimensions).
net.topologyFcn
It can be set to the name of any topology function, including these toolbox
functions:
Topology Functions
gridtop
Gridtop layer topology function.
hextop
Hexagonal layer topology function.
randtop
Random layer topology function.
Custom functions. See Chapter 11, “Advanced Topics” for information on creating
custom topology functions.
12-22
Subobject Properties
Side Effects. Whenever this property is altered, the positions of the layer’s
neurons (net.layers{i}.positions) will be updated.
Plotting. Use plotsom to plot the positions of a layer’s neurons.
For instance, if the first layer neurons of a network are arranged with
dimensions (net.layers{1}.dimensions) of [8 10] and the topology function
(net.layers{1}.topologyFcn) is randtop, the neuron’s positions will be
arranged something like those shown in the plot below.
plotsom(net.layers{1}.positions)
Neuron Positions
6
5
position(2,i)
4
3
2
1
0
0
1
2
3
position(1,i)
4
5
6
transferFcn
This function defines the transfer function use to calculate the ith layer’s
output, given the layer’s net input.
net.layers{i}.transferFcn
12-23
12
Network Object Reference
It can be set to the name of any transfer function, including these toolbox
functions:
Transfer Functions
compet
Competitive transfer function.
hardlim
Hard limit transfer function.
hardlims
Symmetric hard limit transfer function.
logsig
Log sigmoid transfer function.
poslin
Positive linear transfer function.
purelin
Hard limit transfer function.
radbas
Radial basis transfer function.
satlin
Saturating linear transfer function.
satlins
Symmetric saturating linear transfer function.
softmax
Soft max transfer function.
tansig
Hyperbolic tangent sigmoid transfer function.
tribas
Triangular basis transfer function
The transfer function is used to simulate the network when sim is called.
[Y,Pf,Af] = sim(net,P,Pi,Ai)
Custom functions. See Chapter 11, “Advanced Topics” for information on creating
custom transfer functions.
userdata
This property provides a place for users to add custom information to the ith
network layer.
net.layers{i}.userdata
Only one field is predefined. It contains a secret message to all Neural Network
Toolbox users.
net.layers{i}.userdata.note
12-24
Subobject Properties
Outputs
size (read-only)
This property defines the number of elements in the ith layer’s output.
net.outputs{i}.size
It is always set to the size of the ith layer (net.layers{i}.size).
userdata
This property provides a place for users to add custom information to the ith
layer’s output.
net.outputs{i}.userdata
Only one field is predefined. It contains a secret message to all Neural Network
Toolbox users.
net.outputs{i}.userdata.note
Targets
size (read-only)
This property defines the number of elements in the ith layer’s target.
net.targets{i}.size
It is always set to the size of the ith layer (net.layers{i}.size).
userdata
This property provides a place for users to add custom information to the ith
layer’s target.
net.targets{i}.userdata
Only one field is predefined. It contains a secret message to all Neural Network
Toolbox users.
net.targets{i}.userdata.note
12-25
12
Network Object Reference
Biases
initFcn
This property defines the function used to initialize the ith layer’s bias vector,
if the network initialization function is initlay, and the ith layer’s
initialization function is initwb.
net.biases{i}.initFcn
This function can be set to the name of any bias initialization function,
including the toolbox functions:
Bias Initialization Functions
initcon
Conscience bias initialization function.
initzero
Zero weight/bias initialization function.
rands
Symmetric random weight/bias initialization function.
This function will be used to calculate an initial bias vector for the ith layer
(net.b{i}) when init is called, if the network initialization function
(net.initFcn) is initlay, and the ith layer’s initialization function
(net.layers{i}.initFcn) is initwb.
net = init(net)
Custom functions. See Chapter 11, “Advanced Topics” for information on creating
custom initialization functions.
learn
This property defines whether the ith bias vector is to be altered during
training and adaption.
net.biases{i}.learn
It can be set to 0 or 1.
It enables or disables the bias’ learning during calls to either adapt or train.
[net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai)
[net,tr] = train(NET,P,T,Pi,Ai)
12-26
Subobject Properties
learnFcn
This property defines the function used to update the ith layer’s bias vector
during training, if the network training function is trainwb or trainwb1, or
during adaption, if the network adapt function is adaptwb.
net.biases{i}.learnFcn
It can be set to the name of any bias learning function, including these toolbox
functions:
Learning Functions
learncon
Conscience bias learning function.
learngd
Gradient descent weight/bias learning function
learngdm
Grad. descent w/momentum weight/bias learning function
learnp
Perceptron weight/bias learning function.
learnpn
Normalized perceptron weight/bias learning function.
learnwh
Widrow-Hoff weight/bias learning rule
The learning function will update the ith bias vector (net.b{i}) during calls to
train, if the network training function (net.trainFcn) is trainwb or trainwb1,
or during calls to adapt, if the network adapt function (net.adaptFcn) is
adaptwb.
[net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai)
[net,tr] = train(NET,P,T,Pi,Ai)
Custom functions. See Chapter 11, “Advanced Topics” for information on creating
custom learning functions.
Side Effects. Whenever this property is altered the biases’s learning parameters
(net.biases{i}.learnParam) are set to contain the fields and default values of
the new function.
learnParam
This property defines the learning parameters and values for the current
learning function of the ith layer’s bias.
net.biases{i}.learnParam
12-27
12
Network Object Reference
The fields of this property depend on the current learning function
(net.biases{i}.learnFcn). Evaluate the above reference to see the fields of
the current learning function.
Call help on the current learning function to get a description of what each field
means.
help(net.biases{i}.learnFcn)
size (read-only)
This property defines the size of the ith layer’s bias vector.
net.biases{i}.size
It is always set to the size of the ith layer (net.layers{i}.size).
userdata
This property provides a place for users to add custom information to the ith
layer’s bias.
net.biases{i}.userdata
Only one field is predefined. It contains a secret message to all Neural Network
Toolbox users.
net.biases{i}.userdata.note
Input Weights
delays
This property defines a tapped delay line between the jth input and its weight
to the ith layer.
net.inputWeights{i,j}.delays
It must be set to a row vector of increasing 0 or positive integer values.
Side Effects. Whenever this property is altered the weight’s size
(net.inputWeights{i,j}.size) and the dimensions of its weight matrix
(net.IW{i,j}) are updated.
12-28
Subobject Properties
initFcn
This property defines the function used to initialize the weight matrix going to
the ith layer from the jth input, if the network initialization function is
initlay, and the ith layer’s initialization function is initwb.
net.inputWeights{i,j}.initFcn
This function can be set to the name of any weight initialization function,
including these toolbox functions:
Weight Initialization Functions
initzero
Zero weight/bias initialization function.
midpoint
Midpoint weight initialization function.
randnc
Normalized column weight initialization function.
randnr
Normalized row weight initialization function.
rands
Symmetric random weight/bias initialization function.
This function will be used to calculate an initial weight matrix for the weight
going to the ith layer from the jth input (net.IW{i,j}) when init is called, if
the network initialization function (net.initFcn) is initlay, and the ith
layer’s initialization function (net.layers{i}.initFcn) is initwb.
net = init(net)
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom initialization functions.
learn
This property defines whether the weight matrix to the ith layer from the jth
input is to be altered during training and adaption.
net.inputWeights{i,j}.learn
It can be set to 0 or 1.
It enables or disables the weights learning during calls to either adapt or
train.
[net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai)
[net,tr] = train(NET,P,T,Pi,Ai)
12-29
12
Network Object Reference
learnFcn
This property defines the function used to update the weight matrix going to
the ith layer from the jth input during training, if the network training function
is trainwb or trainwb1, or during adaption, if the network adapt function is
adaptwb.
net.inputWeights{i,j}.learnFcn
It can be set to the name of any weight learning function, including these
toolbox functions:
Weight Learning Functions
learngd
Gradient descent weight/bias learning function
learngdm
Grad. descent w/momentum weight/bias learning function
learnh
Hebb weight learning function.
learnhd
Hebb with decay weight learning function.
learnis
Instar weight learning function.
learnk
Kohonen weight learning function.
learnlv1
LVQ1 weight learning function.
learnlv2
LVQ2 weight learning function.
learnos
Outstar weight learning function.
learnp
Perceptron weight/bias learning function.
learnpn
Normalized perceptron weight/bias learning function.
learnsom
Self-organizing map weight learning function.
learnwh
Widrow-Hoff weight/bias learning rule
The learning function will update the weight matrix of the ith layer form the
jth input (net.IW{i,j}) during calls to train, if the network training function
(net.trainFcn) is trainwb or trainwb1, or during calls to adapt, if the network
adapt function (net.adaptFcn) is adaptwb.
[net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai)
[net,tr] = train(NET,P,T,Pi,Ai)
12-30
Subobject Properties
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom learning functions.
learnParam
This property defines the learning parameters and values for the current
learning function of the ith layer’s weight coming from the jth input.
net.inputWeights{i,j}.learnParam
The fields of this property depend on the current learning function
(net.inputWeights{i,j}.learnFcn). Evaluate the above reference to see the
fields of the current learning function.
Call help on the current learning function to get a description of what each field
means.
help(net.inputWeights{i,j}.learnFcn)
size (read-only)
This property defines the dimensions of the ith layer’s weight matrix from the
jth network input.
net.inputWeights{i,j}.size
It is always set to a two element row vector indicating the number of rows and
columns of the associated weight matrix (net.IW{i,j}). The first element is
equal to the size of the ith layer (net.layers{i}.size). The second element is
equal to the product of the length of the weights delay vectors with the size of
the jth input:
length(net.inputWeights{i,j}.delays) * net.inputs{j}.size
userdata
This property provides a place for users to add custom information to the (i,j)th
input weight.
net.inputWeights{i,j}.userdata
Only one field is predefined. It contains a secret message to all Neural Network
Toolbox users.
net.inputWeights{i,j}.userdata.note
12-31
12
Network Object Reference
weightFcn
This property defines the function used to apply the ith layer’s weight from the
jth input to that input.
net.inputWeights{i,j}.weightFcn
It can be set to the name of any weight function, including these toolbox
functions:
Weight Functions
dist
Conscience bias initialization function.
dotprod
Zero weight/bias initialization function.
mandist
Manhattan distance weight function.
negdist
Normalized column weight initialization function.
normprod
Normalized row weight initialization function.
The weight function is used when sim is called to simulate the network.
[Y,Pf,Af] = sim(net,P,Pi,Ai)
Custom functions. See Chapter 11, “Advanced Topics” for information on creating
custom weight functions.
LayerWeights
delays
This property defines a tapped delay line between the jth layer and its weight
to the ith layer.
net.layerWeights{i,j}.delays
It must be set to a row vector of increasing 0 or positive integer values.
initFcn
This property defines the function used to initialize the weight matrix going to
the ith layer from the jth layer, if the network initialization function is
initlay, and the ith layer’s initialization function is initwb.
net.layerWeights{i,j}.initFcn
12-32
Subobject Properties
This function can be set to the name of any weight initialization function,
including the toolbox functions:
Weight and Bias Initialization Functions
initzero
Zero weight/bias initialization function.
midpoint
Midpoint weight initialization function.
randnc
Normalized column weight initialization function.
randnr
Normalized row weight initialization function.
rands
Symmetric random weight/bias initialization function.
This function will be used to calculate an initial weight matrix for the weight
going to the ith layer from the jth layer (net.LW{i,j}) when init is called, if
the network initialization function (net.initFcn) is initlay, and the ith
layer’s initialization function (net.layers{i}.initFcn) is initwb.
net = init(net)
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom initialization functions.
learn
This property defines whether the weight matrix to the ith layer from the jth
layer is to be altered during training and adaption.
net.layerWeights{i,j}.learn
It can be set to 0 or 1.
It enables or disables the weights learning during calls to either adapt or
train.
[net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai)
[net,tr] = train(NET,P,T,Pi,Ai)
learnFcn
This property defines the function used to update the weight matrix going to
the ith layer from the jth layer during training, if the network training function
12-33
12
Network Object Reference
is trainwb or trainwb1, or during adaption, if the network adapt function is
adaptwb.
net.layerWeights{i,j}.learnFcn
It can be set to the name of any weight learning function, including these
toolbox functions:
Learning Functions
learngd
Gradient descent weight/bias learning function
learngdm
Grad. descent w/momentum weight/bias learning function
learnh
Hebb weight learning function.
learnhd
Hebb with decay weight learning function.
learnis
Instar weight learning function.
learnk
Kohonen weight learning function.
learnlv1
LVQ1 weight learning function.
learnlv2
LVQ2 weight learning function.
learnos
Outstar weight learning function.
learnp
Perceptron weight/bias learning function.
learnpn
Normalized perceptron weight/bias learning function.
learnsom
Self-organizing map weight learning function.
learnwh
Widrow-Hoff weight/bias learning rule
The learning function will update the weight matrix of the ith layer form the
jth layer (net.LW{i,j}) during calls to train, if the network training function
(net.trainFcn) is trainwb or trainwb1, or during calls to adapt, if the network
adapt function (net.adaptFcn) is adaptwb.
[net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai)
[net,tr] = train(NET,P,T,Pi,Ai)
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom learning functions.
12-34
Subobject Properties
learnParam
This property defines the learning parameters fields and values for the current
learning function of the ith layer’s weight coming from the jth layer.
net.layerWeights{i,j}.learnParam
The sub-fields of this property depend on the current learning function
(net.layerWeights{i,j}.learnFcn). Evaluate the above reference to see the
fields of the current learning function.
Get help on the current learning function to get a description of what each field
means.
help(net.layerWeights{i,j}.learnFcn)
size (read-only)
This property defines the dimensions of the ith layer’s weight matrix from the
jth layer.
net.layerWeights{i,j}.size
It is always set to a two element row vector indicating the number of rows and
columns of the associated weight matrix (net.LW{i,j}). The first element is
equal to the size of the ith layer (net.layers{i}.size). The second element is
equal to the product of the length of the weights delay vectors with the size of
the jth layer:
length(net.layerWeights{i,j}.delays) * net.layers{j}.size
userdata
This property provides a place for users to add custom information to the (i,j)th
layer weight.
net.layerWeights{i,j}.userdata
Only one field is predefined. It contains a secret message to all Neural Network
Toolbox users.
net.layerWeights{i,j}.userdata.note
12-35
12
Network Object Reference
weightFcn
This property defines the function used to apply the ith layer’s weight from the
jth layer to that layer’s output.
net.layerWeights{i,j}.weightFcn
It can be set to the name of any weight function, including these toolbox
functions:
Weight Functions
dist
Euclidean distance weight function.
dotprod
Dot product weight function.
mandist
Manhattan distance weight function.
negdist
Dot product weight function.
normprod
Normalized dot product weight function.
The weight function is used when sim is called to simulate the network.
[Y,Pf,Af] = sim(net,P,Pi,Ai)
Custom Functions. See Chapter 11, “Advanced Topics” for information on creating
custom weight functions.
12-36
13
Reference
Functions Listed by Network Type . . . . . . . . . . 13-2
Functions by Class . . . . . . . . . . . . . . . . . 13-3
Transfer Functions . . . . . . . . . . . . . . . . 13-13
Transfer Function Graphs
. . . . . . . . . . . . 13-14
Reference Page Headings . . . . . . . . . . . . . 13-18
Functions . . . . . . . . . . . . . . . . . . . . 13-19
13
Reference
Functions Listed by Network Type
Function and type
13-2
assoclr
Associative learning rules
backprop
Backpropagation networks
elman
Elman recurrent networks
hopfield
Hopfield recurrent networks
linnet
Linear networks
lvq
Learning vector quantization
percept
Perceptrons
radbasis
Radial basis networks
selforg
Self-organizing networks
Functions by Class
Functions by Class
Adapt Function
adaptwb
By-weight-and-bias network adaption function.
Analysis Functions
errsurf
Error surface of a single input neuron.
maxlinr
Maximum learning rate for a linear neuron.
Distance Functions
boxdist
Distance between two position vectors
dist
Euclidean distance weight function.
linkdist
Link distance function.
mandist
Manhattan distance weight function.
Layer Initialization Functions
initnw
Nguyen-Widrow layer initialization function.
initwb
By-weight-and-bias layer initialization function
13-3
13
Reference
Learning Functions
learncon
Conscience bias learning function.
learngd
Gradient descent weight/bias learning function
learngdm
Grad. descent w/momentum weight/bias learning function
learnh
Hebb weight learning function.
learnhd
Hebb with decay weight learning function.
learnis
Instar weight learning function.
learnk
Kohonen weight learning function.
learnlv1
LVQ1 weight learning function.
learnlv2
LVQ2 weight learning function.
learnos
Outstar weight learning function.
learnp
Perceptron weight/bias learning function.
learnpn
Normalized perceptron weight/bias learning function.
learnsom
Self-organizing map weight learning function.
learnwh
Widrow-Hoff weight/bias learning rule
Line Search Functions
13-4
srchbac
Backtracking search.
srchbre
Brent's combination golden section/quadratic interpolation.
srchcha
Charalambous' cubic interpolation.
srchgol
Golden section search.
srchhyb
Hybrid bisection/cubic search.
Functions by Class
Net Input Derivative Functions
dnetprod
Product net input derivative function.
dnetsum
Sum net input derivative function.
Net Input Functions
netprod
Product net input function.
netsum
Sum net input function.
Network Initialization Functions
initlay
Layer-by-layer network initialization function.
Network Use
sim
Simulate a neural network.
init
Initialize a neural network.
adapt
Allow a neural network to adapt.
train
Train a neural network.
disp
Display a neural network's properties.
display
Display a neural network variable's name and properties.
13-5
13
Reference
New Networks
network
Create a custom neural network.
newc
Create a competitive layer.
newcf
Create a cascade-forward backpropagation network.
newelm
Create an Elman backpropagation network.
newff
Create a feed-forward backpropagation network.
newfftd
Create a feed-forward input-delay backprop network.
newgrnn
Design a generalized regression neural network.
newhop
Create a Hopfield recurrent network.
newlin
Create a linear layer.
newlind
Design a linear layer.
newlvq
Create a learning vector quantization network
newp
Create a perceptron.
newpnn
Design a probabilistic neural network.
newrb
Design a radial basis network.
newrbe
Design an exact radial basis network.
newsom
Create a self-organizing map.
Performance Derivative Functions
13-6
dmae
Mean absolute error performance derivatives function.
dmse
Mean squared error performance derivatives function.
dmsereg
Mean squared error w/reg performance derivative function.
dsse
Sum squared error performance derivative function.
Functions by Class
Performance Functions
mae
Mean absolute error performance function.
mse
Mean squared error performance function.
msereg
Mean squared error w/reg performance function.
sse
Sum squared error performance function.
Plotting Functions
hinton
Hinton graph of weight matrix.
hintonwb
Hinton graph of weight matrix and bias vector.
plotep
Plot weight and bias position on error surface.
plotes
Plot error surface of single input neuron.
plotpc
Plot classification line on perceptron vector plot.
plotperf
Plot network performance.
plotpv
Plot perceptron input/target vectors.
plotsom
Plot self-organizing map.
plotv
Plot vectors as lines from the origin.
plotvec
Plot vectors with different colors.
13-7
13
Reference
Pre and Post Processing
postmnmx
Unnormalize data which has been norm. by PREMNMX.
postreg
Post-training regression analysis.
poststd
Unnormalize data which has been normalized by PRESTD.
premnmx
Normalize data for maximum of 1 and minimum of -1.
prepca
Principal component analysis on input data.
prestd
Normalize data for unity standard deviation and zero mean.
tramnmx
Transform data with precalculated minimum and max.
trapca
Transform data with PCA matrix computed by PREPCA.
trastd
Transform data with precalc. mean & standard deviation.
Simulink Support
gensim
Generate a Simulink block for neural network simulation.
Topology Functions
13-8
gridtop
Gridtop layer topology function.
hextop
Hexagonal layer topology function.
randtop
Random layer topology function.
Functions by Class
Training Functions
trainbfg
BFGS quasi-Newton backpropagation.
trainbr
Bayesian regularization.
traincgb
Powell-Beale conjugate gradient backpropagation.
traincgf
Fletcher-Powell conjugate gradient backpropagation.
traincgp
Polak-Ribiere conjugate gradient backpropagation.
traingd
Gradient descent backpropagation.
traingda
Gradient descent w/adaptive lr backpropagation.
traingdm
Gradient descent w/momentum backpropagation.
traingdx
Gradient descent w/momentum & adaptive lr backprop.
trainlm
Levenberg-Marquardt backpropagation.
trainoss
One step secant backpropagation.
trainrp
Resilient backpropagation (Rprop)
trainscg
Scaled conjugate gradient backpropagation.
trainwb
By-weight-and-bias network training function.
trainwb1
By-weight-&-bias 1-vector-at-a-time training function.
Transfer Derivative Functions
dhardlim
Hard limit transfer derivative function.
dhardlms
Symmetric hard limit transfer derivative function.
dlogsig
Log sigmoid transfer derivative function.
dposlin
Positive linear transfer derivative function.
13-9
13
Reference
Transfer Derivative Functions
dpurelin
Hard limit transfer derivative function.
dradbas
Radial basis transfer derivative function.
dsatlin
Saturating linear transfer derivative function.
dsatlins
Symmetric saturating linear transfer derivative function.
dtansig
Hyperbolic tangent sigmoid transfer derivative function.
dtribas
Triangular basis transfer derivative function
Transfer Functions
13-10
compet
Competitive transfer function.
hardlim
Hard limit transfer function.
hardlims
Symmetric hard limit transfer function.
logsig
Log sigmoid transfer function.
poslin
Positive linear transfer function.
purelin
Hard limit transfer function.
radbas
Radial basis transfer function.
satlin
Saturating linear transfer function.
satlins
Symmetric saturating linear transfer function.
softmax
Soft max transfer function.
tansig
Hyperbolic tangent sigmoid transfer function.
tribas
Triangular basis transfer function
Functions by Class
Vectors
cell2mat
Combines cell array of matrices into one matrix.
combvec
Create all combinations of vectors.
concur
Create concurrent bias vectors.
con2seq
Converts concurrent vectors to sequential vectors.
copy
Copy matrix or cell array.
ind2vec
Convert indices to vectors.
mat2cell
Break matrix up into cell array of matrices.
minmax
Ranges of matrix rows.
normc
Normalize columns of matrix.
normr
Normalize rows of matrix.
pnormc
Pseudo-normalize columns of matrix.
quant
Discretize value as multiple of a quantity.
seq2con
Converts sequential vectors to concurrent vectors.
sumsqr
Sum squared elements of matrix.
vec2ind
Convert vectors to indices.
Weight and Bias Initialization Functions
initcon
Conscience bias initialization function.
initzero
Zero weight/bias initialization function.
midpoint
Midpoint weight initialization function.
randnc
Normalized column weight initialization function.
randnr
Normalized row weight initialization function.
rands
Symmetric random weight/bias initialization function.
13-11
13
Reference
Weight Derivative Functions
ddotprod
Dot product weight derivative function.
Weight Functions
13-12
dist
Euclidean distance weight function.
dotprod
Dot product weight function.
mandist
Manhattan distance weight function.
negdist
Dot product weight function.
normprod
Normalized dot product weight function.
Transfer Functions
Transfer Functions
Transfer Function
compet
Competitive transfer function.
hardlim
Hard limit transfer function.
hardlims
Symmetric hard limit transfer function
logsig
Log sigmoid transfer function.
poslin
Positive linear transfer function
purelin
Linear transfer function.
radbas
Radial basis transfer function.
satlin
Saturating linear transfer function.
satlins
Symmetric saturating linear transfer function
softmax
Soft max transfer function.
tansig
Hyperbolic tangent sigmoid transfer function.
tribas
Triangular basis transfer function.
C
SM
13-13
13
Reference
Transfer Function Graphs
a
AA
+1
n
0
-1
a = hardlim(n)
Hard Limit Transfer Function
a
+1
n
0
-1
a = hardlims(n)
AA
AA
Symmetric Hard Limit Trans. Funct.
a
+1
n
0
-1
AA
AA
a = logsig(n)
Log-Sigmoid Transfer Function
13-14
Transfer Function Graphs
Transfer Function Graphs (continued)
a
+1
AA
AA
n
0 1
-1
a = poslin(n)
Positive Linear Transfer Funct.
a
+1
n
0
-1
a = purelin(n)
AA
AA
Linear Transfer Function
a
+1
0
-1
n
AA
a = hardlim(n)
Hard Limit Transfer Function
13-15
13
Reference
Transfer Function Graphs (continued)
AA
AA
a
+1
n
-1
0 +1
-1
a = satlin(n)
Satlin Transfer Function
a
+1
n
-1
0 +1
-1
AA
AA
a = satlins(n)
Satlins Transfer Function
a
+1
0
n
-1
a = tansig(n)
Tan-Sigmoid Transfer Function
13-16
Transfer Function Graphs
Transfer Function Graphs (continued)
a
+1
-1
0 +1
n
-1
a = tribas(n)
Triangular Basis Function
13-17
13
Reference
Reference Page Headings
Following is a list of headings used in this section. Not every function will have
all this material, but the material that is included will be ordered as shown.
- Purpose
- Graph and
Symbol
- Syntax
- To Get Help
- Description
- Properties
- Examples
- Network Use
- Algorithm
- Limitations
- Notes
- See Also
- References
13-18
adapt
Purpose
adapt
Allow a neural network to adapt
Syntax
[net,Y,E,Pf,Af] = adapt(net,P,T,Pi,Ai)
To Get Help
Type help network/adapt
Description
[net,Y,E,Pf,Af] = adapt(net,P,T,Pi,Ai) takes,
net - Network.
P
- Network inputs.
T
- Network targets, default = zeros.
Pi
- Initial input delay conditions, default = zeros.
Ai
- Initial layer delay conditions, default = zeros.
and returns the following after applying the adapt function net.adaptFcn with
the adaption parameters net.adaptParam:
net - Updated network.
Y
- Network outputs.
E
- Network errors.
Pf
- Final input delay conditions.
Af
- Final layer delay conditions.
Note that T is optional and only needs to be used for networks that require
targets. Pi and Pf are also optional and only needs to be used for networks that
have input or layer delays.
adapt's signal arguments can have two formats: cell array or matrix.
13-19
adapt
The cell array format is easiest to describe. It is most convenient to be used for
networks with multiple inputs and outputs, and allows sequences of inputs to
be presented:
P
- Ni x TS cell array, each element P{i,ts} is an Ri x Q matrix.
T
- Nt x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Pi - Ni x ID cell array, each element Pi{i,k} is an Ri x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
Y
- NO x TS cell array, each element Y{i,ts} is a Ui x Q matrix.
Pf - Ni x ID cell array, each element Pf{i,k} is an Ri x Q matrix.
Af - Nl x LD cell array, each element Af{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
No = net.numOutputs
Nt = net.numTargets
ID = net.numInputDelays
LD = net.numLayerDelays
TS = Number of time steps
Q
= Batch size
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Ui = net.outputs{i}.size
Vi = net.targets{i}.size
The columns of Pi, Pf, Ai, and Af are ordered from oldest delay condition to
most recent:
Pi{i,k} = input i at time ts = k-ID.
Pf{i,k} = input i at time ts = TS+k-ID.
Ai{i,k} = layer output i at time ts = k-LD.
Af{i,k} = layer output i at time ts = TS+k-LD.
The matrix format can be used if only one time step is to be simulated (TS = 1).
It is convenient for network's with only one input and output, but can be used
with networks that have more.
13-20
adapt
Each matrix argument is found by storing the elements of the corresponding
cell array argument in a single matrix:
P
- (sum of Ri) x Q matrix
T
- (sum of Vi) x Q matrix
Pi - (sum of Ri) x (ID*Q) matrix.
Ai - (sum of Si) x (LD*Q) matrix.
Y
- (sum of Ui) x Q matrix.
Pf - (sum of Ri) x (ID*Q) matrix.
Af - (sum of Si) x (LD*Q) matrix.
Examples
Here two sequences of 12 steps (where T1 is known to depend on P1) are used
to define the operation of a filter.
p1 = {-1 0 1 0 1 1 -1 0 -1 1 0 1};
t1 = {-1 -1 1 1 1 2 0 -1 -1 0 1 1};
Here newlin is used to create a layer with an input range of [-1 1]), one
neuron, input delays of 0 and 1, and a learning rate of 0.5. The linear layer is
then simulated.
net = newlin([-1 1],1,[0 1],0.5);
Here the network adapts for one pass through the sequence.
The network's mean squared error is displayed. (Since this is the first call of
adapt the default Pi is used.)
[net,y,e,pf] = adapt(net,p1,t1);
mse(e)
Note the errors are quite large. Here the network adapts to another 12 time
steps (using the previous Pf as the new initial delay conditions.)
p2 = {1 -1 -1 1 1 -1 0 0 0 1 -1 -1};
t2 = {2 0 -2 0 2 0 -1 0 0 1 0 -1};
[net,y,e,pf] = adapt(net,p2,t2,pf);
mse(e)
13-21
adapt
Here the network adapts for 100 passes through the entire sequence.
p3 = [p1 p2];
t3 = [t1 t2];
net.adaptParam.passes = 100;
[net,y,e] = adapt(net,p3,t3);
mse(e)
The error after 100 passes through the sequence is very small. The network has
adapted to the relationship between the input and target signals.
Algorithm
adapt calls the function indicated by net.adaptFcn, using the adaption
parameter values indicated by net.adaptParam.
Given an input sequence with TS steps the network is updated as follows. Each
step in the sequence of inputs is presented to the network one at a time. The
network's weight and bias values are updated after each step, before the next
step in the sequence is presented. Thus the network is updated TS times.
See Also
13-22
sim, init, train
adaptwb
Purpose
Syntax
adaptwb
By-weight-and-bias network adaption function
[net,Ac,El] = adaptwb(net,Pd,Tl,Ai,Q,TS)
info = adaptwb(code)
Description
adaptwb is a network function which updates each weight and bias according
to its learning function.
adaptwb takes these inputs,
net - Neural network.
Pd
- Delayed inputs.
Tl
- Layer targets.
Ai
- Initial input conditions.
Q
- Batch size.
TS
- Time steps.
After training the network with its weight and bias the learning functions
returns,
net - Updated network.
Ac
- Collective layer outputs.
El
- Layer errors.
Adaption occurs according to the adaptwb's training parameter, shown here
with its default value:
net.adaptparam.passes 1 Number of passes through sequence
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Zij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix or [].
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
Ac - Nl x (LD+TS) cell array, each element Ac{i,k} is an Si x Q matrix.
El - Nl x TS cell array, each element El{i,k} is an Si x Q matrix or [].
13-23
adaptwb
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Zij = Ri * length(net.inputWeights{i,j}.delays)
adaptwb(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Network Use
You can create a standard network that uses adaptwb by calling newp or
newlin.
To prepare a custom network to adapt with adaptwb:
1 Set net.adaptfcn to 'adaptwb'. (This will set net.adaptparam to adaptwb's
default parameters.)
2 Set each net.inputweights{i,j}.learnfcn to a learning function. Set each
net.layerweights{i,j}.learnfcn to a learning function. Set each
net.biases{i}.learnfcn to a learning function. (Weight and bias learning
parameters will automatically be set to default values for the given learning
function.)
To allow the network to adapt:
1 Set net.adaptparam properties to desired values.
2 Set weight and bias learning parameters to desired values.
3 Call adapt.
See newp and newlin for adaption examples.
Algorithm
Each weight and bias is updated according to its learning function after each
step in the input sequence.
See Also
newp, newlin, train
13-24
boxdist
Purpose
boxdist
Box distance function
Syntax
d = boxdist(pos);
Description
boxdist is a layer distance function that is used to find the distances between
the layer's neurons, given their positions.
boxdist(pos) takes one argument,
pos - N x S matrix of neuron positions.
and returns the S x S matrix of distances.
boxdist is most commonly used in conjunction with layers whose topology
function is gridtop.
Examples
Here we define a random matrix of positions for 10 neurons arranged in
three-dimensional space and then find their distances.
pos = rand(3,10);
d = boxdist(pos)
Network Use
You can create a standard network that uses boxdist as a distance function by
calling newsom.
To change a network so that a layer's topology uses boxdist, set
net.layers{i}.distanceFcn to 'boxdist'.
In either case, call sim to simulate the network with boxdist. See newsom for
training and adaption examples.
Algorithm
The box distance D between two position vectors Pi and Pj from a set of S
vectors is:
Dij = max(abs(Pi-Pj))
See Also
sim, dist, mandist, linkdist
13-25
cell2mat
Purpose
cell2mat
Combines a cell array of matrices into one matrix
Syntax
m = cell2mat(c)
Description
m = cell2mat(C)
C - Cell array of matrices: {M11 M12... ; M21 M22... ; ...}
returns
M - Single matrix: [M11 M12 ...; M21 M22... ; ...]
Examples
Examples
c = {[1 2] [3]; [4 5; 6 7] [8; 9]};
m = cell2mat(c)
See Also
13-26
mat2cell
combvec
Purpose
combvec
Create all combinations of vectors
Syntax
combvec(a1,a2)
Description
combvec(A1,A2) takes two inputs,
A1 - Matrix of N1 (column) vectors.
A2 - Matrix of N2 (column) vectors.
and returns a matrix of N1*N2 column vectors, where the columns consist of all
possibilities of A2 vectors, appended to A1 vectors.
Examples
a1 = [1 2 3; 4 5 6];
a2 = [7 8; 9 10];
a3 = combvec(a1,a2)
13-27
compet
Purpose
Syntax
compet
Competitive transfer function
A = compet(N)
info = compet(code)
Description
compet is a transfer function. Transfer functions calculate a layer's output from
its net input.
compet(N) takes one input argument,
N - S x Q matrix of net input (column) vectors.
and returns output vectors with 1 where each net input vector has its
maximum value, and 0 elsewhere.
compet(code) returns information about this function.
These codes are defined:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
compet does not have a derivative function.
In many network paradigms it is useful to have a layer whose neurons compete
for the ability to output a 1. In biology this is done by strong inhibitory
connections between each of the neurons in a layer. The result is that the only
neuron that can respond with appreciable output is the neuron whose net input
is the highest. All other neurons are inhibited so strongly by the winning
neuron that their output’s are negligible.
To model this type of layer efficiently on a computer, a competitive transfer
function is often used. Such a function transforms the net input vector of a
layer of neurons so that the neuron receiving the greatest net input has an
output of 1 and all other neurons have outputs of 0.
13-28
compet
Examples
Here we define a net input vector N, calculate the output, and plot both with
bar graphs.
n = [0; 1; -0.5; 0.5];
a = compet(n);
subplot(2,1,1), bar(n), ylabel('n')
subplot(2,1,2), bar(a), ylabel('a')
Network Use
You can create a standard network that uses compet by calling newc or newpnn.
To change a network so a layer uses compet, set
net.layers{i,j}.transferFcn to 'compet'.
In either case, call sim to simulate the network with compet.
See newc or newpnn for simulation examples.
See Also
sim, softmax
13-29
con2seq
Purpose
con2seq
Convert concurrent vectors to sequential vectors
Syntax
s = con2seq(b)
Description
The neural network toolbox arranges concurrent vectors with a matrix, and
sequential vectors with a cell array (where the second index is the time step).
con2seq and seq2con allow concurrent vectors to be converted to sequential
vectors, and back again.
con2seq(b)takes one input,
b - R x TS matrix.
and returns one output,
S - 1 x TS cell array of R x 1 vectors.
con2seq(b,TS) can also convert multiple batches,
b - N x 1 cell array of matrices with M*TS columns.
TS - Time steps.
and will return,
S - N x TS cell array of matrices with M columns.
Examples
Here a batch of three values is converted to a sequence.
p1 = [1 4 2]
p2 = con2seq(p1)
Here two batches of vectors are converted to two sequences with two time steps.
p1 = {[1 3 4 5; 1 1 7 4]; [7 3 4 4; 6 9 4 1]}
p2 = con2seq(p1,2)
See Also
13-30
seq2con, concur
concur
Purpose
concur
Create concurrent bias vectors
Syntax
concur(b,q)
Description
concur(B,Q)
B - S x 1 bias vector (or Nl x 1 cell array of vectors).
Q - Concurrent size.
Returns an S x B matrix of copies of B (or Nl x 1 cell array of matrices).
Examples
Here concur creates three copies of a bias vector.
b = [1; 3; 2; -1];
concur(b,3)
Network Use
To calculate a layer's net input, the layer's weighted inputs must be combined
with its biases. The following expression calculates the net input for a layer
with the netsum net input function, two input weights, and a bias:
n = netsum(z1,z2,b)
The above expression works if Z1, Z2, and B are all S x 1 vectors. However, if
the network is being simulated by sim (or adapt or train) in response to Q
concurrent vectors, then Z1 and Z2 will be S x Q matrices. Before B can be
combined with Z1 and Z2 we must make Q copies of it.
n = netsum(z1,z2,concur(b,q))
See Also
netsum, netprod, sim, seq2con, con2seq
13-31
ddotprod
Purpose
Syntax
ddotprod
Dot product weight derivative function
dZ_dP = ddotprod('p',W,P,Z)
dZ_dW = ddotprod('w',W,P,Z)
Description
ddotprod is a weight derivative function.
ddotprod('p',W,P,Z) takes three arguments,
W - S x R weight matrix.
