Viability of Artificial Neural Networks in Mobile Health- care Gavin Harper

Viability of Artificial Neural Networks in Mobile Health- care Gavin Harper
Gavin Harper
Viability of Artificial Neural Networks in Mobile Healthcare
Viability of Artificial Neural Networks in Mobile Healthcare
Gavin Harper
Bachelor’s Thesis
Spring 2011
Degree programme in
Business Information Technology
Oulu University of Applied Sciences
Abstract
Oulu University of Applied Sciences
Degree in Business Information Technology
Author(s): Gavin Harper
Title of Bachelor’s thesis: Viability of Artificial Neural Networks in Mobile Healthcare
Supervisor(s): Anu Niva
Term and year of completion: Spring 2011
Number of pages: 43 + 17 appendices
ABSTRACT:
This thesis describes the design process of an artificially intelligent algorithm capable
of differentiating between two discrete electrocardiogram readings (a signal depicting
the electrical activity of the heart at a given point in time). The paper culminates in
a reduced form implementation of one such algorithm utilising the neural computing
paradigm. Algorithms implemented under the neural computing paradigm mimic the ascurrently perceived operation of the brain cells known as neurons. By creating a network
of nodes (neurons) and altering the connections between them, an algorithm can display
rudimentary forms of what may be classified as intelligence by displaying an ability to
recognise patterns in data and adapt to better recall those patterns.
The theoretical section of this paper discusses first those concepts of electrocardiography
that directly relate to the research in this paper. The remainder of the theoretical section
is dedicated to neural computing. By reading this paper, a reader not previously familiar
with the subject should understand the various factions of the artificial intelligence community and both the operation and purpose of neural computing and what role it may
potentially play in future mobile healthcare developments.
Keywords:
Artificial Neural Networks, Electrocardiogram, ECG, Heart
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Contents
1 Introduction
1.1 Research Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Electrocardiograms
2.1 Cardiac Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Interpreting the Electrocardiogram . . . . . . . . . . . . . . . . . . . . .
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3 Neural Networks
3.1 Biological Neural Networks . . . . . . . . .
3.2 Artificial Neural Networks . . . . . . . . .
3.3 Neural Network Infrastructure . . . . . . .
3.3.1 Generic Topology . . . . . . . . . .
3.3.2 Weights, Bias and Directionality . .
3.4 Nonlinear Activation Functions . . . . . .
3.5 Neural Network Operation . . . . . . . . .
3.6 Training and Validating a Neural Network
3.6.1 Backpropagation . . . . . . . . . .
3.6.2 Simulated Annealing . . . . . . . .
3.6.3 Genetic Algorithms . . . . . . . . .
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4 Implementation
4.1 Formal Objective Parametrisation
4.2 Input Data Preparation . . . . .
4.2.1 Sample A - Healthy . . . .
4.2.2 Sample B - Unhealthy . .
4.3 Neural Network Training . . . . .
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5 Results
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6 Conclusion
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7 Discussion
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8 Bibliography
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A Constants and Notations
A.1 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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B GIT Log
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C Makefile
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D Source Code
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List of Figures
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Idealised Representation of PQRST . . . . . . . . . . . . . . . . . . . . .
Generic Neural Network Topology Graph . . . . . . . . . . . . . . . . . .
Adjacency Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inclusion of Weights, Bias and Directionality into the Network Topology
Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Row Matrix for Input and Hidden Layers . . . . . . . . . . . . . . . . . .
Logistic Activation Function . . . . . . . . . . . . . . . . . . . . . . . . .
Logistic Activation Function Plot . . . . . . . . . . . . . . . . . . . . . .
Hyperbolic Tangent Activation Function . . . . . . . . . . . . . . . . . .
Hyperbolic Tangent Activation Function Plot . . . . . . . . . . . . . . .
Calculating the Output of a Neuron . . . . . . . . . . . . . . . . . . . . .
Neural Network Calculation Example Input Matrix . . . . . . . . . . . .
Example of Logistic Activation Function Output - Scaled and Unscaled .
Neural Network Calculation Example Graph . . . . . . . . . . . . . . . .
Root Mean Squared Error (RMSE) . . . . . . . . . . . . . . . . . . . . .
Derivative of the Logistic Activation Function . . . . . . . . . . . . . . .
Step Function for Temperature Decrease in Simulated Annealing . . . . .
Sample A - Healthy Visualisation . . . . . . . . . . . . . . . . . . . . . .
Sample A - Extracted Points . . . . . . . . . . . . . . . . . . . . . . . . .
Sample B - Unhealthy Visualisation . . . . . . . . . . . . . . . . . . . . .
Sample B - Extracted Points . . . . . . . . . . . . . . . . . . . . . . . . .
Simulated Annealing Best Error Function . . . . . . . . . . . . . . . . . .
Simulated Annealing Global Error Function . . . . . . . . . . . . . . . .
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1
Introduction
Mobile healthcare is a rapidly emerging subset of the healthcare industry focusing on
providing medical assistance to those who may be unable, or lack the desire to submit
to traditional healthcare methods carried out at one distinct geographical location such
as a hospital. The premise driving the mobile healthcare industry centres around the
notion that recent advances in technologies such as wireless communication and portable
computation provide a powerful platform for automated procurement, analysis and transmission of patient data such as the activity of the heart at a given point in time. This
paper focuses primarily on developing automated, mobile means of intelligently analysing
the state of a patient’s heart. It is hoped that should the technology powering mobile
healthcare attain suitable accuracy and efficiency, such that wide scale deployment is
possible, these technologies may serve to the betterment of those unable to submit to
traditional medical treatment.
It is exciting to consider a device that may be worn by a patient in an unobtrusive,
discrete manner capable of intelligently analysing the electrical activity of the heart and
alerting at some early stage in the event of an anomaly or onset of arrhythmia. Such
early warning under traditional methods requires the patient to be located in a hospital.
However, if a device could conform to certain expectations such as extensive battery life,
low mass, accurate analysis and high efficiency, it is not unreasonable to suggest that
those at risk to a given heart condition can monitor their health over a long period of
time while attending to their daily life.
Utilising additional technologies such as the Global Positioning System (GPS) and 3G/High
Speed Downlink Packet Access (HSDPA) which may be trivially implemented in small
devices as evidenced by the current generation of mobile phone technology, provides additional benefits. Accessing the internet by means of a cell phone tower allows a mobile
electrocardiogram device to communicate with a centralised server and push updates and
warnings in real-time so ensure that in the event of the onset of arrhythmia or another
undesirable state. This functionality may be coupled with the Global Positioning System to push location updates to the server in the event that medical assistance is required.
This research in this paper is conducted on behalf of Domuset Oy, one such company that
seeks to develop devices similar to those described above. This paper represents a small
subset of the research conducted into developing intelligent signal analysis algorithms
that may be deployed on low powered, portable devices. The research and problem set
7
has been adapted to adhere to requirements that dictate both the spacial restrictions and
the assumed knowledge of the reader. It is such that a reader with little to no background
in mathematics, signal analysis or electrocardiography may still follow the progression of
this paper as the writing style has been intentionally structured to introduce new concepts in a logical and progressive manner.
1.1
Research Methods
Printed material has been employed as the preferential method of information gathering
for those sections relying on mathematical accuracy as there exists many highly respected
texts with a proven track record in the field of adaptive learning algorithms. Journals
have proven to be an invaluable source for obtaining a historical perspective on matters
such as the early history of artificial intelligence. For such topics of research that printed
material was not readily available in the time frame required, electronic sources such as
internet based material were employed. It is important to note that the preference of
printed material in no way represents the opinion that electronic material is sub par or
any less valuable to the research process. Instead it should be noted that a familiarity
with many of the printed sources pertaining to the field of artificial intelligence and adaptive learning algorithms was in existence prior to commencing this research project.
In addition to traditional methods of information gathering, regular meetings with the
commissioner in attendance have proven to be instrumental in directing the research
project. It is the case that many prior methods for analysing the datasets discussed
in this paper have been discussed and over a period of time, optimal solutions became
apparent. The informal nature of these meetings allowed a focus on the flow of ideas and
as such served to simultaneously educate about the subject of electrocardiograms and
the subsequent analysis methods and also to invigorate the need to continue researching
the topics discussed previously. While this paper prefers to focus on a quantitative
approach to data collection and analysis due to the highly nature of the research. It
would, however, be factually inaccurate to state that a qualitative approach had no place
during this research project. Simple encounters such as the meetings described were
highly useful.
1.2
Research Objective
The objective of this research is to develop a proof-of-concept algorithm capable of differentiating between datasets consisting of distinct regions of electrocardiogram readings.
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The resulting algorithm is to display characteristics that may be considered intelligent,
such as pattern recognition and adaptive behaviour. In addition, it is important that the
algorithm be designed in such a way that resources usage and complexity is minimised
allowing deployment in low-powered devices such as mobile telephones and dedicated
hardware.
The scope of this research extends to demonstrating the suitability and in some cases,
the necessity of adaptive learning algorithms for analysing and interpreting nontrivial
datasets such as those obtained from natural processes. This paper will present a subset
of the research into classifying electrocardiogram readings with adaptive learning algorithms carried out for the larger. As such, the problem as it appears in this paper may
be considered to be in reduced form for brevity.
All code samples and algorithms will be presented utilising the C programming language.
C has been chosen due to the inherent portability of the language and the speed benefits
that arise from employing a compiled language. In addition, the scripts required to build
a working implementation of the algorithm will be provided in the form of a GNU Make
Makefile.
Due to the nature of the objective, the proceeding sections will first discuss the required
background knowledge in a logical, progressive and abstracted manner thus disambiguating the notion of adaptive learning software before formally introducing the objective in
a parametrised manner prior to implementation and analysis.
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2
Electrocardiograms
The Electrocardiogram (ECG) is a tool employed for the task of analysing the electrical
activity of the heart. The electrocardiogram was invented by Dutch physiologist Willem
Einthoven in the early 20th century. An early prototype of what eventually lead to
the modern day electrocardiogram was based on a string galvanometer, a tool utilised
for amplifying electric signals over subterranean cables such as those running undersea
(MIT 2005, date of retrieval 03/05/2011). In 1901, a working model utilising a string
galvanometer was produced that was capable of measuring the electrical activity of the
heart. This working model eventually lead to a commercial model being introduced into
the market in 1903. (MIT 2005, date of retrieval 03/05/2011). Einthoven was awarded
a Nobel Prize for the conceptualisation and invention of the Electrocardiogram in 1924
(Cohen, date of retrieval 03/05/2011).
In modern times, typical Electrocardiogram readings are obtained by attaching 12 electrodes onto a patient at strategic points. Cohen, (date of retrieval 03/05/2011) states
that 12 electrodes are attached as no single point is capable of providing a complete
reference of the current state of the heart. However, for the purposes of mobile health
care, it is excessive to attach 12 electrodes to a patient as this severely hampers mobility
and is beyond the expected capabilities of patients to attach correctly all 12 nodes. As a
result, in mobile solutions, far fewer electrodes are typically employed.
2.1
Cardiac Anatomy
The heart is the muscle responsible for circulating oxygen and nutrients contained within
blood to the various cells in the body. The heart is comprised of four chambers, two
atria (singular atrium) and two ventricles. Cohen (date of retrieval 03/05/2011) states
that “[t]he heart is really 2 ‘half hearts,’ the right heart and the left heart, which beat
simultaneously”. Each half contains exactly one atrium and one ventricle, located in the
upper, smaller and lower, larger chambers respectively.