P - R x Q Inputs.
Z - S x Q weighted input.
and returns the S x R derivative dZ/dP.
ddotprod('w',W,P,Z) returns the R x Q derivative dZ/dW.
Examples
Here we define a weight W and input P for an input with 3 elements and a layer
with 2 neurons.
W = [0 -1 0.2; -1.1 1 0];
P = [0.1; 0.6; -0.2];
Here we calculate the weighted input with dotprod, then calculate each
derivative with ddotprod.
Z = dotprod(W,P)
dZ_dP = ddotprod('p',W,P,Z)
dZ_dW = ddotprod('w',W,P,Z)
Algorithm
The derivative of a product of two elements with respect to one element is the
other element.
dZ/dP = W
dZ/dW = P
See Also
13-32
dotprod
dhardlim
Purpose
dhardlim
Derivative of hard limit transfer function
Syntax
dA_dN = dhardlim(N,A)
Description
dhardlim is the derivative function for hardlim.
dhardlim(N,A) takes two arguments,
N - S x Q net input.
A - S x Q output.
and returns the S x Q derivative dA/dN.
Examples
Here we define the net input N for a layer of 3 hardlim neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with hardlim and then the derivative of A
with respect to N.
A = hardlim(N)
dA_dN = dhardlim(N,A)
Algorithm
The derivative of hardlim is calculated as follows:
d = 0
See Also
hardlim
13-33
dhardlms
Purpose
dhardlms
Derivative of symmetric hard limit transfer function
Syntax
dA_dN = dhardlms(N,A)
Description
dhardlms is the derivative function for hardlims.
dhardlms(N,A) takes two arguments,
N - S x Q net input.
A - S x Q output.
and returns the S x Q derivative dA/dN.
Examples
Here we define the net input N for a layer of 3 hardlims neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with hardlims and then the derivative of A
with respect to N.
A = hardlims(N)
dA_dN = dhardlms(N,A)
Algorithm
The derivative of hardlims is calculated as follows:
d = 0
See Also
13-34
hardlims
disp
Purpose
disp
Display a neural network's properties
Syntax
disp(net)
To Get Help
Type help network/disp
Description
disp(net) displays a network's properties.
Examples
Here a perceptron is created and displayed.
net = newp([-1 1; 0 2],3);
disp(net)
See Also
display, sim, init, train, adapt
13-35
display
Purpose
display
Display the name and properties of a neural network variables
Syntax
display(net)
To Get Help
Type help network/disp
Description
display(net) displays a network variable's name and properties.
Examples
Here a perceptron variable is defined and displayed.
net = newp([-1 1; 0 2],3);
display(net)
display is automatically called as follows:
net
See Also
13-36
disp, sim, init, train, adapt
dist
Purpose
Syntax
dist
Euclidean distance weight function
Z = dist(W,P)
df = dist('deriv')
D = dist(pos)
Description
dist is the Euclidean distance weight function. Weight functions apply
weights to an input to get weighted inputs.
dist (W,P) takes these inputs,
W - S x R weight matrix.
P - R x Q matrix of Q input (column) vectors.
and returns the S x Q matrix of vector distances.
dist('deriv') returns '' because dist does not have a derivative function.
dist is also a layer distance function which can be used to find the distances
between neurons in a layer.
dist(pos) takes one argument,
pos - N x S matrix of neuron positions.
and returns the S x S matrix of distances.
Examples
Here we define a random weight matrix W and input vector P and calculate the
corresponding weighted input Z.
W = rand(4,3);
P = rand(3,1);
Z = dist(W,P)
Here we define a random matrix of positions for 10 neurons arranged in three
dimensional space and find their distances.
pos = rand(3,10);
D = dist(pos)
13-37
dist
Network Use
You can create a standard network that uses dist by calling newpnn or
newgrnn.
To change a network so an input weight uses dist, set
net.inputWeight{i,j}.weightFcn to 'dist'.
For a layer weight set net.inputWeight{i,j}.weightFcn to 'dist'.
To change a network so that a layer's topology uses dist, set
net.layers{i}.distanceFcn to 'dist'.
In either case, call sim to simulate the network with dist.
See newpnn or newgrnn for simulation examples.
Algorithm
The Euclidean distance d between two vectors X and Y is:
d = sum((x-y).^2).^0.5
See Also
13-38
sim, dotprod, negdist, normprod, mandist, linkdist
dlogsig
Purpose
dlogsig
Log sigmoid transfer derivative function
Syntax
dA_dN = dlogsig(N,A)
Description
dlogsig is the derivative function for logsig.
dlogsig(N,A) takes two arguments,
N - S x Q net input.
A - S x Q output.
and returns the S x Q derivative dA/dN.
Examples
Here we define the net input N for a layer of 3 tansig neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with logsig and then the derivative of A with
respect to N.
A = logsig(N)
dA_dN = dlogsig(N,A)
Algorithm
The derivative of logsig is calculated as follows:
d = a * (1 - a)
See Also
logsig, tansig, dtansig
13-39
dmae
Purpose
Syntax
dmae
Mean absolute error performance derivative function
dPerf_dE = dmae('e',E,X,perf,PP)
dPerf_dX = dmae('x',E,X,perf,PP)
Description
dmae is the derivative function for mae.
dmae('d',E,X,PERF,PP) takes these arguments,
E
- Matrix or cell array of error vector(s).
X
- Vector of all weight and bias values.
perf - Network performance (ignored).
PP
- Performance parameters (ignored).
and returns the derivative dPerf/dE.
dmae('x',E,X,PERF,PP) returns the derivative dPerf/dX.
Examples
Here we define E and X for a network with one 3-element output and six weight
and bias values.
E = {[1; -2; 0.5]};
X = [0; 0.2; -2.2; 4.1; 0.1; -0.2];
Here we calculate the network's mean absolute error performance, and
derivatives of performance.
perf = mae(E)
dPerf_dE = dmae('e',E,X)
dPerf_dX = dmae('x',E,X)
Note that mae can be called with only one argument and dmae with only three
arguments because the other arguments are ignored. The other arguments
exist so that mae and dmae conform to standard performance function argument
lists.
See Also
13-40
mae
dmse
Purpose
Syntax
dmse
Mean squared error performance derivatives function
dPerf_dE = dmse('e',E,X,perf,PP)
dPerf_dX = dmse('x',E,X,perf,PP)
Description
dmse is the derivative function for mse.
dmse('d',E,X,PERF,PP) takes these arguments,
E
- Matrix or cell array of error vector(s).
X
- Vector of all weight and bias values.
perf - Network performance (ignored).
PP
- Performance parameters (ignored).
and returns the derivative dPerf/dE.
dmse('x',E,X,PERF,PP) returns the derivative dPerf/dX.
Examples
Here we define E and X for a network with one 3-element output and six weight
and bias values.
E = {[1; -2; 0.5]};
X = [0; 0.2; -2.2; 4.1; 0.1; -0.2];
Here we calculate the network's mean squared error performance, and
derivatives of performance.
perf = mse(E)
dPerf_dE = dmse('e',E,X)
dPerf_dX = dmse('x',E,X)
Note that mse can be called with only one argument and dmse with only three
arguments because the other arguments are ignored. The other arguments
exist so that mse and dmse conform to standard performance function argument
lists.
See Also
mse
13-41
dmsereg
Purpose
Syntax
dmsereg
Mean squared error w/reg performance derivative function
dPerf_dE = dmsereg('e',E,X,perf,PP)
dPerf_dX = dmsereg('x',E,X,perf,PP)
Description
dmsereg is the derivative function for msereg.
dmsereg('d',E,X,PERF,PP) takes these arguments,
E
- Matrix or cell array of error vector(s).
X
- Vector of all weight and bias values.
perf - Network performance (ignored).
PP
- mse performance parameter.
where PP defines one performance parameters,
PP.ratio - Relative importance of errors vs. weight and bias values.
and returns the derivative dPerf/dE.
dmsereg('x',E,X,perf) returns the derivative dPerf/dX.
mse has only one performance parameter.
Examples
Here we define an error E and X for a network with one 3-element output and
six weight and bias values.
E = {[1; -2; 0.5]};
X = [0; 0.2; -2.2; 4.1; 0.1; -0.2];
Here the ratio performance parameter is defined so that squared errors are 5
times as important as squared weight and bias values.
pp.ratio = 5/(5+1);
Here we calculate the network's performance, and derivatives of performance.
perf = msereg(E,X,pp)
dPerf_dE = dmsereg('e',E,X,perf,pp)
dPerf_dX = dmsereg('x',E,X,perf,pp)
See Also
13-42
msereg
dnetprod
Purpose
dnetprod
Derivative of net input product function
Syntax
dN_dZ = dnetprod(Z,N)
Description
dnetprod is the net input derivative function for netprod.
dnetprod takes two arguments,
Z - S x Q weighted input.
N - S x Q net input.
and returns the S x Q derivative dN/dZ.
Examples
Here we define two weighted inputs for a layer with three neurons.
Z1 = [0; 1; -1];
Z2 = [1; 0.5; 1.2];
We calculate the layer's net input N with netprod and then the derivative of N
with respect to each weighted input.
N = netprod(Z1,Z2)
dN_dZ1 = dnetprod(Z1,N)
dN_dZ2 = dnetprod(Z2,N)
Algorithm
The derivative a product with respect to any element of that product is the
product of the other elements.
See Also
netsum, netprod, dnetsum
13-43
dnetsum
Purpose
dnetsum
Sum net input derivative function
Syntax
dN_dZ = dnetsum(Z,N)
Description
dnetsum is the net input derivative function for netsum.
dnetsum takes two arguments,
Z - S x Q weighted input.
N - S x Q net input.
and returns the S x Q derivative dN/dZ.
Examples
Here we define two weighted inputs for a layer with three neurons.
Z1 = [0; 1; -1];
Z2 = [1; 0.5; 1.2];
We calculate the layer's net input N with netsum and then the derivative of N
with respect to each weighted input.
N = netsum(Z1,Z2)
dN_dZ1 = dnetsum(Z1,N)
dN_dZ2 = dnetsum(Z2,N)
Algorithm
The derivative of a sum with respect to any element of that sum is always a
ones matrix that is the same size as the sum.
See Also
netsum, netprod, dnetprod
13-44
dotprod
Purpose
Syntax
dotprod
Dot product weight function
Z = dotprod(W,P)
df = dotprod('deriv')
Description
dotprod is the dot product weight function. Weight functions apply weights to
an input to get weighted inputs.
dotprod(W,P) takes these inputs,
W - S x R weight matrix.
P - R x Q matrix of Q input (column) vectors.
and returns the S x Q dot product of W and P.
Examples
Here we define a random weight matrix W and input vector P and calculate the
corresponding weighted input Z.
W = rand(4,3);
P = rand(3,1);
Z = dotprod(W,P)
Network Use
You can create a standard network that uses dotprod by calling newp or
newlin.
To change a network so an input weight uses dotprod, set
net.inputWeight{i,j}.weightFcn to 'dotprod’. For a layer weight, set
net.inputWeight{i,j}.weightFcn to 'dotprod’.
In either case, call sim to simulate the network with dotprod.
See newp and newlin for simulation examples.
See Also
sim, ddotprod, dist, negdist, normprod
13-45
dposlin
Purpose
dposlin
Derivative of positive linear transfer function
Syntax
dA_dN = dposlin(N,A)
Description
dposlin is the derivative function for poslin.
dposlin(N,A) takes two arguments, and returns the S x Q derivative dA/dN.
Examples
Here we define the net input N for a layer of 3 poslin neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with poslin and then the derivative of A with
respect to N.
A = poslin(N)
dA_dN = dposlin(N,A)
Algorithm
The derivative of poslin is calculated as follows:
d = 1, if 0 <= n; 0, Otherwise..
See Also
13-46
poslin
dpurelin
Purpose
dpurelin
Linear transfer derivative function
Syntax
dA_dN = dpurelin(N,A)
Description
dpurelin is the derivative function for logsig.
dpurelin(N,A) takes two arguments,
N - S x Q net input.
A - S x Q output.
and returns the S x Q derivative dA_dN.
Examples
Here we define the net input N for a layer of 3 purelin neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with purelin and then the derivative of A
with respect to N.
A = purelin(N)
dA_dN = dpurelin(N,A)
Algorithm
The derivative of purelin is calculated as follows:
D(i,q) = 1
See Also
purelin
13-47
dradbas
Purpose
dradbas
Derivative of radial basis transfer function
Syntax
dA_dN = dradbas(N,A)
Description
dradbas is the derivative function for radbas.
dradbas(N,A) takes two arguments,
N - S x Q net input.
A - S x Q output.
and returns the S x Q derivative dA/dN.
Examples
Here we define the net input N for a layer of 3 radbas neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with radbas and then the derivative of A with
respect to N.
A = radbas(N)
Algorithm
The derivative of radbas is calculated as follows:
d = -2*n*a
See Also
13-48
radbas
dsatlin
Purpose
dsatlin
Derivative of saturating linear transfer function
Syntax
dA_dN = dsatlin(N,A)
Description
dsatlin is the derivative function for satlin.
dsatlin(N,A) takes two arguments,
N - S x Q net input.
A - S x Q output.
and returns the S x Q derivative dA/dN.
Examples
Here we define the net input N for a layer of 3 satlin neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with satlin and then the derivative of A with
respect to N.
A = satlin(N)
dA_dN = dsatlin(N,A)
Algorithm
The derivative of satlin is calculated as follows:
d = 1, if 0 <= n <= 1; 0, Otherwise..
See Also
satlin
13-49
dsatlins
Purpose
dsatlins
Derivative of symmetric saturating linear transfer function
Syntax
dA_dN = dsatlins(N,A)
Description
dsatlins is the derivative function for satlins.
dsatlins(N,A) takes two arguments,
N - S x Q net input.
A - S x Q output.
and returns the S x Q derivative dA/dN.
Examples
Here we define the net input N for a layer of 3 satlins neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with satlins and then the derivative of A
with respect to N.
A = satlins(N)
dA_dN = dsatlins(N,A)
Algorithm
The derivative of satlins is calculated as follows:
d = 1, if -1 <= n <= 1; 0, Otherwise.
See Also
13-50
satlins
dsse
Purpose
Syntax
dsse
Sum squared error performance derivative function
dPerf_dE = dsse('e',E,X,perf,PP)
dPerf_dX = dsse('x',E,X,perf,PP)
Description
dsse is the derivative function for sse.
dsse('d',E,X,perf,PP) takes these arguments,
E - Matrix or cell array of error vector(s).
X - Vector of all weight and bias values.
perf - Network performance (ignored).
PP - Performance parameters (ignored).
and returns the derivative dPerf_dE.
dsse('x',E,X,perf,PP)returns the derivative dPerf_dX.
Examples
Here we define an error E and X for a network with one 3-element output and
six weight and bias values.
E = {[1; -2; 0.5]};
X = [0; 0.2; -2.2; 4.1; 0.1; -0.2];
Here we calculate the network's sum squared error performance, and
derivatives of performance.
perf = sse(E)
dPerf_dE = dsse('e',E,X)
dPerf_dX = dsse('x',E,X)
Note that sse can be called with only one argument and dsse with only three
arguments because the other arguments are ignored. The other arguments
exist so that sse and dsse conform to standard performance function argument
lists.
See Also
sse
13-51
dtansig
Purpose
dtansig
Hyperbolic tangent sigmoid transfer derivative function
Syntax
dA_dN = dtansig(N,A)
Description
dtansig is the derivative function for tansig.
dtansig(N,A) takes two arguments,
N - S x Q net input.
A - S x Q output.
and returns the S x Q derivative dA/dN.
Examples
Here we define the net input N for a layer of 3 tansig neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with tansig and then the derivative of A with
respect to N.
A = tansig(N)
dA_dN = dtansig(N,A)
Algorithm
The derivative of tansig is calculated as follows:
d = 1-a^2
See Also
13-52
tansig, logsig, dlogsig
dtribas
Purpose
dtribas
Derivative of triangular basis transfer function
Syntax
dA_dN = dtribas(N,A)
Description
dtribas is the derivative function for tribas.
dtribas(N,A) takes two arguments,
N - S x Q net input.
A - S x Q output.
and returns the S x Q derivative dA/dN.
Examples
Here we define the net input N for a layer of 3 tribas neurons.
N = [0.1; 0.8; -0.7];
We calculate the layer's output A with tribas and then the derivative of A with
respect to N.
A = tribas(N)
dA_dN = dtribas(N,A)
Algorithm
The derivative of tribas is calculated as follows:
d = 1, if -1 <= n < 0; -1, if 0 < n <= 1; 0, Otherwise.
See Also
tribas
13-53
errsurf
Purpose
errsurf
Error surface of single input neuron
Syntax
errsurf(p,t,wv,bv,f)
Description
errsurf(p,t,wv,bv,f) takes these arguments,
P
- 1 x Q matrix of input vectors.
T
- 1 x Q matrix of target vectors.
WV - Row vector of values of W.
BV - Row vector of values of B.
F
- Transfer function (string).
and returns a matrix of error values over WV and BV.
Examples
See Also
13-54
p = [-6.0 -6.1 -4.1 -4.0 +4.0 +4.1 +6.0 +6.1];
t = [+0.0 +0.0 +.97 +.99 +.01 +.03 +1.0 +1.0];
wv = -1:.1:1; bv = -2.5:.25:2.5;
es = errsurf(p,t,wv,bv,'logsig');
plotes(wv,bv,ES,[60 30])
plotes
gensim
Purpose
gensim
Generate a simulink block for neural network simulation
Syntax
gensim(net,st)
To Get Help
Type help network/gensim
Description
gensim(net,st) creates a simulink system containing a block which simulates
neural network net.
gensim(net,st) takes these inputs,
net - Neural network.
st
- Sample time (default = 1).
and creates a Simulink system containing a block which simulates neural
network net with a sampling time of st.
If net has no input or layer delays (net.numInputDelays and
net.numLayerDelays are both 0) then you can use -1 for st to get a
continuously sampling network.
Examples
net = newff([0 1],[5 1]);
gensim(net)
13-55
gridtop
Purpose
gridtop
Grid layer topology function
Syntax
pos = gridtop(dim1,dim2,...,dimN)
Description
gridtop calculates neuron positions for layers whose neurons are arranged in
an N dimensional grid.
gridtop(dim1,dim2,...,dimN) takes N arguments,
dimi - Length of layer in dimension i.
and returns an N x S matrix of N coordinate vectors where S is the product of
dim1*dim2*...*dimN.
Examples
This code creates and displays a two-dimensional layer with 40 neurons
arranged in a 8x5 grid.
pos = gridtop(8,5); plotsom(pos)
This code plots the connections between the same neurons, but shows each
neuron at the location of its weight vector. The weights are generated randomly
so the layer is very disorganized as is evident in the following plot.
W = rands(40,2); plotsom(W,dist(pos))
See Also
13-56
hextop, randtop
hardlim
Purpose
Graph and
Symbol
hardlim
Hard limit transfer function
a
+1
0
n
-1
AA
a = hardlim(n)
Hard Limit Transfer Function
Syntax
A = hardlim(N)
info = hardlim(code)
Description
The hard limit transfer function forces a neuron to output a 1 if its net input
reaches a threshold, otherwise it outputs 0. This allows a neuron to make a
decision or classification. It can say yes or no. This kind of neuron is often
trained with the perceptron learning rule.
hardlim is a transfer function. Transfer functions calculate a layer's output
from its net input.
hardlim(N) takes one input,
N - S x Q matrix of net input (column) vectors.
and returns 1 where N is positive, 0 elsewhere.
hardlim(code) returns useful information for each code string,
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
Examples
Here is the code to create a plot of the hardlim transfer function.
n = -5:0.1:5;
a = hardlim(n);
plot(n,a)
13-57
hardlim
Network Use
You can create a standard network that uses hardlim by calling newp.
To change a network so that a layer uses hardlim, set
net.layers{i}.transferFcn to 'hardlim'.
In either case call sim to simulate the network with hardlim.
See newp for simulation examples.
Algorithm
hardlim(n) = 1, if n >= 0; 0 otherwise.
See Also
sim, hardlims
13-58
hardlims
Purpose
Graph and
Symbol
hardlims
Symmetric hard limit transfer function
a
+1
0
n
-1
a = hardlims(n)
AA
AA
Symmetric Hard Limit Trans. Funct.
Syntax
A = hardlims(N)
info = hardlims(code)
Description
The symmetric hard limit transfer function forces a neuron to output a 1 if its
net input reaches a threshold. Otherwise it outputs -1. Like the regular hard
limit function, this allows a neuron to make a decision or classification. It can
say yes or no.
hardlims is a transfer function. Transfer functions calculate a layer's output
from its net input.
hardlims(N) takes one input,
N - S x Q matrix of net input (column) vectors.
and returns 1 where N is positive, -1 elsewhere.
hardlims(code) return useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
Examples
Here is the code to create a plot of the hardlims transfer function.
n = -5:0.1:5;
a = hardlims(n);
plot(n,a)
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hardlims
Network Use
You can create a standard network that uses hardlims by calling newp.
To change a network so that a layer uses hardlims, set
net.layers{i}.transferFcn to 'hardlims'.
In either case call sim to simulate the network with hardlims.
See newp for simulation examples.
Algorithm
hardlim(n) = 1, if n >= 0; -1 otherwise.
See Also
sim, hardlim
13-60
hextop
Purpose
hextop
Hexagonal layer topology function
Syntax
pos = hextop(dim1,dim2,...,dimN)
Description
hextop calculates the neuron positions for layers whose neurons are arranged
in a N dimensional hexagonal pattern.
hextop(dim1,dim2,...,dimN) takes N arguments,
dimi - Length of layer in dimension i.
and returns an N x S matrix of N coordinate vectors where S is the product of
dim1*dim2*...*dimN.
Examples
This code creates and displays a two-dimensional layer with 40 neurons
arranged in a 8x5 hexagonal pattern.
pos = hextop(8,5); plotsom(pos)
This code plots the connections between the same neurons, but shows each
neuron at the location of its weight vector. The weights are generated randomly
so that the layer is very disorganized, as is evident in the following plot.
W = rands(40,2); plotsom(W,dist(pos))
See Also
gridtop, randtop
13-61
hintonw
Purpose
hintonw
Hinton graph of weight matrix
Syntax
hintonw(W,maxw,minw)
Description
hintonw(W,maxw,minw) takes these inputs,
W - S x R weight matrix
maxw - Maximum weight, default = max(max(abs(W))).
minw - Minimum weight, default = M1/100.
and displays a weight matrix represented as a grid of squares.
Each square's area represents a weight’s magnitude. Each square's projection
(color) represents a weight’s sign; inset (red) for negative weights, projecting
(green) for positive.
Examples
W = rands(4,5);
The following code displays the matrix graphically.
hintonw(W)
1
Neuron
2
3
4
1
See Also
13-62
hintonwb
2
3
Input
4
5
hintonwb
Purpose
hintonwb
Hinton graph of weight matrix and bias vector
Syntax
hintonwb(W,b,maxw,minw)
Description
hintonwb(W,B,maxw,minw) takes these inputs,
W - S x R weight matrix.
B - S x 1 bias vector.
maxw - Maximum weight, default = max(max(abs(W))).
minw - Minimum weight, default = M1/100.
and displays a weight matrix and a bias vector represented as a grid of squares.
Each square's area represents a weight’s magnitude. Each square's projection
(color) represents a weight’s sign; inset (red) for negative weights, projecting
(green) for positive. The weights are shown on the left.
Examples
The following code produces the result shown below.
W = rands(4,5);
b = rands(4,1);
hintonwb(W,b)
Neuron
1
2
3
4
0
1
2
3
4
5
Input
See Also
hintonw
13-63
ind2vec
Purpose
ind2vec
Convert indices to vectors
Syntax
vec = ind2vec(ind)
Description
ind2vec and vec2ind allow indices to either be represented by themselves, or
as vectors containing a 1 in the row of the index they represent.
ind2vec(ind) takes one argument,
ind - Row vector of indices.
and returns a sparse matrix of vectors, with one 1 in each column, as indicated
by ind.
Examples
Here four indices are defined and converted to vector representation.
ind = [1 3 2 3]
vec = ind2vec(ind)
See Also
13-64
vec2ind
init
Purpose
init
Initialize a neural network
Syntax
net = init(net)
To Get Help
Type help network/init
Description
init(net) returns neural network net with weight and bias values updated
according to the network initialization function, indicated by net.initFcn,
and the parameter values, indicated by net.initParam.
Examples
Here a perceptron is created with a 2-element input (with ranges of 0 to 1, and
-2 to 2) and 1 neuron. Once it is created we can display the neuron’s weights
and bias.
net = newp([0 1;-2 2],1);
net.iw{1,1}
net.b{1}
Training the perceptron alters its weight and bias values.
P = [0 1 0 1; 0 0 1 1];
T = [0 0 0 1];
net = train(net,P,T);
net.iw{1,1}
net.b{1}
init reinitializes those weight and bias values.
net = init(net);
net.iw{1,1}
net.b{1}
The weights and biases are zeros again, which are the initial values used by
perceptron networks (see newp).
Algorithm
init calls net.initFcn to initialize the weight and bias values according to the
parameter values net.initParam.
Typically, net.initFcn is set to 'initlay' which initializes each layer's
weights and biases according to its net.layers{i}.initFcn.
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init
Backpropagation networks have net.layers{i}.initFcn set to 'initnw',
which calculates the weight and bias values for layer i using the
Nguyen-Widrow initialization method.
Other networks have net.layers{i}.initFcn set to 'initwb', which initializes
each weight and bias with its own initialization function. The most common
weight and bias initialization function is rands, which generates random
values between -1 and 1.
See Also
13-66
sim, adapt, train, initlay, initnw, initwb, rands
initcon
Purpose
initcon
Conscience bias initialization function
Syntax
b = initcon(s,pr)
Description
initcon is a bias initialization function that initializes biases for learning with
the learncon learning function.
initcon (S,PR) takes two arguments,
S
- Number of rows (neurons).
PR - R x 2 matrix of R = [Pmin Pmax], default = [1 1].
and returns an S x 1 bias vector.
Note that for biases R is always 1. initcon could also be used to initialize
weights, but it is not recommended for that purpose.
Examples
Here initial bias values are calculated for a 5 neuron layer.
b = initcon(5)
Network Use
You can create a standard network that uses initcon to initialize weights by
calling newc.
To prepare the bias of layer i of a custom network to initialize with initcon:
1 Set net.initFcn to 'initlay'. (net.initParam will automatically become
initlay's default parameters.)
2 Set net.layers{i}.initFcn to 'initwb'.
3 Set net.biases{i}.initFcn to 'initcon'.
To initialize the network call init. See newc for initialization examples.
Algorithm
learncon updates biases so that each bias value b(i) is a function of the
average output c(i) of the neuron i associated with the bias.
initcon gets initial bias values by assuming that each neuron has responded
to equal numbers of vectors in the "past".
See Also
initwb, initlay, init, learncon
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initlay
Purpose
Syntax
initlay
Layer-by-layer network initialization function
net = initlay(net)
info = initlay(code)
Description
initlay is a network initialization function which initializes each layer i
according to its own initialization function net.layers{i}.initFcn.
initlay(net) takes,
net - Neural network.
and returns the network with each layer updated. initlay(code) returns
useful information for each code string:
'pnames' - Names of initialization parameters.
'pdefaults' - Default initialization parameters.
initlay does not have any initialization parameters.
Network Use
You can create a standard network that uses initlay by calling newp, newlin,
newff, newcf, and many other new network functions.
To prepare a custom network to be initialized with initlay:
1 Set net.initFcn to 'initlay'. (This will set net.initParam to the empty
matrix [ ] since initlay has no initialization parameters.)
2 Set each net.layers{i}.initFcn to a layer initialization function.
(Examples of such functions are initwb and initnw).
To initialize the network call init. See newp and newlin for initialization
examples.
Algorithm
The weights and biases of each layer i are initialized according to
net.layers{i}.initFcn.
See Also
initwb, initnw, init
13-68
initnw
Purpose
initnw
Nguyen-Widrow layer initialization function
Syntax
net = initnw(net,i)
Description
initnw is a layer initialization function which initializes a layer's weights and
biases according to the Nguyen-Widrow initialization algorithm. This
algorithm chooses values in order to distribute the active region of each neuron
in the layer evenly across the layer's input space.
initnw(net,i) takes two arguments,
net - Neural network.
i - Index of a layer.
and returns the network with layer i's weights and biases updated.
Network Use
You can create a standard network that uses initnw by calling newff or newcf.
To prepare a custom network to be initialized with initnw:
1 Set net.initFcn to 'initlay'. (This will set net.initParam to the empty
matrix [ ] since initlay has no initialization parameters.)
2 Set net.layers{i}.initFcn to 'initnw'.
To initialize the network call init. See newff and newcf for training examples.
Algorithm
The Nguyen-Widrow method generates initial weight and bias values for a
layer, so that the active regions of the layer's neurons will be distributed
roughly evenly over the input space.
Advantages over purely random weights and biases are:
1 Few neurons are wasted (since all the neurons are in the input space).
2 Training works faster (since each area of the input space has neurons). The
Nguyen-Widrow method can only be applied to layers...
- with a bias,
- with weights whose "weightFcn" is dotprod,
- with "netInputFcn" set to netsum.
If these conditions are not met then initnw uses rands to initialize the layer's
weights and biases.
See Also
initwb, initlay, init
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initwb
Purpose
initwb
By-weight-and-bias layer initialization function
Syntax
net = initwb(net,i)
Description
initwb is a layer initialization function with initializes a layer's weights and
biases according to their own initialization functions.
initwb(net,i) takes two arguments,
net - Neural network.
i
- Index of a layer.
and returns the network with layer i's weights and biases updated.
Network Use
You can create a standard network that uses initwb by calling newp or newlin.
To prepare a custom network to be initialized with initwb:
1 Set net.initFcn to 'initlay'. (This will set net.initParam to the empty
matrix [ ] since initlay has no initialization parameters.)
2 Set net.layers{i}.initFcn to 'initwb'.
3 Set each net.inputWeights{i,j}.learnFcn to a weight initialization
function. Set each net.layerWeights{i,j}.learnFcn to a weight
initialization function. Set each net.biases{i}.learnFcn to a bias
initialization function. (Examples of such functions are rands and
midpoint.)
To initialize the network call init.
See newp and newlin for training examples.
Algorithm
Each weight (bias) in layer i is set to new values calculated according to its
weight (bias) initialization function.
See Also
initnw, initlay, init
13-70
initzero
Purpose
Syntax
initzero
Zero weight/bias initialization function
W = initzero(S,PR)
b = initzero(S,[1 1])
Description
initzero(S,PR) takes two arguments,
S - Number of rows (neurons).
PR - R x 2 matrix of input value ranges = [Pmin Pmax].
and returns an S x R weight matrix of zeros.
initzero(S,[1 1]) returns S x 1 bias vector of zeros.
Examples
Here initial weights and biases are calculated for a layer with two inputs
ranging over [0 1] and [-2 2], and 4 neurons.
W = initzero(5,[0 1; -2 2])
b = initzero(5,[1 1])
Network Use
You can create a standard network that uses initzero to initialize its weights
by calling newp or newlin.
To prepare the weights and the bias of layer i of a custom network to be
initialized with midpoint:
1 Set net.initFcn to 'initlay'. (net.initParam will automatically become
initlay's default parameters.)
2 Set net.layers{i}.initFcn to 'initwb'.
3 Set each net.inputWeights{i,j}.initFcn to 'initzero'. Set each
net.layerWeights{i,j}.initFcn to 'initzero'. Set each
net.biases{i}.initFcn to 'initzero'.
To initialize the network call init.
See newp or newlin for initialization examples.
See Also
initwb, initlay, init
13-71
learncon
Purpose
Syntax
learncon
Conscience bias learning function
[dB,LS] = learncon(B,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learncon(code)
Description
learncon is the conscience bias learning function used to increase the net input
to neurons which have the lowest average output until each neuron responds
roughly an equal percentage of the time.
learncon(B,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
B
- S x 1 bias vector.
P
- 1x Q ones vector.
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns
dB - S x 1 weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learncon's learning parameter, shown here with
its default value.
LP.lr - 0.001 - Learning rate.
learncon(code) returns useful information for each code string,
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
NNT 2.0 compatibility: The LP.lr described above equals 1 minus the bias time
constant used by trainc in NNT 2.0.
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learncon
Examples
Here we define a random output A, and bias vector W for a layer with 3 neurons.
We also define the learning rate LR.
a = rand(3,1);
b = rand(3,1);
lp.lr = 0.5;
Since learncon only needs these values to calculate a bias change (see
algorithim below), we will use them to do so.
dW = learncon(b,[],[],[],a,[],[],[],[],[],lp,[])
Network Use
To prepare the bias of layer i of a custom network to learn with learncon:
1 Set net.trainFcn to 'trainwb1'. (net.trainParam will automatically
become adaptwb's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
adaptwb's default parameters.)
3 Set net.inputWeights{i}.learnFcn to 'learncon'. Set each
net.layerWeights{i,j}.learnFcn to 'learncon'. (Each weight learning
parameter property will automatically be set to learncon's default
parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (or net.adaptParam) properties as desired.
2 Call train (or adapt).
Algorithm
learncon calculates the bias change db for a given neuron by first updating
each neuron's conscience, i.e. the running average of its output:
c = (1-lr)*c + lr*a
The conscience is then used to compute a bias for the neuron that is greatest
for smaller conscience values.
b = exp(1-log(c)) - b
(Note that learncon is able to recover C each time it is called from the bias
values.)
See Also
learnk, learnos, adaptwb, trainwb, adapt, train
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learngd
Purpose
Syntax
learngd
Gradient descent weight/bias learning function
[dW,LS] = learngd(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
[db,LS] = learngd(b,ones(1,Q),Z,N,A,T,E,gW,gA,D,LP,LS)
info = learngd(code)
Description
learngd is the gradient descent weight/bias learning function.
learngd(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learngd's learning parameter shown here with
its default value.
LP.lr - 0.01 - Learning rate.
learngd(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
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learngd
Examples
Here we define a random gradient gW for a weight going to a layer with 3
neurons, from an input with 2 elements. We also define a learning rate of 0.5.
gW = rand(3,2);
lp.lr = 0.5;
Since learngd only needs these values to calculate a weight change (see
algorithim below), we will use them to do so.
dW = learngd([],[],[],[],[],[],[],gW,[],[],lp,[])
Network Use
You can create a standard network that uses learngd with newff, newcf, or
newelm. To prepare the weights and the bias of layer i of a custom network to
adapt with learngd:
1 Set net.adaptFcn to 'adaptwb'. net.adaptParam will automatically become
trainwb's default parameters.
2 Set each net.inputWeights{i,j}.learnFcn to 'learngd'. Set each
net.layerWeights{i,j}.learnFcn to 'learngd'. Set
net.biases{i}.learnFcn to 'learngd'. Each weight and bias learning
parameter property will automatically be set to learngd's default
parameters.
To allow the network to adapt:
1 Set net.adaptParam properties to desired values.
2 Call adapt with the network.
See newff or newcf for examples.
Algoritm
learngd calculates the weight change dW for a given neuron from the neuron's
input P and error E, and the weight (or bias) learning rate LR, according to the
gradient descent: dw = lr*gW.
See Also
learngdm, newff, newcf, adaptwb, trainwb, adapt, train
13-75
learngdm
Purpose
Syntax
learngdm
Gradient descent w/ momentum weight/bias learning function
[dW,LS] = learngdm(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
[db,LS] = learngdm(b,ones(1,Q),Z,N,A,T,E,gW,gA,D,LP,LS)
info = learngdm(code)
Description
learngdm is the gradient descent with momentum weight/bias learning
function.
learngdm(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learngdm's learning parameters, shown here with
their default values.
LP.lr - 0.01 - Learning rate.
LP.mc - 0.9
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- Momentum constant.
learngdm
learngdm(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
Examples
Here we define a random gradient G for a weight going to a layer with 3
neurons, from an input with 2 elements. We also define a learning rate of 0.5
and momentum constant of 0.8;
gW = rand(3,2);
lp.lr = 0.5;
lp.mc = 0.8;
Since learngdm only needs these values to calculate a weight change (see
algorithim below), we will use them to do so. We will use the default initial
learning state.
ls = [];
[dW,ls] = learngdm([],[],[],[],[],[],[],gW,[],[],lp,ls)
learngdm returns the weight change and a new learning state.
Network Use
You can create a standard network that uses learngdm with newff, newcf, or newelm.
To prepare the weights and the bias of layer i of a custom network to adapt
with learngdm:
1 Set net.adaptFcn to 'adaptwb'. net.adaptParam will automatically become
trainwb's default parameters.