To induce circulation, the heart employs a beating mechanism. The sinus node located in
the right atrium controls the beating of the heart by discharging at a given time, typically
60 − 100/minute. During discharge, an electrical impulse traverses the heart to reach the
atrioventricular node. Cohen (date of retrieval 03/05/2011) describes the atrioventricular node as “a relay point to further propagate the electrical impulse”. Upon such time
that the electrical impulse happens upon the ventricles, the ventricles contract and as a
result of this contraction, blood is carried away from the heart via arteries. (Cohen, date
10
of retrieval 03/05/2011)
2.2
Interpreting the Electrocardiogram
The electrocardiogram machine records visually the electrical activity of the heart as it
surfaces on the epidermis. Readings are typically taken at intervals of 4ms or 250/sec
are are visualised by means of the electrocardiogram trace. The electrocardiogram trace
displays, at any given point in the represented time period, the total strength of the electrical activity of the heart for that point in time. (Ashley et al. 2004, date of retrieval
03/05/2011)
The Electrocardiogram readings contain five discrete points of interest, namely the P, Q,
R, S and T waves. The first point of interest is the P wave. The P wave is the result of
the sinus node depolarising. During this time the left and right atrium are in a state of
contraction. Following the P wave is the QRS Complex (an amalgamation of the Q, R
and S waves), representing the ventricular depolarisation (contraction). The final wave of
interest is the T wave, signifying ventricular repolarisation. Atrial repolarisation is typically engulfed as part of the QRS complex and is not visible on the electrocardiogram strip
under circumstances deemed normal. Figure 1 represents an idealised electrocardiogram
reading of a single depolarisation and repolarisation of the heart, with each constituent
wave emphasised. An additional point of interest is the U wave. This, however, serves
no purpose in this paper so a discussion of this wave is bypassed. The characteristics
of these five, PQRST points aid in diagnosing potential cardiac maladies. (Ashley et al.
2004, date of retrieval 03/05/2011)
Figure 1: Idealised Representation of PQRST
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3
Neural Networks
“I propose to consider the question, ‘Can machines think?’ ” began Turing (1950) in his
seminal paper, Computing Machinery and Intelligence. In this paper, Turing proposed a
test called the “imitation game” (Turing 1950) though this has been since renamed the
Turing Test, that could determine the relative intelligence of a machine by prompting
the machine and analysing the responses. The Turing test is structured as follows: A
researcher communicates with a single machine and a single human, isolated from view
and communicating with written natural language. The researcher will interrogate both
participants and attempt to determine which participant is human and which is machine.
Assuming a machine could consistently dupe a researcher into believing the machine was
a human responding, Turing argued that this would be a display of intelligence.
Turing’s test for machine intelligence was challenged in 1980 by Berkeley professor John
Searl. In his paper, Minds, brains and, programs (1980), Searl asserted that if a machine may pass the Turing test and is able to provide serviceable responses that may be
interpreted as extruding intelligence, there is no guarantee that the same machine understands the responses it generates. To illustrate this, Searl proposed a further thought
experiment named The Chinese Room. In this thought experiment, Searl proposes that,
with no prior knowledge of written Chinese, nor spoken Chinese, he may be locked in a
room and fed sheets of paper containing Chinese symbols and English instructions for
manipulating those symbols. In the event that rules of sufficient quality and detail were
provided, Searl could generate responses that may be indistinguishable from a native
speaker of the Chinese language while not understanding the nature of what was communicated.
These conflicting thought experiments serve to fuel the debate as to what constitutes
true intelligence. Two distinct camps, each with ardent proponents, have arisen in the
artificial intelligence research community. The two camps are strong and weak artificial
intelligence. Strong artificial intelligence, first coined by Searl (1980) when discussing
the Chinese Room thought experiment, promotes the notion that any machine extruding
strong artificial intelligence may “literally ha[ve] cognitive states and that the programs
thereby explain human cognition”. Proponents of strong artificial intelligence envision a
machine that may be able to match or surpass human abilities in areas such as reasoning, planning and learn. Weak artificial intelligence, by contrast, does not aim to match
or surpass human intelligence, but rather to implement characteristics that demonstrate
intelligence in specific situations.
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In addition to strong and weak artificial intelligence, two other factions exist within the
artificial intelligence community, that is those that practise symbolic artificial intelligence
and those that practise connectionist artificial intelligence. The former relates to attaining perceived intelligence by, as described by Tveter (1998, 12), “emphasiz[ing] list
processing and recursion”. Symbolic artificial intelligence is often referred to as top-down
artificial intelligence. An issue, as stated by Mitchell (1999, 4), with Symbolic or topdown artificial intelligence is that “[e]arly on, AI practitioners believed that it would
be straightforward to encode the rules that would confer intelligence on a program”.
Mitchell (1999) continues that “[n]owadays, many AI researchers believe that the “rules”
underlying intelligence are too complex to encode by hand in a “top-down” fashion”.
This notion allowed the rise of connectionist, or bottom-up, artificial intelligence. In the
bottom-up paradigm, “humans write only very simple rules, and complex behaviors such
as intelligence emerge from the massively parallel application and interaction of these
simple rules”.
Artificial neural networks may be classified under the connectionist, bottom-up paradigm
and in this paper, will be portrayed as weak artificial intelligence. For the purposes of
the research contained within paper, it is sufficient that the intelligence quota of the
neural network implementation need not achieve sentience but rather achieve adeptness
in pattern recognition and function approximation.
3.1
Biological Neural Networks
The following description aims to provide the reader with an intuition of the underlying neural processes that will serve as a model for the adaptive artificial neural network
systems described and implemented in the remainder of this paper. It is important to
note that it is unfeasible to model any of the biological processes discussed in this paper
in their entirety due to complexity and problem scope. Instead, desirable behaviourisms
will serve as a model from which to design an intelligent, adaptive system as opposed to
replicating directly the biological model.
The human brain consists largely of neurons. Neurons are described by Masters (1992,
2) as “a single cell capable of a sort of crude computation”. A neuron may receive input
from other neurons and in turn may produce an output that is sent to other neurons. In a
biological neural network, input refers to an electrical or chemical signal that, depending
on the input connection, may serve to excite or inhibit a neuron. A Neuron consists
of many component parts. Input is received over protoplasmic projections that branch,
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similar to a tree, from the soma (cell body) named dendrites and is directed toward the
axon terminal from which the signal is transmitted via the synapses.
“[B]iological neural networks are analog” writes Heaton (2008, 41), “[b]ecause this signal
is analog, the voltage of each signal will vary. If the voltage is within a certain range,
the neuron will fire”. When a neuron fires, it transmits a signal to other neurons. The
decisions made by a neural network are based on whether a given neuron has fired or
remained inert.
3.2
Artificial Neural Networks
Artificial neural networks, hereinafter referred to as neural networks, were first pioneered
by Frank Rosenblatt in 1957 while working at the Cornell University Aeronautical Laboratory. Rosenblatt proposed a single layer neural network dubbed the perceptron. “[The
perceptron] was computationally feasible on the hardware of that time, was based on
biological models, and was capable of learning”, writes Masters (1993, 4). The purpose
of the perceptron was to illustrate the potential for cognitive abilities to be attained by
means of nonlinear units massively interconnected (Schalkoff 1997, 27). The perceptron
had an intriguing quality. Rosenblatt presented an algorithm demonstrating what was
referred to as the “perceptron convergence theorem” (Masters, 1993, 4). The perceptron
convergence theorm demonstrated a training mechanism for the perceptron whereby upon
satisfying the condition that a given set of training weights were learnable, the algorithm
would converge to a series of weights that provide correct response to input data. The
perceptron suffered one fundamental flaw. In order for the perceptron to learn a pattern,
it had to be linearly separable. (Masters 1993, 4)
After an upsurge in neural network research, additional network architectures and topologies were borne. Important architectures include the Hopfield network, the multilayer
feedforward neural network and the self-organising map (SOM). One common trait among
all neural network architectures is the need to be trained before they may serve a purpose.
Training involves exposing a neural network to data that falls within a similar class as the
live, production data it is expected to operate upon. This serves as a distinction between
typical algorithms and computers. As Heaton (2008, 83) writes, “ordinary programs can
only automate repetitive tasks, such as balancing checkbooks or calculating the value of
an interest portfolio. While a program can easily maintain a large collection of images,
it cannot tell us what is illustrated in any of those images. Programs are inherently
unintelligent and uncreative”.
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Neural networks, however, do not provide a ’silver bullet’ for every problem that may
be represented in software or hardware. While it is true that neural networks excel in
many areas, such as classifying data or predicting trends. Heaton (2008, 43) states that
the qualities of neural networks are most prominent when a given problem cannot be
expressed as a series of steps. Neural networks are most beneficial in situations requiring
the processing of unpredictable non linear data, the patterns are subtle and not trivially
distinguishable or the data may be non-absolute (such as consumer opinions) (Masters
1993, 7). In many situations, however, a neural network may serve to the detriment of
performance and accuracy. Such cases may include problems that tend to feature constant
data sets such as an unchanging business rule (Heaton 2008, 43) or the researcher requires
knowledge not just of the result but how the result was derived from the input data.
This last point is fundamental to the operation of neural networks. Neural networks are
trained to learn and identify a given input. A neural network cannot, however, provide
an intuition to the researcher what steps it took to produce a given output (Heaton 2008,
43). This paper is concerned with classifying digital signals. It is not imperative that the
reasoning be deduced providing the classification is accurate.
3.3
Neural Network Infrastructure
There exists many proven neural network topologies that differ from their counterpart
topologies in areas such as size, intention and topology. Common to all neural networks
are certain core components that define the neural computing modus operandi. This
section identifies and discusses these components, demonstrating the operation of neural
networks. Initially, the Perceptron is discussed as a foundation and gradually a generic
neural network topology is built as each required component is discussed and implemented
in place.
3.3.1
Generic Topology
As a result of the inherently connected nature of biological neurons, it becomes trivial
to represent these networks as a graph. Graphs are a powerful tool utilised in the field
of discrete mathematics for visually representing and modeling the relationships between
objects. A simple graph consists of vertices (singular vertex ) and edges. A vertex represents a node in the network. An edge represents a relationship between two vertices.
For the purposes of this paper, neurons are represented by vertices and the relationships
(connections) between nodes are represented as edges.
15
i1
h1
i2
h2
o1
h3
o2
Figure 2: Generic Neural Network Topology Graph
Figure 2 shows a generic neural network structure consisting of three discrete layers. An
input layer, a hidden layer and an output layer. The neurons that comprise the input
layer (denoted in ) perform no processing but instead serves to introduce input data to
the network. Masters (1993, 8) describes the input layer as “a semantic construct used
to represent the input sample”. The hidden layer acts on the data presented by the input
layer, allowing remapping of the input data. The term ‘hidden layer’ is somewhat of a
misnomer as these neurons exist and are of great importance in many neural network
topologies. The term ‘hidden’, refers to a ‘black box’ situation in which the operation of
the hidden layer may be abstracted from view. The output layer is the point at which the
overall mapping of the network input is available (Schalkoff 1997, 158). Heaton (2008,
146) advocates that in situations where the neural network is utilised for classification
(sorting inputs into discrete groups based on inherent characteristics or patterns), one
output neuron should be included for each potential classification group. However, in
such situations where a neural network is employed to perform tasks such as noise reduction, it is beneficial to correlate the number of output neurons with the number of input
neurons, that is on = in .
When transferring a neural network topology to the computational domain, one simple
method for representing a graph is by means of a matrix (plural matrices). Matrices
provide a fundamental tool in linear algebra whereby numerical values may be contained
16
in a rectangular array. The values contained within a matrix are referred to as elements
and may be referenced by their horizontal and vertical position within the matrix. It
is possible to perform many mathematical operations on matrices, many of these will
be discussed in subsequent sections of this paper as it becomes pragmatic to do so. A
matrix of order (dimensions) 7 × 7 is drawn below. Additional notations have been
included that demonstrate referential points. Figure 3 describes an adjacency matrix (a
matrix indicating 1 is a connection exists between two neurons, 0 if no connection exists)
for the network topology shown in figure 3. Note, Ψ symbol serving as an identifier for
the matrix is arbitrary.







Ψ=





0
0
1
1
1
0
0
0
0
1
1
1
0
0
1
1
0
0
0
1
1
1
1
0
0
0
1
1
1
1
0
0
0
1
1
0
0
1
1
1
0
0
0
0
1
1
1
0
0













Figure 3: Adjacency Matrix
3.3.2
Weights, Bias and Directionality
The topology discussed in this section has a topology not dissimilar to the generic topology discussed in earlier sections, however, certain key characteristics will be introduced
and receive a formal definition. A neural network is deemed ‘feedforward’ if each neuron
may only communicate with a neuron or neurons in the next layer (ii may communicate
only with hi , hi may communicate only with oi ). This unidirectional nature of the multilayer feedforward network is represented by means of a directed graph (directionality is
represented by arrows).
In most neural network topologies, merely connecting neurons is not sufficient for achieving a learning capability. Referencing the biological model of a neural network, it is such
that when a neuron fires, it emits an electrical or chemical signal of a given strength based
on the level of excitation exhibited by that neuron. This signal may be implemented in
artificial neural networks by means of weights. A weight is a real value (wij ∈ R) that
is associated with a given connection between two neurons. In the event that a neuron
fires, the connected neurons receive an input equal to the weight of the neuron(s) that
17
have fired. To formally define this notion, Schalkoff (1997, 71) writes “wij represents
the strength of the connection to neuron unit i from (either) neuron unit j or input j”.
Schalkoff (1997, 71) continues that “a large positive value of wij indicates a strong excitory input from unit or input j to unit i, whereas a large negative weight value may be
used to represent a highly inhibitory input”. Weights allow a certain value to be associated to a given neuron. This is of importance when multiple input datum are presented
to a network. Masters (1993, 188) discusses a situation in which one neuron may have
a higher precedence than another neuron due to the contribution that neuron provides
to the output of the network. “[S]uppose we train a neural network to classify people as
overweight or normal weight, based on four variables: Height, Hair Color, Eye Color and
Weight. We may find that Hair Color and Eye Color are not important to the network’s
decision. The other two variables need more careful analysis, though. Certainly Height
alone tells us nothing about obesity. Even Weight alone tells us little. But taken together,
they tell all.”
One special type of weight is a bias. As written by Schalkoff (1997, 86), “[b]iases may be
used, for example, to selectively inhibit the activity of certain neurons”. A bias provides
a constant value of bi = 1.0 into each layer subsequent to the input layer. Inclusion of
a bias into a neural network topology is optional and is dependent on the nature of the
problem encountered.
i2
.8
−2
2
4
5.
2.1
−1
2.
0
o1
1.0
1.
h3
0.9
h2
.9
h1
1.0
b2
1.0
1.
0
2.3
4
0.
1
b1
4.7
.9
−3
4.
9.1
i1
o2
Figure 4: Inclusion of Weights, Bias and Directionality into the Network Topology
18
Heaton (2008, 85) describes the weight matrix as “the ‘memory’ of the neural network,
and will allow the neural network to recall certain patterns when they are presented”.
There exists some notational differences in figure 5 from figure 3. The symbol ∞ replaces
0 for indicating that a connection between two neurons does not exist. This serves the
purpose of conveying that a given connection between two neurons has an infinitely large
weight and thus may not be traversed. Note that while this convention is employed in this
paper, other conventions exist. Additionally, the boolean values that indicate a connection
are replaced with values such that wij ∈ R. The method in which elements are indexed is
not reciprocal. When referencing a connection, rows in the weight matrix represent the
start node and columns represent the end node. Consider briefly the connection between
i1 and h1 . Since this is unidirectional, a value exists at (i1 , h1 ) but not at (h1 , i1 ). A bias,
if it exists, will not be included in the weight matrix as the value is constant (bi = 1.0 )
and if included, the bias of a given layer may be assumed to be connected to each node
in the following layer.







Ψ=





∞
∞
∞
∞
∞
∞
∞
∞
∞
∞
∞
∞
∞
∞

2.3 −3.9 4.1 ∞ ∞

0.4 9.1 4.7 ∞ ∞ 

∞ ∞ ∞ 1.8 2.2 


∞ ∞ ∞ −2.9 2.1 

∞ ∞ ∞ 5.4 0.9 


∞ ∞ ∞ ∞ ∞ 
∞ ∞ ∞ ∞ ∞
Figure 5: Weight Matrix
One consideration to be made when dealing with lower powered devices is handling the
nonreciprocal nature of connections in a feedforward neural network. The weight matrix
for this simple 2 × 3 × 2 network requires 49 allocations for a floating point data type, of
which 12 are required. One possible method of combating this misuse of resources is to
amalgamate the collective weight matrix into a row matrix.
Weight =
h
2.3 −3.9 4.1 0.4 9.1 4.7 1.8 2.2 −2.9 2.1 5.4 0.9
Figure 6: Row Matrix for Input and Hidden Layers
19
i
3.4
Nonlinear Activation Functions
The degree to which a neuron may or may not fire is influenced heavily by an activation
function. In the case of the original Perceptron, the activation function employed resembled a piecewise function. The Perceptron applied a threshold value (t ∈ R) whereby if
the weighted sum of the input exceeded the threshold value, the output from the neuron would be 1, otherwise 0 (Masters 1993, 80). For non-trivial problems, however, the
boolean nature of the Perceptron limits the ability of the neural structure to classify
problems that cannot be represented as a 1 or a 0 but instead rely on fuzzy definitions
(0 ≤ i ≤ 1, i ∈ R or −1 ≤ i ≤ 1, i ∈ R). For this reason, many modern neural network
implementations employ a continuous, nonlinear, real-valued function. Two common activation functions will be discussed.
The most common activation function employed in modern networks is a sigmoid function
named the logistic activation function (Masters 1993, 81). The logistic function scales
the output of the neural network into proper ranges (Heaton 2008, 150). An inherent
property of the logistic activation function is that the possible range of possible y = f (x)
values that may result is 0 ≤ f (x) ≤ 1. The logistic activation function and associated
visual representation can be seen in figures 7 and 8 respectively.
f (x) =
1
1 + e−x
x = Input
Figure 7: Logistic Activation Function
Figure 8: Logistic Activation Function Plot
20
In situations where a neural network is required to produce negative values, other activation functions must be considered. One such function sigmoidal function is the hyperbolic
tangent or tanh(x) function (figure 9. The hyperbolic tangent function is represented
visually in figure 10.
tanh(x) =
ex − e−x
ex + e−x
x = Input
Figure 9: Hyperbolic Tangent Activation Function
Figure 10: Hyperbolic Tangent Activation Function Plot
An important characteristic of sigmoid functions is that a sigmoid function never reach
their theoretical maximum or minimum (Masters 1993, 82). As a result of this, Masters
(1992, 82) suggests that neurons that utilise the logistic activation function be considered
fully activated at values surrounding 0.9 and fully inhibited at values surrounding 0.1.
3.5
Neural Network Operation
The fundamental operation of a neural network may be described with the equation
described in figure 11. Consider the neural network described in figure 14. The operations
carried out to calculate the output of this network follow the operations described in figure
11. The example given describes the series of calculations carried out within a neural
network for one hidden neuron. This process is repeated n times where n is directly
proportional to the number of hidden and output neurons.
21
oi =
n−1
X
!
x i wi
+ bi
i=0
oi = Output
xi = Input
wi = Weight
bi = Bias
Figure 11: Calculating the Output of a Neuron
Figure 12 represents the input to the network for the purposes of this demonstration. As
stated previously, these input values are arbitrary and serve no purpose other than to
demonstrate the reaction of a neural network to some external stimulus over the input
nodes.
Input =
h
5.4 3.5 8.2
i
Figure 12: Neural Network Calculation Example Input Matrix
An additional step that can be incorporated into the calculation of the neural network,
while not strictly necessary, is scaling the input to fall within the range 0 ≤ inputi ≤ 1.
Scaling may be implemented in future neural network calculations as the values in the
implemented neural network in a future section of this paper do not necessarily occur
within the range 0 ≤ inputi ≤ 1. Scaling becomes beneficial as with large input values for
the activation function, the output value may occupy the upper limits of the function and
thus providing less of a variance in output between large samples. One common method
1
. This ensures a range
for linearly scaling the input values is to use the equation input
i
of 0 ≤ inputi 1 that ensures that the ratios between input values are not altered, only
the range. Figure 13 demonstrates the output of passing the unscaled and scaled values
to the logistic activation function. Note that the scaled input allows a wider range of
values to take advantage of the range of the logistic activation function. This difference
is not necessarily visible on a single example, however, scaling may prove beneficial over
the course of network output calculation. The use of input scaling is determined on a
case-by-case basis and should only be implemented if the results of the network deem
that a positive benefit is introduced by the use of scaling.
22
f (6.21) =
1
= 0.99
1 + e−6.21
f (0.16) =
1
= 0.64
1 + e−0.16
Unscaled = 6.21
1
= 0.16
Scaled = U nscaled
Figure 13: Example of Logistic Activation Function Output - Scaled and Unscaled
Figure 14 highlights a simplified neural network with the resulting calculations involved
in determining the output of this network. In this example, the input matrix will be
scaled to ensure the input values fall within the range 0 ≤ inputi ≤ 1.
i1
i2
i3
2.
9
1.4
3
2.
h1
1.8
o1
Figure 14: Neural Network Calculation Example Graph
To calculate the output of this simple network, the input matrix is first scaled to provide
the values (rounded to 2 decimal places)
Input =
h
1
5.4
= 0.19
1
3.5
23
= 0.29
1
8.2
= 0.12
i
Add the product of each input and each weight ((xi wi ) + bi ) with a bias of 1.0
h1 = ((0.19 ∗ 2.3) + (0.29 ∗ 1.4) + (0.12 ∗ 2.9) + 1.0) = 2.19
Applying the logistic activation function to this value provides
1
= 0.90
1 + e−2.19
This new value serves as input to the output layer which in turn is multiplied by the
weight of the connection
f (2.19) =
o1 = ((0.90 ∗ 1.8) + 1.0) = 2.62
Applying the logistic activation function to this value provides the output of the network
for this given path
1
= 0.93
1 + e−2.62
The output for this network and input combination is 0.93 which would, in most circumstances, be interpreted as a value of 1.0. This interpretation is due to the property of
sigmoid functions dictating they only approach their maximum and minimum values of
0 and 1 as limi→±∞ .
f (2.62) =
3.6
Training and Validating a Neural Network
As a core requirement of a neural network, the adaptability of a neural network serves to
distinguish ‘intelligent’ software from conventionally programmed software. The weights
assigned to the neurons in a neural network determine the reaction of the network to
input data samples. A network that has an optimal set of weights for each neuron, such
that the network can accurately classify new (unseen) data to a low margin or error is
considered ready for live use. The process of adjusting these weights is referred to as
training. “One of the most important aspects of biological and artificial neurons is their
learning capability”, writes Cichocki et al. (1994, 47), “whereby synaptic (or connection)
weights are adaptively changed according to an adaptive algorithm”.
Neural networks are typically trained under one of three paradigms, supervised training,
unsupervised and hybrid training. Under the supervised training paradigm, a training set
is assembled that provides the neural network with a complete representation of the inputs
that the network is required to learn (Masters 1993, 9). For each sample in the training
set, an ideal output is included. As various subsets of the training set are presented
24
to the neural network, the weights are adjusted based on how well the neural network
performed and the validity of the result (Heaton 2008, 44). “One pass through the subset of training samples with an updating of the network’s weights is called an epoch”,
writes Masters (1993, 10), “The number of samples in the subset is called the epoch size”.