2 Set each net.inputWeights{i,j}.learnFcn to 'learngdm'. Set each
net.layerWeights{i,j}.learnFcn to 'learngdm'. Set
net.biases{i}.learnFcn to 'learngdm'. Each weight and bias learning
parameter property will automatically be set to learngdm's default
parameters.
To allow the network to adapt:
1 Set net.adaptParam properties to desired values.
2 Call adapt with the network.
See newff or newcf for examples.
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learngdm
Algorithm
learngdm calculates the weight change dW for a given neuron from the neuron's
input P and error E, the weight (or bias) W, learning rate LR, and momentum
constant MC, according to gradient descent with momentum:
dW = mc*dWprev + (1-mc)*lr*gW
The previous weight change dWprev is stored and read from the learning
state LS.
See Also
13-78
learngd, newff, newcf, adaptwb, trainwb, adapt, train
learnh
Purpose
Syntax
learnh
Hebb weight learning rule
[dW,LS] = learnh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnh(code)
Description
learnh is the Hebb weight learning function.
learnh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learnh's learning parameter, shown here with its
default value.
LP.lr - 0.01 - Learning rate.
learnh(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
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learnh
Examples
Here we define a random input P and output A for a layer with a 2-element
input and 3 neurons. We also define the learning rate LR.
p = rand(2,1);
a = rand(3,1);
lp.lr = 0.5;
Since learnh only needs these values to calculate a weight change (see
algorithim below), we will use them to do so.
dW = learnh([],p,[],[],a,[],[],[],[],[],lp,[])
Network Use
To prepare the weights and the bias of layer i of a custom network to learn with
learnh:
1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become
trainwb's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnh'. Set each
net.layerWeights{i,j}.learnFcn to 'learnh'. Each weight learning
parameter property will automatically be set to learnh's default
parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (net.adaptParam) properties to desired values.
2 Call train (adapt).
Algorithm
learnh calculates the weight change dW for a given neuron from the neuron's
input P, output A, and learning rate LR according to the Hebb learning rule:
dw =
lr*a*p'
See Also
learnhd, adaptwb, trainwb, adapt, train
References
Hebb, D.O., The Organization of Behavior, New York: Wiley, 1949.
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learnhd
Purpose
Syntax
learnhd
Hebb with decay weight learning rule
[dW,LS] = learnhd(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnhd(code)
Description
learnhd is the Hebb weight learning function.
learnhd(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learnhd's learning parameters shown here with
default values.
LP.dr - 0.01 - Decay rate.
LP.lr - 0.1
- Learning rate.
learnhd(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
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learnhd
Examples
Here we define a random input P, output A, and weights W for a layer with a
2-element input and 3 neurons. We also define the decay and learning rates.
p = rand(2,1);
a = rand(3,1);
w = rand(3,2);
lp.dr = 0.05;
lp.lr = 0.5;
Since learnhd only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
dW = learnhd(w,p,[],[],a,[],[],[],[],[],lp,[])
Network Use
To prepare the weights and the bias of layer i of a custom network to learn with
learnhd:
1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become
trainwb's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnhd'. Set each
net.layerWeights{i,j}.learnFcn to 'learnhd'. (Each weight learning
parameter property will automatically be set to learnhd's default
parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (net.adaptParam) properties to desired values.
2 Call train (adapt).
Algorithm
learnhd calculates the weight change dW for a given neuron from the neuron's
input P, output A, decay rate DR, and learning rate LR according to the Hebb
with decay learning rule:
dw =
See Also
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lr*a*p' - dr*w
learnh, adaptwb, trainwb, adapt, train
learnis
Purpose
Syntax
learnis
Instar weight learning function
[dW,LS] = learnis(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnis(code)
Description
learnis is the instar weight learning function.
learnis(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learnis's learning parameter, shown here with
its default value.
LP.lr - 0.01 - Learning rate.
learnis(code) return useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
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learnis
Examples
Here we define a random input P, output A, and weight matrix W for a layer with
a 2-element input and 3 neurons. We also define the learning rate LR.
p = rand(2,1);
a = rand(3,1);
w = rand(3,2);
lp.lr = 0.5;
Since learnis only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
dW = learnis(w,p,[],[],a,[],[],[],[],[],lp,[])
Network Use
To prepare the weights and the bias of layer i of a custom network so that it
can learn with learnis:
1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become
trainwb's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnis'. Set each
net.layerWeights{i,j}.learnFcn to 'learnis'. (Each weight learning
parameter property will automatically be set to learnis's default
parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (net.adaptParam) properties to desired values.
2 Call train (adapt).
Algorithm
learnis calculates the weight change dW for a given neuron from the neuron's
input P, output A, and learning rate LR according to the instar learning rule:
dw =
lr*a*(p'-w)
See Also
learnk, learnos, adaptwb, trainwb, adapt, train
References
Grossberg, S., Studies of the Mind and Brain, Drodrecht, Holland: Reidel
Press, 1982.
13-84
learnk
Purpose
Syntax
learnk
Kohonen weight learning function
[dW,LS] = learnk(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnk(code)
Description
learnk is the Kohonen weight learning function.
learnk(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learnk's learning parameter, shown here with its
default value.
LP.lr - 0.01 - Learning rate.
learnk(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
13-85
learnk
Examples
Here we define a random input P, output A, and weight matrix W for a layer with
a 2-element input and 3 neurons. We also define the learning rate LR.
p = rand(2,1);
a = rand(3,1);
w = rand(3,2);
lp.lr = 0.5;
Since learnk only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
dW = learnk(w,p,[],[],a,[],[],[],[],[],lp,[])
Network Use
To prepare the weights of layer i of a custom network to learn with learnk:
1 Set net.trainFcn to 'trainwb1'. (net.trainParam will automatically
become trainwb1's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb1's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnk'. Set each
net.layerWeights{i,j}.learnFcn to 'learnk'. (Each weight learning
parameter property will automatically be set to learnk's default
parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (or net.adaptParam) properties as desired.
2 Call train (or adapt).
Algorithm
learnk calculates the weight change dW for a given neuron from the neuron's
input P, output A, and learning rate LR according to the Kohonen learning rule:
dw = lr*(p'-w), if a ~= 0; = 0, otherwise.
See Also
learnis, learnos, adaptwb, trainwb, adapt, train
References
Kohonen, T., Self-Organizing and Associative Memory, New York:
Springer-Verlag, 1984.
13-86
learnlv1
Purpose
Syntax
learnlv1
LVQ1 weight learning function
[dW,LS] = learnlv1(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnlv1(code)
Description
learnlv1 is the LVQ1 weight learning function.
learnlv1(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R weight gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x R neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learnlv1's learning parameter shown here with
its default value.
LP.lr - 0.01 - Learning rate.
learnlv1(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
needg' - Returns 1 if this function uses gW or gA.
13-87
learnlv1
Examples
Here we define a random input P, output A, weight matrix W, and output
gradient gA for a layer with a 2-element input and 3 neurons.
We also define the learning rate LR.
p = rand(2,1);
w = rand(3,2);
a = compet(negdist(w,p));
gA = [-1;1; 1];
lp.lr = 0.5;
Since learnlv1 only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
dW = learnlv1(w,p,[],[],a,[],[],[],gA,[],lp,[])
Network Use
You can create a standard network that uses learnlv1 with newlvq. To prepare
the weights of layer i of a custom network to learn with learnlv1:
1 Set net.trainFcn to trainwb1'. (net.trainParam will automatically become
trainwb1's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb1's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnlv1'. Set each
net.layerWeights{i,j}.learnFcn to 'learnlv1'. (Each weight learning
parameter property will automatically be set to learnlv1's default
parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (or net.adaptParam) properties as desired.
2 Call train (or adapt).
Algorithm
learnlv1 calculates the weight change dW for a given neuron from the neuron's
input P, output A, output gradient gA and learning rate LR, according to the
LVQ1 rule, given i the index of the neuron whose output a(i) is 1:
dw(i,:) = +lr*(p-w(i,:)) if gA(i) = 0;= -lr*(p-w(i,:)) if gA(i) = -1
See Also
13-88
learnlv2, adaptwb, trainwb, adapt, train
learnlv2
Purpose
Syntax
learnlv2
LVQ2 weight learning function
[dW,LS] = learnlv2(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnlv2(code)
Description
learnlv2 is the LVQ2 weight learning function.
learnlv2(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R weight gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learnlv1's learning parameter, shown here with
its default value.
LP.lr - 0.01 - Learning rate.
learnlv2(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
13-89
learnlv2
Examples
Here we define a sample input P, output A, weight matrix W, and output
gradient gA for a layer with a 2-element input and 3 neurons.
We also define the learning rate LR.
p = rand(2,1);
w = rand(3,2);
n = negdist(w,p);
a = compet(n);
gA = [-1;1; 1];
lp.lr = 0.5;
Since learnlv2 only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
dW = learnlv2(w,p,[],n,a,[],[],[],gA,[],lp,[])
Network Use
You can create a standard network that uses learnlv2 with newlvq.
To prepare the weights of layer i of a custom network to learn with learnlv2:
1 Set net.trainFcn to 'trainwb1'. (net.trainParam will automatically
become trainwb1's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb1's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnlv2'. Set each
net.layerWeights{i,j}.learnFcn to 'learnlv2'. (Each weight learning
parameter property will automatically be set to learnlv2's default
parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (or net.adaptParam) properties as desired.
2 Call train (or adapt).
Algorithm
learnlv2 calculates the weight change dW for a given neuron from the neuron's
input P, output A, output gradient gA and learning rate LR according to the LVQ2
rule, given i the index of the neuron whose output a(i) is 1:
dw(i,:) = +lr*(p-w(i,:)) if gA(i) = 0; = -lr*(p-w(i,:))
13-90
learnlv2
If gA(i) = -1; if gA(i) is -1 then the index j is found of the neuron with the
greatest net input n(k), from the neurons whose gA(k) is 1. This neuron's
weights are updated as follows:
dw(j,:) = +lr*(p-w(i,:))
See Also
learnlv1, adaptwb, trainwb, adapt, train
13-91
learnos
Purpose
Syntax
learnos
Outstar weight learning function
[dW,LS] = learnos(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnos(code)
Description
learnos is the outstar weight learning function.
learnos(W,P,Z,N,A,T,E,G,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R weight gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learnos's learning parameter, shown here with
its default value.
LP.lr - 0.01 - Learning rate.
learnos(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
13-92
learnos
Examples
Here we define a random input P, output A, and weight matrix W for a layer with
a 2-element input and 3 neurons. We also define the learning rate LR.
p = rand(2,1);
a = rand(3,1);
w = rand(3,2);
lp.lr = 0.5;
Since learnos only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
dW = learnos(w,p,[],[],a,[],[],[],[],[],lp,[])
Network Use
To prepare the weights and the bias of layer i of a custom network to learn with
learnos:
1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become
trainwb's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnos'. Set each
net.layerWeights{i,j}.learnFcn to 'learnos'. (Each weight learning
parameter property will automatically be set to learnos's default
parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (net.adaptParam) properties to desired values.
2 Call train (adapt).
Algorithm
learnos calculates the weight change dW for a given neuron from the neuron's
input P, output A, and learning rate LR according to the outstar learning rule:
dw =
lr*(a-w)*p'
See Also
learnis, learnk, adaptwb, trainwb, adapt, train
References
Grossberg, S., Studies of the Mind and Brain, Drodrecht, Holland: Reidel
Press, 1982.
13-93
learnp
Purpose
Syntax
learnp
Perceptron weight/bias learning function
[dW,LS] = learnp(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
[db,LS] = learnp(b,ones(1,Q),Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnp(code)
Description
learnp is the perceptron weight/bias learning function.
learnp(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R weight gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
learnp(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
13-94
learnp
Examples
Here we define a random input P and error E to a layer with a 2-element input
and 3 neurons.
p = rand(2,1);
e = rand(3,1);
Since learnp only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
dW = learnp([],p,[],[],[],[],e,[],[],[],[],[])
Network Use
You can create a standard network that uses learnp with newp.
To prepare the weights and the bias of layer i of a custom network to learn with
learnp:
1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become
trainwb's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnp'. Set each
net.layerWeights{i,j}.learnFcn to 'learnp'. Set
net.biases{i}.learnFcn to 'learnp'. (Each weight and bias learning
parameter property will automatically become the empty matrix since
learnp has no learning parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (net.adaptParam) properties to desired values.
2 Call train (adapt).
See newp for adaption and training examples.
Algorithm
learnp calculates the weight change dW for a given neuron from the neuron's
input P and error E according to the perceptron learning rule:
dw = 0, if e =
= p', if e = 1
= -p', if e = -1
0
13-95
learnp
This can be summarized as:
dw = e*p'
See Also
learnpn, newp, adaptwb, trainwb, adapt, train
References
Rosenblatt, F., Principles of Neurodynamics, Washington D.C.: Spartan Press,
1961.
13-96
learnpn
Purpose
Syntax
learnpn
Normalized perceptron weight/bias learning function
[dW,LS] = learnpn(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnpn(code)
Description
learnpn is a weight/bias learning function. It can result in faster learning than
learnp when input vectors have widely varying magnitudes.
learnpn(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R weight gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
learnpn(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
13-97
learnpn
Examples
Here we define a random input P and error E to a layer with a 2-element input
and 3 neurons.
p = rand(2,1);
e = rand(3,1);
Since learnpn only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
dW = learnpn([],p,[],[],[],[],e,[],[],[],[],[])
Network Use
You can create a standard network that uses learnpn with newp.
To prepare the weights and the bias of layer i of a custom network to learn with
learnpn:
1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become
trainwb's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnpn'. Set each
net.layerWeights{i,j}.learnFcn to 'learnpn'. Set
net.biases{i}.learnFcn to 'learnpn'. (Each weight and bias learning
parameter property will automatically become the empty matrix since
learnpn has no learning parameters.)
To train the network (or enable it to adapt):
1 Set net.trainParam (net.adaptParam) properties to desired values.
2 Call train (adapt).
See newp for adaption and training examples.
Algorithm
learnpn calculates the weight change dW for a given neuron from the neuron's
input P and error E according to the normalized perceptron learning rule:
pn = p / sqrt(1 + p(1)^2 + p(2)^2) + ... + p(R)^2)
dw = 0,
if e = 0
= pn', if e = 1
= -pn', if e = -1
13-98
learnpn
The expression for dW can be summarized as:
dw = e*pn'
Limitations
Perceptrons do have one real limitation. The set of input vectors must be
linearly separable if a solution is to be found. That is, if the input vectors with
targets of 1 cannot be separated by a line or hyperplane from the input vectors
associated with values of 0, the perceptron will never be able to classify them
correctly.
See Also
learnp, newp, adaptwb, trainwb, adapt, train
13-99
learnsom
Purpose
Syntax
learnsom
Self-organizing map weight learning function
[dW,LS] = learnsom(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnsom(code)
Description
learnsom is the self-organizing map weight learning function.
learnsom(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R weight gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learnsom's learning parameter, shown here with
its default value.
LP.order_lr
13-100
0.9
Ordering phase learning rate.
LP.order_steps
1000
Ordering phase steps.
LP.tune_lr
0.02
Tuning phase learning rate.
LP.tune_nd
1
Tuning phase neighborhood distance.
learnsom
learnpn(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
Examples
Here we define a random input P, output A, and weight matrix W, for a layer
with a 2-element input and 6 neurons. We also calculate positions and
distances for the neurons which are arranged in a 2x3 hexagonal pattern. Then
we define the four learning parameters.
p = rand(2,1);
a = rand(6,1);
w = rand(6,2);
pos = hextop(2,3);
d = linkdist(pos);
lp.order_lr = 0.9;
lp.order_steps = 1000;
lp.tune_lr = 0.02;
lp.tune_nd = 1;
Since learnsom only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
ls = [];
[dW,ls] = learnsom(w,p,[],[],a,[],[],[],[],d,lp,ls)
Network Use
You can create a standard network that uses learnsom with newsom.
1 Set net.trainFcn to 'trainwb1'. (net.trainParam will automatically
become trainwb1's default parameters.)
2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become
trainwb1's default parameters.)
3 Set each net.inputWeights{i,j}.learnFcn to 'learnsom'. Set each
net.layerWeights{i,j}.learnFcn to 'learnsom'. Set
net.biases{i}.learnFcn to 'learnsom'. (Each weight learning parameter
property will automatically be set to learnsom's default parameters.)
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learnsom
To train the network (or enable it to adapt):
1 Set net.trainParam (net.adaptParam) properties to desired values.
2 Call train (adapt).
Algorithm
learnsom calculates the weight change dW for a given neuron from the neuron's
input P, activation A2, and learning rate LR:
dw =
lr*a2*(p'-w)
where the activation A2 is found from the layer output A and neuron distances
D and the current neighborhood size ND:
a2(i,q) = 1,
if a(i,q) = 1
= 0.5, if a(j,q) = 1 and D(i,j) <= nd
= 0,
otherwise
The learning rate LR and neighborhood size NS are altered through two phases:
an ordering phase and a tuning phase.
The ordering phases lasts as many steps as LP.order_steps. During this
phase LR is adjusted from LP.order_lr down to LP.tune_lr, and ND is adjusted
from the maximum neuron distance down to 1. It is during this phase that
neuron weights are expected to order themselves in the input space consistent
with the associated neuron positions.
During the tuning phase LR decreases slowly from LP.tune_lr and ND is always
set to LP.tune_nd. During this phase the weights are expected to spread out
relatively evenly over the input space while retaining their topological order
found during the ordering phase.
See Also
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adaptwb, trainwb, adapt, train
learnwh
Purpose
Syntax
Description
learnwh
Widrow-Hoff weight/bias learning function
[dW,LS] = learnwh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
[db,LS] = learnwh(b,ones(1,Q),Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnwh(code)
learnwh is the Widrow-Hoff weight/bias learning function, and is also known
as the delta or least mean squared (LMS) rule.
learnwh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,
W
- S x R weight matrix (or S x 1 bias vector).
P
- R x Q input vectors (or ones(1,Q)).
Z
- S x Q weighted input vectors.
N
- S x Q net input vectors.
A
- S x Q output vectors.
T
- S x Q layer target vectors.
E
- S x Q layer error vectors.
gW - S x R weight gradient with respect to performance.
gA - S x Q output gradient with respect to performance.
D
- S x S neuron distances.
LP - Learning parameters, none, LP = [].
LS - Learning state, initially should be = [].
and returns,
dW - S x R weight (or bias) change matrix.
LS - New learning state.
Learning occurs according to learnwh's learning parameter shown here with
its default value.
LP.lr - 0.01 - Learning rate.
learnwh(code) returns useful information for each code string:
'pnames' - Names of learning parameters.
'pdefaults' - Default learning parameters.
'needg' - Returns 1 if this function uses gW or gA.
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learnwh
Examples
Here we define a random input P and error E to a layer with a 2-element input
and 3 neurons. We also define the learning rate LR learning parameter.
p = rand(2,1);
e = rand(3,1);
lp.lr = 0.5;
Since learnwh only needs these values to calculate a weight change (see
algorithm below), we will use them to do so.
dW = learnwh([],p,[],[],[],[],e,[],[],[],lp,[])
Network Use
You can create a standard network that uses learnwh with newlin.
To prepare the weights and the bias of layer i of a custom network to learn with
learnwh:
1 Set net.trainFcn to 'trainwb'. net.trainParam will automatically become
trainwb's default parameters.
2 Set net.adaptFcn to 'adaptwb'. net.adaptParam will automatically become
trainwb's default parameters.
3 Set each net.inputWeights{i,j}.learnFcn to 'learnwh'. Set each
net.layerWeights{i,j}.learnFcn to 'learnwh'. Set
net.biases{i}.learnFcn to 'learnwh'.
Each weight and bias learning parameter property will automatically be set to
learnwh's default parameters.
To train the network (or enable it to adapt):
1 Set net.trainParam (net.adaptParam) properties to desired values.
2 Call train(adapt).
See newlin for adaption and training examples.
Algorithm
learnwh calculates the weight change dW for a given neuron from the neuron's
input P and error E, and the weight (or bias) learning rate LR, according to the
Widrow-Hoff learning rule:
dw = lr*e*pn'
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learnwh
See Also
newlin, adaptwb, trainwb, adapt, train
References
Widrow, B., and M. E. Hoff, “Adaptive switching circuits,” 1960 IRE WESCON
Convention Record, New York IRE, pp. 96-104, 1960.
Widrow B. and S. D. Sterns, Adaptive Signal Processing, New York:
Prentice-Hall, 1985.
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linkdist
Purpose
linkdist
Link distance function
Syntax
d = linkdist(pos)
Description
linkdist is a layer distance function used to find the distances between the
layer's neurons given their positions.
linkdist(pos) takes one argument,
pos - N x S matrix of neuron positions.
and returns the S x S matrix of distances.
Examples
Here we define a random matrix of positions for 10 neurons arranged in three
dimensional space and find their distances.
pos = rand(3,10);
D = linkdist(pos)
Network Use
You can create a standard network that uses linkdist as a distance function
by calling newsom.
To change a network so that a layer's topology uses linkdist, set
net.layers{i}.distanceFcn to 'linkdist’.
In either case, call sim to simulate the network with dist. See newsom for
training and adaption examples.
Algorithm
The link distance D between two position vectors Pi and Pj from a set of S
vectors is:
Dij = 0, if i==j
= 1, if (sum((Pi-Pj).^2)).^0.5 is <= 1
= 2, if k exists, Dik = Dkj = 1
= 3, if k1, k2 exist, Dik1 = Dk1k2 = Dk2j = 1.
= N, if k1..kN exist, Dik1 = Dk1k2 = ...= DkNj = 1
= S, if none of the above conditions apply.
See Also
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sim, dist, mandist
logsig
Purpose
Graph and
Symbol
logsig
Log sigmoid transfer function
a
+1
n
0
-1
AA
AA
a = logsig(n)
Log-Sigmoid Transfer Function
Syntax
A = logsig(N)
info = logsig(code)
Description
logsig is a transfer function. Transfer functions calculate a layer's output from
its net input.
logsig(N) takes one input,
N - S x Q matrix of net input (column) vectors.
and returns each element of N squashed between 0 and 1.
logsig(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
Examples
Here is the code to create a plot of the logsig transfer function.
n = -5:0.1:5;
a = logsig(n);
plot(n,a)
Network Use
You can create a standard network that uses logsig by calling newff or newcf.
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logsig
To change a network so a layer uses logsig set net.layers{i}.transferFcn
to 'logsig'.
In either case, call sim to simulate the network with purelin.
See newff or newcf for simulation examples.
Algorithm
logsig(n) = 1 / (1 + exp(-n))
See Also
sim, dlogsig, tansig
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mae
Purpose
Syntax
mae
Mean absolute error performance function
perf = mae(e,x,pp)
perf = mae(e,net,pp)
info = mae(code)
Description
mae is a network performance function.
mae(E,X,PP) takes from one to three arguments,
E - Matrix or cell array of error vector(s).
X
- Vector of all weight and bias values (ignored).
PP - Performance parameters (ignored).
and returns the mean absolute error.
The errors E can be given in cell array form,
E - Nt x TS cell array, each element E{i,ts} is a Vi x Q matrix or[].
or as a matrix,
E - (sum of Vi) x Q matrix
where
Nt = net.numTargets
TS = Number of time steps
Q = Batch size
Vi = net.targets{i}.size
mae(E,net,PP) can take an alternate argument to X,
net - Neural network from which X can be obtained (ignored).
mae(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
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mae
Examples
Here a perceptron is created with a 1-element input ranging from -10 to 10, and
one neuron.
net = newp([-10 10],1);
Here the network is given a batch of inputs P. The error is calculated by
subtracting the output A from target T. Then the mean absolute error is
calculated.
p = [-10 -5 0 5 10];
t = [0 0 1 1 1];
y = sim(net,p)
e = t-y
perf = mae(e)
Note that mae can be called with only one argument because the other
arguments are ignored. mae supports those arguments to conform to the
standard performance function argument list.
Network Use
You can create a standard network that uses mae with newp.
To prepare a custom network to be trained with mae, set net.performFcn to
'mae'. This will automatically set net.performParam to the empty matrix [], as
mae has no performance parameters.
In either case, calling train or adapt will result in mae being used to calculate
performance.
See newp for examples.
See Also
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mse, msereg, dmae
mandist
Purpose
Syntax
mandist
Manhattan distance weight function
Z = mandist(W,P)
df = mandist('deriv')
D = mandist(pos);
Description
mandist is the Manhattan distance weight function. Weight functions apply
weights to an input to get weighted inputs.
mandist(W,P) takes these inputs,
W - S x R weight matrix.
P - R x Q matrix of Q input (column) vectors.
and returns the S x Q matrix of vector distances.
mandist('deriv') returns '' because mandist does not have a derivative
function.
mandist is also a layer distance function which can be used to find the distances
between neurons in a layer.
mandist(pos) takes one argument,
pos - An S row matrix of neuron positions.
and returns the S x S matrix of distances.
Examples
Here we define a random weight matrix W and input vector P and calculate the
corresponding weighted input Z.
W = rand(4,3);
P = rand(3,1);
Z = mandist(W,P)
Here we define a random matrix of positions for 10 neurons arranged in three
dimensional space and then find their distances.
pos = rand(3,10);
D = mandist(pos)
Network Use
You can create a standard network that uses mandist as a distance function by
calling newsom.
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mandist
To change a network so an input weight uses mandist, set
net.inputWeight{i,j}.weightFcn to 'mandist’. For a layer weight set
net.inputWeight{i,j}.weightFcn to 'mandist'.
To change a network so a layer's topology uses mandist, set
net.layers{i}.distanceFcn to 'mandist'.
In either case, call sim to simulate the network with dist. See newpnn or
newgrnn for simulation examples.
Algorithm
The Manhattan distance D between two vectors X and Y is:
D = sum(abs(x-y))
See Also
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sim, dist, linkdist
mat2cell
Purpose
mat2cell
Break matrix up into a cell array of matrices
Syntax
mat2cell(M,R,C);
Description
mat2cell(M,R,C) takes three arguments,
M - row x col matrix.
R - Vector of row sizes (should sum to row).
C - Vector of col sizes (should sum to col).
and returns a cell array of matrices, found using R and C.
Examples
See Also
M = [1 2 3 4; 5 6 7 8; 9 10 11 12];
C = mat2cell(M,[1 2],[2 2])
cell2mat
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maxlinlr
Purpose
maxlinlr
Maximum learning rate for a linear layer
Syntax
lr = maxlinlr(P)
lr = maxlinlr(P,'bias')
Description
maxlinlr is used to calculate learning rates for newlin.
maxlinlr(P) takes one argument,
P - R x Q matrix of input vectors.
and returns the maximum learning rate for a linear layer without a bias that
is to be trained only on the vectors in P.
maxlinlr(P,'bias') returns the maximum learning rate for a linear layer
with a bias.
Examples
Here we define a batch of 4 2-element input vectors and find the maximum
learning rate for a linear layer with a bias.
P = [1 2 -4 7; 0.1 3 10 6];
lr = maxlinlr(P,'bias')
See Also
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linnet, newlin, newlind
midpoint
Purpose
midpoint
Midpoint weight initialization function
Syntax
W = midpoint(S,PR)
Description
midpoint is a weight initialization function that sets weight (row) vectors to
the center of the input ranges.
midpoint(S,PR) takes two arguments,
S
- Number of rows (neurons).
PR - R x 2 matrix of input value ranges = [Pmin Pmax].
and returns an S x R matrix with rows set to (Pmin+Pmax)'/2.
Examples
Here initial weight values are calculated for a 5 neuron layer with input
elements ranging over [0 1] and [-2 2].
W = midpoint(5,[0 1; -2 2])
Network Use
You can create a standard network that uses midpoint to initialize weights by
calling newc.
To prepare the weights and the bias of layer i of a custom network to initialize
with midpoint:
1 Set net.initFcn to 'initlay'. (net.initParam will automatically become
initlay's default parameters.)
2 Set net.layers{i}.initFcn to 'initwb'.
3 Set each net.inputWeights{i,j}.initFcn to 'midpoint'. Set each
net.layerWeights{i,j}.initFcn to 'midpoint';
To initialize the network call init.
See Also
initwb, initlay, init
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minmax
Purpose
minmax
Ranges of matrix rows
Syntax
pr = minmax(P)
Description
minmax(P) takes one argument,
PR - R x Q matrix.
and returns the R x 2 matrix PR of minimum and maximum values for each row
of M.
Examples
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p = [0 1 2; -1 -2 -0.5]
pr = minmax(p)
mse
Purpose
Syntax
mse
Mean squared error performance function
perf = mse(e,x,pp)
perf = mse(e,net,pp)
info = mse(code)
Description
mse is a network performance function. It measures the network’s performance
according to the mean of squared errors.
mse(E,X,PP) takes from one to three arguments,
E
- Matrix or cell array of error vector(s).
X
- Vector of all weight and bias values (ignored).
PP - Performance parameters (ignored).
and returns the mean squared error.
mse(E,net,PP) can take an alternate argument to X,
net - Neural network from which X can be obtained (ignored).
mse(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here a two layer feed-forward network is created with a 1-element input
ranging from -10 to 10, four hidden tansig neurons, and one purelin output
neuron.
net = newff([-10 10],[4 1],{'tansig','purelin'});
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mse
Here the network is given a batch of inputs P. The error is calculated by
subtracting the output A from target T. Then the mean squared error is
calculated.
p = [-10 -5 0 5 10];
t = [0 0 1 1 1];
y = sim(net,p)
e = t-y
perf = mse(e)
Note that mse can be called with only one argument because the other
arguments are ignored. mse supports those ignored arguments to conform to
the standard performance function argument list.
Network Use
You can create a standard network that uses mse with newff, newcf, or newelm.
To prepare a custom network to be trained with mse, set net.performFcn to
'mse'. This will automatically set net.performParam to the empty matrix [], as
mse has no performance parameters.
In either case, calling train or adapt will result in mse being used to calculate
performance.
See newff or newcf for examples.
See Also
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msereg, mae, dmse
msereg
Purpose
Syntax
msereg
Mean squared error w/reg performance function
perf = mse(e,x,pp)
perf = mse(e,net,pp)
info = mse(code)
Description
msereg is a network performance function. It measures network performance
as the weight sum of two factors: the mean squared error and the mean
squared weight and bias values.
msereg(E,X,PP) takes from three arguments,
E
- Matrix or cell array of error vector(s).
X
- Vector of all weight and bias values.
PP - Performance parameter.
where PP defines one performance parameters,
PP.ratio - Relative importance of errors vs. weight and bias values.
and returns the sum of mean squared errors (times PP.ratio) with the mean
squared weight and bias values (times 1-PP.ratio).
The errors E can be given in cell array form,
E - Nt x TS cell array, each element E{i,ts} is an Vi x Q matrix or [].
or as a matrix,
E - (sum of Vi) x Q matrix
where
Nt = net.numTargets
TS = Number of time steps
Q
= Batch size
Vi = net.targets{i}.size
mse(E,net) takes an alternate argument to X and PP,
net - Neural network from which X and PP can be obtained.
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msereg
mse(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here a two layer feed-forward is created with a 1-element input ranging from
-2 to 2, four hidden tansig neurons, and one purelin output neuron.
net = newff([-2 2],[4 1]
{'tansig','purelin'},'trainlm','learngdm','msereg');
Here the network is given a batch of inputs P. The error is calculated by
subtracting the output A from target T. Then the mean squared error is
calculated using a ratio of 20/(20+1). (Errors are 20 times as important as
weight and bias values).
p = [-2 -1 0 1 2];
t = [0 1 1 1 0];
y = sim(net,p)
e = t-y
net.performParam.ratio = 20/(20+1);
perf = msereg(e,net)
Network Use
You can create a standard network that uses msereg with newff, newcf, or
newelm.
To prepare a custom network to be trained with msereg, set net.performFcn
to 'msereg'. This will automatically set net.performParam to msereg's default
performance parameters.
In either case, calling train or adapt will result in msereg being used to
calculate performance.
See newff or newcf for examples.
See Also
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mse, mae, dmsereg
negdist
Purpose
Syntax
negdist
Negative distance weight function
Z = negdist(W,P)
df = negdist('deriv')
Description
negdist is a weight function. Weight functions apply weights to an input to get
weighted inputs.
negdist(W,P) takes these inputs,
W - S x R weight matrix.
P - R x Q matrix of Q input (column) vectors.
and returns the S x Q matrix of negative vector distances.
negdist('deriv') returns '' because negdist does not have a derivative
function.
Examples
Here we define a random weight matrix W and input vector P and calculate the
corresponding weighted input Z.
W = rand(4,3);
P = rand(3,1);
Z = negdist(W,P)
Network Use
You can create a standard network that uses negdist by calling newc or
newsom.
To change a network so an input weight uses negdist, set
net.inputWeight{i,j}.weightFcn to 'negdist’. For a layer weight set
net.inputWeight{i,j}.weightFcn to 'negdist’.
In either case, call sim to simulate the network with negdist. See newc or
newsom for simulation examples.
Algorithm
negdist returns the negative Euclidean distance:
z = -sqrt(sum(w-p)^2)
See Also
sim, dotprod, dist
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netprod
Purpose
Syntax
netprod
Product net input function
N = netprod(Z1,Z2,...)
df = netprod('deriv')
Description
netprod is a net input function. Net input functions calculate a layer's net
input by combining its weighted inputs and biases.
netprod(Z1,Z2,...,Zn)takes,
Zi - S x Q matrices.
and returns an element-wise sum of Zi's.
netprod('deriv') returns netprod's derivative function.
Examples
Here netprod combines two sets of weighted input vectors (which we have
defined ourselves).
z1 = [1 2 4;3 4 1];
z2 = [-1 2 2; -5 -6 1];
n = netprod(z1,z2)
Here netprod combines the same weighted inputs with a bias vector. Because
Z1 and Z2 each contain three concurrent vectors, three concurrent copies of B
must be created with concur so that all sizes match up.
b = [0; -1];
n = netprod(z1,z2,concur(b,3))
Network Use
You can create a standard network that uses netprod by calling newpnn or
newgrnn.
To change a network so that a layer uses netprod, set
net.layers{i}.netInputFcn to 'netprod'.
In either case, call sim to simulate the network with netprod. See newpnn or
newgrnn for simulation examples.
See Also
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sim, dnetprod, netsum, concur.
netsum
Purpose
Syntax
netsum
Sum net input function
N = netsum(Z1,Z2,...)
df = netsum('deriv')
Description
netsum is a net input function. Net input functions calculate a layer's net input
by combining its weighted inputs and biases.
netsum(Z1,Z2,...,Zn) takes any number of inputs,
Zi - S x Q matrices,
and returns N, the element-wise sum of Zi's.
netsum('deriv') returns netsum's derivative function.
Examples
Here netsum combines two sets of weighted input vectors (which we have
defined ourselves).
z1 = [1 2 4;3 4 1];
z2 = [-1 2 2; -5 -6 1];
n = netsum(z1,z2)
Here netsum combines the same weighted inputs with a bias vector. Because
Z1 and Z2 each contain three concurrent vectors, three concurrent copies of B
must be created with concur so that all sizes match up.
b = [0; -1];
n = netsum(z1,z2,concur(b,3))
Network Use
You can create a standard network that uses netsum by calling newp or newlin.
To change a network so a layer uses netsum, set net.layers{i}.netInputFcn
to 'netsum'.
In either case, call sim to simulate the network with netsum. See newp or
newlin for simulation examples.
See Also
sim, dnetprod, netprod, concur
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network
Purpose
Syntax
network
Create a custom neural network
net = network
net = network(numInputs,numLayers,biasConnect,inputConnect,
layerConnect,outputConnect,targetConnect)
To Get Help
Type help network/network
Description
network creates new custom networks. It is used to create networks that are
then customized by functions such as newp, newlin, newff, etc.
network takes these optional arguments (shown with default values):
numInputs
- Number of inputs, 0.
numLayers
- Number of layers, 0.
biasConnect
- numLayers-by-1 Boolean vector, zeros.
inputConnect
- numLayers-by-numInputs Boolean matrix, zeros.
layerConnect
- numLayers-by-numLayers Boolean matrix, zeros.
outputConnect - 1-by-numLayers Boolean vector, zeros.
targetConnect - 1-by-numLayers Boolean vector, zeros.
and returns,
net - New network with the given property values.
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network
Properties
Architecture properties:
net.numInputs: 0 or a positive integer.
Number of inputs.
net.numLayers: 0 or a positive integer.
Number of layers.
net.biasConnect: numLayer-by-1 Boolean vector.
If net.biasConnect(i) is 1 then the layer i has a bias and net.biases{i}
is a structure describing that bias.
net.inputConnect: numLayer-by-numInputs Boolean vector.
If net.inputConnect(i,j) is 1 then layer i has a weight coming from
input j and net.inputWeights{i,j} is a structure describing that weight.
net.layerConnect: numLayer-by-numLayers Boolean vector.