Supervised training is the most common method for training a neural network (Heaton
2008, 44), however, there exists many situations in which ideal outputs are not available.
Under the unsupervised training paradigm, a similar training set typical of the range of
inputs is collected however no ideal output is presented to the network. Masters (1993,
10) describes the unsupervised training process as “[t]he process of training the network
[, ] letting it discover salient features of the training set, and using these features to group
the inputs that it (the network) finds distinct”.
Heaton (2008, 45) describes hybrid training as “combin[ing] aspects of both supervised
and unsupervised learning”. One such example of hybrid learning is reinforcement learning. During training that is carried out by reinforcement learning, a training set is
presented to the network but no ideal outputs are provided. Instead, the neural network
is informed manually as to the validity of the response it generated. (Masters 1993, 10)
Training a neural network is a non-trivial task. Masters (1993, 94) states “Training a
practical network is usually a numerical nightmare”. In most cases of neural network
training, whether supervised or unsupervised, an initial weight matrix is populated with
random values (Heaton 2008, 123). The optimum weight matrix is achieved when the
neural network output error is minimised. Error is loosely defined as the difference
between expected output and actual output. Heaton (2008, 123) advocates calculating
the Root Mean Squared Error at such time when all samples in the training set have been
presented to the network. Figure 15 describes the formula for the Root Mean Square Error
calculation.
v
u n−1
u1X
(ti − oi )2
RM SE = t
n i=0
n = Number of Epochs
ti = Expected Output
oi = Actual Output
Figure 15: Root Mean Squared Error (RMSE)
25
Several issues may arise in training, including that of false minima. False minima are
those points where a training algorithm may settle with a given weight matrix and assume
that the error achieved is the lowest error possible. It is the goal of any training algorithm
to locate the global minima, the set of weights that provide the lowest error possible. Local minima are those points that are preceded by a download slope (decreasing error)
and resolve to a slope of near zero (that is, any attempts to ‘escape’ this minima are met
by slopes that provide a worse-case error). It is in these points that a naive training algorithm may land. The local minima is by definition, not the optimal minimum error. To
combat this phenomenon, specialised algorithms for training neural networks to optimise
while avoiding local minima have been introduced. Three such algorithms are discussed
for the remainder of this section.
At such time that the training of a neural network is deemed complete, the network must
then be validated. Network validation requires that the network is presented with unseen
data to ensure correct operation (Heaton 2008, 45). Masters (1993, 10) stipulates that
“[i]n many respects, proper validation is more important than proper training”. During
training, a network may ‘overfit’ the data, as Masters (1993, 10) describes, “[r]ather than
learning the basic structure of the data, enabling it to generalize well, it learns irrelevant
details of the individual cases. In such cases that the validation phase performs poorly,
Heaton (2008, 46) suggests that potential cause may include unique validation data not
represented by the training set. The process of validating a neural network will be
discussed in further detail once the neural network has been implemented. Postponing a
formal analysis of the validation stage until such time is beneficial as it is possible to track
the progress of the neural network training algorithms by means of gathering diagnostic
data from the neural network.
3.6.1
Backpropagation
Training by backpropagation is typically associated with the training of multilayer feedforward neural networks, not least because the backpropagation training method was, as
Masters (1993, 100) states, “the first practical method for training a multiplelayer feedforward network”. Masters (1993, 100) attributes the resurgence of interest into the field
of artificial neural networks to the backpropagation training method. The backpropagation algorithm, as described by Masters (1993, 100) is a gradient descent algorithm, the
operation of which is discussed for the remainder of this section.
During the training process of a neural network, the network is presented with a training set. The training set, as discussed in previous sections, contains a selection of data
26
representative to the data that a neural network is expected to encounter during normal
operation. The purpose of training a neural network is to minimise the error produced
when a network is exposed to data. Training by backpropagation seeks to minimise this
error by modifying the weights and thresholds associated with the neural network. The
backpropagation algorithm works by first calculating the error of the output layer and
determining the contribution of each neuron in the layer to the error returned. It follows
that neurons with a higher weight value contribute more significantly and thus are to be
adjusted more. This process then moves through the preceeding layers and adjusts each
layer in turn. (Heaton 2008, 162-163)
For backpropagation to operate, the activation function must be differentiable. Differentiation represents the instantaneous rate of change of a given function as the input to the
function changes. It is not required for the reader to be familiar with differential calculus
to understand this paper as the derivative of any function is provided for the reader. The
importance of derivatives with regard to training by backpropagation are described by
Masters (1993, 95). “Derivatives are priceless to the numerical analyst who must optimize
a function. If we know the partial derivative of the error with respect to each weight, we
know (at least on a local scale) which way the weights must move in order to reduce the
error”. Figure 16 demonstrates the derivative for the logistic activation function.
1
dy
= f 0 (x) = f (x)(1 − f (x))
−x
dx 1 + e
x = Input
Figure 16: Derivative of the Logistic Activation Function
There are two additional important parameters to consider when training by backpropagation, learning rate and momentum. The learning rate is the rate at which the values
are adjusted. A higher learning rate results in more radical adjustments being made in
a shorter number of iterations. The momentum determines the degree to which training
from a previous iteration may affect the current iteration. (Heaton 2008, 421)
3.6.2
Simulated Annealing
Annealing is a process employed in the field of metallurgy. The annealing process involves
heating metal to a high temperature, at which point, the atoms have a greater degree
of freedom than at lower temperatures. During the cooling process, the temperature is
27
gradually lowered allowing the atoms to form stable structures. If the temperature is
lowered too rapidly (rapid quenching), anomalies in the structures formed by the atoms
may occur and thus the metal may be defective or brittle. Simulated annealing, as pioneered in the mid 1970s by Scott Kirkpatrick, applies these principles to such problems
as function optimisation or searching a nonlinear search space. Simulated annealing, in
addition, serves simultaneously as a highly effective training algorithm for artificial neural
networks. The remainder of this section will discuss in more detail the operation of the
simulated annealing process with regards to artificial neural networks. (Masters 1993,
118)
The simulated annealing algorithm operates on the weight matrix of an artificial neural
network attempting to produce an optimal set of weights that minimise the error of the
network. Press et al. (1999, 451) state that “[t]he basic ideas of simulated annealing are
also applicable to optimization problems with continuous N-dimensional control spaces,
e.g., finding the (ideally global) minimum of some function f (x)”. A typical simulated
annealing algorithm consists of two main processes (Heaton 2008, 201). A brief natural
language description of the simulated annealing algorithm will be discussed as it pertains to artificial neural networks, utilising annealing nomenclature where required for
the purposes of maintaining a relationship between the physical process and the simulated process.
Prior to discussing the operation of the simulated annealing algorithm, it is advantageous to first discuss the key parameters in use. The first parameter to be discussed
is the temperature parameter. The temperature value represents the standard deviation
of the employed random number generator (Masters 1993, 119). Initially, a high standard deviation is chosen to allow a large degree of potential values for the elements of
the weight matrix to assume. This standard deviation is gradually reduced. Another
important parameter to consider is the current best solution. The current best solution
represents the seed of the random number generator that has performed most desirable
(that is, produced the lowest neural network error) weight matrix and is replaced in such
cases that another matrix produces a lower error. A final parameter to define prior to
discussion of the operation of the algorithm is the lower bound temperature. The lower
bound temperature provides a value determining the minimum temperature the algorithm may operate at. This may serve as a failsafe to prevent an infinite loop. (Masters
1993, 119)
The operation of the simulated annealing algorithm is simple (Press et al. 1999, 444).
28
An initial randomised weight matrix is produced as per the range permitted by the current temperature. The initial weight matrix is marked as the current best solution and
provided to the artificial neural network. Upon completion of the required training data,
the error is compared to that generated by the current best solution. In such cases that
the error shows improvement over the existing current best solution then the previous
solution is replaced with the solution containing more favourable characteristics. For
each temperature, a predetermined number of iterations may occur. After this number
of iterations has been met, the temperature is reduced and the process repeats, using
the current best solution for each temperature as a central point for the random number
generator. This process continues until such time that the lower bound temperature is
met. (Heaton 2008, 201)
One important calculation performed in the simulated annealing process is the rate at
which the temperature decreases. It is often beneficial to calculate a non-static step
value to ensure that the temperature decreases in such a way that a similar number of
steps may occur for all ranges of start and end temperatures. The equation commonly
employed for calculating the step value for temperature decrease can be found in figure
17. (Heaton 2008, 204).
(
step = e
)
ln start
end
cycles−1
start = Temperature Upper Bound
end = Temperature Lower Bound
cycles = Number of cycles
Figure 17: Step Function for Temperature Decrease in Simulated Annealing
It should be noted that simulated annealing is the preferred training method for the
artificial neural networks deployed as part of this research.
3.6.3
Genetic Algorithms
Genetic algorithms are a prominent subset of the field of evolutionary computing. Evolutionary computing is an adaptive paradigm, mimicking natural processes such as natural
selection and applying those principles to complex computational problems that may
extrude large run-times under conventional methods. Genetic algorithms were originally devised by John Holland, along with his students and colleagues at the Univer29
sity of Michigan in the 1960’s (Mitchell 1999, 2). As Mitchell (1999, 2) states, Genetic
algorithms were initially intended to model biological evolution and adaptation, their
contributions to problems in computer science such as optimisation and searching were
incidental. Genetic algorithms, similarly to all evolutionary computational models, begin
with a random population of candidate solutions. These samples combine and mutate,
allowing those candidate solutions that provide superior character traits (read: better
solutions) to continue to reproduce new candidate solutions (Heaton 2008, 177). This
section first discusses the biological processes that will be modelled, then applies that
understanding to computationally feasible algorithms that allow the searching nonlinear
spaces for optimal threshold and weight values for a neural network.
A typical genetic algorithm begins by creating a population that are typically initialised
to random values. Each member of the population represents one potential solution to the
problem being studied. Upon successful generation of a population, the fitness of each
solution is tested to determine to what degree a given solution performed. Following
Darwinian theory of evolution, those potential solutions that performed most desirably
have a higher probability of reproducing new potential solutions. Those solutions that reproduce share material defined by a random cutoff point, additionally, random mutation
occurs to ensure new material is introduced. Mutations may or may not provide desirable
characteristics, however, this is a moot point as the fitness verification will determine if
these mutations proved successful. (Heaton 2008, 176-177)
30
4
Implementation
This section discusses the implementation specifics of an artificial neural network capable
of differentiating between two conditions of the heart. Prior to specific implementation
details being discussed, the objective detailed in section 1.2 will be expanded upon,
formally defining the problem. It is deemed beneficial to the reader to formally introduce
the objective at this later stage as the reader, assuming a logical and progressive order
of reading, is now aware of the theoretical constructs that underpin the research in this
paper. Promptly following this formal objective parametrisation is a discussion regarding
data collection techniques. This section will discuss the source of the data and how it is
to be prepared for exposure to the neural network. The construction and training of the
neural network, including relevant C code samples are included and discussed detailing
the construction and operation of the neural network as it is implemented and trained.
4.1
Formal Objective Parametrisation
The implementation stage of this research project will continue in three discrete parts.
First, two samples will be obtained from Physiobank ATM, one representing a healthy
signal and serving as a control, and another representing an onset of arrhythmia. Specific
points from each sample are to be extracted. To ensure a large, consistent training and
validation set, further input samples from each case are to be randomly generated based
on ranges of the respective values over a 10 second interval from the original samples.
The values obtained will represent the peak values of the P, Q, R, S and T waves.
The input values, coupled with expected output, are to be presented to the neural network. The neural network will employ simulated annealing as the training method.
Training will occur in cycles and iterations until such time that the weight matrix in use
by the network is sufficiently optimised, that is, the error is reduced to an acceptable rate.
The final stage of involves validating a new dataset against the trained neural network.
The outcome expected is that the neural network can differentiate between the healthy
signal and the unhealthy signal by means of outputting close to 0 when the healthy signal
is detected and close to 1 if characteristics of the unhealthy signal are detected.
It is the purpose of the algorithm to identify and react to the diminished P wave of
the ‘unhealthy’ sample. If the neural network is able to assimilate this diminished P
wave characteristic into a weight matrix and provide a ≥ 70% detection accuracy, this
algorithm will be considered successful.
31
4.2
Input Data Preparation
The data generated is to serve as training data and as validation data so it follows that
two discrete batches from each sample set are to be generated. To ensure consistent sample sets that may scale in size in the event that current training data proves inadequate or
of insufficient length, a sample of peak values from the relevant data points and random
values are to be generated within the maximum and minimum range of the sample values
taken. This mimics potential fluctuations in a live signal within a small range that fall
within an expected range for that given signal.
The data points of interested are peak values from the P, Q, R, S and T waves. These
values are to be collected and hard coded into the resulting software implementation to
alleviate system incompatibilities between systems such as Unix based systems (Linux,
BSD, Macintosh) and Windows based systems. Included in each given sample in the
training set is an expected value from which the accuracy of the classification made by
the neural network may be measured.
4.2.1
Sample A - Healthy
The sample chosen to serve as the control sample, serving as the baseline from which
deviations are considered unhealthy. This electrocardiogram readings chosen to represent
the healthy control sample were obtained from Physiobank ATM, with the particular
sample set being the Challenge 2010 Test Set C sample set. Only the readings from the
II node were used for generating a base set of training and validation data. Noteworthy
points in this sample include the existence of well-defined and formed P and T waves and
an obviously discernible QRS complex.
Figure 18: Sample A - Healthy Visualisation
32
Figure 19: Sample A - Extracted Points
4.2.2
Sample B - Unhealthy
The unhealthy sample set employed to serve as an anomalous reading also originated from
the Physiobank ATM database, specifically the Intracardiac Atrian Fibrillation Database.
Similarly to Sample A, only the II node readings were obtained. This sample serves as
an apt pattern for neural network classification due to the reduced P wave and elevated
QRS complex peaks.
Figure 20: Sample B - Unhealthy Visualisation
33
Figure 21: Sample B - Extracted Points
4.3
Neural Network Training
The training algorithm employed for this neural network implementation was the simulated annealing training method. Simulated annealing was chosen for the interesting
properties it extrudes. Figure 22 provides provides a visual representation of the decrease
in error from those weight matrices generated during the simulated annealing training
algorithm that displayed increasingly optimal weight values. Figure 23 represents the
error fluctuation over the entire training cycle. Displayed below is the output from the
neural network algorithm that generated these error function plots. It is encouraging to
note that the resulting accuracy of the network was 95.3%. A behaviour of this particular
implementation that may be considered somewhat noteworthy is that while the simulated
annealing training method produced an accuracy far in excess of the anticipated 70%,
the algorithm ceased to optimise after a relatively short period. This is likely due to the
trivial nature of the training data, and as a result, such positive results should not be
expected so quickly from larger networks with considerably more complex input patterns.
Arch% bin/ecgnn
|------------------------------------------|
| PQRST Analysis Artificial Neural Network |
| Thesis Work by Gavin Harper 08.05.2011 |
|------------------------------------------|
(New Best Error: 0.68201713), (Temp: 400.00)
(New Best Error: 0.67112941), (Temp: 400.00)
34
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Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Best
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
Error:
0.65418777),
0.65175272),
0.64616436),
0.59962134),
0.54911686),
0.53239386),
0.49881412),
0.49048498),
0.48953939),
0.48944191),
0.48889536),
0.46721348),
0.33996150),
0.22568789),
0.21086273),
0.17777691),
0.13902444),
0.09757576),
0.05510005),
0.04694315),
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
(Temp:
|------Training Accuracy------|
(Best: 95.3057%)
|---Optimal Training Matrix---|
([00]: 3.51)
([01]: -5.86)
([02]: -5.35)
([03]: 1.69)
([04]: 9.15)
([05]: -1.81)
([06]: -0.09)
([07]: 1.87)
([08]: 1.29)
([09]: 1.10)
([10]: 2.80)
([11]: 3.84)
35
400.00)
400.00)
400.00)
400.00)
400.00)
400.00)
400.00)
400.00)
400.00)
400.00)
400.00)
398.98)
387.78)
387.78)
387.78)
386.77)
386.77)
386.77)
386.77)
386.77)
([12]:
([13]:
([14]:
([15]:
([16]:
([17]:
15.66)
-2.61)
-6.26)
17.80)
-9.56)
1.87)
Figure 22: Simulated Annealing Best Error Function
Figure 23: Simulated Annealing Global Error Function
36
5
Results
The results gathered during the training stage of the neural network implementation far
surpass the original expectations placed upon the network in the design stage. Upon
verifying the simulated annealing training algorithm operated correctly and efficiently,
verification measures were then introduced. To verify the correct operation of the network, newly generated samples that had not been previously exposed to the network
were introduced into the topology. To reiterate the purpose of the verification of a neural
network, verification ensures that the network has identified generic patterns and formed
generic rules with regards to the data, as opposed to learning only the training set. In this
case, four tests were set up with threshold values of 70%. In almost all cases, the neural
network correctly identified and differentiated between both samples at an accuracy and
certainty of ≥ 70%. With such a trivial training set and small neural network topology,
difficulties arise in determining if the neural network is overfitting data. However, in
cases such as this, the results show definite potential.
|------Neural Validation------|
Entering a subset of Sample A (Expecting 0.0)
Actual: 0.12734469
PASS
Entering a subset of Sample B (Expecting 1.0)
Actual: 0.77359891
PASS
Entering a subset of Sample A (Expecting 0.0)
Actual: 0.16785368
PASS
Entering a subset of Sample B (Expecting 1.0)
Actual: 0.90986073
PASS
Detailed above is an except from a training cycle demonstrating correct identification
to within 70%. One important consideration is the fundamental nature of stochastic
37
algorithms. The results are likely to fluctuate between different executions and as a result,
the figures obtained on one complete training cycle will vary from those in subsequent
cycles. These results do not, however, conclusively prove or disprove the benefits or
necessities of artificial neural networks in mobile health care beyond the trivial example
discussed in this paper. Further research should be carried out, implementing artificial
neural networks in relevant use cases before any commercial product should be introduced
into the market. In many cases, the success of the simulated annealing training algorithm
varied at run-time with values ranging from 50% to 90%, however this was remedied
by better implementing the function that inflicted random modifications to the weight
matrix. This research should be considered successful as the algorithm performed most
admirably, exceeding original expectations.
38
6
Conclusion
Neural computing provides a solid foundation for designing digital signal analysis algorithms. Compared to traditional methods, there exists some initial complexity in the
training and validation stages that may deter those unfamiliar with neural computing
from considering a neural network as a candidate solution to a problem. An argument
that may be deduced from this paper, however is more apparent with larger networks
searching larger spaces, is that the accuracy and adaptability provided by a well-designed
and well-trained neural network can provide significant benefits over longer periods of
time.
While other, perfectly capable, neural network topologies exist such as the Self-Organising
Map (SOM) or the Hopfield Network, the decision to utilise the multilayer feedforward
neural network topology for this paper was borne from this specific topology serving as
the archetypal neural network structure. Lessons learned from the multilayer feedforward
neural network apply across a wide number of disciplines and topologies in neural computing and it serves as a sound basis from which neural computing may be introduced to
readers unfamiliar with the paradigm.
It is important to note that the example provided in this paper does not constitute a realworld electrocardiogram analysis algorithm. It is not enough to present a neural network
with three peak values from the P wave, QRS complex and T wave and expect that under
laboratory conditions a suitable detection rate with a real-time signal could be achieved.
This research intends to demonstrate the characteristics of neural computing that make
it a suitable candidate to the problem of analysing a digital signal for anomalies. This
paper, however, has suitably demonstrated that given appropriate conditions (to be discussed in a subsequent paragraph), a neural network is able to classify and identify data
presented in a coherent manner to a high degree of accuracy.
To elaborate further on what may be deemed appropriate conditions, it is beneficial to
consider first the nature of a digital electrocardiogram signal and then how a neural
network operates. It can be stated that the challenge in analysing any digital signal is
appropriately and consistently quantifying desired characteristics of the signal so they
can be presented to a neural network in a predictable manner. To achieve this, the
signal must be filtered and operated on using further mathematical techniques that are
beyond the scope of this paper though are employed in the continuation of this research
project post-thesis. Obtaining desired data from a real time signal, which may essentially
39
be considered as a one dimensional matrix of floating point numbers, is no trivial task.
The success or lack thereof of this project depends entirely on an appropriate filtering
technique being designed and implemented.
One final consideration is the suitability of artificial neural networks for mobile healthcare.
Based soley on the knowledge that while this paper has demonstrated the adaptability
and classification qualities of artificial neural networks, this example does not constitute
a real-world case and as a result, should undergo further testing and experimentation
before a sound recommendation can be made. This scepticism is based entirely on a lack
of empirical data supporting the viability of artificial neural networks in a real-world case,
but in no way seeks to cast down on the viability if, and only if, further specialised testing
is carried out. It is the intention of this author to continue researching this matter and
over a period of some years, correctly implement a neural computing solution capable of
adapting to real-word cases.
40
7
Discussion
This paper and associated research has posed some curious and exciting challenges over
the past weeks. It is always intriguing and holds a certain appeal to tackle challenges
beyond that which is typically encountered at that given point in time with regards to
education. All research and study into the topics discussed within the confines of this
paper have been based entirely on a rigorous self-educating and experimentation process
that has provided much insight into an area of computer science and engineering not
previously considered.
This paper represents a subset of the research into developing artificially intelligent signal analysis software capable of differentiating electrocardiogram readings for Domuset
Oy. Immediately after submission of this paper, the research process will resume with
the shortcomings exemplified in this paper being met with new methods and approaches
learned since the formalisation of the thesis topic. Research into the field of artificial intelligence and signal analysis provide exciting avenues for self-betterment and challenges
beyond what one would normally encounter at this stage in the education process. It is
such that this research project will continue, directly or indirectly relating to further studies, for some years to come. It is hoped that the problem of machine based, intelligent
electrocardiogram analysis can be continued beyond Master of Science level education
and into the realm of doctorate degrees.
In spite of the relative success in demonstrating the viability of artificial neural networks
for mobile medical analysis, it is still such that the problem tackled in this paper was
a reduced, trivialised form of the challenges faced external to this thesis. The respective fields of Artificial Intelligence, Signal Analysis and Electrocardiography are vast and
interesting topics too large to be covered with any consistent accuracy in a short paper
such as this. One challenge encountered during the writing of this paper was approaching
what is considered a vastly complex subject in such a way that the relevant information
may be provided to a reader with no background knowledge in artificial intelligence, signal analysis nor electrocardiography. It is the case that many fundamental operations in
Neural Networks, to isolate one example, are constructed from multiple smaller, simpler
parts. To cover all of these parts in any detail including tracing back the theory behind
each part to prevent assumed knowledge being applied proved a challenge.
41
8
Bibliography
Ashley, E. A. & Niebauer, J. 2004. Conquering the ECG - Cardiology Explained. Date
of retrieval 03/05/2011.
http://www.ncbi.nlm.nih.gov/books/NBK2214/
Cohen, J. S. Year Unknown. Electrocardiogram (ECG). Date of retrieval 03/05/2011.
http://www.emedicinehealth.com/electrocardiogram ecg/article em.htm
Cichocki, A & Unbehauen, R. 1994. Neural Networks for optimisation and signal porcessing. Stuttgart: John Wiley & Sons Ltd. & B. G. Teubner.