If net.layerConnect(i,j) is 1 then layer i has a weight coming from
layer j and net.layerWeights{i,j} is a structure describing that weight.
net.outputConnect: 1-by-numLayers Boolean vector.
If net.outputConnect(i) is 1 then the network has an output from layer
i and net.outputs{i} is a structure describing that output.
net.targetConnect: 1-by-numLayers Boolean vector.
If net.outputConnect(i) is 1 then the network has a target from layer i
and net.targets{i} is a structure describing that target.
net.numOutputs: 0 or a positive integer. Read only.
Number of network outputs according to net.outputConnect.
net.numTargets: 0 or a positive integer. Read only.
Number of targets according to net.targetConnect.
net.numInputDelays: 0 or a positive integer. Read only.
Maximum input delay according to all net.inputWeight{i,j}.delays.
net.numLayerDelays: 0 or a positive number. Read only.
Maximum layer delay according to all net.layerWeight{i,j}.delays.
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network
Subobject structure properties:
net.inputs: numInputs-by-1 cell array.
net.inputs{i} is a structure defining input i:
net.layers: numLayers-by-1 cell array.
net.layers{i} is a structure defining layer i:
net.biases: numLayers-by-1 cell array.
If net.biasConnect(i) is 1, then net.biases{i} is a structure defining
the bias for layer i.
net.inputWeights: numLayers-by-numInputs cell array.
If net.inputConnect(i,j) is 1, then net.inputWeights{i,j} is a
structure defining the weight to layer i from input j.
net.layerWeights: numLayers-by-numLayers cell array.
If net.layerConnect(i,j) is 1, then net.layerWeights{i,j} is a
structure defining the weight to layer i from layer j.
net.outputs: 1-by-numLayers cell array.
If net.outputConnect(i) is 1, then net.outputs{i} is a structure
defining the network output from layer i.
net.targets: 1-by-numLayers cell array.
If net.targetConnect(i) is 1, then net.targets{i} is a structure
defining the network target to layer i.
Function properties:
net.adaptFcn: name of a network adaption function or ''.
net.initFcn: name of a network initialization function or ''.
net.performFcn: name of a network performance function or ''.
net.trainFcn: name of a network training function or ''.
Parameter properties:
net.adaptParam: network adaption parameters.
net.initParam: network initialization parameters.
net.performParam: network performance parameters.
net.trainParam: network training parameters.
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network
Weight and bias value properties:
net.IW: numLayers-by-numInputs cell array of input weight values.
net.LW: numLayers-by-numLayers cell array of layer weight values.
net.b: numLayers-by-1 cell array of bias values.
Other properties:
net.userdata: structure you can use to store useful values.
Examples
Here is the code to create a network without any inputs and layers, and then
set its number of inputs and layer to 1 and 2 respectively.
net = network
net.numInputs = 1
net.numLayers = 2
Here is the code to create the same network with one line of code.
net = network(1,2)
Here is the code to create a 1 input, 2 layer, feed-forward network. Only the
first layer will have a bias. An input weight will connect to layer 1 from input
1. A layer weight will connect to layer 2 from layer 1. Layer 2 will be a network
output, and have a target.
net = network(1,2,[1;0],[1; 0],[0 0; 1 0],[0 1],[0 1])
We can then see the properties of subobjects as follows:
net.inputs{1}
net.layers{1}, net.layers{2}
net.biases{1}
net.inputWeights{1,1}, net.layerWeights{2,1}
net.outputs{2}
net.targets{2}
We can get the weight matrices and bias vector as follows:
net.iw.{1,1}, net.iw{2,1}, net.b{1}
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network
We can alter the properties of any of these subobjects. Here we change the
transfer functions of both layers:
net.layers{1}.transferFcn = 'tansig';
net.layers{2}.transferFcn = 'logsig';
Here we change the number of elements in input 1 to 2, by setting each
element’s range:
net.inputs{1}.range = [0 1; -1 1];
Next we can simulate the network for a 2-element input vector:
p = [0.5; -0.1];
y = sim(net,p)
See Also
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sim
newc
Purpose
newc
Create a competitive layer
Syntax
net = newc(PR,S,KLR,CLR)
Description
Competitive layers are used to solve classification problems.
net = newc(PR,S,KLR,CLR) takes these inputs,
PR - R x 2 matrix of min and max values for R input elements.
S
- Number of neurons.
KLR - Kohonen learning rate, default = 0.01.
CLR - Conscience learning rate, default = 0.001.
and returns a new competitive layer.
Properties
Competitive layers consist of a single layer with the negdist weight function,
netsum net input function, and the compet transfer function.
The layer has a weight from the input, and a bias.
Weights and biases are initialized with midpoint and initcon.
Adaption and training are done with adaptwb and trainwb1, which both update
weight and bias values with the learnk and learncon learning functions.
Examples
Here is a set of four two-element vectors P.
P = [.1 .8
.1 .9; .2 .9 .1 .8];
To competitive layer can be used to divide these inputs into two classes. First
a two neuron layer is created with two input elements ranging from 0 to 1, then
it is trained.
net = newc([0 1; 0 1],2);
net = train(net,P);
The resulting network can then be simulated and its output vectors converted
to class indices.
Y = sim(net,P)
Yc = vec2ind(Y)
See Also
sim, init, adapt, train, adaptwb, trainwb1
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newcf
Purpose
newcf
Trainable cascade-forward backpropagation network
Syntax
net = newcf(Pr,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF)
Description
newcf(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) takes,
PR
- R x 2 matrix of min and max values for R input elements.
Si
- Size of ith layer, for Nl layers.
TFi - Transfer function of ith layer, default = 'tansig'.
BTF - Backprop network training function, default = 'traingd'.
BLF - Backprop weight/bias learning function, default = 'learngdm'.
PF
- Performance function, default = 'mse'.
and returns an N layer cascade-forward backprop network.
The transfer functions TFi can be any differentiable transfer function such as
tansig, logsig, or purelin.
The training function BTF can be any of the backprop training functions such
as trainlm, trainbfg, trainrp, traingd, etc.
WARNING: trainlm is the default training function because it is very fast, but
it requires a lot of memory to run. If you get an out-of-memory error when
training try doing one of these:
1 Slow trainlm training, but reduce memory requirements by setting
net.trainParam.mem_reduc to 2 or more. (See help trainlm.)
2 Use trainbfg, which is slower but more memory-efficient than trainlm.
3 Use trainrp which is slower but more memory-efficient than trainbfg.
The learning function BLF can be either of the backpropagation learning
functions such as learngd, or learngdm.
The performance function can be any of the differentiable performance
functions such as mse or msereg.
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5 6 7 8 9 10];
T = [0 1 2 3 4 3 2 1 2 3 4];
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newcf
Here a two-layer cascade-forward network is created. The network's input
ranges from [0 to 10]. The first layer has five tansig neurons, the second layer
has one purelin neuron. The trainlm network training function is to be used.
net = newcf([0 10],[5 1],{'tansig' 'purelin'});
Here the network is simulated and its output plotted against the targets.
Y = sim(net,P);
plot(P,T,P,Y,'o')
Here the network is trained for 50 epochs. Again the network's output is
plotted.
net.trainParam.epochs = 50;
net = train(net,P,T);
Y = sim(net,P);
plot(P,T,P,Y,'o')
Algorithm
Cascade-forward networks consist of Nl layers using the dotprod weight
function, netsum net input function, and the specified transfer functions.
The first layer has weights coming from the input. Each subsequent layer has
weights coming from the input and all previous layers. All layers have biases.
The last layer is the network output.
Each layer's weights and biases are initialized with initnw.
Adaption is done with adaptwb which updates weights with the specified
learning function. Training is done with the specified training function.
Performance is measured according to the specified performance function.
See Also
newff, newelm, sim, init, adapt, train
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newelm
Purpose
newelm
Create an Elman backpropagation network
Syntax
net = newelm(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF)
Description
newelm(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) takes several
arguments,
PR
- R x 2 matrix of min and max values for R input elements.
Si
- Size of ith layer, for Nl layers.
TFi - Transfer function of ith layer, default = 'tansig'.
BTF - Backprop network training function, default = 'traingdx'.
BLF - Backprop weight/bias learning function, default = 'learngdm'.
PF
- Performance function, default = 'mse'.
and returns an Elman network.
The training function BTF can be any of the backprop training functions such
as trainlm, trainbfg, trainrp, traingd, etc.
WARNING: trainlm is the default training function because it is very fast, but
it requires a lot of memory to run. If you get an "out-of-memory" error when
training try doing one of these:
1 Slow trainlm training, but reduce memory requirements by setting
net.trainParam.mem_reduc to 2 or more. (See help trainlm.)
2 Use trainbfg, which is slower but more memory-efficient than trainlm.
3 Use trainrp which is slower but more memory-efficient than trainbfg.
The learning function BLF can be either of the backpropagation learning
functions such as learngd, or learngdm.
The performance function can be any of the differentiable performance
functions such as mse or msereg.
Examples
Here is a series of Boolean inputs P, and another sequence T, which is 1
wherever P has had two 1's in a row.
P = round(rand(1,20));
T = [0 (P(1:end-1)+P(2:end) == 2)];
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newelm
We would like the network to recognize whenever two 1's occur in a row. First
we arrange these values as sequences.
Pseq = con2seq(P);
Tseq = con2seq(T);
Next we create an Elman network whose input varies from 0 to 1, and has five
hidden neurons and 1 output.
net = newelm([0 1],[10 1],{'tansig','logsig'});
Then we train the network with a mean squared error goal of 0.1, and simulate
it.
net = train(net,Pseq,Tseq);
Y = sim(net,Pseq)
Algorithm
Elman networks consist of Nl layers using the dotprod weight function, netsum
net input function, and the specified transfer functions.
The first layer has weights coming from the input. Each subsequent layer has
a weight coming from the previous layer. All layers except the last have a
recurrent weight. All layers have biases. The last layer is the network output.
Each layer's weights and biases are initialized with initnw.
Adaption is done with adaptwb which updates weights with the specified
learning function. Training is done with the specified training function.
Performance is measured according to the specified performance function.
See Also
newff, newcf, sim, init, adapt, train
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newff
Purpose
newff
Create a feed-forward backpropagation network
Syntax
net = newff(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF)
Description
newff(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) takes,
PR
- R x 2 matrix of min and max values for R input elements.
Si
- Size of ith layer, for Nl layers.
TFi - Transfer function of ith layer, default = 'tansig'.
BTF - Backprop network training function, default = 'traingdx'.
BLF - Backprop weight/bias learning function, default = 'learngdm'.
PF
- Performance function, default = 'mse'.
and returns an N layer feed-forward backprop network.
The transfer functions TFi can be any differentiable transfer function such as
tansig, logsig, or purelin.
The training function BTF can be any of the backprop training functions such
as trainlm, trainbfg, trainrp, traingd, etc.
WARNING: trainlm is the default training function because it is very fast, but
it requires a lot of memory to run. If you get an "out-of-memory" error when
training try doing one of these:
1 Slow trainlm training, but reduce memory requirements by setting
net.trainParam.mem_reduc to 2 or more. (See help trainlm.)
2 Use trainbfg, which is slower but more memory-efficient than trainlm.
3 Use trainrp which is slower but more memory-efficient than trainbfg.
The learning function BLF can be either of the backpropagation learning
functions such as learngd, or learngdm.
The performance function can be any of the differentiable performance
functions such as mse or msereg.
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5 6 7 8 9 10];
T = [0 1 2 3 4 3 2 1 2 3 4];
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newff
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has five tansig neurons, the second layer has one
purelin neuron. The trainlm network training function is to be used.
net = newff([0 10],[5 1],{'tansig' 'purelin'});
Here the network is simulated and its output plotted against the targets.
Y = sim(net,P);
plot(P,T,P,Y,'o')
Here the network is trained for 50 epochs. Again the network's output is
plotted.
net.trainParam.epochs = 50;
net = train(net,P,T);
Y = sim(net,P);
plot(P,T,P,Y,'o')
Algorithm
Feed-forward networks consist of Nl layers using the dotprod weight function,
netsum net input function, and the specified transfer functions.
The first layer has weights coming from the input. Each subsequent layer has
a weight coming from the previous layer. All layers have biases. The last layer
is the network output.
Each layer's weights and biases are initialized with initnw.
Adaption is done with adaptwb which updates weights with the specified
learning function. Training is done with the specified training function.
Performance is measured according to the specified performance function.
See Also
newcf, newelm, sim, init, adapt, train
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newfftd
Purpose
newfftd
Create a feed-forward input-delay backprop network
Syntax
net = newfftd(PR,ID,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF)
Description
newfftd(PR,ID,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) takes,
PR
- R x 2 matrix of min and max values for R input elements.
ID
- Input delay vector.
Si
- Size of ith layer, for Nl layers.
TFi - Transfer function of ith layer, default = 'tansig'.
BTF - Backprop network training function, default = 'traingdx'.
BLF - Backprop weight/bias learning function, default = 'learngdm'.
PF
- Performance function, default = 'mse'.
and returns an N layer feed-forward backprop network.
The transfer functions TFi can be any differentiable transfer function such as
tansig, logsig, or purelin.
The training function BTF can be any of the backprop training functions such
as trainlm, trainbfg, trainrp, traingd, etc.
WARNING: trainlm is the default training function because it is very fast, but
it requires a lot of memory to run. If you get an "out-of-memory" error when
training try doing one of these:
1 Slow trainlm training, but reduce memory requirements by setting
net.trainParam.mem_reduc to 2 or more. (See help trainlm.)
2 Use trainbfg, which is slower but more memory-efficient than trainlm.
3 Use trainrp which is slower but more memory-efficient than trainbfg.
The learning function BLF can be either of the backpropagation learning
functions such as learngd, or learngdm.
The performance function can be any of the differentiable performance
functions such as mse or msereg.
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newfftd
Examples
Here is a problem consisting of an input sequence P and target sequence T that
can be solved by a network with one delay.
P = {1 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1};
T = {1 -1 0 1 0 -1 1 -1 0 0 0 1 0 -1 0 1};
Here a two-layer feed-forward network is created with input delays of 0 and 1.
The network's input ranges from [0 to 1]. The first layer has five tansig
neurons, the second layer has one purelin neuron. The trainlm network
training function is to be used.
net = newfftd([0 1],[0 1],[5 1],{'tansig' 'purelin'});
Here the network is simulated.
Y = sim(net,P)
Here the network is trained for 50 epochs. Again the network's output is
calculated.
net.trainParam.epochs = 50;
net = train(net,P,T);
Y = sim(net,P)
Algorithm
Feed-forward networks consist of Nl layers using the dotprod weight function,
netsum net input function, and the specified transfer functions.
The first layer has weights coming from the input with the specified input
delays. Each subsequent layer has a weight coming from the previous layer. All
layers have biases. The last layer is the network output.
Each layer's weights and biases are initialized with initnw.
Adaption is done with adaptwb which updates weights with the specified
learning function. Training is done with the specified training function.
Performance is measured according to the specified performance function.
See Also
newcf, newelm, sim, init, adapt, train
13-137
newgrnn
Purpose
newgrnn
Design a generalized regression neural network
Syntax
net = newgrnn(P,T,spread)
Description
Generalized regression neural networks are a kind of radial basis network that
is often used for function approximation. grnn’s can be designed very quickly.
newgrnn(P,T,spread) takes three inputs,
P - R x Q matrix of Q input vectors.
T - S x Q matrix of Q target class vectors.
spread - Spread of radial basis functions, default = 1.0.
and returns a new generalized regression neural network.
The larger the spread, is the smoother the function approximation will be. To
fit data very closely, use a spread smaller than the typical distance between
input vectors. To fit the data more smoothly, use a larger spread.
Properties
newgrnn creates a two layer network. The first layer has radbas neurons,
calculates weighted inputs with dist and net input with netprod. The second
layer has purelin neurons, calculates weighted input with normprod and net
inputs with netsum. Only the first layer has biases.
newgrnn sets the first layer weights to P', and the first layer biases are all set
to 0.8326/spread, resulting in radial basis functions that cross 0.5 at weighted
inputs of +/- spread. The second layer weights W2 are set to T.
Examples
Here we design a radial basis network given inputs P and targets T.
P = [1 2 3];
T = [2.0 4.1 5.9];
net = newgrnn(P,T);
Here the network is simulated for a new input.
P = 1.5;
Y = sim(net,P)
See Also
sim, newrb, newrbe, newpnn
References
Wasserman, P.D., Advanced Methods in Neural Computing, New York: Van
Nostrand Reinhold, pp. 155-61, 1993.
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newhop
Purpose
newhop
Create a Hopfield recurrent network
Syntax
net = newhop(T)
Description
Hopfield networks are used for pattern recall.
newhop(T) takes one input argument,
T - R x Q matrix of Q target vectors. (Values must be +1 or -1.)
and returns a new Hopfield recurrent neural network with stable points at the
vectors in T.
Properties
Hopfield networks consist of a single layer with the dotprod weight function,
netsum net input function, and the satlins transfer function.
The layer has a recurrent weight from itself and a bias.
Examples
Here we create a Hopfield network with two three-element stable points T.
T = [-1 -1 1; 1 -1 1]';
net = newhop(T);
Below we check that the network is stable at these points by using them as
initial layer delay conditions. If the network is stable we would expect that the
outputs Y will be the same. (Since Hopfield networks have no inputs, the second
argument to sim is Q = 2 when using matrix notation).
Ai = T;
[Y,Pf,Af] = sim(net,2,[],Ai);
Y
To see if the network can correct a corrupted vector, run the following code
which simulates the Hopfield network for five timesteps. (Since Hopfield
networks have no inputs, the second argument to sim is {Q TS} = [1 5] when
using cell array notation.)
Ai = {[-0.9; -0.8; 0.7]};
[Y,Pf,Af] = sim(net,{1 5},{},Ai);
Y{1}
If you run the above code, Y{1} will equal T(:,1) if the network has managed
to convert the corrupted vector Ai to the nearest target vector.
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newhop
Algorithm
Hopfield networks are designed to have stable layer outputs as defined by user
supplied targets. The algorithm minimizes the number of unwanted stable
points.
See Also
sim, satlins
References
Li, J., A. N. Michel, and W. Porod, "Analysis and synthesis of a class of neural
networks: linear systems operating on a closed hypercube," IEEE Transactions
on Circuits and Systems, vol. 36, no. 11, pp. 1405-1422, November 1989.
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newlin
Purpose
Syntax
newlin
Create a linear layer
net = newlin(PR,S,ID,LR)
new = newlin
Description
Linear layers are often used as adaptive filters for signal processing and
prediction.
newlin(PR,S,ID,LR) takes these arguments,
PR - R x 2 matrix of min and max values for R input elements.
S
- Number of elements in the output vector.
ID - Input delay vector, default = [0].
LR - Learning rate, default = 0.01.
and returns a new linear layer.
net = newlin(PR,S,0,P) takes an alternate argument,
P
- Matrix of input vectors.
and returns a linear layer with the maximum stable learning rate for learning
with inputs P.
Call newlin without input arguments to define the network's attributes in a
dialog window.
Examples
This code creates a single input (range of [-1 1] linear layer with one neuron,
input delays of 0 and 1, and a learning rate of 0.01. It is simulated for an input
sequence P1.
net = newlin([-1 1],1,[0 1],0.01);
P1 = {0 -1 1 1 0 -1 1 0 0 1};
Y = sim(net,P1)
Here targets T1 are defined and the layer adapts to them. (Since this is the first
call to adapt, the default input delay conditions are used.)
T1 = {0 -1 0 2 1 -1 0 1 0 1};
[net,Y,E,Pf] = adapt(net,P1,T1); Y
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newlin
Here the linear layer continues to adapt for a new sequence using the previous
final conditions PF as initial conditions.
P2 = {1 0 -1 -1 1 1 1 0 -1};
T2 = {2 1 -1 -2 0 2 2 1 0};
[net,Y,E,Pf] = adapt(net,P2,T2); Y
Here we initialize the layer's weights and biases to new values.
net = init(net);
Here we train the newly initialized layer on the entire sequence for 200 epochs
to an error goal of 0.1.
P3 = [P1 P2];
T3 = [T1 T2];
net.trainParam.epochs = 200;
net.trainParam.goal = 0.1;
net = train(net,P3,T3);
Y = sim(net,[P1 P2])
Algorithm
Linear layers consist of a single layer with the dotprod weight function, netsum
net input function, and purelin transfer function.
The layer has a weight from the input and a bias.
Weights and biases are initialized with initzero.
Adaption and training are done with adaptwb and trainwb, which both update
weight and bias values with learnwh. Performance is measured with mse.
See Also
13-142
newlind, sim, init, adapt, train
newlind
Purpose
Syntax
newlind
Design a linear layer
net = newlind(P,T)
new = newlind
Description
newlind(P,T) takes two input arguments,
P - R x Q matrix of Q input vectors.
T - S x Q matrix of Q target class vectors.
and returns a linear layer designed to output T (with minimum sum square
error) given input P.
Call newlind without input arguments to define the network's attributes in a
dialog window.
Examples
We would like a linear layer that outputs T given P for the following definitions.
P = [1 2 3];
T = [2.0 4.1 5.9];
Here we use newlind to design such a network and check its response.
net = newlind(P,T);
Y = sim(net,P)
Algorithm
newlind calculates weight W and bias B values for a linear layer from inputs P
and targets T by solving this linear equation in the least squares sense:
[W b] * [P; ones] = T
See Also
sim, newlin
13-143
newlvq
Purpose
newlvq
Create a learning vector quantization network
Syntax
net = newlvq(PR,S1,PC,LR,LF)
Description
LVQ networks are used to solve classification problems.
net = newlvq(PR,S1,PC,LR,LF) takes these inputs,
PR - R x 2 matrix of min and max values for R input elements.
S1 - Number of hidden neurons.
PC - S2 element vector of typical class percentages.
LR - Learning rate, default = 0.01.
LF - Learning function, default = 'learnlv2'.
returns a new LVQ network.
The learning function LF can be learnlv1 or learnlv2.
Properties
newlvq creates a two layer network. The first layer uses the compet transfer
function, calculates weighted inputs with negdist, and net input with netsum.
The second layer has purelin neurons, calculates weighted input with dotprod
and net inputs with netsum. Neither layer has biases.
First layer weights are initialized with midpoint. The second layer weights are
set so that each output neuron i has unit weights coming to it from PC(i)
percent of the hidden neurons.
Adaption and training are done with adaptwb and trainwb1, which both update
the first layer weights with the specified learning functions.
Examples
The input vectors P and target classes Tc below define a classification problem
to be solved by an LVQ network.
P = [-3 -2 -2 0 0 0 0 +2 +2 +3; ...
0 +1 -1 +2 +1 -1 -2 +1 -1 0];
Tc = [1 1 1 2 2 2 2 1 1 1];
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newlvq
The target classes Tc are converted to target vectors T. Then, an LVQ network
is created (with inputs ranges obtained from P, 4 hidden neurons, and class
percentages of 0.6 and 0.4) and is trained.
T = ind2vec(Tc);
net = newlvq(minmax(P),4,[.6 .4]);
net = train(net,P,T);
The resulting network can be tested.
Y = sim(net,P)
Yc = vec2ind(Y)
See Also
sim, init, adapt, train, adaptwb, trainwb1, learnlv1, learnlv2
13-145
newp
Purpose
newp
Create a perceptron
Syntax
net = newp(pr,s,tf,lf)
Description
Perceptrons are used to solve simple (i.e. linearly separable) classification
problems.
net = newp(PR,S,TF,LF) takes these inputs,
PR - R x 2 matrix of min and max values for R input elements.
S
- Number of neurons.
TF - Transfer function, default = 'hardlim'.
LF - Learning function, default = 'learnp'.
and returns a new perceptron.
The transfer function TF can be hardlim or hardlims. The learning function LF
can be learnp or learnpn.
Call newp without input arguments to define the network's attributes in a
dialog window.
Properties
Perceptrons consist of a single layer with the dotprod weight function, the
netsum net input function, and the specified transfer function.
The layer has a weight from the input and a bias.
Weights and biases are initialized with initzero.
Adaption and training are done with adaptwb and trainwb, which both update
weight and bias values with the specified learning function. Performance is
measured with mae.
Examples
This code creates a perceptron layer with one 2-element input (ranges [0 1] and
[-2 2]) and one neuron. (Supplying only two arguments to newp results in the
default perceptron learning function learnp being used.)
net = newp([0 1; -2 2],1);
Here we simulate the network to a sequence of inputs P.
P1 = {[0; 0] [0; 1] [1; 0] [1; 1]};
Y = sim(net,P1)
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newp
Here we define a sequence of targets T (together P and T define the operation of
an AND gate), and then let the network adapt for 10 passes through the
sequence. We then simulate the updated network.
T1 = {0 0 0 1};
net.adaptParam.passes = 10;
net = adapt(net,P1,T1);
Y = sim(net,P1)
Now we define a new problem, an OR gate, with batch inputs P and targets T.
P2 = [0 0 1 1; 0 1 0 1];
T2 = [0 1 1 1];
Here we initialize the perceptron (resulting in new random weight and bias
values), simulate its output, train for a maximum of 20 epochs, and then
simulate it again.
net = init(net);
Y = sim(net,P2)
net.trainParam.epochs = 20;
net = train(net,P2,T2);
Y = sim(net,P2)
Notes
Perceptrons can classify linearly separable classes in a finite amount of time.
If input vectors have a large variance in their lengths, the learnpn can be
faster than learnp.
See Also
sim, init, adapt, train, hardlim, hardlims, learnp, learnpn
13-147
newpnn
Purpose
newpnn
Design a probabilistic neural network
Syntax
net = newpnn(P,T,spread)
Description
Probabilistic neural networks are a kind of radial basis network suitable for
classification problems.
net = newpnn(P,T,spread)takes two or three arguments,
P - R x Q matrix of Q input vectors.
T - S x Q matrix of Q target class vectors.
spread - Spread of radial basis functions, default = 0.1.
and returns a new probabilistic neural network.
If spread is near zero the network will act as a nearest neighbor classifier. As
spread becomes larger the designed network will take into account several
nearby design vectors.
Examples
Here a classification problem is defined with a set of inputs P and class indices
Tc.
P = [1 2 3 4 5 6 7];
Tc = [1 2 3 2 2 3 1];
Here the class indices are converted to target vectors, and a PNN is designed
and tested.
T = ind2vec(Tc)
net = newpnn(P,T);
Y = sim(net,P)
Yc = vec2ind(Y)
Algorithm
newpnn creates a two layer network. The first layer has radbas neurons, and
calculates its weighted inputs with dist, and its net input with netprod. The
second layer has compet neurons, and calculates its weighted input with
dotprod and its net inputs with netsum. Only the first layer has biases.
newpnn sets the first layer weights to P', and the first layer biases are all set to
0.8326/spread resulting in radial basis functions that cross 0.5 at weighted
inputs of +/- spread. The second layer weights W2 are set to T.
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newpnn
See Also
sim, ind2vec, vec2ind, newrb, newrbe, newgrnn
References
Wasserman, P.D., Advanced Methods in Neural Computing, New York: Van
Nostrand Reinhold, pp. 35-55, 1993.
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newrb
Purpose
newrb
Design a radial basis network
Syntax
net = newrb(P,T,goal,spread)
Description
Radial basis networks can be used to approximate functions. newrb adds
neurons to the hidden layer of a radial basis network until it meets the
specified mean squared error goal.
newrb(P,T,goal,spread) takes two to four arguments,
P - R x Q matrix of Q input vectors.
T - S x Q matrix of Q target class vectors.
goal - Mean squared error goal, default = 0.0.
spread - Spread of radial basis functions, default = 1.0.
and returns a new radial basis network.
The larger that spread is, the smoother the function approximation will be. Too
large a spread means a lot of neurons will be required to fit a fast changing
function. Too small a spread means many neurons will be required to fit a
smooth function, and the network may not generalize well. Call newrb with
different spreads to find the best value for a given problem.
Examples
Here we design a radial basis network given inputs P and targets T.
P = [1 2 3];
T = [2.0 4.1 5.9];
net = newrb(P,T);
Here the network is simulated for a new input.
P = 1.5;
Y = sim(net,P)
Algorithm
13-150
newrb creates a two layer network. The first layer has radbas neurons, and
calculates its weighted inputs with dist, and its net input with netprod. The
second layer has purelin neurons, and calculates its weighted input with
dotprod and its net inputs with netsum. Both layers have biases.
newrb
Initially the radbas layer has no neurons. The following steps are repeated
until the network's mean squared error falls below goal.
1 The network is simulated.
2 The input vector with the greatest error is found.
3 A radbas neuron is added with weights equal to that vector.
4 The purelin layer weights are redesigned to minimize error.
See Also
sim, newrbe, newgrnn, newpnn
13-151
newrbe
Purpose
newrbe
Design an exact radial basis network
Syntax
net = newrbe(P,T,spread)
Description
Radial basis networks can be used to approximate functions. newrbe very
quickly designs a radial basis network with zero error on the design vectors.
newrbe(P,T,spread) takes two or three arguments,
P - R x Q matrix of Q input vectors.
T - S x Q matrix of Q target class vectors.
spread - Spread of radial basis functions, default = 1.0.
and returns a new exact radial basis network.
The larger the spread is, the smoother the function approximation will be. Too
large a spread can cause numerical problems.
Examples
Here we design a radial basis network given inputs P and targets T.
P = [1 2 3];
T = [2.0 4.1 5.9];
net = newrbe(P,T);
Here the network is simulated for a new input.
P = 1.5;
Y = sim(net,P)
Algorithm
newrbe creates a two layer network. The first layer has radbas neurons, and
calculates its weighted inputs with dist, and its net input with netprod. The
second layer has purelin neurons, and calculates its weighted input with
dotprod and its net inputs with netsum. Both layers have biases.
newrbe sets the first layer weights to P', and the first layer biases are all set to
0.8326/spread, resulting in radial basis functions that cross 0.5 at weighted
inputs of +/- spread.
The second layer weights IW{2,1} and biases b{2} are found by simulating the
first layer outputs A{1}, and then solving the following linear expression:
[W{2,1} b{2}] * [A{1}; ones] = T
See Also
13-152
sim, newrb, newgrnn, newpnn
newsom
Purpose
newsom
Create a self-organizing map
Syntax
net = newsom(PR,[d1,d2,...],tfcn,dfcn,olr,osteps,tlr,tnd)
Description
Competitive layers are used to solve classification problems.
net = newsom (PR,[D1,D2,...],TFCN,DFCN,OLR,OSTEPS,TLR,TND) takes,
PR - R x 2 matrix of min and max values for R input elements.
I - Size of ith layer dimension, defaults = [5 8].
TFCN - Topology function, default ='hextop'.
DFCN - Distance function, default ='linkdist'.
OLR - Ordering phase learning rate, default = 0.9.
OSTEPS - Ordering phase steps, default = 1000.
TLR - Tuning phase learning rate, default = 0.02;
TND - Tuning phase neighborhood distance, default = 1.
and returns a new self-organizing map.
The topology function TFCN can be hextop, gridtop, or randtop. The distance
function can be linkdist, dist, or mandist.
Properties
Simms consist of a single layer with the negdist weight function, netsum net
input function, and the compet transfer function.
The layer has a weight from the input, but no bias. The weight is initialized
with midpoint.
Adaption and training are done with adaptwb and trainwb1, which both update
the weight with learnsom.
Examples
The input vectors defined below are distributed over an 2-dimension input
space varying over [0 2] and [0 1]. This data will be used to train a SOM with
dimensions [3 5].
P = [rand(1,400)*2; rand(1,400)];
net = newsom([0 2; 0 1],[3 5]);
plotsom(net.layers{1}.positions)
13-153
newsom
Here the SOM is trained and the input vectors are plotted with the map which
the SOM's weights have formed.
net = train(net,P);
plot(P(1,:),P(2,:),'.g','markersize',20)
hold on
plotsom(net.iw{1,1},net.layers{1}.distances)
hold off
See Also
13-154
sim, init, adapt, train, adaptwb, trainwb1
nncopy
Purpose
nncopy
Copy matrix or cell array
Syntax
nncopy(x,m,n)
Description
nncopy(X,M,N) takes two arguments,
X - R x C matrix (or cell array).
M - Number of vertical copies.
N - Number of horizontal copies.
and returns a new (R*M) x (C*N) matrix (or cell array).
Examples
x1
y1
x2
y2
=
=
=
=
[1 2 3; 4 5 6];
nncopy(x1,3,2)
{[1 2]; [3; 4; 5]}
nncopy(x2,2,3)
13-155
nnt2c
Purpose
nnt2c
Update NNT 2.0 competitive layer to NNT 3.0
Syntax
net = nnt2c(pr,w,klr,clr)
Description
nnt2c(PR,W,KLR,CLR) takes these arguments,
PR
- R x 2 matrix of min and max values for R input elements.
W
- S x R weight matrix.
KLR - Kohonen learning rate, default = 0.01.
CLR - Conscience learning rate, default = 0.001.
and returns a competitive layer.
Once a network has been updated it can be simulated, initialized, or trained
with sim, init, adapt, and train.
See Also
13-156
newc
nnt2elm
Purpose
nnt2elm
Update NNT 2.0 Elman backpropagation network to NNT 3.0
Syntax
net = nnt2elm(pr,w1,b1,w2,b2,btf,blf,pf)
Description
nnt2elm(PR,W1,B1,W2,B2,BTF,BLF,PF) takes these arguments,
PR - R x 2 matrix of min and max values for R input elements.
W1 - S1 x (R+S1) weight matrix.
B1 - S1 x 1 bias vector.
W2 - S2 x S1 weight matrix.
B2 - S2 x 1 bias vector.
BTF - Backprop network training function, default = 'traingdx'.
BLF - Backprop weight/bias learning function, default = 'learngdm'.
PF
- Performance function, default = 'mse'.
and returns a feed-forward network.
The training function BTF can be any of the backprop training functions such
as traingd, traingdm, traingda, and traingdx. Large step-size algorithms,
such as trainlm, are not recommended for Elman networks.
The learning function BLF can be either of the backpropagation learning
functions such as learngd or learngdm.
The performance function can be any of the differentiable performance
functions such as mse or msereg.
Once a network has been updated it can be simulated, initialized, adapted, or
trained with sim, init, adapt, and train.
See Also
newelm
13-157
nnt2ff
Purpose
nnt2ff
Update NNT 2.0 feed-forward network to NNT 3.0
Syntax
net = nnt2ff(pr,{w1 w2 ...},{b1 b2 ...},{tf1 tf2 ...},btf,blr,pf)
Description
nnt2ff(PR,{W1 W2 ...},{B1 B2 ...},{TF1 TF2 ...},BTF,BLR,PF) takes
these arguments,
PR
- R x 2 matrix of min and max values for R input elements.
Wi
- Weight matrix for the ith layer.
Bi
- Bias vector for the ith layer.
TFi - Transfer function of ith layer, default = 'tansig'.
BTF - Backprop network training function, default = 'traingdx'.
BLF - Backprop weight/bias learning function, default = 'learngdm'.
PF
- Performance function, default = 'mse'.
and returns a feed-forward network.
The training function BTF can be any of the backprop training functions such
as traingd, traingdm, traingda, traingdx or trainlm.
The learning function BLF can be either of the backpropagation learning
functions such as learngd or learngdm.
The performance function can be any of the differentiable performance
functions such as mse or msereg.
Once a network has been updated it can be simulated, initialized, adapted, or
trained with sim, init, adapt, and train.
See Also
13-158
newff, newcf, newfftd, newelm
nnt2hop
Purpose
nnt2hop
Update NNT 2.0 Hopfield recurrent network to NNT 3.0
Syntax
net = nnt2p(w,b)
Description
nnt2hop (W,B) takes these arguments,
W - S x S weight matrix.
B - S x 1 bias vector
and returns a perceptron.
Once a network has been updated it can be simulated, initialized, adapted, or
trained with sim, init, adapt, and train.
See Also
newhop
13-159
nnt2lin
Purpose
nnt2lin
Update NNT 2.0 linear layer to NNT 3.0
Syntax
net = nnt2lin(pr,w,b,lr)
Description
nnt2lin(PR,W,B) takes these arguments,
PR - R x 2 matrix of min and max values for R input elements.
W
- S x R weight matrix.
B
- S x 1 bias vector
LR - Learning rate, default = 0.01;
and returns a linear layer.
Once a network has been updated it can be simulated, initialized, adapted, or
trained with sim, init, adapt, and train.
See Also
13-160
newlin
nnt2lvq
Purpose
nnt2lvq
Update NNT 2.0 learning vector quantization network to NNT 3.0
Syntax
net = nnt2lvq(pr,w1,w2,lr,lf)
Description
nnt2lvq(PR,W1,W2,LR,LF) takes these arguments,
PR - R x 2 matrix of min and max values for R input elements.
W1 - S1 x R weight matrix.
W2 - S2 x S1 weight matrix.
LR - Learning rate, default = 0.01.
LF - Learning function, default = 'learnlv2'.
and returns a radial basis network.
The learning function LF can be learnlv1 or learnlv2.
Once a network has been updated it can be simulated, initialized, adapted, or
trained with sim, init, adapt, and train.
See Also
newlvq
13-161
nnt2p
Purpose
nnt2p
Update NNT 2.0 perceptron to NNT 3.0
Syntax
net = nnt2p(pr,w,b,tf,lf)
Description
nnt2p(PR,W,B,TF,LF) takes these arguments,
PR - R x 2 matrix of min and max values for R input elements.
W
- S x R weight matrix.
B
- S x 1 bias vector.
TF - Transfer function, default = 'hardlim'.
LF - Learning function, default = 'learnp'.
and returns a perceptron.
The transfer function TF can be hardlim or hardlims. The learning function LF
can be learnp or learnpn.
Once a network has been updated it can be simulated, initialized, adapted, or
trained with sim, init, adapt, and train.
See Also
13-162
newp
nnt2rb
Purpose
nnt2rb
Update NNT 2.0 radial basis network to NNT 3.0
Syntax
net = nnt2rb(pr,w1,b1,w2,b2)
Description
nnt2rb(PR,W1,B1,W2,B2) takes these arguments,
PR - R x 2 matrix of min and max values for R input elements.
W1 - S1 x R weight matrix.
B1 - S1 x 1 bias vector.
W2 - S2 x S1 weight matrix.
B2 - S2 x 1 bias vector.
and returns a radial basis network.
Once a network has been updated it can be simulated, initialized, adapted, or
trained with sim, init, adapt, and train.
See Also
newrb, newrbe, newgrnn, newpnn
13-163
nnt2som
Purpose
Syntax
Description
nnt2som
Update NNT 2.0 self-organizing map to NNT 3.0
net = nnt2som(pr,[d1 d2 ...],w,olr,osteps,tlr,tnd)
nnt2som(PR,[D1,D2,...],W,OLR,OSTEPS,TLR,TND) takes these arguments,
PR
- R x 2 matrix of min and max values for R input elements.
Di
- Size of ith layer dimension.
W
- S x R weight matrix.
OLR - Ordering phase learning rate, default = 0.9.
OSTEPS - Ordering phase steps, default = 1000.
TLR - Tuning phase learning rate, default = 0.02;
TND - Tuning phase neighborhood distance, default = 1.
and returns a self-organizing map.
nnt2som assumes that the self-organizing map has a grid topology (gridtop)
using link distances (linkdist). This corresponds with the nbman neighborhood
function in NNT 2.0.
The new network will only output 1 for the neuron with the greatest net input.
In NNT 2.0 the network would also output 0.5 for that neuron's neighbors.
Once a network has been updated it can be simulated, initialized, adapted, or
trained with sim, init, adapt, and train.
See Also
13-164
newsom
normc
Purpose
normc
Normalize the columns of a matrix
Syntax
normc(M)
Description
normc(M) normalizes the columns of M to a length of 1.
Examples
See Also
m = [1 2; 3 4];
normc(m)
ans =
0.3162
0.4472
0.9487
0.8944
normr
13-165
normprod
Purpose
Syntax
normprod
Normalized dot product weight function
Z = normprod(W,P)
df = normprod('deriv')
Description
normprod is a weight function. Weight functions apply weights to an input to
get weighted inputs.
normprod(W,P) takes these inputs,
W - S x R weight matrix.
P - R x Q matrix of Q input (column) vectors.
and returns the S x Q matrix of normalized dot products.
normprod('deriv') returns '' because normprod does not have a derivative
function.
Examples
Here we define a random weight matrix W and input vector P and calculate the
corresponding weighted input Z.
W = rand(4,3);
P = rand(3,1);
Z = normprod(W,P)
Network Use
You can create a standard network that uses normprod by calling newgrnn.
To change a network so an input weight uses normprod, set
net.inputWeight{i,j}.weightFcn to 'normprod’. For a layer weight set
net.inputWeight{i,j}.weightFcn to 'normprod’.
In either case call sim to simulate the network with normprod. See newgrnn for
simulation examples.
Algorithm
normprod returns the dot product normalized by the sum of the input vector
elements.
z = w*p/sum(p)
See Also
13-166
sim, dotprod, negdist, dist
normr
Purpose
normr
Normalize the rows of a matrix
Syntax
normr(M)
Description
normr(M) normalizes the columns of M to a length of 1.
Examples
See Also
m = [1 2; 3 4];
normr(m)
ans =
0.4472
0.6000
0.8944
0.8000
normc
13-167
plotep
Purpose
plotep
Plot a weight-bias position on an error surface
Syntax
h = plotep(w,b,e)
h = plotep(w,b,e,h)
Description
plotep is used to show network learning on a plot already created by plotes.
plotep(W,B,E) takes these arguments,
W - Current weight value.
B - Current bias value.
E - Current error.
and returns a vector H, containing information for continuing the plot.
plotep(W,B,E,H) continues plotting using the vector H returned by the last call
to plotep.
H contains handles to dots plotted on the error surface, so they can be deleted
next time, as well as points on the error contour, so they can be connected.
See Also
13-168
errsurf, plotes
plotes
Purpose
plotes
Plot the error surface of a single input neuron
Syntax
plotes(wv,bv,es,v)
Description
plotes(WV,BV,ES,V) takes these arguments,
WV - 1 x N row vector of values of W.
BV - 1 x M row vector of values of B.
ES - M x N matrix of error vectors.
V
- View, default = [-37.5, 30].
and plots the error surface with a contour underneath.
Calculate the error surface ES with errsurf.
Examples
See Also
p = [3 2];
t = [0.4 0.8];
wv = -4:0.4:4; bv = wv;
ES = errsurf(p,t,wv,bv,'logsig');
plotes(wv,bv,ES,[60 30])
errsurf
13-169
plotpc
Purpose
plotpc
Plot a classification line on a perceptron vector plot
Syntax
plotpc(W,b)
Description
plotpc(W,B) takes these inputs,
W - S x R weight matrix (R must be 3 or less).
B - S x 1 bias vector.
and returns a handle to a plotted classification line.
plotpc(W,B,H) takes these inputs,
H - Handle to last plotted line.
and deletes the last line before plotting the new one.
This function does not change the current axis and is intended to be called after
plotpv.
Examples
The code below defines and plots the inputs and targets for a perceptron:
p = [0 0 1 1; 0 1 0 1];
t = [0 0 0 1];
plotpv(p,t)
The following code creates a perceptron with inputs ranging over the values in
P, assigns values to its weights and biases, and plots the resulting classification
line.
net = newp(minmax(p),1);
net.iw{1,1} = [-1.2 -0.5];
net.b{1} = 1;
plotpc(net.iw{1,1},net.b{1})
See Also
13-170
plotpv
plotperf
Purpose
plotperf
Plot network performance
Syntax
plotperf(tr,goal,name,epoch)
Description
plotperf(TR,goal,name,epoch) takes these inputs,
TR - Training record returned by train.
goal - Performance goal, default = NaN.
name - Training function name, default = ''.
epoch - Number of epochs, default = length of training record.
and plots the training performance, and if available, the performance goal,
validation performance, and test performance.
Examples
Here are 8 input values P and associated targets T, plus a like number of
validation inputs VV.P and targets VV.T.
P = 1:8; T = sin(P);
VV.P = P; VV.T = T+rand(1,8)*0.1;
The code below creates a network and trains it on this problem.
net = newff(minmax(P),[4 1],{'tansig','tansig'});
[net,tr] = train(net,P,T,[],[],VV);
During training plotperf was called to display the training record. You can
also call plotperf directly with the final training record TR, as shown below.
plotperf(tr)
13-171
plotpv
Purpose
plotpv
Plot perceptron input/target vectors
Syntax
plotpv(p,t)
Description
plotpv(P,T) take these inputs,
P - R x Q matrix of input vectors (R must be 3 or less).
T - S x Q matrix of binary target vectors (S must be 3 or less).
and plots column vectors in P with markers based on T.
plotpv(P,T,V) takes an additional input,
V - Graph limits = [x_min x_max y_min y_max]
and plots the column vectors with limits set by V.
Examples
The code below defines and plots the inputs and targets for a perceptron:
p = [0 0 1 1; 0 1 0 1];
t = [0 0 0 1];
plotpv(p,t)
The following code creates a perceptron with inputs ranging over the values in
P, assigns values to its weights and biases, and plots the resulting classification
line.
net = newp(minmax(p),1);
net.iw{1,1} = [-1.2 -0.5];
net.b{1} = 1;
plotpc(net.iw{1,1},net.b{1})
See Also
13-172
plotpc
plotv
Purpose
plotv
Plot vectors as lines from the origin
Syntax
plotv(m,t)
Description
plotv(M,T) takes two inputs,
M - R x Q matrix of Q column vectors with R elements.
T - (optional) the line plotting type, default = '-'.
and plots the column vectors of M.
R must be 2 or greater. If R is greater than two, only the first two rows of M are
used for the plot.
Examples
plotv([-.4 0.7 .2; -0.5 .1 0.5],'-')
13-173
plotvec
Purpose
plotvec
Plot vectors with different colors
Syntax
plotvec(x,c,m)
Description
plotvec(X,C,M) takes these inputs,
X - Matrix of (column) vectors.
C - Row vector of color coordinate.
M - Marker, default = '+'.
and plots each ith vector in X with a marker M and using the ith value in C as
the color coordinate.
plotvec(X) only takes a matrix X and plots each ith vector in X with marker
'+' using the index i as the color coordinate.
Examples
13-174
x = [0 1 0.5 0.7; -1 2 0.5 0.1];
c = [1 2 3 4];
plotvec(x,c)
pnormc
Purpose
pnormc
Pseudo-normalize columns of a matrix
Syntax
pnormc(x,r)
Description
pnormc(M,R) takes these arguments,
X - M x N matrix.
R - (optional) radius to normalize columns to, default = 1.
and returns X with an additional row of elements, which results in new column
vector lengths of R.
WARNING: For this function to work properly, the columns of X must
originally have vector lengths less than R.
Examples
See Also
x = [0.1 0.6; 0.3 0.1];
y = pnormc(x)
normc, normr
13-175
poslin
Purpose
Graph and
Symbol
poslin
Positive linear transfer function
a
+1
AA
AA
n
0 1
-1
a = poslin(n)
Positive Linear Transfer Funct.
Syntax
A = poslin(N)
info = poslin(code)
Description
poslin is a transfer function. Transfer functions calculate a layer's output from
its net input.
poslin(N) takes one input,
N - S x Q matrix of net input (column) vectors.
and returns the maximum of 0 and each element of N.
poslin(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
Examples
Here is the code to create a plot of the poslin transfer function.
n = -5:0.1:5;
a = poslin(n);
plot(n,a)
13-176
poslin
Network Use
To change a network so that a layer uses poslin, set
net.layers{i}.transferFcn to 'poslin'.
Call sim to simulate the network with poslin.
Algorithm
poslin(n) = n, if n >= 0; = 0, if n <= 0.
See Also
sim, purelin, satlin, satlins
13-177
postmnmx
Purpose
Syntax
postmnmx
Postprocess data which has been preprocessed by premnmx
[p,t] = postmnmx(pn,minp,maxp,tn,mint,maxt)
[p] = postmnmx(pn,minp,maxp)
Description
postmnmx postprocesses the network training set which was preprocessed by
premnmx. It converts the data back into unnormalized units.
postmnmx takes these inputs,
PN
- R x Q matrix of normalized input vectors.
minp- R x 1 vector containing minimums for each P.
maxp- R x 1 vector containing maximums for each P.
TN
- S x Q matrix of normalized target vectors.
mint- S x 1 vector containing minimums for each T.
maxt- S x 1 vector containing maximums for each T.
and returns,
P - R x Q matrix of input (column) vectors.
T - R x Q matrix of target vectors.
Examples
In this example we normalize a set of training data with premnmx, create and
train a network using the normalized data, simulate the network, unnormalize
the output of the network using postmnmx, and perform a linear regression
between the network outputs (unnormalized) and the targets to check the
quality of the network training.
p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93];
t = [-0.08 3.4 -0.82 0.69 3.1];
[pn,minp,maxp,tn,mint,maxt] = premnmx(p,t);
net = newff(minmax(pn),[5 1],{'tansig' 'purelin'},'trainlm');
net = train(net,pn,tn);
an = sim(net,pn);
[a] = postmnmx(an,mint,maxt);
[m,b,r] = postreg(a,t);
Algorithm
See Also
13-178
p = 0.5(pn+1)*(maxp-minp) + minp;
premnmx, prepca, poststd
postreg
Purpose
postreg
Postprocess the trained network response with a linear regression
Syntax
[m,b,r] = postreg(A,T)
Description
postreg postprocesses the network training set by performing a linear
regression between each element of the network response and the
corresponding target.
postreg(A,T) takes these inputs,
A - 1 x Q array of network outputs. One element of the network output.
T - 1 x Q array of targets. One element of the target vector.
and returns,
M - Slope of the linear regression.
B - Y intercept of the linear regression.
R - Regression R-value. R=1 means perfect correlation.
Examples
In this example we normalize a set of training data with prestd, perform a
principal component transformation on the normalized data, create and train
a network using the pca data, simulate the network, unnormalize the output
of the network using poststd, and perform a linear regression between the
network outputs (unnormalized) and the targets to check the quality of the
network training.
p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93];
t = [-0.08 3.4 -0.82 0.69 3.1];
[pn,meanp,stdp,tn,meant,stdt] = prestd(p,t);
[ptrans,transMat] = prepca(pn,0.02);
net = newff(minmax(ptrans),[5 1],{'tansig''purelin'},'trainlm');
net = train(net,ptrans,tn);
an = sim(net,ptrans);
a = poststd(an,meant,stdt);
[m,b,r] = postreg(a,t);
Algorithm
Performs a linear regression between the network response and the target, and
then computes the correlation coefficient (R-value) between the network
response and the target.
See Also
premnmx, prepca
13-179
poststd
Purpose
Syntax
poststd
Postprocess data which has been preprocessed by prestd
[p,t] = poststd(pn,meanp,stdp,tn,meant,stdt)
[p] = poststd(pn,meanp,stdp)
Description
poststd postprocesses the network training set which was preprocessed by
prestd. It converts the data back into unnormalized units.
poststd takes these inputs,
PN
- R x Q matrix of normalized input vectors.
meanp - R x 1 vector containing standard deviations for each P.
stdp
- R x 1 vector containing standard deviations for each P.
TN
- S x Q matrix of normalized target vectors.
meant - S x 1 vector containing standard deviations for each T.
stdt
- S x 1 vector containing standard deviations for each T.
and returns,
P - R x Q matrix of input (column) vectors.
T - S x Q matrix of target vectors.
Examples
In this example we normalize a set of training data with prestd, create and
train a network using the normalized data, simulate the network, unnormalize
the output of the network using poststd, and perform a linear regression
between the network outputs (unnormalized) and the targets to check the
quality of the network training.
p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93];
t = [-0.08 3.4 -0.82 0.69 3.1];
[pn,meanp,stdp,tn,meant,stdt] = prestd(p,t);
net = newff(minmax(pn),[5 1],{'tansig' 'purelin'},'trainlm');
net = train(net,pn,tn);
an = sim(net,pn);
a = poststd(an,meant,stdt);
[m,b,r] = postreg(a,t);
Algorithm
See Also
13-180
p = stdp*pn + meanp;
premnmx, prepca, postmnmx, prestd
premnmx
Purpose
Syntax
premnmx
Preprocess data so that minimum is -1 and maximum is 1
[pn,minp,maxp,tn,mint,maxt] = premnmx(p,t)
[pn,minp,maxp] = premnmx(p)
Description
premnmx preprocesses the network training set by normalizing the inputs and
targets so that they fall in the interval [-1,1].
premnmx(P,T) takes these inputs,
P - R x Q matrix of input (column) vectors.
T - S x Q matrix of target vectors.
and returns,
PN
- R x Q matrix of normalized input vectors.
minp- R x 1 vector containing minimums for each P.
maxp- R x 1 vector containing maximums for each P.
TN
- S x Q matrix of normalized target vectors.
mint- S x 1 vector containing minimums for each T.
maxt- S x 1 vector containing maximums for each T.
Examples
Here is the code to normalize a given data set so that the inputs and targets
will fall in the range [-1,1].
p = [-10 -7.5 -5 -2.5 0 2.5 5 7.5 10];
t = [0 7.07 -10 -7.07 0 7.07 10 7.07 0];
[pn,minp,maxp,tn,mint,maxt] = premnmx(p,t);
If you just want to normalize the input,
[pn,minp,maxp] = premnmx(p);
Algorithm
See Also
pn = 2*(p-minp)/(maxp-minp) - 1;
prestd, prepca, postmnmx
13-181
prepca
Purpose
prepca
Principal component analysis
Syntax
[ptrans,transMat] = prepca(P,min_frac)
Description
prepca preprocesses the network input training set by applying a principal
component analysis. This analysis transforms the input data so that the
elements of the input vector set will be uncorrelated. In addition, the size of the
input vectors may be reduced by retaining only those components which
contribute more than a specified fraction (min_frac) of the total variation in
the data set.
prepca(p,min_frac) takes these inputs
P - R x Q matrix of centered input (column) vectors.
min_frac - Minimum fraction variance component to keep.
and returns
ptrans - Transformed data set.
transMat - Transformation matrix.
Examples
Here is the code to perform a principal component analysis and retain only
those components which contribute more than 2 percent to the variance in the
data set. prestd is called first to create zero mean data, which are needed for
prepca.
p=[-1.5 -0.58 0.21 -0.96 -0.79; -2.2 -0.87 0.31 -1.4
[pn,meanp,stdp] = prestd(p);
[ptrans,transMat] = prepca(pn,0.02);
-1.2];
Since the second row of p is almost a multiple of the first row, this example will
produce a transformed data set which contains only one row.
Algorithm
13-182
This routine uses singular value decomposition to compute the principal
components. The input vectors are multiplied by a matrix whose rows consist
of the eigenvectors of the input covariance matrix. This produces transformed
input vectors whose components are uncorrelated and ordered according to the
magnitude of their variance.
prepca
Those components which contribute only a small amount to the total variance
in the data set are eliminated. It is assumed that the input data set has already
been normalized so that it has a zero mean. The function prestd can be used
to normalize the data.
See Also
prestd, premnmx
References
Jolliffe, I.T.,Principal Component Analysis, New York: Springer-Verlag, 1986.
13-183
prestd
Purpose
Syntax
prestd
Preprocess data so that its mean is 0 and the standard deviation is 1
[pn,meanp,stdp,tn,meant,stdt] = prestd(p,t)
[pn,meanp,stdp] = prestd(p)
Description
prestd preprocesses the network training set by normalizing the inputs and
targets so that they have means of zero and standard deviations of 1.
prestd(p,t) takes these inputs,
p - R x Q matrix of input (column) vectors.
t - S x Q matrix of target vectors.
and returns,
pn
- R x Q matrix of normalized input vectors.
meanp - R x 1 vector containing standard deviations for each P.
stdp
- R x 1 vector containing standard deviations for each P.
tn
- S x Q matrix of normalized target vectors.
meant - S x 1 vector containing standard deviations for each T.
stdt
Examples
- S x 1 vector containing standard deviations for each T.
Here is the code to normalize a given data set so that the inputs and targets
will have means of zero and standard deviations of 1.
p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93];
t = [-0.08 3.4 -0.82 0.69 3.1];
[pn,meanp,stdp,tn,meant,stdt] = prestd(p,t);
If you just want to normalize the input,
[pn,meanp,stdp] = prestd(p);
Algorithm
See Also
13-184
pn = (p-meanp)/stdp;
premnmx, prepca
purelin
Purpose
purelin
Linear transfer function
Syntax
A = purelin(N)
info = purelin(code)
Description
purelin is a transfer function. Transfer functions calculate a layer's output
from its net input.
purelin(N) takes one input,
N - S x Q matrix of net input (column) vectors.
and returns N.
purelin(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
Examples
Here is the code to create a plot of the purelin transfer function.
n = -5:0.1:5;
a = purelin(n);
plot(n,a)
Network Use
You can create a standard network that uses purelin by calling newlin or
newlind.
To change a network so a layer uses purelin, set
net.layers{i}.transferFcn to 'purelin'.
In either case, call sim to simulate the network with purelin. See newlin or
newlind for simulation examples.
Algorithm
See Also
purelin(n) = n
sim, dpurelin, satlin, satlins
13-185
quant
Purpose
quant
Discretize values as multiples of a quantity
Syntax
quant(X,q)
Description
quant(X,q) takes two inputs,
X - Matrix, vector or scalar.
Q - Minimum value.
and returns values in X rounded to nearest multiple of Q.
Examples
13-186
x = [1.333 4.756 -3.897];
y = quant(x,0.1)
radbas
Purpose
radbas
Radial basis transfer function
Syntax
A = radbas(N)
info = radbas(code)
Description
radbas is a transfer function. Transfer functions calculate a layer's output from
its net input.
radbas(N) takes one input,
N - S x Q matrix of net input (column) vectors.
and returns each element of N passed through a radial basis function.
radbas(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
Examples
Here we create a plot of the radbas transfer function.
n = -5:0.1:5;
a = radbas(n);
plot(n,a)
Network Use
You can create a standard network that uses radbas by calling newpnn or
newgrnn.
To change a network so that a layer uses radbas, set
net.layers{i}.transferFcn to 'radbas'.
In either case, call sim to simulate the network with radbas. See newpnn or
newgrnn for simulation examples.
Algorithm
radbas(N) calculates its output with according to:
a = exp(-n2)
See Also
sim, tribas, dradbas
13-187
randnc
Purpose
randnc
Normalized column weight initialization function
Syntax
W = randnc(S,PR)
W = randnc(S,R)
Description
randnc is a weight initialization function.
randnc(S,P) takes two inputs,
S
- Number of rows (neurons).
PR - R x 2 matrix of input value ranges = [Pmin Pmax].
and returns an S x R random matrix with normalized columns.
Can also be called as randnc(S,R).
Examples
A random matrix of four normalized three-element columns is generated:
M = randnc(3,4)
M =
–0.6007
–0.4715
–0.7628
–0.6967
–0.2395
0.5406
See Also
13-188
randnr
–0.2724
–0.9172
–0.2907
0.5596
0.7819
0.2747
randnr
Purpose
randnr
Normalized row weight initialization function
Syntax
W = randnr(S,PR)
W = randnr(S,R)
Description
randnr is a weight initialization function.
randnr(S,P) takes two inputs,
S
- Number of rows (neurons).
PR - R x 2 matrix of input value ranges = [Pmin Pmax].
and returns an S x R random matrix with normalized rows.
Can also be called as randnr(S,R).
Examples
A matrix of three normalized four-element rows is generated:
M = randnr(3,4)
M =
0.9713
0.0800
0.8228
0.0338
–0.3042
–0.5725
See Also
–0.1838
0.1797
0.5436
–0.1282
0.5381
0.5331
randnc
13-189
rands
Purpose
rands
Symmetric random weight/bias initialization function
Syntax
W = rands(S,PR)
M = rands(S,R)
v = rands(S);
Description
rands is a weight/bias initialization function.
rands(S,PR) takes,
S
- Number of neurons.
PR - R x 2 matrix of R input ranges.
and returns an S-by-R weight matrix of random values between -1 and 1.
rands(S,R) returns an S-by-R matrix of random values. rands(S) returns an
S-by-1 vector of random values.
Examples
Here three sets of random values are generated with rands.
rands(4,[0 1; -2 2])
rands(4)
rands(2,3)
Network Use
To prepare the weights and the bias of layer i of a custom network to be
initialized with rands:
1 Set net.initFcn to 'initlay'. (net.initParam will automatically become
initlay's default parameters.)
2 Set net.layers{i}.initFcn to 'initwb'.
3 Set each net.inputWeights{i,j}.initFcn to 'rands'. Set each
net.layerWeights{i,j}.initFcn to 'rands'. Set each
net.biases{i}.initFcn to 'rands'.
To initialize the network call init.
See Also
13-190
randnr, randnc, initwb, initlay, init
randtop
Purpose
randtop
Random layer topology function
Syntax
pos = randtop(dim1,dim2,...,dimN)
Description
randtop calculates the neuron positions for layers whose neurons are arranged
in an N dimensional random pattern.
randtop(dim1,dim2,...,dimN)) takes N arguments,
dimi - Length of layer in dimension i.
and returns an N x S matrix of N coordinate vectors, where S is the product of
dim1*dim2*...*dimN.
Examples
This code creates and displays a two-dimensional layer with 192 neurons
arranged in a 16x12 random pattern.
pos = randtop(16,12); plotsom(pos)
This code plots the connections between the same neurons, but shows each
neuron at the location of its weight vector. The weights are generated randomly
so that the layer is very unorganized, as is evident in the plot.
W = rands(192,2); plotsom(W,dist(pos))
See Also
gridtop, hextop
13-191
satlin
Purpose
satlin
Saturating linear transfer function
Syntax
A = satlin(N)
info = satlin(code)
Description
satlin is a transfer function. Transfer functions calculate a layer's output from
its net input.
satlin(N) takes one input,
N - S x Q matrix of net input (column) vectors,
and returns values of N truncated into the interval [-1, 1].
satlin(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
Examples
Here is the code to create a plot of the satlin transfer function.
n = -5:0.1:5;
a = satlin(n);
plot(n,a)
Network Use
To change a network so that a layer uses satlin, set
net.layers{i}.transferFcn to 'satlin'.
Call sim to simulate the network with satlin. See newhop for simulation
examples.
Algorithm
satlin(n) = 0, if n <= 0; n, if 0 <= n <= 1; 1, if 1 <= n.
See Also
sim, poslin, satlins, purelin
13-192
satlins
Purpose
satlins
Symmetric saturating linear transfer function
Syntax
A = satlins(N)
info = satlins(code)
Description
satlins is a transfer function. Transfer functions calculate a layer's output
from its net input.
satlins(N) takes one input,
N - S x Q matrix of net input (column) vectors.
and returns values of N truncated into the interval [-1, 1].
satlins(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
Examples
Here is the code to create a plot of the satlins transfer function.
n = -5:0.1:5;
a = satlins(n);
plot(n,a)
Network Use
You can create a standard network that uses satlins by calling newhop.
To change a network so that a layer uses satlins, set
net.layers{i}.transferFcn to 'satlins'.
In either case, call sim to simulate the network with satlins. See newhop for
simulation examples.
Algorithm
satlins(n) = -1, if n <= -1; n, if -1 <= n <= 1; 1, if 1 <= n.
See Also
sim, satlin, poslin, purelin
13-193
seq2con
Purpose
seq2con
Converts sequential vectors to concurrent vectors
Syntax
b = seq2con(s)
Description
The neural network toolbox represents batches of vectors with a matrix, and
sequences of vectors with multiple columns of a cell array.
seq2con and con2seq allow concurrent vectors to be converted to sequential
vectors, and back again.
seq2con(S) takes one input,
S - N x TS cell array of matrices with M columns.
and returns,
B - N x 1 cell array of matrices with M*TS columns.
Examples
Here three sequential values are converted to concurrent values.
p1 = {1 4 2}
p2 = seq2con(p1)
Here two sequences of vectors over three time steps are converted to concurrent
vectors.
p1 = {[1; 1] [5; 4] [1; 2]; [3; 9] [4; 1] [9; 8]}
p2 = seq2con(p1)
See Also
13-194
con2seq, concur
sim
Purpose
Syntax
sim
Simulate a neural network
[Y,Pf,Af] = sim(net,P,Pi,Ai)
[Y,Pf,Af] = sim(net,{Q TS},Pi,Ai)
[Y,Pf,Af] = sim(net,Q,Pi,Ai)
To Get Help
Type help network/sim
Description
sim simulates neural networks.
[Y,Pf,Af] = sim(net,P,Pi,Ai) takes,
net - Network.
P
- Network inputs.
Pi
- Initial input delay conditions, default = zeros.
Ai
- Initial layer delay conditions, default = zeros.
and returns,
Y
- Network outputs.
Pf
- Final input delay conditions.
Af
- Final layer delay conditions.
Note that arguments Pi, Ai, Pf, and Af are optional and need only be used for
networks that have input or layer delays.
sim's signal arguments can have two formats: cell array or matrix.
The cell array format is easiest to describe. It is most convenient for networks
with multiple inputs and outputs, and allows sequences of inputs to be
presented:
P
- Ni x TS cell array, each element P{i,ts} is an Ri x Q matrix.
Pi - Ni x ID cell array, each element Pi{i,k} is an Ri x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
Y
- NO x TS cell array, each element Y{i,ts} is a Ui x Q matrix.
Pf - Ni x ID cell array, each element Pf{i,k} is an Ri x Q matrix.
Af - Nl x LD cell array, each element Af{i,k} is an Si x Q matrix.
13-195
sim
where
Ni = net.numInputs
Nl = net.numLayers,
No = net.numOutputs
D
= net.numInputDelays
LD = net.numLayerDelays
TS = Number of time steps
Q
= Batch size
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Ui = net.outputs{i}.size
The columns of Pi, Ai, Pf, and Af are ordered from oldest delay condition to
most recent:
Pi{i,k} = input i at time ts=k-ID.
Pf{i,k} = input i at time ts=TS+k-ID.
Ai{i,k} = layer output i at time ts=k-LD.
Af{i,k} = layer output i at time ts=TS+k-LD.
The matrix format can be used if only one time step is to be simulated (TS =
1). It is convenient for networks with only one input and output, but can also
be used with networks that have more.
Each matrix argument is found by storing the elements of the corresponding
cell array argument into a single matrix:
P
- (sum of Ri) x Q matrix
Pi - (sum of Ri) x (ID*Q) matrix.
Ai - (sum of Si) x (LD*Q) matrix.
Y
- (sum of Ui) x Q matrix.
Pf - (sum of Ri) x (ID*Q) matrix.
Af - (sum of Si) x (LD*Q) matrix.
13-196
sim
[Y,Pf,Af] = sim(net,{Q TS},Pi,Ai) is used for networks which do not have
an input, such as Hopfield networks, when cell array notation is used.
[Y,Pf,Af] = sim(net,Q,Pi,Ai) is used for networks which do not have an
input, such as Hopfield networks, when matrix notation is used.
Examples
Here newp is used to create a perceptron layer with a 2-element input (with
ranges of [0 1]), and a single neuron.
net = newp([0 1;0 1],1);
Here the perceptron is simulated for an individual vector, a batch of 3 vectors,
and a sequence of 3 vectors.
p1 = [.2; .9]; a1 = sim(net,p1)
p2 = [.2 .5 .1; .9 .3 .7]; a2 = sim(net,p2)
p3 = {[.2; .9] [.5; .3] [.1; .7]}; a3 = sim(net,p3)
Here newlind is used to create a linear layer with a 3-element input, 2 neurons.
net = newlin([0 2;0 2;0 2],2,[0 1]);
Here the linear layer is simulated with a sequence of 2 input vectors using the
default initial input delay conditions (all zeros).
p1 = {[2; 0.5; 1] [1; 1.2; 0.1]};
[y1,pf] = sim(net,p1)
Here the layer is simulated for 3 more vectors using the previous final input
delay conditions as the new initial delay conditions.
p2 = {[0.5; 0.6; 1.8] [1.3; 1.6; 1.1] [0.2; 0.1; 0]};
[y2,pf] = sim(net,p2,pf)
Here newelm is used to create an Elman network with a 1-element input, and
a layer 1 with 3 tansig neurons followed by a layer 2 with 2 purelin neurons.
Because it is an Elman network it has a tap delay line with a delay of 1 going
from layer 1 to layer 1.
net = newelm([0 1],[3 2],{'tansig','purelin'});
13-197
sim
Here the Elman network is simulated for a sequence of 3 values using default
initial delay conditions.
p1 = {0.2 0.7 0.1};
[y1,pf,af] = sim(net,p1)
Here the network is simulated for 4 more values, using the previous final delay
conditions as the new initial delay conditions.
p2 = {0.1 0.9 0.8 0.4};
[y2,pf,af] = sim(net,p2,pf,af)
Algorithm
sim uses these properties to simulate a network net.
net.numInputs, net.numLayers
net.outputConnect, net.biasConnect
net.inputConnect, net.layerConnect
These properties determine the network's weight and bias values, and the
number of delays associated with each weight:
net.inputWeights{i,j}.value
net.layerWeights{i,j}.value
net.layers{i}.value
net.inputWeights{i,j}.delays
net.layerWeights{i,j}.delays
These function properties indicate how sim applies weight and bias values to
inputs to get each layer's output:
net.inputWeights{i,j}.weightFcn
net.layerWeights{i,j}.weightFcn
net.layers{i}.netInputFcn
net.layers{i}.transferFcn
See Chapter 2 for more information on network simulation.
See Also
13-198
init, adapt, train
softmax
Purpose
Syntax
softmax
Soft max transfer function
A = softmax(N)
info = softmax(code)
Description
softmax is a transfer function. Transfer functions calculate a layer's output
from its net input.
softmax(N) takes one input argument,
N - S x Q matrix of net input (column) vectors.
and returns output vectors with elements between 0 and 1, but with their size
relations intact.
softmax('code') returns information about this function.
These codes are defined:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
compet does not have a derivative function.
Examples
Here we define a net input vector N, calculate the output, and plot both with
bar graphs.
n = [0; 1; -0.5; 0.5];
a = softmax(n);
subplot(2,1,1), bar(n), ylabel('n')
subplot(2,1,2), bar(a), ylabel('a')
Network Use
To change a network so that a layer uses softmax, set
net.layers{i,j}.transferFcn to 'softmax'.
Call sim to simulate the network with softmax. See newc or newpnn for
simulation examples.
See Also
sim, compet
13-199
srchbac
Purpose
srchbac
One-dimensional minimization using backtracking
Syntax
[a,gX,perf,retcode,delta,tol] =
srchbac(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
Description
srchbac is a linear search routine. It searches in a given direction to locate the
minimum of the performance function in that direction. It uses a technique
called backtracking.
srchbac(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,TOL,ch_perf)
takes these inputs,
net - Neural network.
X
- Vector containing current values of weights and biases.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
dX
- Search direction vector.
gX
- Gradient vector.
perf - Performance value at current X.
dperf - Slope of performance value at current X in direction of dX.
delta - Initial step size.
tol - Tolerance on search.
ch_perf - Change in performance on previous step.
and returns,
A - Step size which minimizes performance.
gX - Gradient at new minimum point.
perf - Performance value at new minimum point.
retcode - Return code which has three elements. The first two elements
correspond to the number of function evaluations in the two stages of the
search. The third element is a return code. These will have different
13-200
srchbac
meanings for different search algorithms. Some may not be used in this
function.
0 - normal; 1 - minimum step taken;
2 - maximum step taken; 3 - beta condition not met.
delta - New initial step size. Based on the current step size.
tol - New tolerance on search.
Parameters used for the backstepping algorithm are:
alpha
- Scale factor which determines sufficient reduction in perf.
beta
- Scale factor which determines sufficiently large step size.
low_lim - Lower limit on change in step size.
up_lim
- Upper limit on change in step size.
maxstep - Maximum step length.
minstep - Minimum step length.
scale_tol - Parameter which relates the tolerance tol to the initial step
size delta. Usually set to 20.
The defaults for these parameters are set in the training function which calls
it. See traincgf, traincgb, traincgp, trainbfg, trainoss.
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
13-201
srchbac
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The traincgf network training function and the
srchbac search function are to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf');
a = sim(net,p)
Train and Retest the Network
net.trainParam.searchFcn = 'srchbac';
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
Network Use
You can create a standard network that uses srchbac with newff, newcf, or
newelm.
To prepare a custom network to be trained with traincgf, using the line
search function srchbac:
1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's
default parameters.
2 Set net.trainParam.searchFcn to 'srchbac'.
The srchbac function can be used with any of the following training functions:
traincgf, traincgb, traincgp, trainbfg, trainoss.
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srchbac
Algorithm
srchbac locates the minimum of the performance function in the search
direction dX, using the backtracking algorithm described on page 126 and 328
of Dennis and Schnabel. (Numerical Methods for Unconstrained Optimization
and Nonlinear Equations 1983).
See Also
srchbrc, srchcha, srchgol, srchhyb
References
Dennis, J. E., and R. B. Schnabel, Numerical Methods for Unconstrained
Optimization and Nonlinear Equations, Englewood Cliffs, NJ: Prentice-Hall,
1983.
13-203
srchbre
Purpose
srchbre
One-dimensional interval location using Brent's method
Syntax
[a,gX,perf,retcode,delta,tol] =
srchbre(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
Description
srchbre is a linear search routine. It searches in a given direction to locate the
minimum of the performance function in that direction. It uses a technique
called Brent’s technique.
srchbre(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
takes these inputs,
net - Neural network.