Cormen, T. H. & Leiserson, C. E. & Rivest, R. L. & Stein, C. 2009. Introduction to
Algorithms. Third Edition. Massachusetts: The MIT Press.
Croft, A. & Davison, R. & Hargreaves, M. 2003. Engineering Mathematics: A Foundation for Electronic, Electrical, Communications and Systems Engineers. Third Edition.
Essex: Pearson Education Limited.
Haykin, S. 1999. Neural Networks A Comprehensive Foundation. Second Edition. New
Jersey: Prentice-Hall, Inc.
Heaton, J. 2008. Introduction to Neural Networks with Java. Second Edition. St Louis:
Heaton Research, Inc.
Masters, T. 1993. Practical Neural Network Recipes in C++. San Diego: Academic
Press, Inc.
MIT. 2005. Inventor of the Week: Archive. Date of retrieval 03/05/2011.
http://web.mit.edu/invent/iow/einthoven.html
Mitchell, M. 1999. An Introduction to Genetic Algorithms.
sachusetts: The MIT Press.
Fifth Printing.
Mas-
Peterson, J. & Kynor, D. 2003. Intracardiac Atrial Fibrillation Database. Date of retrieval 08/05/2011.
http://www.physionet.org/physiobank/database/iafdb/
42
Physionet. 2010. Challenge 2010 Test Set C. Date of retrieval 08/05/2011.
http://www.physionet.org/physiobank/database/challenge/2010/set-c/
Press, W. H. & Teukolsky, S. A. & Vetterling, W. T. & Flannery, B. T. 1999. Numerical
Recipes in C: The Art of Scientific Computing. Second Edition. Cambridge: The Press
Syndicate of the University of Cambridge.
Schalkoff, R. J. 1997. Artificial Neural Networks. New York: McGraw-Hill
Searle, J. R. 1980. Minds, brains, and programs. Behavioral and Brain Sciences. 3 (3).
417-424.
Sedgewick, R. 1992. Algorithms in C++. Massachusetts: Addison-Wesley Publishing
Company, Inc.
Turing, A. M. 1950. Computing Machinery and Intelligence. Mind. 59 (236), 433-460.
Tveter, D. R. 1998. The Pattern Recognition Basis of Artificial Intelligence. Los Alamitos: IEEE Computer Society Press.
43
A
A.1
Constants and Notations
Constants
e = 2.71828 . . .
∞ = Infinity
A.2
Notations
α A value represented by some non-numerical character
αn The ‘nth’ value contained within α
αβ The product of a value, α multiplied by itself β times
√
α Denotes the square root of α
αβ The product (multiplication) of two values, α and beta
α
β
The division of two values, α divided by β
α > β The value of α is greater than the value of β
α < β The value of α is less than the value of β
α ≥ β The value of α is greater than or equal to the value of β
α ≤ β The value of α is less than or equal to the value of β
α 6= β The value of α is not equal to the value of β
0 ≤ α ≤ 1 The value of α is in the range 0 → 1
Pn
i=0
αi βi Denotes the summation of the product for all αi and βi in the range i → n
44
∈ Denotes set membership. α ∈ A is read as α is contained within some set A
f (x) Denotes some function that receives an independent variable x and outputs exactly
one dependent variable y
f 0 (x) Denotes the derivative of some function f (x)
dy
f (x)
dx
Denotes the derivative of some function f (x)
ln (α) The base of the natural logarithm is e. The natural logarithm of α is the power e
is raised to to equal α
limx→∞ The limit of a function as x tends to infinity or some value
"
1 0
0 1
A.3
#
Denotes some 2 × 2 matrix.
Number Systems
R The set of all Real Numbers (x ∈ R : −∞ ≤ x ≤ ∞)
45
B
GIT Log
commit e 0 b 9 d 4 b 5 1 f 4 9 9 2 3 0 1 a 8 e c e 0 4 4 5 b 4 a 7 b 2 6 1 d a e 7 5 3
Author : Gavin Harper <[email protected] . oamk . f i >
Date :
Mon May 9 1 9 : 5 0 : 2 7 2011 +0300
F i n a l Commit
commit 1 e c 3 d 0 3 c d 0 8 8 0 2 7 9 1 b c 4 3 d 3 c 5 2 7 a 7 8 d 2 4 f d 6 e 8 3 4
Author : Gavin Harper <[email protected] . oamk . f i >
Date :
Mon May 9 1 6 : 4 7 : 5 9 2011 +0300
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commit a c b 0 3 d 5 5 7 0 6 d 2 1 2 5 f 6 2 9 7 0 d b 4 c 2 4 3 a 4 2 f 8 6 e 2 2 7 d
Author : Gavin Harper <[email protected] . oamk . f i >
Date :
Mon May 9 0 5 : 5 3 : 2 1 2011 +0300
P e d a n t i c OCD k i c k e d i n . Had t o update t h e v a l i d a t i o n h e a d i n g
commit e 7 a 0 b f 8 6 5 f 3 e 3 6 d c 0 d 5 2 2 3 e 2 8 5 b c 3 1 d 4 e 0 1 7 1 e 2 2
Author : Gavin Harper <[email protected] . oamk . f i >
Date :
Mon May 9 0 5 : 3 4 : 4 0 2011 +0300
F i n a l v e r s i o n − commented
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Author : Gavin Harper <[email protected] . oamk . f i >
Date :
Mon May 9 0 2 : 5 5 : 3 1 2011 +0300
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commit 6 c 0 a 5 0 8 9 3 6 c 8 9 8 1 0 c 3 0 a c 4 5 8 6 8 0 1 5 e 7 5 3 b 0 4 2 b 4 f
Author : Gavin Harper <[email protected] . oamk . f i >
Date :
Mon May 9 0 2 : 4 0 : 0 7 2011 +0300
Network seems t o be h a n d l i n g t r a i n i n g data w e l l
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Author : Gavin Harper <[email protected] . oamk . f i >
Date :
Mon May 9 0 0 : 2 9 : 2 1 2011 +0300
General s t a b i l i t y updates
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Author : Gavin Harper <[email protected] . oamk . f i >
Date :
Mon May 9 0 0 : 2 8 : 3 1 2011 +0300
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Author : Gavin Harper <[email protected] . oamk . f i >
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Mon May 9 0 0 : 2 8 : 1 0 2011 +0300
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commit 3 e 1 9 a 4 e 1 9 d 1 3 0 a c f 0 0 d c 9 f 2 9 3 f 0 e b 6 a 6 f 9 f 2 e 8 b 4
Author : Gavin Harper <[email protected] . oamk . f i >
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Sun May 8 1 8 : 4 6 : 3 9 2011 +0300
Fixed i n c o n s i s t e n c y i n e r r o r d i s p l a y
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Sun May 8 1 6 : 2 7 : 0 8 2011 +0300
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Author : Gavin Harper <[email protected] . oamk . f i >
Date :
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commit 2 e f 1 7 f e 1 0 f 2 8 c a d b 4 d 4 e 9 2 2 b d 7 2 8 1 1 9 f 9 5 4 9 2 e 7 a
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Sun May 8 1 4 : 2 1 : 5 9 2011 +0300
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Sun May 8 1 4 : 2 0 : 2 7 2011 +0300
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Date :
Sun May 8 1 4 : 2 0 : 0 9 2011 +0300
Repo i n i t i a l i s a t i o n
48
C
Makefile
#
# M a k e f i l e f o r T h e s i s Work : PQRST A n a l y s i s ( M u l t i l a y e r Feedforward Neural Network )
#
# Gavin Harper − F8827
#
# Compiler : Gnu GCC 4 . 6 . 0 20110415
#
CC=g c c
CFLAGS= −lm −O3 −Wall
OBJ = o b j / main . o o b j / f n g e t w e i g h t m a t r i x . o o b j / f n c a l c u l a t e n e t w o r k . o \
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obj / fnannealstep . o obj / f n g e t e r r o r . o obj / fnannealshake . o \
obj / fngetrmse . o
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FNCALCULATENETWORK = f n c a l c u l a t e n e t w o r k . h f n c a l c u l a t e n e t w o r k . c
FNACTIVATIONFUNCTION = f n a c t i v a t i o n f u n c t i o n . h f n a c t i v a t i o n f u n c t i o n . c
FNGETRANDOM = fngetrandom . h fngetrandom . c
FNANNEAL = f n a n n e a l . h f n a n n e a l . c
FNANNEALSTEP = f n a n n e a l s t e p . h f n a n n e a l s t e p . c
FNANNEALSHAKE = f n a n n e a l s h a k e . h f n a n n e a l s h a k e . c
FNGETERROR = f n g e t e r r o r . h f n g e t e r r o r . c
FNGETRMSE = f n g e t r m s e . h f n g e t r m s e . c
a l l : ecgnn
ecgnn : $ (OBJ)
$ (CC) $ (CFLAGS) $ (OBJ) −o b i n / ecgnn
o b j / main . o : main . c $ (FNANNEAL)
$ (CC) $ (CFLAGS) −c main . c −o o b j / main . o
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$ (CC) $ (CFLAGS) −c f n g e t w e i g h t m a t r i x . c −o o b j / f n g e t w e i g h t m a t r i x . o
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o b j / f n g e t e r r o r . o : $ (FNGETERROR)
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D
Source Code
####################
#∗∗∗∗∗ main . c ∗∗∗∗∗#
####################
#include <s t d i o . h>
#include < s t d l i b . h>
#include ” f n a n n e a l . h”
i n t main ( i n t a r g c , char ∗ a r g v [ ] )
{
p r i n t f ( ”|−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−|\n” ) ;
p r i n t f ( ” | PQRST A n a l y s i s A r t i f i c i a l N e u r a l Network | \ n” ) ;
printf (” |
T h e s i s Work by Gavin Harper 0 8 . 0 5 . 2 0 1 1
| \ n” ) ;
p r i n t f ( ”|−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−|\n” ) ;
p r i n t f ( ” \n” ) ;
fnAnneal ( ) ;
return 0 ;
}
########################
#∗∗∗∗∗ f n a n n e a l . h ∗∗∗∗∗#
########################
#i f n d e f FNANNEAL H