X
- Vector containing current values of weights and biases.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
dX
- Search direction vector.
gX
- Gradient vector.
perf - Performance value at current X.
dperf - Slope of performance value at current X in direction of dX.
delta - Initial step size.
tol - Tolerance on search.
ch_perf - Change in performance on previous step.
and returns,
A - Step size which minimizes performance.
gX - Gradient at new minimum point.
perf - Performance value at new minimum point.
retcode - Return code, which has three elements. The first two elements
correspond to the number of function evaluations in the two stages of the
search. The third element is a return code. These will have different
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srchbre
meanings for different search algorithms. Some may not be used in this
function.
0 - normal; 1 - minimum step taken;
2 - maximum step taken; 3 - beta condition not met.
delta - New initial step size. Based on the current step size.
tol - New tolerance on search.
Parameters used for the brent algorithm are:
alpha - Scale factor which determines sufficient reduction in perf.
beta
- Scale factor which determines sufficiently large step size.
bmax
- Largest step size.
scale_tol - Parameter which relates the tolerance tol to the initial step
size delta. Usually set to 20.
The defaults for these parameters are set in the training function which calls
it. See traincgf, traincgb, traincgp, trainbfg, trainoss.
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
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srchbre
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The traincgf network training function and the
srchbac search function are to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf');
a = sim(net,p)
Train and Retest the Network
net.trainParam.searchFcn = 'srchbre';
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
Network Use
You can create a standard network that uses srchbre with newff, newcf, or
newelm.
To prepare a custom network to be trained with traincgf, using the line
search function srchbre:
1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's
default parameters.
2 Set net.trainParam.searchFcn to 'srchbre'.
The srchbre function can be used with any of the following training functions:
traincgf, traincgb, traincgp, trainbfg, trainoss.
Algorithm
srchbre brackets the minimum of the performance function in the search
direction dX, using Brent's algorithm described on page 46 of Scales
(Introduction to Non-Linear Estimation 1985). It is a hybrid algorithm based
on the golden section search and the quadratic approximation.
See Also
srchbac, srchcha, srchgol, srchhyb
References
Scales, L. E., Introduction to Non-Linear Optimization, New York:
Springer-Verlag, 1985.
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srchcha
Purpose
srchcha
One-dimensional minimization using the method of Charalambous
Syntax
[a,gX,perf,retcode,delta,tol] =
srchcha(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
Description
srchcha is a linear search routine. It searches in a given direction to locate the
minimum of the performance function in that direction. It uses a technique
based on the method of Charalambous.
srchcha(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
takes these inputs,
net - Neural network.
X
- Vector containing current values of weights and biases.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
dX
- Search direction vector.
gX
- Gradient vector.
perf - Performance value at current X.
dperf - Slope of performance value at current X in direction of dX.
delta - Initial step size.
tol - Tolerance on search.
ch_perf - Change in performance on previous step.
and returns,
A - Step size which minimizes performance.
gX - Gradient at new minimum point.
perf - Performance value at new minimum point.
retcode - Return code, which has three elements. The first two elements
correspond to the number of function evaluations in the two stages of the
search. The third element is a return code. These will have different
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srchcha
meanings for different search algorithms. Some may not be used in this
function.
0 - normal; 1 - minimum step taken;
2 - maximum step taken; 3 - beta condition not met.
delta - New initial step size. Based on the current step size.
TOL - New tolerance on search.
Parameters used for the Charalambous algorithm are:
alpha - Scale factor which determines sufficient reduction in perf.
beta
- Scale factor which determines sufficiently large step size.
gama
- Parameter to avoid small reductions in performance. Usually set to 0.1.
scale_tol - Parameter which relates the tolerance tol to the initial step
size delta. Usually set to 20.
The defaults for these parameters are set in the training function which calls
it. See traincgf, traincgb, traincgp, trainbfg, trainoss.
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
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srchcha
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The traincgf network training function and the
srchcha search function are to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf');
a = sim(net,p)
Train and Retest the Network
net.trainParam.searchFcn = 'srchcha';
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
Network Use
You can create a standard network that uses srchcha with newff, newcf, or
newelm.
To prepare a custom network to be trained with traincgf, using the line search
function srchcha:
1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's
default parameters.
2 Set net.trainParam.searchFcn to 'srchcha'.
The srchcha function can be used with any of the following training functions:
traincgf, traincgb, traincgp, trainbfg, trainoss.
Algorithm
srchcha locates the minimum of the performance function in the search
direction dX, using an algorithm based on the method described in
Charalambous (IEEE Proc. vol. 139, no. 3, June 1992).
See Also
srchbac, srchbre, srchgol, srchhyb
References
Charalambous, C.,“Conjugate gradient algorithm for efficient training of
artificial neural networks,” IEEE Proceedings, vol. 139, no. 3, pp. 301–310,
1992.
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srchgol
Purpose
srchgol
One-dimensional minimization using golden section search
Syntax
[a,gX,perf,retcode,delta,tol] =
srchgol(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
Description
srchgol is a linear search routine. It searches in a given direction to locate the
minimum of the performance function in that direction. It uses a technique
called the golden section search.
srchgol(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
takes these inputs,
net - Neural network.
X
- Vector containing current values of weights and biases.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
dX
- Search direction vector.
gX
- Gradient vector.
perf - Performance value at current X.
dperf - Slope of performance value at current X in direction of dX.
delta - Initial step size.
tol - Tolerance on search.
ch_perf - Change in performance on previous step.
and returns,
A - Step size which minimizes performance.
gX - Gradient at new minimum point.
perf - Performance value at new minimum point.
retcode - Return code, which has three elements. The first two elements
correspond to the number of function evaluations in the two stages of the
search. The third element is a return code. These will have different
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srchgol
meanings for different search algorithms. Some may not be used in this
function.
0 - normal; 1 - minimum step taken;
2 - maximum step taken; 3 - beta condition not met.
delta - New initial step size. Based on the current step size.
tol - New tolerance on search.
Parameters used for the golden section algorithm are:
alpha - Scale factor which determines sufficient reduction in perf.
bmax
- Largest step size.
scale_tol - Parameter which relates the tolerance tol to the initial step
size delta. Usually set to 20.
The defaults for these parameters are set in the training function which calls
it. See traincgf, traincgb, traincgp, trainbfg, trainoss.
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
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srchgol
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The traincgf network training function and the
srchgol search function are to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf');
a = sim(net,p)
Train and Retest the Network
net.trainParam.searchFcn = 'srchgol';
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
Network Use
You can create a standard network that uses srchgol with newff, newcf, or
newelm.
To prepare a custom network to be trained with traincgf, using the line search
function srchgol:
1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's
default parameters.
2 Set net.trainParam.searchFcn to 'srchgol'.
The srchgol function can be used with any of the following training functions:
traincgf, traincgb, traincgp, trainbfg, trainoss.
Algorithm
srchgol locates the minimum of the performance function in the search
direction dX, using the golden section search. It is based on the algorithm as
described on page 33 of Scales (Introduction to Non-Linear Estimation 1985).
See Also
srchbac, srchbre, srchcha, srchhyb
References
Scales, L. E.,Introduction to Non-Linear Optimization, New York:
Springer-Verlag, 1985.
13-212
srchhyb
Purpose
srchhyb
One-dimensional minimization using a hybrid bisection-cubic search
Syntax
[a,gX,perf,retcode,delta,tol] =
srchhyb(net,X,P,T,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
Description
srchhyb is a linear search routine. It searches in a given direction to locate the
minimum of the performance function in that direction. It uses a technique
that is a combination of a bisection and a cubic interpolation.
srchhyb(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
takes these inputs,
net - Neural network.
X
- Vector containing current values of weights and biases.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
dX
- Search direction vector.
gX
- Gradient vector.
perf - Performance value at current X.
dperf - Slope of performance value at current X in direction of dX.
delta - Initial step size.
tol - Tolerance on search.
ch_perf - Change in performance on previous step.
and returns,
A - Step size which minimizes performance.
gX - Gradient at new minimum point.
perf - Performance value at new minimum point.
retcode - Return code, which has three elements. The first two elements
correspond to the number of function evaluations in the two stages of the
search. The third element is a return code. These will have different
13-213
srchhyb
meanings for different search algorithms. Some may not be used in this
function.
0 - normal; 1 - minimum step taken;
2 - maximum step taken; 3 - beta condition not met.
delta - New initial step size. Based on the current step size.
tol - New tolerance on search.
Parameters used for the hybrid bisection-cubic algorithm are:
alpha - Scale factor which determines sufficient reduction in perf.
beta
- Scale factor which determines sufficiently large step size.
bmax
- Largest step size.
scale_tol - Parameter which relates the tolerance tol to the initial step
size delta. Usually set to 20.
The defaults for these parameters are set in the training function which calls
it. See traincgf, traincgb, traincgp, trainbfg, trainoss.
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
13-214
srchhyb
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The traincgf network training function and the
srchhyb search function are to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf');
a = sim(net,p)
Train and Retest the Network
net.trainParam.searchFcn = 'srchhyb';
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
Network Use
You can create a standard network that uses srchhyb with newff, newcf, or
newelm.
To prepare a custom network to be trained with traincgf, using the line search
function srchhyb:
1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's
default parameters.
2 Set net.trainParam.searchFcn to 'srchhyb'.
The srchhyb function can be used with any of the following training functions:
traincgf, traincgb, traincgp, trainbfg, trainoss.
Algorithm
srchhyb locates the minimum of the performance function in the search
direction dX, using the hybrid bisection-cubic interpolation algorithm described
on page 50 of Scales (Introduction to Non-Linear Estimation 1985).
See Also
srchbac, srchbre, srchcha, srchgol
References
Scales, L. E., Introduction to Non-Linear Optimization, New York:
Springer-Verlag, 1985.
13-215
sse
Purpose
Syntax
sse
Sum squared error performance function
perf = sse(e,x,pp)
perf = sse(e,net,pp)
info = sse(code)
Description
sse is a network performance function. It measures performance according to
the sum of squared errors.
sse(E,X,PP) takes from one to three arguments,
E
- Matrix or cell array of error vector(s).
X
- Vector of all weight and bias values (ignored).
PP - Performance parameters (ignored).
and returns the sum squared error.
sse(E,net,PP) can take an alternate argument to X,
net - Neural network from which X can be obtained (ignored).
sse(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here a two layer feed-forward is created with a 1-element input ranging from
-10 to 10, four hidden tansig neurons, and one purelin output neuron.
net = newff([-10 10],[4 1],{'tansig','purelin'});
Here the network is given a batch of inputs P. The error is calculated by
subtracting the output A from target T. Then the sum squared error is
calculated.
13-216
sse
p = [-10 -5 0 5 10];
t = [0 0 1 1 1];
y = sim(net,p)
e = t-y
perf = sse(e)
Note that sse can be called with only one argument because the other
arguments are ignored. sse supports those arguments to conform to the
standard performance function argument list.
Network Use
To prepare a custom network to be trained with sse set net.performFcn to
'sse'. This will automatically set net.performParam to the empty matrix [], as
sse has no performance parameters.
Calling train or adapt will result in sse being used to calculate performance.
See Also
dsse
13-217
sumsqr
Purpose
sumsqr
Sum squared elements of a matrix
Syntax
sumsqr(m)
Description
sumsqr(M) returns the sum of the squared elements in M.
Examples
13-218
s = sumsqr([1 2;3 4])
tansig
Purpose
Graph and
Symbol
tansig
Hyperbolic tangent sigmoid transfer function
a
+1
0
n
-1
a = tansig(n)
Tan-Sigmoid Transfer Function
Syntax
A = tansig(N)
info = tansig(code)
Description
tansig is a transfer function. Transfer functions calculate a layer's output from
its net input.
tansig(N) takes one input,
N - S x Q matrix of net input (column) vectors.
and returns each element of N squashed between -1 and 1.
tansig(code) return useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
tansig is named after the hyperbolic tangent which has the same shape.
However, tanh may be more accurate and is recommended for applications that
require the hyperbolic tangent.
Examples
Here is the code to create a plot of the tansig transfer function.
n = -5:0.1:5;
a = tansig(n);
plot(n,a)
13-219
tansig
Network Use
You can create a standard network that uses tansig by calling newff or newcf.
To change a network so a layer uses tansig set
net.layers{i,j}.transferFcn to 'tansig'.
In either case, call sim to simulate the network with tansig. See newff or
newcf for simulation examples.
Algorithm
tansig(N) calculates its output according to:
n = 2/(1+exp(-2*n))-1
This is mathematically equivalent to tanh(N). It differs in that it runs faster
than the MATLAB implementation of tanh, but the results can have very small
numerical differences. This function is a good trade off for neural networks,
where speed is important and the exact shape of the transfer function is not.
See Also
sim, dtansig, logsig
References
Vogl, T. P., J.K. Mangis, A.K. Rigler, W.T. Zink, and D.L. Alkon, “Accelerating
the convergence of the backpropagation method,” Biological Cybernetics, vol.
59, pp. 257-263, 1988.
13-220
train
Purpose
Syntax
train
Train a neural network
[net,tr] = train(NET,P,T,Pi,Ai)
[net,tr] = train(NET,P,T,Pi,Ai,VV,TV)
To Get Help
Type help network/train
Description
train trains a network net according to net.trainFcn and net.trainParam.
train(net,P,T,Pi,Ai) takes,
net - Network.
P
- Network inputs.
T
- Network targets, default = zeros.
Pi
- Initial input delay conditions, default = zeros.
Ai
- Initial layer delay conditions, default = zeros.
and returns,
net - New network.
TR
- Training record (epoch and perf).
Note that T is optional and need only be used for networks that require targets.
Pi and Pf are also optional and need only be used for networks that have input
or layer delays.
train's signal arguments can have two formats: cell array or matrix.
The cell array format is easiest to describe. It is most convenient for networks
with multiple inputs and outputs, and allows sequences of inputs to be
presented:
P
- Ni x TS cell array, each element P{i,ts} is an Ri x Q matrix.
T
- Nt x TS cell array, each element P{i,ts} is an Vi x Q matrix.
Pi - Ni x ID cell array, each element Pi{i,k} is an Ri x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
13-221
train
where
Ni = net.numInputs
Nl = net.numLayers
Nt = net.numTargets
ID = net.numInputDelays
LD = net.numLayerDelays
TS = Number of time steps
Q
= Batch size
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
The columns of Pi, Pf, Ai, and Af are ordered from the oldest delay condition
to the most recent:
Pi{i,k} = input i at time ts=k-ID.
Pf{i,k} = input i at time ts=TS+k-ID.
Ai{i,k} = layer output i at time ts=k-LD.
Af{i,k} = layer output i at time ts=TS+k-LD.
The matrix format can be used if only one time step is to be simulated (TS = 1).
It is convenient for network's with only one input and output, but can be used
with networks that have more.
Each matrix argument is found by storing the elements of the corresponding
cell array argument into a single matrix:
P
- (sum of Ri) x Q matrix
T
- (sum of Vi) x Q matrix
Pi - (sum of Ri) x (ID*Q) matrix.
Ai - (sum of Si) x (LD*Q) matrix.
13-222
train
train(net,P,T,Pi,Ai,VV,TV) takes optional structures of validation and test
vectors,
VV.P,
TV.P - Validation/test inputs.
VV.T,
VV.T
- Validation/test targets, default = zeros.
VV.Pi, VV.Pi - Validation/test initial input delay conditions, default= zeros.
VV.Ai, VV.Ai - Validation/test layer delay conditions, default = zeros.
The validation vectors are used to stop training early if further training on the
primary vectors will hurt generalization to the validation vectors. Test vector
performance can be used to measure how well the network generalizes beyond
primary and validation vectors. If VV.T, VV.Pi, or VV.Ai are set to an empty
matrix or cell array, default values will be used. The same is true for VT.T,
VT.Pi, VT.Ai.
Examples
Here input P and targets T define a simple function which we can plot:
p = [0 1 2 3 4 5 6 7 8];
t = [0 0.84 0.91 0.14 -0.77 -0.96 -0.28 0.66 0.99];
plot(p,t,'o')
Here newff is used to create a two layer feed forward network. The network
will have an input (ranging from 0 to 8), followed by a layer of 10 tansig
neurons, followed by a layer with 1 purelin neuron. trainlm backpropagation
is used. The network is also simulated.
net = newff([0 8],[10 1],{'tansig' 'purelin'},'trainlm');
y1 = sim(net,p)
plot(p,t,'o',p,y1,'x')
Here the network is trained for up to 50 epochs to a error goal of 0.01, and then
resimulated.
net.trainParam.epochs = 50;
net.trainParam.goal = 0.01;
net = train(net,p,t);
y2 = sim(net,p)
plot(p,t,'o',p,y1,'x',p,y2,'*')
Algorithm
train calls the function indicated by net.trainFcn, using the adaption
parameter values indicated by net.trainParam.
13-223
train
Typically one epoch of training is defined as a single presentation of all input
vectors to the network. The network is then updated according to the results of
all those presentations.
Training occurs until a maximum number of epochs occurs, the performance
goal is met, or any other stopping condition of the function net.trainFcn
occurs.
Some training functions depart from this norm by presenting only one input
vector (or sequence) each epoch. An input vector (or sequence) is chosen
randomly each epoch from concurrent input vectors (or sequences). newc and
newsom return networks that use trainwb1, a training function that presents
only one input vector.
See Also
13-224
sim, init, adapt
trainbfg
Purpose
Syntax
trainbfg
BFGS quasi-Newton backpropagation
[net,tr] = trainbfg(net,Pd,Tl,Ai,Q,TS,VV)
info = trainbfg(code)
Description
trainbfg is a network training function that updates weight and bias values
according to the BFGS quasi-Newton method.
trainbfg(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
TV
- Either empty matrix [] or structure of test vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf -
Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
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trainbfg
Training occurs according to the trainbfg's training parameters, shown here
with their default values:
net.trainParam.epochs
100
net.trainParam.show
25
net.trainParam.goal
0
net.trainParam.time
inf
net.trainParam.min_grad
1e-6
net.trainParam.max_fail
5
net.trainParam.searchFcn
'srchcha'
Maximum number of epochs to train
Epochs between showing progress
Performance goal
Maximum time to train in seconds
Minimum performance gradient
Maximum validation failures
Name of line search routine to use.
Parameters related to line search methods (not all used for all methods):
net.trainParam.scal_tol
20
Divide into delta to determine tolerance for linear search.
net.trainParam.alpha
0.001
Scale factor which determines sufficient reduction in perf.
net.trainParam.beta
0.1
Scale factor which determines sufficiently large step size.
net.trainParam.delta
0.01
Initial step size in interval location step.
net.trainParam.gama
0.1
Parameter to avoid small reductions in performance. Usually set to 0.1.
(See use in srch_cha.)
net.trainParam.low_lim
0.1 Lower limit on change in step size.
net.trainParam.up_lim
0.5 Upper limit on change in step size.
net.trainParam.maxstep
100 Maximum step length.
net.trainParam.minstep
1.0e-6 Minimum step length.
net.trainParam.bmax
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26 Maximum step size.
trainbfg
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
If TV is not [], it must be a structure of validation vectors,
TV.PD - Validation delayed inputs.
TV.Tl - Validation layer targets.
TV.Ai - Validation initial input conditions.
TV.Q
- Validation batch size.
TV.TS - Validation time steps.
which is used to test the generalization capability of the trained network.
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trainbfg
trainbfg(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The trainbfg network training function is to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'trainbfg');
a = sim(net,p)
Train and Retest the Network
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
See newff, newcf, and newelm for other examples
Network Use
You can create a standard network that uses trainbfg with newff, newcf, or
newelm.
To prepare a custom network to be trained with trainbfg:
1 Set net.trainFcn to 'trainbfg'. This will set net.trainParam to trainbfg's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with trainbfg.
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trainbfg
Algorithm
trainbfg can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to the following:
X = X + a*dX;
where dX is the search direction. The parameter a is selected to minimize the
performance along the search direction. The line search function searchFcn is
used to locate the minimum point. The first search direction is the negative of
the gradient of performance. In succeeding iterations the search direction is
computed according to the following formula:
dX = -H\gX;
where gX is the gradient and H is an approximate Hessian matrix. See page 119
of Gill, Murray & Wright (Practical Optimization 1981) for a more detailed
discussion of the BFGS quasi-Newton method.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
See Also
newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp,
traincgf, traincgb, trainscg, traincgp, trainoss.
References
Gill, P. E.,W. Murray, and M. H. Wright, Practical Optimization, New York:
Academic Press, 1981.
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trainbr
Purpose
Syntax
trainbr
Bayesian Regulation backpropagation
[net,tr] = trainbr(net,Pd,Tl,Ai,Q,TS,VV)
info = trainbr(code)
Description
trainbr is a network training function that updates the weight and bias values
according to Levenberg-Marquardt optimization. It minimizes a combination of
squared errors and weights, and then determines the correct combination so as
to produce a network which generalizes well. The process is called Bayesian
regularization.
trainbr(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
TR.mu
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- Adaptive mu value.
trainbr
Training occurs according to the trainlm's training parameters, shown here
with their default values:
100 Maximum number of epochs to train
net.trainParam.epochs
0 Performance goal
net.trainParam.goal
net.trainParam.mu
0.005 Marquardt adjustment parameter
net.trainParam.mu_dec
0.1 Decrease factor for mu
net.trainParam.mu_inc
10 Increase factor for mu
net.trainParam.mu_max
1e-10 Maximum value for mu
net.trainParam.max_fail
5 Maximum validation failures
net.trainParam.mem_reduc
1
Factor to use for memory/speed trade-off
net.trainParam.min_grad 1e-10 Minimum performance gradient
net.trainParam.show
25 Epochs between showing progress
net.trainParam.time
inf Maximum time to train in seconds
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
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trainbr
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is normally used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row. This early stopping is not used for trainbr, but the validation
performance is computed for analysis purposes if VV is not [].
trainbr(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here is a problem consisting of inputs p and targets t that we would like to
solve with a network. It involves fitting a noisy sine wave.
p = [-1:.05:1];
t = sin(2*pi*p)+0.1*randn(size(p));
Here a two-layer feed-forward network is created. The network's input ranges
from [-1 to 1]. The first layer has 20 tansig neurons, the second layer has one
purelin neuron. The trainbr network training function is to be used. The plot
of the resulting network output should show a smooth response, without
overfitting.
Create a Network
net=newff([-1 1],[20,1],{'tansig','purelin'},'trainbr');
Train and Test the Network
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net = train(net,p,t);
a = sim(net,p)
plot(p,a,p,t,'+')
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trainbr
Network Use
You can create a standard network that uses trainbr with newff, newcf, or
newelm.
To prepare a custom network to be trained with trainbr:
1 Set net.trainFcn to 'trainlm'. This will set net.trainParam to trainbr's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with trainbr.
See newff, newcf, and newelm for examples.
Algorithm
trainbr can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Bayesian regularization minimizes a linear combination of squared errors and
weights. It also modifies the linear combination so that at the end of training
the resulting network has good generalization qualities. See MacKay (Neural
Computation, vol. 4, no. 3, 1992, pp. 415-447) and Foresee and Hagan
(Proceedings of the International Joint Conference on Neural Networks, June,
1997) for more detailed discussions of Bayesian regularization.
This Bayesian regularization takes place within the Levenberg-Marquardt
algorithm. Backpropagation is used to calculate the Jacobian jX of
performance perf with respect to the weight and bias variables X. Each
variable is adjusted according to Levenberg-Marquardt,
jj = jX * jX
je = jX * E
dX = -(jj+I*mu) \ je
where E is all errors and I is the identity matrix.
The adaptive value mu is increased by mu_inc until the change shown above
results in a reduced performance value. The change is then made to the
network and mu is decreased by mu_dec.
The parameter mem_reduc indicates how to use memory and speed to calculate
the Jacobian jX. If mem_reduc is 1, then trainlm runs the fastest, but can
require a lot of memory. Increase mem_reduc to 2, cuts some of the memory
required by a factor of two, but slows trainlm somewhat. Higher values
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trainbr
continue to decrease the amount of memory needed and increase the training
times.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 mu exceeds mu_max.
6 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
See Also
newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp,
traincgf, traincgb, trainscg, traincgp, trainoss
References
Foresee, F. D., and M. T. Hagan, “Gauss-Newton approximation to Bayesian
regularization,” Proceedings of the 1997 International Joint Conference on
Neural Networks, 1997.
MacKay, D. J. C., “Bayesian interpolation,” Neural Computation, vol. 4, no. 3,
pp. 415-447, 1992.
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traincgb
Purpose
Syntax
traincgb
Conjugate gradient backpropagation with Powell-Beale restarts
[net,tr] = traincgb(net,Pd,Tl,Ai,Q,TS,VV)
info = traincgb(code)
Description
traincgb is a network training function that updates weight and bias values
according to the conjugate gradient backpropagation with Powell-Beale
restarts.
traincgb(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
TV
- Either empty matrix [] or structure of test vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
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traincgb
Training occurs according to the traincgb's training parameters, shown here
with their default values:
net.trainParam.epochs
100
net.trainParam.show
25
net.trainParam.goal
0
net.trainParam.time
inf
net.trainParam.min_grad
1e-6
net.trainParam.max_fail
5
net.trainParam.searchFcn
'srchcha'
Maximum number of epochs to train
Epochs between showing progress
Performance goal
Maximum time to train in seconds
Minimum performance gradient
Maximum validation failures
Name of line search routine to use.
Parameters related to line search methods (not all used for all methods):
net.trainParam.scal_tol
20
Divide into delta to determine tolerance for linear search.
net.trainParam.alpha
0.001
Scale factor which determines sufficient reduction in perf.
net.trainParam.beta
0.1
Scale factor which determines sufficiently large step size.
net.trainParam.delta
0.01
Initial step size in interval location step.
net.trainParam.gama
0.1
Parameter to avoid small reductions in performance. Usually set to 0.1.
(See use in srch_cha.)
net.trainParam.low_lim
0.1 Lower limit on change in step size.
net.trainParam.up_lim
0.5 Upper limit on change in step size.
net.trainParam.maxstep
100 Maximum step length.
net.trainParam.minstep
1.0e-6 Minimum step length.
net.trainParam.bmax
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26 Maximum step size.
traincgb
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
If TV is not [], it must be a structure of validation vectors,
TV.PD - Validation delayed inputs.
TV.Tl - Validation layer targets.
TV.Ai - Validation initial input conditions.
TV.Q
- Validation batch size.
TV.TS - Validation time steps.
which is used to test the generalization capability of the trained network.
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traincgb
traincgb(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The traincgb network training function is to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'traincgb');
a = sim(net,p)
Train and Retest the Network
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
See newff, newcf, and newelm for other examples.
Network Use
You can create a standard network that uses traincgb with newff, newcf, or
newelm.
To prepare a custom network to be trained with traincgb:
1 Set net.trainFcn to 'traincgb'. This will set net.trainParam to traincgb's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with traincgb.
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traincgb
Algorithm
traincgb can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to the following:
X = X + a*dX;
where dX is the search direction. The parameter a is selected to minimize the
performance along the search direction. The line search function searchFcn is
used to locate the minimum point. The first search direction is the negative of
the gradient of performance. In succeeding iterations the search direction is
computed from the new gradient and the previous search direction according
to the formula:
dX = -gX + dX_old*Z;
where gX is the gradient. The parameter Z can be computed in several different
ways. The Powell-Beale variation of conjugate gradient is distinguished by two
features. First, the algorithm uses a test to determine when to reset the search
direction to the negative of the gradient. Second, the search direction is
computed from the negative gradient, the previous search direction, and the
last search direction before the previous reset. See Powell, Mathematical
Programming, Vol. 12 (1977) pp. 241-254, for a more detailed discussion of the
algorithm.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
See Also
newff, newcf, traingdm, traingda, traingdx, trainlm, traincgp,
traincgf, traincgb, trainscg, trainoss, trainbfg
References
Powell, M. J. D.,“Restart procedures for the conjugate gradient method,”
Mathematical Programming, vol. 12, pp. 241-254, 1977.
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traincgf
Purpose
Syntax
traincgf
Conjugate gradient backpropagation with Fletcher-Reeves updates
[net,tr] = traincgf(net,Pd,Tl,Ai,Q,TS,VV)
info = traincgf(code)
Description
traincgf is a network training function that updates weight and bias values
according to the conjugate gradient backpropagation with Fletcher-Reeves
updates.
traincgf(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
TV
- Either empty matrix [] or structure of test vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
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traincgf
Training occurs according to the traincgf's training parameters, shown here
with their default values:
net.trainParam.epochs
100
net.trainParam.show
25
net.trainParam.goal
0
net.trainParam.time
inf
net.trainParam.min_grad
1e-6
net.trainParam.max_fail
5
net.trainParam.searchFcn
'srchcha'
Maximum number of epochs to train
Epochs between showing progress
Performance goal
Maximum time to train in seconds
Minimum performance gradient
Maximum validation failures
Name of line search routine to use
Parameters related to line search methods (not all used for all methods):
net.trainParam.scal_tol
20
Divide into delta to determine tolerance for linear search.
net.trainParam.alpha
0.001
Scale factor which determines sufficient reduction in perf.
net.trainParam.beta
0.1
Scale factor which determines sufficiently large step size.
net.trainParam.delta
0.01
Initial step size in interval location step.
net.trainParam.gama
0.1
Parameter to avoid small reductions in performance. Usually set to 0.1.
(See use in srch_cha.)
net.trainParam.low_lim
0.1 Lower limit on change in step size.
net.trainParam.up_lim
0.5 Upper limit on change in step size.
net.trainParam.maxstep
100 Maximum step length.
net.trainParam.minstep
1.0e-6 Minimum step length.
net.trainParam.bmax
26 Maximum step size.
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traincgf
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
If TV is not [], it must be a structure of validation vectors,
TV.PD - Validation delayed inputs.
TV.Tl - Validation layer targets.
TV.Ai - Validation initial input conditions.
TV.Q
- Validation batch size.
TV.TS - Validation time steps.
which is used to test the generalization capability of the trained network.
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traincgf
traincgf(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The traincgf network training function is to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf');
a = sim(net,p)
Train and Retest the Network
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
See newff, newcf, and newelm for other examples.
Network Use
You can create a standard network that uses traincgf with newff, newcf, or
newelm.
To prepare a custom network to be trained with traincgf:
1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with traincgf.
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traincgf
Algorithm
traincgf can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to the following:
X = X + a*dX;
where dX is the search direction. The parameter a is selected to minimize the
performance along the search direction. The line search function searchFcn is
used to locate the minimum point. The first search direction is the negative of
the gradient of performance. In succeeding iterations the search direction is
computed from the new gradient and the previous search direction, according
to the formula:
dX = -gX + dX_old*Z;
where gX is the gradient. The parameter Z can be computed in several different
ways. For the Fletcher-Reeves variation of conjugate gradient it is computed
according to
Z=normnew_sqr/norm_sqr;
where norm_sqr is the norm square of the previous gradient and normnew_sqr
is the norm square of the current gradient. See page 78 of Scales (Introduction
to Non-Linear Optimization 1985) for a more detailed discussion of the
algorithm.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
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traincgf
See Also
newff, newcf, traingdm, traingda, traingdx, trainlm, traincgp,
traincgb, trainscg, traincgp, trainoss, trainbfg
References
Scales, L. E.,Introduction to Non-Linear Optimization, New York:
Springer-Verlag, 1985.
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traincgp
Purpose
Syntax
traincgp
Conjugate gradient backpropagation with Polak-Ribiere updates
[net,tr] = traincgp(net,Pd,Tl,Ai,Q,TS,VV)
info = traincgp(code)
Description
traincgp is a network training function that updates weight and bias values
according to the conjugate gradient backpropagation with Polak-Ribiere
updates.
traincgp(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
TV
- Either empty matrix [] or structure of test vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
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traincgp
Training occurs according to the traincgp's training parameters shown here
with their default values:
net.trainParam.epochs
100
net.trainParam.show
25
net.trainParam.goal
0
net.trainParam.time
inf
net.trainParam.min_grad
1e-6
net.trainParam.max_fail
5
net.trainParam.searchFcn
'srchcha'
Maximum number of epochs to train
Epochs between showing progress
Performance goal
Maximum time to train in seconds
Minimum performance gradient
Maximum validation failures
Name of line search routine to use
Parameters related to line search methods (not all used for all methods):
net.trainParam.scal_tol
20
Divide into delta to determine tolerance for linear search.
net.trainParam.alpha
0.001
Scale factor which determines sufficient reduction in perf.
net.trainParam.beta
0.1
Scale factor which determines sufficiently large step size.
net.trainParam.delta
0.01
Initial step size in interval location step.
net.trainParam.gama
0.1
Parameter to avoid small reductions in performance. Usually set to 0.1.
(See use in srch_cha.)
net.trainParam.low_lim
0.1 Lower limit on change in step size.
net.trainParam.up_lim
0.5 Upper limit on change in step size.
net.trainParam.maxstep
100 Maximum step length.
net.trainParam.minstep
1.0e-6 Minimum step length.
net.trainParam.bmax
26 Maximum step size.
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traincgp
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
If TV is not [], it must be a structure of validation vectors,
TV.PD - Validation delayed inputs.
TV.Tl - Validation layer targets.
TV.Ai - Validation initial input conditions.
TV.Q
- Validation batch size.
TV.TS - Validation time steps.
which is used to test the generalization capability of the trained network.
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traincgp
traincgp(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The traincgp network training function is to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'traincgp');
a = sim(net,p)
Train and Retest the Network
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
See newff, newcf, and newelm for other examples.
Network Use
You can create a standard network that uses traincgp with newff, newcf, or
newelm.
To prepare a custom network to be trained with traincgp:
1 Set net.trainFcn to 'traincgp'. This will set net.trainParam to traincgp's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with traincgp.
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traincgp
Algorithm
traincgp can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to the following:
X = X + a*dX;
where dX is the search direction. The parameter a is selected to minimize the
performance along the search direction. The line search function searchFcn is
used to locate the minimum point. The first search direction is the negative of
the gradient of performance. In succeeding iterations the search direction is
computed from the new gradient and the previous search direction according
to the formula:
dX = -gX + dX_old*Z;
where gX is the gradient. The parameter Z can be computed in several different
ways. For the Polak-Ribiere variation of conjugate gradient it is computed
according to:
Z = ((gX - gX_old)'*gX)/norm_sqr;
where norm_sqr is the norm square of the previous gradient and gX_old is the
gradient on the previous iteration. See page 78 of Scales (Introduction to
Non-Linear Optimization 1985) for a more detailed discussion of the algorithm.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
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traincgp
See Also
newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp,
traincgf, traincgb, trainscg, trainoss, trainbfg
References
Scales, L. E.,Introduction to Non-Linear Optimization, New York:
Springer-Verlag, 1985.
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traingd
Purpose
Syntax
traingd
Gradient descent backpropagation
[net,tr] = traingd(net,Pd,Tl,Ai,Q,TS,VV)
info = traingd(code)
Description
traingd is a network training function that updates weight and bias values
according to gradient descent.
traingd(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either an empty matrix [] or a structure of validation vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
Training occurs according to the traingd's training parameters shown here
with their default values:
net.trainParam.epochs
net.trainParam.goal
net.trainParam.lr
net.trainParam.max_fail
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10
0
0.01
5
Maximum number of epochs to train
Performance goal
Learning rate
Maximum validation failures
net.trainParam.min_grad 1e-10
Minimum performance gradient
net.trainParam.show
25
Epochs between showing progress
net.trainParam.time
inf
Maximum time to train in seconds
traingd
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which are used to stop training early if the network performance on the
validation vectors fails to improve or remain the same for max_fail epochs in
a row.
traingd(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Network Use
You can create a standard network that uses traingd with newff, newcf, or
newelm.
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traingd
To prepare a custom network to be trained with traingd:
1 Set net.trainFcn to 'traingd'. This will set net.trainParam to traingd's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with traingd.
See newff, newcf, and newelm for examples.
Algorithm
traingd can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to gradient descent:
dX = lr * dperf/dX
Training stops when any of these conditions occurs:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
See Also
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newff, newcf, traingdm, traingda, traingdx, trainlm
traingda
Purpose
Syntax
traingda
Gradient descent with adaptive lr backpropagation
[net,tr] = traingda(net,Pd,Tl,Ai,Q,TS,VV)
info = traingda(code)
Description
traingda is a network training function that updates weight and bias values
according to gradient descent with adaptive learning rate.
traingda(NET,Pd,Tl,Ai,Q,TS,VV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
TR.lr
- Adaptive learning rate.
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traingda
Training occurs according to the traingda's training parameters, shown here
with their default values:
net.trainParam.epochs
10
net.trainParam.goal
0
Maximum number of epochs to train
Performance goal
net.trainParam.lr
0.01
Learning rate
net.trainParam.lr_inc
1.05
Ratio to increase learning rate
net.trainParam.lr_dec
0.7
Ratio to decrease learning rate
net.trainParam.max_fail
5
Maximum validation failures
net.trainParam.max_perf_inc 1.04
Maximum performance increase
net.trainParam.min_grad
Minimum performance gradient
1e-10
net.trainParam.show
net.trainParam.time
25
Epochs between showing progress
inf Maximum time to train in seconds
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
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traingda
which are used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
traingda(code) return useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Network Use
You can create a standard network that uses traingda with newff, newcf, or
newelm.
To prepare a custom network to be trained with traingda:
1 Set net.trainFcn to 'traingda'. This will set net.trainParam to traingda's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with traingda.
See newff, newcf, and newelm for examples.
Algorithm
traingda can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance dperf with
respect to the weight and bias variables X. Each variable is adjusted according
to gradient descent:
dX = lr*dperf/dX
At each epoch, if performance decreases toward the goal, then the learning rate
is increased by the factor lr_inc. If performance increases by more than the
factor max_perf_inc, the learning rate is adjusted by the factor lr_dec and the
change, which increased the performance, is not made.
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traingda
Training stops when any of these conditions occurs:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
See Also
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newff, newcf, traingd, traingdm, traingdx, trainlm
traingdm
Purpose
Syntax
traingdm
Gradient descent w/momentum backpropagation
[net,tr] = traingdm(net,Pd,Tl,Ai,Q,TS,VV)
info = traingdm(code)
Description
traingdm is a network training function that updates weight and bias values
according to gradient descent with adaptive learning rate.
traingdm(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
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traingdm
TR.lr - adaptive learning rate. Training occurs according to the traingdm's
training parameters shown here with their default values:
net.trainParam.epochs
10
net.trainParam.goal
net.trainParam.lr
0
0.01
net.trainParam.max_fail
5
net.trainParam.mc
net.trainParam.min_grad
0.9
1e-10
net.trainParam.show
net.trainParam.time
25
Maximum number of epochs to train
Performance goal
Learning rate
Maximum validation failures
Momentum constant.
Minimum performance gradient
Epochs between showing progress
inf Maximum time to train in seconds
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
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traingdm
traingdm(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Network Use
You can create a standard network that uses traingdm with newff, newcf, or
newelm.
To prepare a custom network to be trained with traingdm:
1 Set net.trainFcn to 'traingdm'. This will set net.trainParam to traingdm's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with traingdm.
See newff, newcf, and newelm for examples.
Algorithm
traingdm can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to gradient descent with momentum,
dX = mc*dXprev + lr*mc*dperf/dX
where dXprev is the previous change to the weight or bias.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increase more than max_fail times since the
last time it decreased (when using validation).
See Also
newff, newcf, traingd, traingda, traingdx, trainlm
13-261
traingdx
Purpose
Syntax
traingdx
Gradient descent w/momentum & adaptive lr backpropagation
[net,tr] = traingdx(net,Pd,Tl,Ai,Q,TS,VV)
info = traingdx(code)
Description
traingdx is a network training function that updates weight and bias values
according to gradient descent momentum and an adaptive learning rate.
traingdx(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
TR.lr
13-262
- Adaptive learning rate.
traingdx
Training occurs according to the traingdx's training parameters shown here
with their default values:
net.trainParam.epochs
10
net.trainParam.goal
0
Maximum number of epochs to train
Performance goal
net.trainParam.lr
0.01
Learning rate
net.trainParam.lr_inc
1.05
Ratio to increase learning rate
net.trainParam.lr_dec
0.7
Ratio to decrease learning rate
net.trainParam.max_fail
5
Maximum validation failures
net.trainParam.max_perf_inc 1.04
Maximum performance increase
net.trainParam.mc
Momentum constant.
net.trainParam.min_grad
0.9
1e-10
net.trainParam.show
net.trainParam.time
25
Minimum performance gradient
Epochs between showing progress
inf Maximum time to train in seconds
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
13-263
traingdx
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
traingdx(code) return useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Network Use
You can create a standard network that uses traingdx with newff, newcf, or
newelm.
To prepare a custom network to be trained with traingdx:
1 Set net.trainFcn to 'traingdx'. This will set net.trainParam to traingdx's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with traingdx.
See newff, newcf, and newelm for examples.
Algorithm
traingdx can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to gradient descent with momentum,
dX = mc*dXprev + lr*mc*dperf/dX
where dXprev is the previous change to the weight or bias.
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traingdx
For each epoch, if performance decreases toward the goal, then the learning
rate is increased by the factor lr_inc. If performance increases by more than
the factor max_perf_inc, the learning rate is adjusted by the factor lr_dec and
the change, which increased the performance, is not made.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increase more than max_fail times since the
last time it decreased (when using validation).
See Also
newff, newcf, traingd, traingdm, traingda, trainlm
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trainlm
Purpose
trainlm
Levenberg-Marquardt backpropagation
Syntax
[net,tr] = trainlm(net,Pd,Tl,Ai,Q,TS,VV)
info = trainlm(code)
Description
trainlm is a network training function that updates weight and bias values
according to Levenberg-Marquardt optimization.
trainlm(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
TR.mu
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- Adaptive mu value.
trainlm
Training occurs according to the trainlm's training parameters shown here
with their default values:
net.trainParam.epochs
10
net.trainParam.goal
net.trainParam.lr
0
0.01
Maximum number of epochs to train
Performance goal
Learning rate
Maximum validation failures
net.trainParam.max_fail
5
net.trainParam.mem_reduc
1 Factor to use for memory/speed
trade off
net.trainParam.min_grad 1e-10
Minimum performance gradient
net.trainParam.show
25
Epochs between showing progress
net.trainParam.time
inf
Maximum time to train in seconds
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
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trainlm
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
trainlm(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Network Use
You can create a standard network that uses trainlm with newff, newcf, or
newelm.
To prepare a custom network to be trained with trainlm:
1 Set net.trainFcn to 'trainlm'. This will set net.trainParam to trainlm's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with trainlm.
See newff, newcf, and newelm for examples.
Algorithm
trainlm can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate the Jacobian jX of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to Levenberg-Marquardt,
jj = jX * jX
je = jX * E
dX = -(jj+I*mu) \ je
where E is all errors and I is the identity matrix.
The adaptive value mu is increased by mu_inc until the change above results in
a reduced performance value. The change is then made to the network and mu
is decreased by mu_dec.
The parameter mem_reduc indicates how to use memory and speed to calculate
the Jacobian jX. If mem_reduc is 1, then trainlm runs the fastest, but can
require a lot of memory. Increasing mem_reduc to 2, cuts some of the memory
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trainlm
required by a factor of two, but slows trainlm somewhat. Higher values
continue to decrease the amount of memory needed and increase training
times.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 mu exceeds mu_max.
6 Validation performance has increase more than max_fail times since the
last time it decreased (when using validation).
See Also
newff, newcf, traingd, traingdm, traingda, traingdx
13-269
trainoss
Purpose
Syntax
trainoss
One step secant backpropagation
[net,tr] = trainoss(net,Pd,Tl,Ai,Q,TS,VV)
info = trainoss(code)
Description
trainoss is a network training function that updates weight and bias values
according to the one step secant method.
trainoss(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
TV
- Either empty matrix [] or structure of test vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
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trainoss
Training occurs according to the trainoss's training parameters, shown here
with their default values:
net.trainParam.epochs
100
net.trainParam.show
25
net.trainParam.goal
0
net.trainParam.time
inf
net.trainParam.min_grad
1e-6
net.trainParam.max_fail
5
net.trainParam.searchFcn
'srchcha'
Maximum number of epochs to train
Epochs between showing progress
Performance goal
Maximum time to train in seconds
Minimum performance gradient
Maximum validation failures
Name of line search routine to use
Parameters related to line search methods (not all used for all methods):
net.trainParam.scal_tol
20
Divide into delta to determine tolerance for linear search.
net.trainParam.alpha
0.001
Scale factor which determines sufficient reduction in perf.
net.trainParam.beta
0.1
Scale factor which determines sufficiently large step size.
net.trainParam.delta
0.01
Initial step size in interval location step.
net.trainParam.gama
0.1
Parameter to avoid small reductions in performance. Usually set to 0.1.
(See use in srch_cha.)
net.trainParam.low_lim
0.1 Lower limit on change in step size.
net.trainParam.up_lim
0.5 Upper limit on change in step size.
net.trainParam.maxstep
100 Maximum step length.
net.trainParam.minstep
1.0e-6 Minimum step length.
net.trainParam.bmax
26 Maximum step size.
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trainoss
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
If TV is not [], it must be a structure of validation vectors,
TV.PD - Validation delayed inputs.
TV.Tl - Validation layer targets.
TV.Ai - Validation initial input conditions.
TV.Q
- Validation batch size.
TV.TS - Validation time steps.
which is used to test the generalization capability of the trained network.
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trainoss
trainoss(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The trainoss network training function is to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'trainoss');
a = sim(net,p)
Train and Retest the Network
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
See newff, newcf, and newelm for other examples.
Network Use
You can create a standard network that uses trainoss with newff, newcf, or
newelm.
To prepare a custom network to be trained with trainoss:
1 Set net.trainFcn to 'trainoss'. This will set net.trainParam to trainoss's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with trainoss.
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trainoss
Algorithm
trainoss can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to the following:
X = X + a*dX;
where dX is the search direction. The parameter a is selected to minimize the
performance along the search direction. The line search function searchFcn is
used to locate the minimum point. The first search direction is the negative of
the gradient of performance. In succeeding iterations the search direction is
computed from the new gradient and the previous steps and gradients
according to the following formula:
dX = -gX + Ac*X_step + Bc*dgX;
where gX is the gradient, X_step is the change in the weights on the previous
iteration, and dgX is the change in the gradient from the last iteration. See
Battiti (Neural Computation, vol. 4, 1992, pp. 141-166) for a more detailed
discussion of the one step secant algorithm.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
See Also
newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp,
traincgf, traincgb, trainscg, traincgp, trainbfg
References
R. Battiti, “First and second order methods for learning: Between steepest
descent and Newton’s method,” Neural Computation, vol. 4, no. 2, pp. 141–166,
1992.
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trainrp
Purpose
Syntax
trainrp
RPROP backpropagation
[net,tr] = trainrp(net,Pd,Tl,Ai,Q,TS,VV)
info = trainrp(code)
Description
trainrp is a network training function that updates weight and bias values
according to the resilient backpropagation algorithm (RPROP).
trainrp(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
TV
- Either empty matrix [] or structure of test vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
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trainrp
Training occurs according to the trainrp's training parameters shown here
with their default values:
net.trainParam.epochs
100
net.trainParam.show
25
net.trainParam.goal
0
net.trainParam.time
inf
net.trainParam.min_grad
1e-6
net.trainParam.max_fail
5
net.trainParam.lr
0.01
Maximum number of epochs to train
Epochs between showing progress
Performance goal
Maximum time to train in seconds
Minimum performance gradient
Maximum validation failures
Learning rate
net.trainParam.delt_inc
1.2
Increment to weight change
net.trainParam.delt_dec
0.5
Decrement to weight change
net.trainParam.delta0
0.07
Initial weight change
net.trainParam.deltamax
50.0
Maximum weight change
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
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trainrp
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
If TV is not [], it must be a structure of validation vectors,
TV.PD - Validation delayed inputs.
TV.Tl - Validation layer targets.
TV.Ai - Validation initial input conditions.
TV.Q
- Validation batch size.
TV.TS - Validation time steps.
which is used to test the generalization capability of the trained network.
trainrp(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The trainrp network training function is to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'trainrp');
a = sim(net,p)
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trainrp
Train and Retest the Network
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
See newff, newcf, and newelm for other examples.
Network Use
You can create a standard network that uses trainrp with newff, newcf, or
newelm.
To prepare a custom network to be trained with trainrp:
1 Set net.trainFcn to 'trainrp'. This will set net.trainParam to trainrp's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with trainrp.
Algorithm
trainrp can train any network as long as its weight, net input, and transfer
functions have derivative functions.
Backpropagation is used to calculate derivatives of performance perf with
respect to the weight and bias variables X. Each variable is adjusted according
to the following:
dX = deltaX.*sign(gX);
where the elements of deltaX are all initialized to delta0 and gX is the
gradient. At each iteration the elements of deltaX are modified. If an element
of gX changes sign from one iteration to the next, then the corresponding
element of deltaX is decreased by delta_dec. If an element of gX maintains the
same sign from one iteration to the next, then the corresponding element of
deltaX is increased by delta_inc. See Reidmiller and Braun, Proceedings of
the IEEE International Conference on Neural Networks,, 1993, pp. 586-591.
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trainrp
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
See Also
newff, newcf, traingdm, traingda, traingdx, trainlm, traincgp,
traincgf, traincgb, trainscg, trainoss, trainbfg
References
Riedmiller, M., and H. Braun, “A direct adaptive method for faster
backpropagation learning: The RPROP algorithm,” Proceedings of the IEEE
International Conference on Neural Networks, San Francisco,1993.
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trainscg
Purpose
Syntax
trainscg
Scaled conjugate gradient backpropagation
[net,tr] = trainscg(net,Pd,Tl,Ai,Q,TS,VV)
info = trainscg(code)
Description
trainscg is a network training function that updates weight and bias values
according to the scaled conjugate gradient method.
trainscg(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
TV
- Either empty matrix [] or structure of test vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
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trainscg
Training occurs according to the trainscg's training parameters shown here
with their default values:
net.trainParam.epochs
100
net.trainParam.show
25
net.trainParam.goal
0
net.trainParam.time
inf
net.trainParam.min_grad
1e-6
net.trainParam.max_fail
5
net.trainParam.sigma
5.0e-5
net.trainParam.lambda
5.0e-7
Maximum number of epochs to train
Epochs between showing progress
Performance goal
Maximum time to train in seconds
Minimum performance gradient
Maximum validation failures
Determines change in weight for
second derivative approximation.
Parameter for regulating the
indefiniteness of the Hessian.
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
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trainscg
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
If TV is not [], it must be a structure of validation vectors,
TV.PD - Validation delayed inputs.
TV.Tl - Validation layer targets.
TV.Ai - Validation initial input conditions.
TV.Q
- Validation batch size.
TV.TS - Validation time steps.
which is used to test the generalization capability of the trained network.
trainscg(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Examples
Here is a problem consisting of inputs P and targets T that we would like to
solve with a network.
P = [0 1 2 3 4 5];
T = [0 0 0 1 1 1];
Here a two-layer feed-forward network is created. The network's input ranges
from [0 to 10]. The first layer has two tansig neurons, and the second layer
has one logsig neuron. The trainscg network training function is to be used.
Create and Test a Network
net = newff([0 5],[2 1],{'tansig','logsig'},'trainscg');
a = sim(net,p)
Train and Retest the Network
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
See newff, newcf, and newelm for other examples.
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trainscg
Network Use
You can create a standard network that uses trainscg with newff, newcf, or
newelm.
To prepare a custom network to be trained with trainscg:
1 Set net.trainFcn to 'trainscg'. This will set net.trainParam to trainscg's
default parameters.
2 Set net.trainParam properties to desired values.
In either case, calling train with the resulting network will train the network
with trainscg.
Algorithm
trainscg can train any network as long as its weight, net input, and transfer
functions have derivative functions. Backpropagation is used to calculate
derivatives of performance perf with respect to the weight and bias variables
X.
The scaled conjugate gradient algorithm is based on conjugate directions, as in
traincgp, traincgf and traincgb, but this algorithm does not perform a line
search at each iteration. See Moller (Neural Networks, vol. 6, 1993,
pp.525-533) for a more detailed discussion of the scaled conjugate gradient
algorithm.
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
3 Performance has been minimized to the goal.
4 The performance gradient falls below mingrad.
5 Validation performance has increased more than max_fail times since the
last time it decreased (when using validation).
See Also
newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp,
traincgf, traincgb, trainbfg, traincgp, trainoss
References
Moller, M. F., “A scaled conjugate gradient algorithm for fast supervised
learning,” Neural Networks, vol. 6, pp. 525-533, 1993.
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trainwb
Purpose
Syntax
trainwb
By-weight-&-bias 1-vector-at-a-time training function
[net,tr] = trainwb(net,Pd,Tl,Ai,Q,TS,VV)
info = trainwb(code)
Description
trainwb is a network training function that updates weight and bias values
according to Levenberg-Marquardt optimization.
trainwb(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.vperf - Validation performance.
TR.tperf - Test performance.
Training occurs according to the trainwb1's training parameters, shown here
with their default values:
net.trainParam.epochs
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100 Maximum number of epochs to train
net.trainParam.goal
0
Performance goal
net.trainParam.max_fail
5
Maximum validation failures
net.trainParam.show
25
Epochs between showing progress
net.trainParam.time
inf
Maximum time to train in seconds
trainwb
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
If VV is not [], it must be a structure of validation vectors,
VV.PD - Validation delayed inputs.
VV.Tl - Validation layer targets.
VV.Ai - Validation initial input conditions.
VV.Q
- Validation batch size.
VV.TS - Validation time steps.
which is used to stop training early if the network performance on the
validation vectors fails to improve or remains the same for max_fail epochs in
a row.
trainwb(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Network Use
You can create a standard network that uses trainwb with newp or newlin.
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trainwb
To prepare a custom network to be trained with trainwb:
1 Set net.trainFcn to 'trainwb'. (This will set net.trainParam to trainwb's
default parameters.)
2 Set each net.inputWeights{i,j}.learnFcn to a learning function. Set each
net.layerWeights{i,j}.learnFcn to a learning function. Set each
net.biases{i}.learnFcn to a learning function. (Weight and bias learning
parameters will automatically be set to default values for the given learning
function.)
To train the network:
1 Set net.trainParam properties to desired values.
2 Set weight and bias learning parameters to desired values.
3 Call train.
See newp and newlin for training examples.
Algorithm
Each weight and bias updates according to its learning function after each
epoch (one pass through the entire set of input vectors).
Training stops when any of these conditions occur:
1 The maximum number of epochs (repetitions) is reached.
2 Performance has been minimized to the goal.
3 The maximum amount of time has been exceeded.
4 Validation performance has increase more than max_fail times since the
last time it decreased (when using validation).
See Also
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newp, newlin, train
trainwb1
Purpose
Syntax
trainwb1
By-weight-and-bias network training function
[net,tr] = trainwb1(net,Pd,Tl,Ai,Q,TS,VV)
info = trainwb1(code)
Description
trainwb1 is a network training function which updates each weight and bias
according to its learning function. At each epoch trainwb1 randomly chooses
just one input vector (or sequence) to present to the network.
trainwb1(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs,
net - Neural network.
Pd
- Delayed input vectors.
Tl
- Layer target vectors.
Ai
- Initial input delay conditions.
Q
- Batch size.
TS
- Time steps.
VV
- Either empty matrix [] or structure of validation vectors.
and returns,
net - Trained network.
TR
- Training record of various values over each epoch:
TR.epoch - Epoch number.
TR.perf
- Training performance.
TR.index - Index of presented input vector (or sequence).
Training occurs according to the trainwb1's training parameters shown here
with their default values:
net.trainParam.epochs
100 Maximum number of epochs to train
net.trainParam.show
25
Epochs between showing progress
net.trainParam.time
inf
Maximum time to train in seconds
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trainwb1
Dimensions for these variables are:
Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix.
Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix.
Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix.
where
Ni = net.numInputs
Nl = net.numLayers
LD = net.numLayerDelays
Ri = net.inputs{i}.size
Si = net.layers{i}.size
Vi = net.targets{i}.size
Dij = Ri * length(net.inputWeights{i,j}.delays)
trainwb1 does not implement validation or test vectors, so arguments VV and
TV are ignored.
trainwb1(code) returns useful information for each code string:
'pnames' - Names of training parameters.
'pdefaults' - Default training parameters.
Network Use
You can create a standard network that uses trainwb1 with newc or newsom.
To prepare a custom network to be trained with trainwb1:
1 Set net.trainFcn to 'trainwb1'. (This will set net.trainParam to
trainwb1's default parameters.)
2 Set each net.inputWeights{i,j}.learnFcn to a learning function. Set each
net.layerWeights{i,j}.learnFcn to a learning function. Set each
net.biases{i}.learnFcn to a learning function. (Weight and bias learning
parameters will automatically be set to default values for the given learning
function.)
To train the network:
1 Set net.trainParam properties to desired values.
2 Set weight and bias learning parameters to desired values.
3 Call train.
See newc and newsom for training examples.
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trainwb1
Algorithm
For each epoch a vector (or sequence) is chosen randomly and presented to the
network and then the weight and bias values are updated accordingly.
Training stops when any of these conditions are met:
1 The maximum number of epochs (repetitions) is reached.
2 The maximum amount of time has been exceeded.
See Also
newp, newlin, train
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tramnmx
Purpose
tramnmx
Transform data using a precalculated min and max
Syntax
[pn] = tramnmx(p,minp,maxp)
Description
tramnmx transforms the network input set using minimum and maximum
values that were previously computed by premnmx. This function needs to be
used when a network has been trained using data normalized by premnmx. All
subsequent inputs to the network need to be transformed using the same
normalization.
tramnmx(P,minp, maxp)takes these inputs
P
- R x Q matrix of input (column) vectors.
minp- R x 1 vector containing original minimums for each input.
maxp- R x 1 vector containing original maximums for each input.
and returns,
PN
Examples
- R x Q matrix of normalized input vectors
Here is the code to normalize a given data set, so that the inputs and targets
will fall in the range [-1,1], using premnmx, and also code to train a network
with the normalized data.
p = [-10 -7.5 -5 -2.5 0 2.5 5 7.5 10];
t = [0 7.07 -10 -7.07 0 7.07 10 7.07 0];
[pn,minp,maxp,tn,mint,maxt] = premnmx(p,t);
net = newff(minmax(pn),[5 1],{'tansig' 'purelin'},'trainlm');
net = train(net,pn,tn);
If we then receive new inputs to apply to the trained network, we will use
tramnmx to transform them first. Then the transformed inputs can be used to
simulate the previously trained network. The network output must also be
unnormalized using postmnmx.
p2 = [4 -7];
[p2n] = tramnmx(p2,minp,maxp);
an = sim(net,pn);
[a] = postmnmx(an,mint,maxt);
Algorithm
See Also
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pn = 2*(p-minp)/(maxp-minp) - 1;
premnmx, prestd, prepca, trastd, trapca
trapca
Purpose
trapca
Principal component transformation
Syntax
[Ptrans] = trapca(P,TransMat)
Description
trapca preprocesses the network input training set by applying the principal
component transformation that was previously computed by prepca. This
function needs to be used when a network has been trained using data
normalized by prepca. All subsequent inputs to the network need to be
transformed using the same normalization.
trapca(P,TransMat) takes these inputs,
P - R x Q matrix of centered input (column) vectors.
TransMat - Transformation matrix.
and returns,
Ptrans - Transformed data set.
Examples
Here is the code to perform a principal component analysis and retain only
those components which contribute more than 2 percent to the variance in the
data set. prestd is called first to create zero mean data, which are needed for
prepca.
p = [-1.5 -0.58 0.21 -0.96 -0.79; -2.2 -0.87 0.31 -1.4 -1.2];
t = [-0.08 3.4 -0.82 0.69 3.1];
[pn,meanp,stdp,tn,meant,stdt] = prestd(p,t);
[ptrans,transMat] = prepca(pn,0.02);
net = newff(minmax(ptrans),[5 1],{'tansig''purelin'},'trainlm');
net = train(net,ptrans,tn);
If we then receive new inputs to apply to the trained network, we will use
trastd and trapca to transform them first. Then the transformed inputs can
be used to simulate the previously trained network. The network output must
also be unnormalized using poststd.
p2 = [1.5 -0.8;0.05 -0.3];
[p2n] = trastd(p2,meanp,stdp);
[p2trans] = trapca(p2n,TransMat)
an = sim(net,p2trans);
[a] = poststd(an,meant,stdt);
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trapca
Algorithm
See Also
13-292
Ptrans = TransMat*P;
prestd, premnmx, prepca, trastd, tramnmx
trastd
Purpose
trastd
Preprocess data using a precalculated mean and standard deviation
Syntax
[pn] = trastd(p,meanp,stdp)
Description
trastd preprocesses the network training set using the mean and standard
deviation that were previously computed by prestd. This function needs to be
used when a network has been trained using data normalized by prestd. All
subsequent inputs to the network need to be transformed using the same
normalization.
trastd(P,T) takes these inputs,
P
- R x Q matrix of input (column) vectors.
meanp - R x 1 vector containing the original means for each input.
stdp
- R x 1 vector containing the original standard deviations for each
input.
and returns,
PN - R x Q matrix of normalized input vectors.
Examples
Here is the code to normalize a given data set so that the inputs and targets
will have means of zero and standard deviations of 1.
p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93];
t = [-0.08 3.4 -0.82 0.69 3.1];
[pn,meanp,stdp,tn,meant,stdt] = prestd(p,t);
net = newff(minmax(pn),[5 1],{'tansig' 'purelin'},'trainlm');
net = train(net,pn,tn);
If we then receive new inputs to apply to the trained network, we will use
trastd to transform them first. Then the transformed inputs can be used to
simulate the previously trained network. The network output must also be
unnormalized using poststd.
p2 = [1.5 -0.8;0.05 -0.3];
[p2n] = trastd(p2,meanp,stdp);
an = sim(net,pn);
[a] = poststd(an,meant,stdt);
Algorithm
See Also
pn = (p-meanp)/stdp;
premnmx, prepca, prestd, trapca, tramnmx
13-293
tribas
Purpose
Syntax
tribas
Triangular basis transfer function
A = tribas(N)
info = tribas(code)
Description
tribas is a transfer function. Transfer functions calculate a layer's output from
its net input.
tribas(N) takes one input,
N - S x Q matrix of net input (column) vectors.
and returns each element of N passed through a radial basis function.
tribas(code) returns useful information for each code string:
'deriv' - Name of derivative function.
'name' - Full name.
'output' - Output range.
'active' - Active input range.
Examples
Here we create a plot of the tribas transfer function.
n = -5:0.1:5;
a = tribas(n);
plot(n,a)
Network Use
To change a network so that a layer uses tribas, set
net.layers{i}.transferFcn to 'tribas'.
Call sim to simulate the network with tribas.
Algorithm
tribas(N) calculates its output with according to:
tribas(n) = 1-abs(n), if -1 <= n <= 1; = 0, otherwise.
See Also
13-294
sim, radbas
vec2ind
Purpose
vec2ind
Convert vectors to indices
Syntax
ind = vec2ind(vec)
Description
ind2vec and vec2ind allow indices to be represented either by themselves or as
vectors containing a 1 in the row of the index they represent.
vec2ind(vec) takes one argument,
vec - Matrix of vectors, each containing a single 1.
and returns the indices of the 1's.
Examples
Here four vectors (containing only one 1 each) are defined and the indices of the
1's are found.
vec = [1 0 0 0; 0 0 1 0; 0 1 0 1]
ind = vec2ind(vec)
See Also
ind2vec
13-295
vec2ind
13-296
A
Glossary
A
Glossary
ADALINE - an acronym for a linear neuron: ADAptive LINear Element.
adaption - a function that proceeds through the specified sequence of inputs,
calculating the output, error and network adjustment for each input vector in
the sequence as the inputs are presented.
adaptive learning rate - a learning rate that is adjusted according to an
algorithm during training to minimize training time.
adaptive filter - a network that contains delays and whose weights are adjusted
after each new input vector is presented. The network “adapts” to changes in the
input signal properties if such occur. This kind of filter is used in long distance
telephone lines to cancel echoes.
architecture - a description of the number of the layers in a neural network,
each layer’s transfer function, the number of neurons per layer, and the
connections between layers.
backpropagation learning rule -a learning rule in which weights and biases
are adjusted by error derivative (delta) vectors backpropagated through the
network. Backpropagation is commonly applied to feedforward multilayer
networks. Sometimes this rule is called the generalized delta rule.
backtracking search - linear search routine which begins with a step
multiplier of 1 and then backtracks until an acceptable reduction in the
performance is obtained.
batch - a matrix of input (or target) vectors applied to the network
“simultaneously”. Changes to the network weights and biases are made just
once for the entire set of vectors in the input matrix.
batching - the process of presenting a matrix (batch) of input vectors for
simultaneous calculation of a matrix of output vectors and/or new weights and
biases.
Bayesian framework - assumes that the weights and biases of the network
are random variables with specified distributions.
BFGS quasi-Newton algorithm - a variation of Newton’s optimization
algorithm, in which an approximation of the Hessian matrix is obtained from
gradients computed at each iteration of the algorithm.
bias - a neuron parameter that is summed with the neuron’s weighted inputs
and passed through the neuron’s transfer function to generate the neuron’s
output.
A-2
bias vector - a column vector of bias values for a layer of neurons.
Brent’s search - a linear search which is a hybrid combination of the golden
section search and a quadratic interpolation.
Charalambous’ search - a hybrid line search that uses a cubic interpolation,
together with a type of sectioning.
classification - an association of an input vector with a particular target vector.
competitive layer - a layer of neurons in which only the neuron with
maximum net input has an output of 1 and all other neurons output 0. Neurons
compete with each other for the right to respond to a given input vector.
competitive learning - the unsupervised training of a competitive layer with
the instar rule or Kohonen rule. Individual neurons learn to become feature
detectors. After training, the layer categorizes input vectors among its
neurons.
competitive transfer function - accepts a net input vector for a layer and
returns neuron outputs of 0 for all neurons except for the “winner,” the neuron
associated with the most positive element of the net input n.
concurrent input vectors - name given to a matrix of input vectors that are to be
presented to a network “simultaneously.” All the vectors in the matrix will be used
in making just one set of changes in the weights and biases.
conjugate gradient algorithm -in the conjugate gradient algorithms a
search is performed along conjugate directions, which produces generally
faster convergence than a search along the steepest descent directions.
connection - a one-way link between neurons in a network.
connection strength - the strength of a link between two neurons in a
network. The strength, often called weight, determines the effect that one
neuron has on another.
cycle - a single presentation of an input vector, calculation of output and new
weights and biases.
dead neurons - a competitive layer neuron that never won any competition
during training and so has not become a useful feature detector. Dead neurons
do not respond to any of the training vectors.
decision boundary - a line, determined by the weight and bias vectors, for
which the net input n is zero.
A-3
A
Glossary
delta rule - the Widrow-Hoff rule.
delta vector - the delta vector for a layer is the derivative of a network’s output
error with respect to that layer’s net input vector.
distance - the distance between neurons, calculated from their positions with
a distance function.
early stopping - a technique based on dividing the data into three subsets. The
first subset is the training set used for computing the gradient and updating
the network weights and biases. The second subset is the validation set. When
the validation error increases for a specified number of iterations, the training
is stopped, and the weights and biases at the minimum of the validation error
are returned. The third set is the test set. It is used to verify the network
design.
epoch - the presentation of the set of training (input and/or target) vectors to
a network and the calculation of new weights and biases. Note that training
vectors may be presented one at a time or all together in a batch.
error jumping -a sudden increase in a network’s sum-squared error during
training. This is often due to too large a learning rate.
error ratio - a training parameter used with adaptive learning rate and
momentum training of backpropagation networks.
error vector - the difference between a network’s output vector in response to
an input vector and an associated target output vector.
feedback network - a network with connections from a layer’s output to that
layer’s input. The feedback connection may be direct or pass through several
layers.
feedforward network - a layered network in which each layer only receives
inputs from previous layers.
Fletcher-Reeves update - a method developed by Fletcher and Reeves for
computing a set of conjugate directions. These directions are used as search
directions as part of a conjugate gradient optimization procedure.
function approximation - the task performed by a network trained to respond
to inputs with an approximation of a desired function.
generalization - an attribute of a network whose output for a new input vector
tends to be close to outputs for similar input vectors in its training set.
A-4
generalized regression network - approximates a continuous function to an
arbitrary accuracy, given a sufficient number of hidden neurons.
global minimum - the lowest value of a function over the entire range of its
input parameters. Gradient descent methods adjust weights and biases in
order to find the global minimum of error for a network.
golden section search - a linear search which does not require the
calculation of the slope. The interval containing the minimum of the
performance is subdivided at each iteration of the search, and one subdivision
is eliminated at each iteration.
gradient descent - the process of making changes to weights and biases, where
the changes are proportional to the derivatives of network error with respect to
those weights and biases. This is done to minimize network error.
hard limit transfer function - a transfer that maps inputs greater-than or
equal-to 0 to 1, and all other values to 0.
Hebb learning rule - historically the first proposed learning rule for neurons.
Weights are adjusted proportional to the product of the outputs of pre- and
post-weight neurons.
hidden layer - a layer of a network that is not connected to the network output.
(For instance, the first layer of a two layer feedforward network.)
home neuron - a neuron at the center of a neighborhood.
hybrid bisection-cubicsearch - a line search that combines bisection and
cubic interpolation.
input layer - a layer of neurons receiving inputs directly from outside the
network.
initialization - the process of setting the network weights and biases to their
original values.
input space - the range of all possible input vectors.
input vector - a vector presented to the network.
input weights - the weights connecting network inputs to layers.
input weight vector - the row vector of weights going to a neuron.
Jacobian matrix - contains the first derivatives of the network errors with
respect to the weights and biases.
A-5
A
Glossary
Kohonen learning rule - a learning rule that trains selected neuron’s weight
vectors to take on the values of the current input vector.
layer - a group of neurons having connections to the same inputs and sending
outputs to the same destinations.
layer diagram - a network architecture figure showing the layers and the
weight matrices connecting them. Each layer’s transfer function is indicated
with a symbol. Sizes of input, output, bias and weight matrices are shown.
Individual neurons and connections are not shown. (See Chapter 2.)
layer weights - the weights connecting layers to other layers. Such weights
need to have non-zero delays if they form a recurrent connection (i.e. a loop).
learning - the process by which weights and biases are adjusted to achieve
some desired network behavior.
learning rate - a training parameter that controls the size of weight and bias
changes during learning.
learning rules - methods of deriving the next changes that might be made in
a network OR a procedure for modifying the weights and biases of a network.
Levenberg-Marquardt - an algorithm that trains a neural network 10 to 100
faster than the usual gradient descent backpropagation method. It will always
compute the approximate Hessian matrix, which has dimensions n × n .
line search function - procedure for searching along a given search direction
(line) to locate the minimum of the network performance.
linear transfer function - a transfer function that produces its input as its
output.
link distance - the number of links, or steps, that must be taken to get to the
neuron under consideration.
local minimum - the minimum of a function over a limited range of input
values. A local minimum may not be the global minimum.
log-sigmoid transfer function - a squashing function of the form shown below
that maps the input to the interval (0,1). (The toolbox function is logsig.)
1
f (n) = -----------------1 + e –n
A-6
Manhattan distance - The Manhattan distance between two vectors x and y
is calculated as:
D = sum(abs(x-y))
maximum performance increase - the maximum amount by which the
performance is allowed to increase in one iteration of the variable learning rate
training algorithm.
maximum step size - the maximum step size allowed during a linear search.
The magnitude of the weight vector is not allowed to increase by more than this
maximum step size in one iteration of a training algorithm.
mean square error function - the performance function that calculates the
average squared error between the network outputs a and the target outputs t.
momentum - a technique often used to make it less likely for a
backpropagation networks to get caught in a shallow minima.
momentum constant - A training parameter that controls how much
“momentum” is used.
mu parameter - the initial value for the scalar µ.
neighborhood - a group of neurons within a specified distance of a particular
neuron. The neighborhood is specified by the indices for all of the neurons that
lie within a radius d of the winning neuron i∗ :
N i ( d ) = { j, d ij ≤ d }
net input vector - the combination, in a layer, of all the layer’s weighted input
vectors with its bias.
neuron - the basic processing element of a neural network. Includes weights
and bias, a summing junction and an output transfer function. Artificial
neurons, such as those simulated and trained with this toolbox, are
abstractions of biological neurons.
neuron diagram - a network architecture figure showing the neurons and the
weights connecting them. Each neuron’s transfer function is indicated with a
symbol.
ordering phase - period of training during which neuron weights are expected
to order themselves in the input space consistent with the associated neuron
positions.
A-7
A
Glossary
output layer - a layer whose output is passed to the world outside the network.
output vector - the output of a neural network. Each element of the output
vector is the output of a neuron.
output weight vector - the column vector of weights coming from a neuron or
input. (See outstar learning rule.)
outstar learning rule - a learning rule that trains an neuron’s (or input’s)
output weight vector to take on the values of the current output vector of the
post-weight layer. Changes in the weights are proportional to the neuron’s
output.
overfitting - a case in which the error on the training set is driven to a very
small value, but when new data is presented to the network, the error is large.
pass - each traverse through all of the training input and target vectors.
pattern - a vector.
pattern association - the task performed by a network trained to respond with
the correct output vector for each presented input vector.
pattern recognition - the task performed by a network trained to respond
when an input vector close to a learned vector is presented. The network
“recognizes” the input as one of the original target vectors.
performance function - commonly the mean squared error of the network
outputs. However, the toolbox also considers other performance functions.
Type nnets and look under performance functions.
perceptron - a single-layer network with a hard limit transfer function. This
network is often trained with the perceptron learning rule.
perceptron learning rule - a learning rule for training single-layer hard limit
networks. It is guaranteed to result in a perfectly functioning network in finite
time given that the network is capable of doing so.