#define FNANNEAL H
#include <f l o a t . h>
#include <s t d i o . h>
#include
#include
#include
#include
#include
#include
#include
” f n a n n e a l s t e p . h”
” f n a n n e a l s h a k e . h”
” f n g e t w e i g h t m a t r i x . h”
” f n g e t e r r o r . h”
” f n g e t r m s e . h”
” f n c a l c u l a t e n e t w o r k . h”
” fngetrandom . h”
/∗ Train t h e n e t w o r k u s i n g s i m u l a t e d a n n e a l i n g . Due t o s i m p l i c i t y ,
network v a l i d a t i o n i s i n c l u d e d in t h i s f u n c t i o n to reduce
u n n e c e s s a r y f i l e s ∗/
void f n A n n e a l ( ) ;
#endif
51
########################
#∗∗∗∗∗ f n a n n e a l . c ∗∗∗∗∗#
########################
#include ” f n a n n e a l . h”
void f n A n n e a l ( )
{
int e S i z e = 2 0 ;
int nCycles = 4 0 ;
double bErr = DBL MAX;
double sTemp = 4 0 0 . 0 ;
double eTemp = 2 0 0 . 0 ;
double cTemp = sTemp ;
double s t e p ;
double c E r r ;
double oNet ;
/∗
/∗
/∗
/∗
/∗
/∗
/∗
/∗
/∗
S i z e o f e r r o r m a t r i x ∗/
Number o f c y c l e s p e r t e m p e r a t u r e ∗/
High o r i g i n a l b e s t e r r o r ∗/
S t a r t i n g t e m p e r a t u r e ∗/
Ending t e m p e r a t u r e ∗/
Current Temperature ∗/
Value by which temperaure i s d e c r e a s e d ∗/
Current e r r o r ∗/
Output o f Neural Network C a l c u l a t i o n ∗/
double ∗wMtx = fnGetWeightMatrix ( ) ; /∗ Weight m a t r i x ∗/
double ∗bMtx = ( double ∗ ) m a l l o c ( 3 0 ∗ s i z e o f ( double ) ) ; /∗ B e s t m a t r i x ∗/
double ∗eMtx = ( double ∗ ) m a l l o c ( e S i z e ∗ s i z e o f ( double ) ) ; /∗ Error m a t r i x ∗/
i n i t r a n d ( ) ; /∗ S e e d s random number g e n e r a t o r ∗/
while ( cTemp > eTemp )
{
int i ;
f o r ( i = 0 ; i < n C y c l e s ; i ++)
{
fnAnnealShake ( sTemp , cTemp , wMtx) ; /∗ A l t e r t h e w e i g h t m a t r i x ∗/
/∗ C a l c u l a t e o u t p u t t h e n e r r o r o f n e u r a l n e t w o r k ∗/
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 9 , 0 . 4 1 , 1 . 3 9 , 0 . 2 2 , 0 . 6 ) ;
eMtx [ 0 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.03 , −0.21 , 2 . 2 2 , −0.29 , 0 . 5 4 ) ;
eMtx [ 1 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 9 , 0 . 4 1 , 1 . 3 9 , 0 . 2 3 , 0 . 6 1 ) ;
eMtx [ 2 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.05 , −0.2 , 2 . 4 2 , −0.2 , 0 . 3 7 ) ;
eMtx [ 3 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 5 , 0 . 4 1 , 1 . 4 3 , 0 . 2 3 , 0 . 6 1 ) ;
eMtx [ 4 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.08 , −0.16 , 2 . 3 6 , −0.25 , 0 . 4 4 ) ;
eMtx [ 5 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 7 , 0 . 4 1 , 1 . 4 , 0 . 2 5 , 0 . 6 5 ) ;
eMtx [ 6 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.09 , −0.17 , 2 . 2 , −0.24 , 0 . 4 6 ) ;
eMtx [ 7 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 6 , 0 . 4 3 , 1 . 3 9 , 0 . 2 4 , 0 . 6 4 ) ;
eMtx [ 8 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.05 , −0.19 , 2 . 2 , −0.22 , 0 . 5 3 ) ;
eMtx [ 9 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 7 , 0 . 4 , 1 . 4 , 0 . 2 3 , 0 . 6 2 ) ;
eMtx [ 1 0 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.08 , −0.15 , 2 . 1 9 , −0.26 , 0 . 5 1 ) ;
eMtx [ 1 1 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
52
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 7 , 0 . 4 3 , 1 . 4 1 , 0 . 2 6 , 0 . 6 3 ) ;
eMtx [ 1 2 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.05 , −0.23 , 2 . 2 5 , −0.22 , 0 . 3 9 ) ;
eMtx [ 1 3 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 5 , 0 . 4 1 , 1 . 3 9 , 0 . 2 5 , 0 . 6 4 ) ;
eMtx [ 1 4 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.05 , −0.21 , 2 . 2 5 , −0.3 , 0 . 4 7 ) ;
eMtx [ 1 5 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 7 , 0 . 4 3 , 1 . 4 2 , 0 . 2 5 , 0 . 6 4 ) ;
eMtx [ 1 6 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.06 , −0.24 , 2 . 2 , −0.25 , 0 . 4 1 ) ;
eMtx [ 1 7 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , 0 . 5 8 , 0 . 4 1 , 1 . 4 , 0 . 2 2 , 0 . 6 4 ) ;
eMtx [ 1 8 ] = f n G e t E r r o r ( oNet , 0 . 0 ) ;
oNet = f n C a l c u l a t e N e t w o r k (wMtx , −0.05 , −0.24 , 2 . 3 8 , −0.3 , 0 . 4 9 ) ;
eMtx [ 1 9 ] = f n G e t E r r o r ( oNet , 1 . 0 ) ;
/∗ Return RMSE ∗/
c E r r = fnGetRMSE ( eMtx , e S i z e ) ;
/∗ R e p l a c e c u r r e n t e r r o r w i t h b e s t i f c u r r e n t i s l o w e r ∗/
i f ( c E r r < bErr )
{
bErr = c E r r ;
/∗ Dump c u r r e n t m a t r i x i n t o b e s t m a t r i x ∗/
int i ;
f o r ( i = 0 ; i < 3 0 ; i ++)
{
bMtx [ i ] = wMtx [ i ] ;
}
p r i n t f ( ” (New Best E r r o r : %.8 f ) , (Temp : %.2 f ) \n” , bErr , cTemp ) ;
}
/∗ Write t h e g l o b a l and b e s t e r r o r v a l u e s f o r p l o t t i n g ∗/
FILE ∗ f p ;
f p = f o p e n ( ” dgn / g l o b a l . e r r ” , ” a ” ) ;
f p r i n t f ( fp , ” %.8 f \n” , c E r r ) ;
f c l o s e ( fp ) ;
}
/∗ C a l c u l a t e t e m p e r a t u r e d e c r e a s e ∗/
s t e p = f n A n n e a l S t e p ( sTemp , eTemp , n C y c l e s ) ;
cTemp −= s t e p ;
}
/∗ S t a t i s t i c s ∗/
p r i n t f ( ” \n” ) ;
p r i n t f ( ”|−−−−−−T r a i n i n g Accuracy −−−−−−|\n” ) ;
p r i n t f ( ” ( Best : %.4 f \%)\n” , ( 1 0 0 . 0 − ( bErr ∗ 1 0 0 . 0 ) ) ) ;
p r i n t f ( ” \n” ) ;
p r i n t f ( ”|−−−Optimal T r a i n i n g Matrix −−−|\n” ) ;
int j ;
f o r ( j = 0 ; j < 1 8 ; j ++)
{
i f ( j < 10)
{
53
p r i n t f ( ” ([0% d ] : %.2 f ) \n” , j , bMtx [ j ] ) ;
}
else
{
p r i n t f ( ” ([% d ] : %.2 f ) \n” , j , bMtx [ j ] ) ;
}
}
/∗ V a l i d a t i o n ∗/
/∗ 70% Accuracy e x p e c t e d . This i s somewhat h i g h f o r such a
s m a l l t r a i n i n g s e t and n e t w o r k s i z e b u t i t seems t o p a s s
c o n s i s t e n t l y ∗/
p r i n t f ( ” \n” ) ;
p r i n t f ( ”|−−−−−−N e u r a l V a l i d a t i o n −−−−−−|\n” ) ;
p r i n t f ( ” \n” ) ;
p r i n t f ( ” E n t e r i n g a s u b s e t o f Sample A ( E x p e c t i n g 0 . 0 ) \n” ) ;
oNet = f n C a l c u l a t e N e t w o r k ( bMtx , 0 . 5 5 , 0 . 4 3 , 1 . 4 1 , 0 . 2 6 , 0 . 6 1 ) ;
p r i n t f ( ” A c t u a l : %.8 f \n” , oNet ) ;
i f ( oNet <= 0 . 3 )
{
p r i n t f ( ”PASS\n” ) ;
}
else
{
p r i n t f ( ”FAIL\n” ) ;
}
p r i n t f ( ” \n” ) ;
p r i n t f ( ” E n t e r i n g a s u b s e t o f Sample B ( E x p e c t i n g 1 . 0 ) \n” ) ;
oNet = f n C a l c u l a t e N e t w o r k ( bMtx , −0.02 , −0.17 , 2 . 4 , −0.26 , 0 . 3 8 ) ;
p r i n t f ( ” A c t u a l : %.8 f \n” , oNet ) ;
i f ( oNet >= 0 . 7 )
{
p r i n t f ( ”PASS\n” ) ;
}
else
{
p r i n t f ( ”FAIL\n” ) ;
}
p r i n t f ( ” \n” ) ;
p r i n t f ( ” E n t e r i n g a s u b s e t o f Sample A ( E x p e c t i n g 0 . 0 ) \n” ) ;
oNet = f n C a l c u l a t e N e t w o r k ( bMtx , 0 . 5 7 , 0 . 4 2 , 1 . 4 3 , 0 . 2 5 , 0 . 6 5 ) ;
p r i n t f ( ” A c t u a l : %.8 f \n” , oNet ) ;
i f ( oNet <= 0 . 3 )
{
p r i n t f ( ”PASS\n” ) ;
}
else
{
p r i n t f ( ”FAIL\n” ) ;
}
p r i n t f ( ” \n” ) ;
54
p r i n t f ( ” E n t e r i n g a s u b s e t o f Sample B ( E x p e c t i n g 1 . 0 ) \n” ) ;
oNet = f n C a l c u l a t e N e t w o r k ( bMtx , −0.05 , −0.2 , 2 . 2 7 , −0.29 , 0 . 5 1 ) ;
p r i n t f ( ” A c t u a l : %.8 f \n” , oNet ) ;
i f ( oNet >= 0 . 7 )
{
p r i n t f ( ”PASS\n” ) ;
}
else
{
p r i n t f ( ”FAIL\n” ) ;
}
p r i n t f ( ” \n” ) ;
f r e e (wMtx) ;
f r e e ( eMtx ) ;
}
#############################
#∗∗∗∗∗ f n a n n e a l s h a k e . h ∗∗∗∗∗#
#############################
#i f n d e f FNANNEALSHAKE H
#define FNANNEALSHAKE H
#include ” fngetrandom . h”
/∗ Shake t h e w e i g h t m a t r i x e l e m e n t s t o g e n e r a t e randomness ∗/
void fnAnnealShake ( double sTemp , double cTemp , double ∗wMtx) ;
#endif
#############################
#∗∗∗∗∗ f n a n n e a l s h a k e . c ∗∗∗∗∗#
#############################
#include ” f n a n n e a l s h a k e . h”
void fnAnnealShake ( double sTemp , double cTemp , double ∗wMtx)
{
int i ;
f o r ( i = 0 ; i < 3 0 ; i ++)
{
/∗ Determine random v a l u e change ∗/
double d e l t a = 0 . 5 − fnGetRandom ( ) ;
d e l t a ∗= ( cTemp / sTemp ) ;
wMtx [ i ] = wMtx [ i ] + d e l t a ;
}
}
55
############################
#∗∗∗∗∗ f n a n n e a l s t e p . h ∗∗∗∗∗#
############################
#i f n d e f FNANNEALSTEP H
#define FNANNEALSTEP H
#include < s t d l i b . h>
#include <math . h>
/∗ Reduce t e m p e r a t u r e v a l u e l o g a r i t h m i c a l l y ∗/
double f n A n n e a l S t e p ( double sTemp , double eTemp , i n t n C y c l e s ) ;
#endif
############################
#∗∗∗∗∗ f n a n n e a l s t e p . c ∗∗∗∗∗#