Polak-Ribiére update - a method developed by Polak and Ribiére for
computing a set of conjugate directions. These directions are used as search
directions as part of a conjugate gradient optimization procedure.
positive linear transfer function - a transfer function that produces an output
of zero for negative inputs and an output equal to the input for positive inputs.
postprocessing - converts normalized outputs back into the same units which
were used for the original targets.
A-8
Powell-Beale restarts - a method developed by Powell and Beale for
computing a set of conjugate directions. These directions are used as search
directions as part of a conjugate gradient optimization procedure. This
procedure also periodically resets the search direction to the negative of the
gradient.
preprocessing - perform some transformation of the input or target data
before it is presented to the neural network.
principal component analysis - orthogonalize the components of network
input vectors. This procedure can also reduce the dimension of the input
vectors by eliminating redundant components.
quasi-Newton algorithm - class of optimization algorithm based on
Newton’s method. An approximate Hessian matrix is computed at each
iteration of the algorithm based on the gradients.
radial basis networks - a neural network that can be designed directly by fitting
special response elements where they will do the most good.
radial basis transfer function - the transfer function for a radial basis neuron
is:
radbas ( n ) = e
–n
2
regularization - involves modifying the performance function, which is
normally chosen to be the sum of squares of the network errors on the training
set, by adding some fraction of the squares of the network weights.
resilient backpropagation - a training algorithm that eliminates the harmful
effect of having a small slope at the extreme ends of the sigmoid “squashing”
transfer functions.
saturating linear transfer function - a function that is linear in the interval
(-1,+1) and saturates outside this interval to -1 or +1. (The toolbox function is
satlin.)
scaled conjugate gradient algorithm - avoids the time consuming line
search of the standard conjugate gradient algorithm.
sequential input vectors - a set of vectors that are to be presented to a
network “one after the other.” The network weights and biases are adjusted on
the presentation of each input vector.
A-9
A
Glossary
sigma parameter - determines the change in weight for the calculation of the
approximate Hessian matrix in the scaled conjugate gradient algorithm.
sigmoid - monotonic S-shaped function mapping numbers in the interval
(-∞,∞) to a finite interval such as (-1,+1) or (0,1).
simulation - takes the network input p, and the network object net, and
returns the network outputs a.
spread constant - the distance an input vector must be from a neuron’s weight
vector to produce an output of 0.5.
squashing function - a monotonic increasing function that takes input values
between -∞ and +∞ and returns values in a finite interval.
star learning rule - a learning rule that trains a neuron’s weight vector to take
on the values of the current input vector. Changes in the weights are
proportional to the neuron’s output.
sum-squared error - The sum of squared differences between the network
targets and actual outputs for a given input vector or set of vectors.
supervised learning - a learning process in which changes in a network’s
weights and biases are due to the intervention of any external teacher. The
teacher typically provides output targets.
symmetric hard limit transfer function - a transfer that maps inputs
greater-than or equal-to 0 to +1, and all other values to -1.
symmetric saturating linear transfer function - produces the input as its
output as long as the input i in the range -1 to 1. Outside that range the output is
-1 and +1 respectively.
tan-sigmoid transfer function - a squashing function of the form shown below
that maps the input to the interval (-1,1). (The toolbox function is tansig.)
1
f (n) = -----------------1 + e –n
tapped delay line - a sequential set of delays with outputs available at each delay
output.
target vector - the desired output vector for a given input vector.
topology functions - ways to arrange the neurons in a grid, box, hexagonal, or
random topology.
A-10
training - a procedure whereby a network is adjusted to do a particular job
training vector - an input and/or target vector used to train a network.
transfer function - the function that maps a neuron’s (or layer’s) net output n
to its actual output.
tuning phase - period of SOFM training during which weights are expected to
spread out relatively evenly over the input space while retaining their
topological order found during the ordering phase.
underdetermined system - a system that has more variables than constraints.
unsupervised learning - a learning process in which changes in a network’s
weights and biases are not due to the intervention of any external teacher.
Commonly changes are a function of the current network input vectors, output
vectors, and previous weights and biases.
update - make a change in weights and biases. The update can occur after
presentation of a single input vector or after accumulating changes over
several input vectors.
weighted input vector - the result of applying a weight to a layer's input,
whether it is a network input or the output of another layer.
weight matrix - a matrix containing connection strengths from a layer’s inputs
to its neurons. The element wi,j of a weight matrix W refers to the connection
strength from input j to neuron i.
Widrow-Hoff learning rule - a learning rule used to trained single layer linear
networks. This rule is the predecessor of the backpropagation rule and is
sometimes referred to as the delta rule.
A-11
A
Glossary
A-12
B
Notation
Mathematical Notation for Equations and Figures
Basic Concepts . . . . . . . . . . . . . . . . .
Language . . . . . . . . . . . . . . . . . . . .
Weight Matrices . . . . . . . . . . . . . . . . .
Layer Notation . . . . . . . . . . . . . . . . .
Figure and Equation Examples . . . . . . . . . . .
Mathematics and Code Equivalents
.
.
.
.
.
.
.
.
.
.
.
.
B-2
B-2
B-2
B-2
B-2
B-3
. . . . . . . . . B-4
B
Notation
Mathematical Notation for Equations and Figures
Basic Concepts
Scalars-small italic letters.....a,b,c
Vectors - small bold non-italic letters.....a,b,c
Matrices - capital BOLD non-italic letters.....A,B,C
Language
Vector means a column of numbers.
Weight Matrices
Scalar Element w i, j ( t )
i - row, j - column, t - time or iteration
Matrix W ( t )
Column Vector w j ( t )
Row Vector iw ( t ) ...vector made of ith row of weight matrix W
Bias Vector
Scalar Element b i ( t )
Vector b ( t )
Layer Notation
A single superscript will be used to identify elements of layer. For instance, the
net input of layer 3 would be shown as n3.
Superscripts k, l will be used to identify the source (l) connection and the
destination (k) connection of layer weight matrices ans input weight matrices.
For instance, the layer weight matrix from layer 2 to layer 4 would be shown
as LW4,2.
B-2
Mathematical Notation for Equations and Figures
Input Weight Matrix IW
k, l
Layer Weight Matrix LW
k, l
Figure and Equation Examples
The following figure, taken from Chapter 11, “Advanced Topics,” illustrates
notation used in such advanced figures.
Inputs
p1(k)
2x1
Layers 1 and 2
AA
AA
AA
AAAA
AA
AAAA
AA
IW1,1
4x2
1
Layer 3
n1(k)
4x1
TDL
1
4
0,1
IW2,1
3 x (2*2)
p2(k)
5x1
5
TDL
1
IW2,2
3 x (1*5)
IW3,1
4x1
1x4
a1(k) = tansig (IW1,1p1(k) +b1)
TDL
LW3,3
1 x (1*1)
a1(k)
b1
4x1
2
AA
AA
A
A
AA
AA
A
AA AA
AA
AA
A
AA
AA
A
AA
A
A
1
b3
1x1
a2(k)
n2(k)
3x1
3x1
Outputs
n3(k)
a3(k)
y2(k)
1x1
1x1
1x1
1
LW3,2
1x3
y1(k)
3x1
3
a2(k) = logsig (IW2,1 [p1(k);p1(k-1) ]+ IW2,2p2(k-1)) a3(k)=purelin(LW3,3a3(k-1)+IW3,1 a1 (k)+b3+LW3,2a2 (k))
B-3
B
Notation
Mathematics and Code Equivalents
The transition from mathematics to code or vice versa can be made with the aid
of a few rules. They are listed here for future reference.
To change from Mathematics notation to MATLAB notation the user needs to:
• Change superscripts to cell array indices
1
For example, p → p { 1 }
• Change subscripts to parentheses indices
1
For example, p 2 → p ( 2 ) , and p 2 → p { 1 } ( 2 )
• Change parentheses indices to a second cell array index
1
For example, p ( k – 1 ) → p { 1, k – 1 }
• Change mathematics operators to MATLAB operators and toolbox functions
For example, ab → a*b
The following Equations illustrate the notation used in figures.
n = w 1, 1 p 1 + w 1, 2 p 2 + ... + w 1, R p R + b
w 1, 1 w 1, 2 … w 1, R
W =
w 2, 1 w 2, 2 … w 2, R
w S, 1 w S, 2 … w S, R
B-4
C
Bibliography
C
Bibliography
[Batt92] Battiti, R., “First and second order methods for learning: Between
steepest descent and Newton’s method,” Neural Computation, vol. 4, no. 2, pp.
141–166, 1992.
[Beal72] Beale, E. M. L., “A derivation of conjugate gradients,” in F. A.
Lootsma, ed., Numerical methods for nonlinear optimization, London:
Academic Press, 1972.
[Bren73] Brent, R. P., Algorithms for Minimization Without Derivatives,
Englewood Cliffs, NJ: Prentice-Hall, 1973.
[Caud89] Caudill, M., Neural Networks Primer, San Francisco, CA: Miller
Freeman Publications, 1989.
This collection of papers from the AI Expert Magazine gives an excellent
introduction to the field of neural networks. The papers use a minimum of
mathematics to explain the main results clearly. Several good suggestions for
further reading are included.
[CaBu92] Caudill, M., and C. Butler, Understanding Neural Networks:
Computer Explorations, Vols. 1 and 2, Cambridge, MA: the MIT Press, 1992.
This is a two volume workbook designed to give students “hands on” experience
with neural networks. It is written for a laboratory course at the senior or
first-year graduate level. Software for IBM PC and Apple Macintosh computers
is included. The material is well written, clear and helpful in understanding a
field that traditionally has been buried in mathematics.
[Char92] Charalambous, C.,“Conjugate gradient algorithm for efficient
training of artificial neural networks,” IEEE Proceedings, vol. 139, no. 3, pp.
301–310, 1992.
[ChCo91] Chen, S., C. F. N. Cowan, and P. M. Grant, “Orthogonal least
squares learning algorithm for radial basis function networks,” IEEE
Transactions on Neural Networks, vol. 2, no. 2, pp. 302-309, 1991.
This paper gives an excellent introduction to the field of radial basis functions.
The papers use a minimum of mathematics to explain the main results clearly.
Several good suggestions for further reading are included.
[DARP88] DARPA Neural Network Study, Lexington, MA: M.I.T. Lincoln
Laboratory, 1988.
This book is a compendium of knowledge of neural networks as they were
known to 1988. It presents the theoretical foundations of neural networks and
C-2
discusses their current applications. It contains sections on associative
memories, recurrent networks, vision, speech recognition, and robotics.
Finally, it discusses simulation tools and implementation technology.
[DeSc83] Dennis, J. E., and R. B. Schnabel, Numerical Methods for
Unconstrained Optimization and Nonlinear Equations, Englewood Cliffs, NJ:
Prentice-Hall, 1983.
[Elma90] Elman, J. L.,“Finding structure in time,” Cognitive Science, vol. 14,
pp. 179-211, 1990.
This paper is a superb introduction to the Elman networks described in
Chapter 10, “Recurrent Networks.”
[FlRe64] Fletcher, R., and C. M. Reeves, “Function minimization by conjugate
gradients,” Computer Journal, vol. 7, pp. 149-154, 1964.
[FoHa97] Foresee, F. D., and M. T. Hagan, “Gauss-Newton approximation to
Bayesian regularization,” Proceedings of the 1997 International Joint
Conference on Neural Networks,, pages 1930-1935, 1997.
[GiMu81] Gill, P. E., W. Murray, and M. H. Wright, Practical Optimization,
New York: Academic Press, 1981.
[Gros82] Grossberg, S., Studies of the Mind and Brain, Drodrecht, Holland:
Reidel Press, 1982.
This book contains articles summarizing Grossberg’s theoretical
psychophysiology work up to 1980. Each article contains a preface explaining
the main points.
[HaMe94] Hagan, M. T., and M. Menhaj, “Training feedforward networks with
the Marquardt algorithm,” IEEE Transactions on Neural Networks, vol. 5, no. 6, pp.
989–993, 1994.
This paper reports the first development of the Levenberg-Marquardt
algorithm for neural networks. It describes the theory and application of the
algorithm, which trains neural networks at a rate 10 to 100 times faster than
the usual gradient descent backpropagation method.
[HDB96] Hagan, M. T., H. B. Demuth, and M. H. Beale, Neural Network
Design, Boston, MA: PWS Publishing, 1996.
[Hebb49] Hebb, D. O., The Organization of Behavior, New York: Wiley, 1949.
C-3
C
Bibliography
This book proposed neural network architectures and the first learning rule.
The learning rule is used to form a theory of how collections of cells might form
a concept.
[Himm72] Himmelblau, D. M., Applied Nonlinear Programming, New York:
McGraw-Hill, 1972.
[Joll86] Jolliffe, I. T., Principal Component Analysis, New York:
Springer-Verlag, 1986.
[Koho87] Kohonen, T., Self-Organization and Associative Memory, 2nd
Edition, Berlin: Springer-Verlag, 1987.
This book analyzes several learning rules. The Kohonen learning rule is then
introduced and embedded in self-organizing feature maps. Associative
networks are also studied.
[LiMi89] Li, J., A. N. Michel, and W. Porod, “Analysis and synthesis of a class
of neural networks: linear systems operating on a closed hypercube,” IEEE
Transactions on Circuits and Systems, vol. 36, no. 11, pp. 1405-1422, 1989.
This paper discusses a class of neural networks described by first order linear
differential equations that are defined on a closed hypercube. The systems
considered retain the basic structure of the Hopfield model but are easier to
analyze and implement. The paper presents an efficient method for
determining the set of asymptotically stable equilibrium points and the set of
unstable equilibrium points. Examples are presented. The method of Li et. al.
is implemented in Chapter 9 of this Users Guide.
[Lipp87] Lippman, R. P., “An introduction to computing with neural nets,”
IEEE ASSP Magazine, pp. 4-22, 1987.
This paper gives an introduction to the field of neural nets by reviewing six
neural net models that can be used for pattern classification.The paper shows
how existing classification and clustering algorithms can be performed using
simple components that are like neurons. This is a highly readable paper.
[MacK92] MacKay, D. J. C., “Bayesian interpolation,” Neural Computation,
vol. 4, no. 3, pp. 415-447, 1992.
[McPi43] McCulloch, W. S., and W. H. Pitts, “A logical calculus of ideas
immanent in nervous activity,” Bulletin of Mathematical Biophysics, vol. 5, pp.
115-133, 1943.
C-4
A classic paper that describes a model of a neuron that is binary and has a fixed
threshold. A network of such neurons can perform logical operations.
[Moll93] Moller, M. F., “A scaled conjugate gradient algorithm for fast
supervised learning,” Neural Networks, vol. 6, pp. 525-533, 1993.
[NgWi89] Nguyen, D., and B. Widrow, “The truck backer-upper: An example
of self-learning in neural networks,” Proceedings of the International Joint
Conference on Neural Networks, vol 2, pp. 357-363, 1989.
This paper describes a two-layer network that first learned the truck dynamics
and then learned how to back the truck to a specified position at a loading dock.
To do this, the neural network had to solve a highly nonlinear control systems
problem.
[NgWi90] Nguyen, D., and B. Widrow, “Improving the learning speed of
2-layer neural networks by choosing initial values of the adaptive weights,”
Proceedings of the International Joint Conference on Neural Networks, vol 3,
pp. 21-26, 1990.
Nguyen and Widrow demonstrate that a 2-layer sigmoid/linear network can be
viewed as performing a piecewise linear approximation of any learned
function. It is shown that weights and biases generated with certain
constraints will result in an initial network better able to form a function
approximation of an arbitrary function. Use of the Nguyen-Widrow (instead of
purely random) initial conditions often shortens training time by more than an
order of magnitude.
[Powe77] Powell, M. J. D., “Restart procedures for the conjugate gradient
method,” Mathematical Programming, vol. 12, pp. 241-254, 1977.
[Pulu92] N. Purdie, E.A. Lucas and M.B. Talley, "Direct measure of total
cholesterol and its distribution among major serum lipoproteins," Clinical
Chemistry, vol. 38, no. 9, pp. 1645-1647, 1992.
[RiBr93] Riedmiller, M., and H. Braun, “A direct adaptive method for faster
backpropagation learning: The RPROP algorithm,” Proceedings of the IEEE
International Conference on Neural Networks, 1993.
[Rose61] Rosenblatt, F., Principles of Neurodynamics, Washington D.C.:
Spartan Press, 1961.
This book presents all of Rosenblatt’s results on perceptrons. In particular, it
presents his most important result, the perceptron learning theorem.
C-5
C
Bibliography
[RuHi86a] Rumelhart, D. E., G. E. Hinton, and R. J. Williams, “Learning
internal representations by error propagation,”, in D. E. Rumelhart and J. L.
McClelland, eds. Parallel Data Processing, vol.1, Cambridge, MA: The M.I.T.
Press, pp. 318-362, 1986.
This is a basic reference on backpropagation.
[RuHi86b] Rumelhart, D. E., G. E. Hinton, and R. J. Williams, “Learning
representations by back-propagating errors,” Nature, vol. 323, pp. 533–536,
1986.
[RuMc86] Rumelhart, D. E., J. L. McClelland, and the PDP Research Group,
eds., Parallel Distributed Processing, Vols. 1 and 2, Cambridge, MA: The M.I.T.
Press, 1986.
These two volumes contain a set of monographs that present a technical
introduction to the field of neural networks. Each section is written by different
authors. These works present a summary of most of the research in neural
networks to the date of publication.
[Scal85] Scales, L. E., Introduction to Non-Linear Optimization, New York:
Springer-Verlag, 1985.
[VoMa88] Vogl, T. P., J. K. Mangis, A. K. Rigler, W. T. Zink, and D. L. Alkon,
“Accelerating the convergence of the backpropagation method,” Biological
Cybernetics, vol. 59, pp. 256-264, 1988.
Backpropagation learning can be speeded up and made less sensitive to small
features in the error surface such as shallow local minima by combining
techniques such as batching, adaptive learning rate, and momentum.
[Wass93] Wasserman, P. D., Advanced Methods in Neural Computing, New
York: Van Nostrand Reinhold, 1993.
[WiHo60] Widrow, B., and M. E. Hoff, “Adaptive switching circuits,” 1960 IRE
WESCON Convention Record, New York IRE, pp. 96-104, 1960.
[WiSt85] Widrow, B., and S. D. Sterns, Adaptive Signal Processing, New York:
Prentice-Hall, 1985.
This is a basic paper on adaptive signal processing.
C-6
D
Demonstrations and
Applications
Tables of Demonstrations and Applications .
Chapter 2 Neuron Model & Network Architectures .
Chapter 3 Perceptrons . . . . . . . . . . . .
Chapter 4 Adaptive Linear Filters . . . . . . . .
Chapter 5 Backpropagation . . . . . . . . . .
Chapter 6 Radial Basis Networks . . . . . . . .
Chapter 7 Self-Organizing Networks . . . . . . .
Chapter 8 Learning Vector Quantization . . . . .
Chapter 9 Recurrent Networks . . . . . . . . .
Chapter 10 Applications . . . . . . . . . . . .
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D-2
D-2
D-2
D-3
D-3
D-4
D-4
D-4
D-5
D-5
D
Demonstrations and Applications
Tables of Demonstrations and Applications
Chapter 2 Neuron Model & Network Architectures
File Name
Page
Simple neuron and transfer functions
nnd2n1
page 2-6
Neuron with vector input
nnd2n2
page 2-9
File Name
Page
Decision Boundaries
nnd4db
page 3-5
Perceptron learning rule. Pick boundaries
nnd4pr
page 3-14
Classification with a 2-input perceptron
demop1
page 3-20
Outlier input vectors
demop4
page 3-22
Normalized perceptron rule
demop5
page 3-22
Linearly non-separable vectors
demop6
page 3-21
Chapter 3 Perceptrons
D-2
Tables of Demonstrations and Applications
Chapter 4 Adaptive Linear Filters
File Name
Page
Pattern association showing error surface
demolin1
page 4-10
Training a linear neuron
demolin2
page 4-15
Linear classification system
nnd10lc
page 4-15
Adaptive noise cancellation, Toolbox Example
demolin8
page 4-10
Adaptive noise cancellation in airplane cockpit
nnd10nc
page 4-21
Linear fit of nonlinear problem
demolin4
page 4-15
Underdetermined problem
demolin5
page 4-25
Linearly dependent problem
demolin6
page 4-26
Too large a learning rate
demolin7
page 4-26
File Name
Page
Generalization
nnd11gn
page 5-37
Steepest descent backpropagation
nnd12sd1
page 5-13
nnd12mo
page 5-15
nnd12vl
page 5-17
nnd12cg
page 5-22
nnd12m
page 5-33
Chapter 5 Backpropagation
Momentum backpropagation
Variable learning rate backpropagation
Conjugate gradient backpropagation
Marquardt backpropagation
D-3
D
Demonstrations and Applications
Chapter 6 Radial Basis Networks
File Name
Page
Radial basis approximation
demorb1
page 6-8
Radial basis underlapping neurons
demorb3
page 6-8
Radial basis overlapping neurons
demorb4
page 6-8
GRNN Function Approximation
demogrn1
page 6-11
PNN Classification
demopnn1
page 6-14
Chapter 7 Self-Organizing Networks
File Name
Page
Competitive learning
democ1
page 7-9
One-Dimensional Self-organizing map
demosm1
page 7-25
Two-Dimensional Self-organizing map
demosm2
page 7-25
Chapter 8 Learning Vector Quantization
Learning vector quantization
D-4
File Name
Page
demolvq1
page 8-12
Tables of Demonstrations and Applications
Chapter 9 Recurrent Networks
File Name
Page
Hopfield two neuron design
demohop1
page 9-15
Hopfield unstable equilibria
demohop2
page 9-15
Hopfield three neuron design
demohop3
page 9-15
Hopfield spurious stable points
demohop4
page 9-15
File Name
Page
Linear design
applin1
page 10-3
Adaptive linear prediction
applin2
page 10-7
Linear system identification
applin3
page 10-11
Adaptive linear system identification
applin4
page 10-15
Elman amplitude detection
appelm1
page 10-19
Nonlinear control system identification
appcs1
page 10-24
Model Reference Control
appcs2
page 10-30
Character Recognition
appcr1
page 10-38
Chapter 10 Applications
D-5
D
Demonstrations and Applications
D-6
E
Simulink
Block Set . . . . . . .
Transfer Function Blocks
Net Input Blocks . . . .
Weight Blocks . . . . .
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E-2
E-3
E-3
E-4
Block Generation . . . . . . . . . . . . . . . . . . E-5
Example . . . . . . . . . . . . . . . . . . . . . . E-5
Exercises . . . . . . . . . . . . . . . . . . . . . . E-7
E
Simulink
Block Set
The Neural Network Toolbox provides a set of blocks you can use to build
neural networks in Simulink or which can be used by the function gensim to
generate the Simulink version of any network you have created in MATLAB.
Bring up the Neural Network Toolbox block set with this command:
neural
The result will be a window, which contains three blocks. Each of these block
contains additional blocks.
E-2
Block Set
Transfer Function Blocks
Double click on the Transfer Functions block in the Neural window to bring
up a window containing several transfer function blocks.
Each of these blocks takes a net input vector and generates a corresponding
output vector whose dimensions are the same as the input vector.
Net Input Blocks
Double click on the Net Input Functions block in the Neural window to
bring up a window containing two net input function blocks.
Each of these blocks takes any number of weighted input vectors, weight layer
output vectors, and bias vectors, and returns a net input vector.
E-3
E
Simulink
Weight Blocks
Double click on the Weight Functions block in the Neural window to bring
up a window containing three weight function blocks.
Each of these blocks takes a neuron’s weight vector and applies it to an input
vector (or a layer output vector) to get a weighted input value for a neuron.
It is important to note that the blocks above expect the neuron’s weight vector
to be defined as a column vector. This is because Simulink signals can be
column vectors, but cannot be matrices or row vectors.
It is also important to note that because of this limitation you will have to
create S weight function blocks, one for each row, to implement a weight matrix
going to a layer with S neurons.
This contrasts with the other two kinds of blocks. Only one net input function
and one transfer function block are required for each layer.
E-4
Block Generation
Block Generation
The function gensim generates block descriptions of networks so you can
simulate them in Simulink.
gensim(net,st)
The second argument to gensim determines the sample time, which is normally
chosen to be some positive real value.
If a network has no delays associated with its input weights or layer weights
this value can be set to -1. A value of -1 tells gensim to generate a network with
continuous sampling.
Example
Here is a simple problem defining a set of inputs p and corresponding targets t:
p = [1 2 3 4 5];
t = [1 3 5 7 9];
The code below designs a linear layer to solve this problem.
net = newlind(p,t)
We can test the network on our original inputs with sim.
y = sim(net,p)
The results returned show the network has solved the problem.
y =
1.0000
3.0000
5.0000
7.0000
9.0000
Call gensim as follows to generate a Simulink version of the network:
gensim(net,-1)
The second argument is -1 so the resulting network block will sample
continuously.
The call to gensim results in the following screen appearing. It contains a
Simulink system consisting of the linear network connected to a sample input
and a scope.
E-5
E
Simulink
To test the network double click on the Input 1 block at left.
The input block is actually a standard Constant block. Change the constant
value from the initial randomly generated value to 2, then select Close.
Select Start from the Simulation menu. Simulink will momentarily pause as
it simulates the system.
When the simulation is over double click the scope at the right to see the
following display of the network’s response.
E-6
Block Generation
Note that the output is 3, which is the correct output for an input of 2.
Exercises
Here are a couple of exercises you might try.
Changing Input Signal
Replace the constant input block with a signal generator from the standard
Simulink block set Sources. Simulate the system and view the network’s
response.
Discrete Sample Time
Recreate the network, but with a discrete sample time of 0.5, instead of
continuous sampling.
gensim(net,0.5)
Again replace the constant input with a signal generator. Simulate the system
and view the network’s response.
E-7
E
Simulink
E-8
Index
A
ADALINE network
decision boundary 4-6
Adaption 5-9
custom function 11-32
definition 3-15
function 12-10
parameters 12-13
Adaptive filter 4-17
example 4-18
noise cancellation example 4-21
prediction application 10-7
prediction example 4-20
system identification application 10-15
training 2-20
Adaptive linear networks 4-2
Amplitude detection 10-19
Applications
adaptive filtering 4-16
aerospace 1-16
automotive 1-16
banking 1-16
defense 1-16
electronics 1-16
entertainment 1-17
financial 1-17
insurance 1-17
manufacturing 1-17
medical 1-17, 5-49
oil and gas exploration 1-17
robotics 1-17
speech 1-18
telecommunications 1-18
transportation 1-18
Architecture
bias connection 11-6, 12-3
input connection 11-6, 12-4
input delays 12-5
layer connection 11-6, 12-4
layer delays 12-6
number of inputs 11-5, 12-2
number of layers 11-5, 12-2
number of outputs 11-7, 12-5
number of targets 11-7, 12-5
output connection 11-7, 12-4
target connection 11-7, 12-5
B
Backpropagation 5-2
algorithm 5-9
example 5-49
Backtracking search 5-28
Batch training 2-20, 2-22, 5-12
dynamic networks 2-24
static networks 2-22
Bayesian framework 5-39
BFGS quasi-Newton algorithm 5-29
Bias
connection 11-6
definition 2-4
initialization function 12-26
learning 12-26
learning function 12-27
learning parameters 12-27
subobject 11-11, 12-26
value 11-13, 12-16
Box distance 7-17
Brent’s search 5-27
I-1
Index
C
Cell array 2-3, 2-17, 2-18, 2-19, 2-21, 2-22, 2-24,
11-8, 11-9, 11-13, 11-14, 11-29
Charalambous’ search 5-28
Classification 6-12
input vectors 3-4
linear 4-13
regions 3-5
Code
mathematical equivalents 2-2, 2-12
perceptron network 3-7
writing 2-7
Competitive layer 7-3
Competitive neural network 7-4
example 7-8
Competitive transfer function 6-12, 7-3, 7-19
Concurrent inputs 2-15, 2-18
Conjugate gradient algorithm 5-20
Fletcher-Reeves update 5-21
Polak-Ribiere update 5-22
Powell-Beale restarts 5-24
scaled 5-25
Custom
neural network 11-2
D
Dead neurons 7-5
Decision boundary 4-6
definition 3-5
Demonstrations
appcs1 10-24
appelm1 10-19
applin3 10-11
applin4 10-15
definition 1-4
demohop1 9-15
I-2
demohop2 9-15
demolin4 4-25
demorb4 6-8
nnd10lc 4-15
nnd11gn 5-37
nnd12cg 5-22
nnd12m 5-33
nnd12mo 5-15
nnd12sd1 5-13, 5-26
nnd12vl 5-17
Distance 7-10, 7-16
box 7-17
custom function 11-40
Euclidean 7-16
link 7-18
Manhattan 7-18
tuning phase 7-20
Dynamic network 2-16, 2-18
Dynamic networks
training 2-22, 2-24
E
Early stopping 1-9, 5-41
Elman network 9-3
recurrent connection 9-3
Euclidean distance 7-16
F
Feedforward network 5-5
Finite impulse response filter 4-17
Fletcher-Reeves update 5-21
Index
G
Generalization 5-37
regularization 5-38
Generalized regression network 1-11, 6-9
Golden section search 5-26
Gradient descent algorithm 5-2, 5-9
batch 5-12, 5-33
with momentum 5-11, 5-13
Gridtop topology 7-12
H
Hard limit transfer function 2-5, 2-28, 3-4
Heuristic techniques 5-16
Hextop topology 7-12
Hidden layer
definition 2-14
Home neuron 7-16
Hopfield network
architecture 9-9
design equilibrium point 9-10
solution trajectories 9-15
stable equilibrium point 9-11
target equilibrium points 9-11
Hybrid bisection-cubic search 5-27
I
Identification
adaptive 10-15
linear 10-11
nonlinear system application 10-24
Incremental training 2-20
Initial step size function 5-19
Initialization
additional functions 11-17
custom function 11-26
definition 3-9
function 5-7, 12-10
parameters 12-13
Input
concurrent 2-15
connection 11-6
number 11-5
range 12-17
size 12-17
subobject 11-8, 11-9, 12-17
Input vector
classification 3-4
dimension reduction 5-46
distance 7-10
outlier 3-21
topology 7-10
Input weights
definition 2-12
Inputs
concurrent 2-18
sequential 2-15, 2-16
Installation
guide 1-4
J
Jacobian matrix 5-32
K
Kohonen learning rule 7-5
I-3
Index
L
Lambda parameter 5-25
Layer
connection 11-6
dimensions 12-18
distance function 12-19
distances 12-19
initialization function 12-20
net input function 12-20
number 11-5
positions 12-21
size 12-22
subobject 12-18
topology function 12-22
transfer function 12-23
Layer weights
definition 2-12
Learning rate 5-10
adaptive 5-17
maximum stable 4-12
optimal 5-16
ordering phase 7-20
too large 4-26
tuning phase 7-20
Learning rules 3-2
custom 11-36
Hebb 11-18
Hebb with decay 11-18
instar 11-18
Kohonen 7-5
outstar 11-18
supervised learning 3-11
unsupervised learning 3-11
Widrow-Hoff 4-2, 4-5, 4-11, 5-2
Learning vector quantization 8-2
creation 8-5
learning rule 8-9
I-4
LVQ network 8-3
subclasses 8-3
target classes 8-3
union of two sub-classes 8-7
Least mean square error 4-9
Levenberg-Marquardt algorithm 5-31
definition 1-6
reduced memory 5-33
Line search function 5-21
backtracking search 5-28
Brent’s search 5-27
Charalambous’ search 5-28
Golden section search 5-26
hybrid bisection-cubic search 5-27
Linear networks
design 4-10
Linear transfer function 2-6, 2-28, 4-4, 5-4
Linearly dependent vectors 4-26
Link distance 7-18
Log-sigmoid transfer function 2-6, 2-29, 5-3
M
MADALINE 4-5
Manhattan distance 7-18
Maximum performance increase 5-14
Maximum step size 5-19
Mean square error function 5-8
least 4-9
Memory reduction 5-34
Model reference control 10-30
Momentum constant 5-11
Mu parameter 5-32
Index
N
Neighborhood 7-10
Net input function
custom 11-22
Network
definition 11-5
dynamic 2-16, 2-18
static 2-15
Network Function 11-12
Network layer
competitive 7-3
definition 2-8
Neural network
adaptive linear 4-2
competitive 7-4
custom 11-2
definition 1-2
feedforward 5-5
generalized regression 6-9
multiple layer 2-13, 4-23, 5-2
one layer 2-10, 3-6, 4-5
probabilistic 6-12
radial basis 6-2
self organizing 7-2
self-organizing feature map 7-10
Neuron
dead (not allocated) 7-5
definition 2-4
home 7-16
Newton’s method 5-32
Normalization
inputs and targets 5-44
mean and standard deviation 5-45
Notation
abbreviated 2-7, 4-24
layer 2-13
. mathematical 2-2
transfer function symbols 2-6, 2-9
Numerical optimization 5-16
O
One step secant algorithm 5-30
Ordering phase learning rate 7-20
Outlier input vector 3-21
Output
connection 11-7
number 11-7
size 12-25
subobject 11-10, 12-25
Output layer
definition 2-14
linear 5-5
Overdetermined systems 4-25
Overfitting 5-37
P
Pass
definition 3-15
pattern recognition 10-38
Perceptron learning rule 3-2, 3-12
normalized 3-22
Perceptron network 3-2
code 3-7
creation 3-3
limitations 3-21
I-5
Index
S
Performance function 12-11
custom 11-34
modified 5-38
parameters 12-13
Polak-Ribiere update 5-22
Postprocessing 5-44
Post-training analysis 5-47
Powell-Beale restarts 5-24
Preprocessing 5-44
Principal component analysis 5-46
Probabilistic neural network 6-12
design 6-13
Q
Quasi-Newton algorithm 1-7, 5-28
BFGS 1-7, 5-29
R
Radial basis
design 6-11
efficient network 6-7
function 6-2
network 6-2
network design 6-5
Radial basis transfer function 6-4
Randtop topology 7-12
Recurrent connection 9-3
Recurrent networks 9-2
Regularization 1-9, 5-38
automated 5-39
Resilient backpropagation 5-18
training algorithm 1-7
I-6
Self-organizing feature map (SOFM)
network 7-10
neighborhood 7-10
one-dimensional example 7-25
two-dimensional example 7-27
Self-organizing networks 7-2
Sequential inputs 2-15, 2-16
S-function 13-2
Sigma parameter 5-25
Simulation 5-8
definition 3-8
Simulink
generating networks E-5
NNT block set E-2
Simulink support 1-13
Speed comparison 1-8, 5-35
Spread constant 6-5
Squashing functions 5-18
Static network 2-15
Static networks
batch training 2-22
training 2-20
Subobject
bias 11-11, 12-8, 12-26
input 11-8, 11-9, 12-7, 12-17
layer 12-7, 12-18
output 11-10, 12-7, 12-25
target 11-10, 12-8, 12-25
weight 11-11, 12-9, 12-28, 12-32
Supervised learning 3-11
target output 3-11
training set 3-11
Index
T
Tan-sigmoid transfer function 5-4
Tapped delay line 4-16
Target
connection 11-7
number 11-7
size 12-25
subobject 11-10, 12-25
Target output 3-11
Topologies
gridtop 7-12
hextop 7-12
randtop 7-12
Topologies for SOFM neuron locations 7-12
Topology 7-10
custom function 11-39
Training 5-8
batch 2-20, 5-12
competitive networks 7-6
custom function 11-29
definition 2-4, 3-2
efficient 5-44
faster 5-16
function 12-11
incremental 2-20, 5-9
ordering phase 7-22
parameters 12-14
post-training analysis 5-47
self organizing feature map 7-22
styles 2-20
tuning phase 7-22
Training set 3-11
Training styles 2-20
Training with noise 10-41
Transfer function
competitive 6-12, 7-3, 7-19
custom 11-20
definition 2-4
derivatives 5-4
hard limit 2-5, 3-4
linear 4-4, 5-4
log-sigmoid 2-6, 2-29, 5-3
radial basis 6-4
saturating linear 11-17
soft maximum 11-17
tan-sigmoid 5-4
triangular basis 11-17
Transformation matrix 5-46
Tuning phase learning rate 7-20
Tuning phase neighborhood distance 7-20
U
Underdetermined systems 4-25
Unsupervised learning 3-11
V
Variable learning rate algorithm 5-17
Vectors
linearly dependent 4-26
I-7
Index
W
Weight
definition 2-4
delays 12-28, 12-32
initialization function 12-29, 12-32
learning 12-29, 12-33
learning function 12-30, 12-33
learning parameters 12-31, 12-35
size 12-31, 12-35
subobject 11-11, 12-28, 12-32
value 11-13, 12-14, 12-15
weight function 12-32, 12-36
I-8
Weight function
custom 11-24
Weight matrix
definition 2-10
Widrow-Hoff learning rule 4-2, 4-5, 4-11, 5-2
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