############################
#include ” f n a n n e a l s t e p . h”
double f n A n n e a l S t e p ( double sTemp , double eTemp , i n t n C y c l e s )
{
return exp ( ( l o g ( sTemp / eTemp ) ) / ( ( double ) n C y c l e s − 1 ) ) ;
}
#################################
#∗∗∗∗∗ f n g e t w e i g h t m a t r i x . h ∗∗∗∗∗#
#################################
#i f n d e f FNGETWEIGHTMATRIX h
#define FNGETWEIGHTMATRIX h
#include < s t d l i b . h>
#include ” fngetrandom . h”
/∗ Return a random w e i g h t m a t r i x t o b e g i n t r a i n i n g ∗/
double ∗ fnGetWeightMatrix ( ) ;
#endif
56
#################################
#∗∗∗∗∗ f n g e t w e i g h t m a t r i x . c ∗∗∗∗∗#
#################################
#include ” f n g e t w e i g h t m a t r i x . h”
double ∗ fnGetWeightMatrix ( )
{
int i ;
/∗ 30 i s t h e w e i g h t m a t r i x s i z e . Yes , t h i s s h o u l d be
a #d e f i n e b u t i t i s 3AM ∗/
double ∗wMtx = ( double ∗ ) m a l l o c ( 3 0 ∗ s i z e o f ( double ) ) ;
f o r ( i = 0 ; i < 3 0 ; i ++)
{
wMtx [ i ] = fnGetRandom ( ) ;
}
return wMtx ;
}
##################################
#∗∗∗∗∗ f n c a l c u l a t e n e t w o r k . h ∗∗∗∗∗#
##################################
#i f n d e f FNCALCULATENETWORK H
#define FNCALCULATENETWORK H
#include <s t d i o . h>
#include < s t d l i b . h>
#include ” f n a c t i v a t i o n f u n c t i o n . h”
/∗ C a l c u l a t e n e u r a l n e t w o r k o u t p u t ∗/
double f n C a l c u l a t e N e t w o r k ( double ∗wMtx , double i 1 , double i 2 \
double i 3 , double i 4 , double i 5 ) ;
#endif
##################################
#∗∗∗∗∗ f n c a l c u l a t e n e t w o r k . c ∗∗∗∗∗#
##################################
#include ” f n c a l c u l a t e n e t w o r k . h”
double f n C a l c u l a t e N e t w o r k ( double ∗wMtx ,
double i 3 , double i 4 , double i 5 )
{
double h1 = 0 . 0 ; /∗ Hidden Neuron 1
double h2 = 0 . 0 ; /∗ Hidden Neuron 2
double h3 = 0 . 0 ; /∗ e t c ∗/
double h4 = 0 . 0 ;
double h5 = 0 . 0 ;
double o1 = 0 . 0 ; /∗ Output Neuron 1
double i 1 , double i 2 , \
∗/
∗/
∗/
57
/∗ Get p r o d u c t o f a l l i n p u t v a l u e s and r e s p e c t i v e w e i g h t s
t h e n s q u a s h w i t h t h e l o g i s t i c a c t i v a t i o n f u n c t i o n ∗/
h1 = ( ( i 1 ∗ wMtx [ 0 ] ) + ( i 2 + wMtx [ 1 ] ) + ( i 3 + wMtx [ 2 ] ) + \
( i 4 ∗ wMtx [ 3 ] ) + ( i 5 ∗ wMtx [ 4 ] ) + 1 . 0 ) ;
h1 = f n A c t i v a t i o n F u n c t i o n ( h1 ) ;
h2 = ( ( i 1 ∗ wMtx [ 5 ] ) + ( i 2 + wMtx [ 6 ] ) + ( i 3 + wMtx [ 7 ] ) + \
( i 4 ∗ wMtx [ 8 ] ) + ( i 5 ∗ wMtx [ 9 ] ) + 1 . 0 ) ;
h2 = f n A c t i v a t i o n F u n c t i o n ( h2 ) ;
h3 = ( ( i 1 ∗ wMtx [ 1 0 ] ) + ( i 2 + wMtx [ 1 1 ] ) + ( i 3 + wMtx [ 1 2 ] ) + \
( i 4 ∗ wMtx [ 1 3 ] ) + ( i 5 ∗ wMtx [ 1 4 ] ) + 1 . 0 ) ;
h3 = f n A c t i v a t i o n F u n c t i o n ( h3 ) ;
h4 = ( ( i 1 ∗ wMtx [ 1 5 ] ) + ( i 2 + wMtx [ 1 6 ] ) + ( i 3 + wMtx [ 1 7 ] ) + \
( i 4 ∗ wMtx [ 1 8 ] ) + ( i 5 ∗ wMtx [ 1 9 ] ) + 1 . 0 ) ;
h4 = f n A c t i v a t i o n F u n c t i o n ( h4 ) ;
h5 = ( ( i 1 ∗ wMtx [ 2 0 ] ) + ( i 2 + wMtx [ 2 1 ] ) + ( i 3 + wMtx [ 2 2 ] ) + \
( i 4 ∗ wMtx [ 2 3 ] ) + ( i 5 ∗ wMtx [ 2 4 ] ) + 1 . 0 ) ;
h5 = f n A c t i v a t i o n F u n c t i o n ( h5 ) ;
o1 = ( ( h1 ∗ wMtx [ 2 5 ] ) + ( h2 ∗ wMtx [ 2 6 ] ) + ( h3 ∗ wMtx [ 2 7 ] ) + \
( h4 ∗ wMtx [ 2 8 ] ) + ( h5 ∗ wMtx [ 2 9 ] ) + 1 . 0 ) ;
o1 = f n A c t i v a t i o n F u n c t i o n ( o1 ) ;
return o1 ;
}
####################################
#∗∗∗∗∗ f n a c t i v a t i o n f u n c t i o n . h ∗∗∗∗∗#
####################################
#i f n d e f FNACTIVATIONFUNCTION H
#define FNACTIVATIONFUNCTION H
#include < s t d l i b . h>
#include <math . h>
/∗ Squash neuron o u t p u t v a l u e s t o t h e ran ge 0 . . 1 ∗/
double f n A c t i v a t i o n F u n c t i o n ( double nVal ) ;
#endif
58
####################################
#∗∗∗∗∗ f n a c t i v a t i o n f u n c t i o n . c ∗∗∗∗∗#
####################################
#include ” f n a c t i v a t i o n f u n c t i o n . h”
double f n A c t i v a t i o n F u n c t i o n ( double nVal )
{
return ( ( 1 . 0 ) / ( 1 . 0 + exp(−nVal ) ) ) ;
}
##########################
#∗∗∗∗∗ f n g e t e r r o r . h ∗∗∗∗∗#
##########################
#i f n d e f FNGETERROR H
#define FNGETERROR H
/∗ Return s i m p l e e r r o r − D i f f e r e n c e b e t w e e n a c t u a l and e x p e c t e d o u t p u t ∗/
double f n G e t E r r o r ( double a c t u a l , double e x p e c t e d ) ;
#endif
##########################
#∗∗∗∗∗ f n g e t e r r o r . c ∗∗∗∗∗#
##########################
#include ” f n g e t e r r o r . h”
double f n G e t E r r o r ( double a c t u a l , double e x p e c t e d )
{
return ( a c t u a l − e x p e c t e d ) ;
}
#########################
#∗∗∗∗∗ f n g e t r m s e . h ∗∗∗∗∗#
#########################
#i f n d e f FNGETRMSE H
#define FNGETRMSE H
#include <math . h>
/∗ Return Root Mean Squared Error f o r t r a i n i n g c y c l e ∗/
double fnGetRMSE ( double ∗eMtx , i n t n C y c l e s ) ;
#endif
59
#########################
#∗∗∗∗∗ f n g e t r m s e . c ∗∗∗∗∗#
#########################
#include ” f n g e t r m s e . h”
double fnGetRMSE ( double ∗eMtx , i n t n C y c l e s )
{
double sigma = 0 . 0 ;
int i ;
f o r ( i = 0 ; i < n C y c l e s ; i ++)
{
sigma += pow ( eMtx [ i ] , 2 ) ;
}
return s q r t ( sigma / n C y c l e s ) ;
}
###########################
#∗∗∗∗∗ fngetrandom . h ∗∗∗∗∗#
###########################
#i f n d e f FNGETRANDOM H
#define FNGETRANDOM H
#include < s t d l i b . h>
#include <f l o a t . h>
#include <time . h>
/∗ I n i t random s e e d ∗/
void i n i t r a n d ( ) ;
/∗ Get random f l o a t ∗/
double fnGetRandom ( ) ;
#endif
###########################
#∗∗∗∗∗ fngetrandom . c ∗∗∗∗∗#
###########################
#include ” fngetrandom . h”
void i n i t r a n d ( )
{
/∗ Seed random number g e n e r a t o r t o i n c r e a s e random d i s t r i b u t i o n ∗/
s r a n d ( ( unsigned ) ( time ( 0 ) ) ) ;
}
double fnGetRandom ( )
{
/∗ Return a random d o u b l e p r e c i s i o n v a l u e ∗/
return ( ( double ) rand ( ) / RAND MAX) ;
}
60
#############################
#∗∗∗∗∗ w r i t e s a m p l e s . py ∗∗∗∗∗#
#############################
#! / u s r / b i n / python2
# Python s c r i p t t o g e n e r a t e C f u n c t i o n c a l l s f o r s e n d i n g
# g e n e r a t e d ECG v a l u e s t o t h e n e u r a l n e t w o r k
from random import ∗
# Import random f u n c t i o n a l i t y
f = open ( ’Smp−T . e c g ’ , ’w ’ )
# Dump v a l u e s t o a t e x t t i l e
j = 0
# Loop c o u n t e r t o e n s u r e t h a t 40 v a l u e s may be w r i t t e n
f o r i in r a n g e ( 2 0 ) :
p
q
r
s
t
o
=
=
=
=
=
=
s t r ( round ( u n i f o r m ( 0 . 5 5 , 0 . 6 2 )
s t r ( round ( u n i f o r m ( 0 . 4 0 , 0 . 4 6 )
s t r ( round ( u n i f o r m ( 1 . 3 8 , 1 . 4 5 )
s t r ( round ( u n i f o r m ( 0 . 2 6 , 0 . 2 1 )
s t r ( round ( u n i f o r m ( 0 . 6 0 , 0 . 6 6 )
str (0.00)
# Expected output
,
,
,
,
,
2) )
2) )
2) )
2) )
2) )
#
#
#
#
#
Healthy
Healthy
Healthy
Healthy
Healthy
P
Q
R
S
T
wave
wave
wave
wave
wave
ran ge
r ange
ra nge
ran ge
ra nge
# Generate C f u n c t i o n c a l l s
c a l c s t r = ” oNet = f n C a l c u l a t e N e t w o r k (wMtx , ” + s t r ( p ) + ” , ” + s t r ( q ) + \
” , ” + s t r ( r ) + ” , ” + s t r ( s ) + ” , ” + s t r ( t ) + ” ) ; ” + ” \n”
f . write ( c a l c s t r )
e r r s t r = ”eMtx [ ” + s t r ( j ) + ” ] = f n G e t E r r o r ( oNet , 0 . 0 ) ; ” + ” \n”
f . w r i t e ( e r r s t r ) # Append h e a l t h y f u n c t i o n c a l l t o f i l e
j += 1
p
q
r
s
t
o
=
=
=
=
=
=
s t r ( round ( u n i f o r m ( − 0 . 0 9 , −0.02) , 2 ) )
s t r ( round ( u n i f o r m ( − 0 . 2 6 , −0.14) , 2 ) )
s t r ( round ( u n i f o r m ( 2 . 1 9 , 2 . 4 2 ) , 2 ) )
s t r ( round ( u n i f o r m ( − 0 . 2 , −0.3) , 2 ) )
s t r ( round ( u n i f o r m ( 0 . 3 7 , 0 . 5 5 ) , 2 ) )
str (1.00)
# Expected output
#
#
#
#
#
U n h e a l t h y <e c g wave> ran ge
...
...
...
...
# Generate C f u n c t i o n c a l l s
c a l c s t r = ” oNet = f n C a l c u l a t e N e t w o r k (wMtx , ” + s t r ( p ) + ” , ” + s t r ( q ) + \
” , ” + s t r ( r ) + ” , ” + s t r ( s ) + ” , ” + s t r ( t ) + ” ) ; ” + ” \n”
f . write ( c a l c s t r )
e r r s t r = ”eMtx [ ” + s t r ( j ) + ” ] = f n G e t E r r o r ( oNet , 1 . 0 ) ; ” + ” \n”
f . w r i t e ( e r r s t r ) # Append u n h e a l t h y f u n c t i o n c a l l t o f i l e
j += 1
f . c l o s e ( ) # C l o s e f i l e stream
61
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