Concepts of Soft Computing: Fuzzy and ANN with Programming

206 Pages·2019·9.061 MB·English

by  Snehashish Chakraverty, Deepti Moyi Sahoo, Nisha Rani Mahato

Snehashish Chakraverty ·
Deepti Moyi Sahoo · Nisha Rani Mahato
Concepts
of Soft
Computing
Fuzzy and ANN with Programming
Concepts of Soft Computing
Snehashish Chakraverty
(cid:129)
Deepti Moyi Sahoo Nisha Rani Mahato
(cid:129)
Concepts of Soft Computing
Fuzzy and ANN with Programming
123
SnehashishChakraverty DeeptiMoyi Sahoo
Department ofMathematics Department ofMathematics,
National Institute ofTechnology Rourkela Schoolof Science
Rourkela, Odisha, India O.P. JindalUniversity
Punjipathra, India
NishaRani Mahato
Department ofMathematics
National Institute ofTechnology Rourkela
Rourkela, Odisha, India
ISBN978-981-13-7429-6 ISBN978-981-13-7430-2 (eBook)
https://doi.org/10.1007/978-981-13-7430-2
LibraryofCongressControlNumber:2019935543
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Preface
In the early 1990s, soft computing (SC) gained popularity due to its efficiency in
handling problems with imprecision, approximation, decision making, and uncer-
tainty. Unlike earlier hard computing techniques which dealt with precision, soft
computing techniques consider imprecision to obtain rigorous, computationally
efficient,androbustresults.Thefascinatingabilityofsoftcomputingtechniquesto
deal with human intelligence, reasoning, analysis, learning, imprecision, and mul-
titasking has made it favorable in various scientific and technological disciplines,
viz.computerscience,mathematics,controltheory,structuralengineering,medical,
management, psychology, etc. Generally, the principal constituents of soft com-
puting include techniques focusing on fuzzy system, evolutionary computation,
machine learning, and probabilistic reasoning. In this regard, this book starts with
anintroductorychapterforabriefknowledgeaboutsoftcomputinganditsvarious
constituents.
Among the various constituents discussed above, fuzzy systems and machine
learninghaveacquiredmuchattentioninsolvingreal-worldproblemsdealingwith
imprecisionorapproximations.Thereexistseveralbooksthatdealwithoneormore
constituents of soft computing. But, two major concepts, viz. fuzzy set theory and
neural network, form the basic platforms offuzzy systems and machine learning,
respectively. Inthisregard,thecomputational programmingparadigm ofthefuzzy
settheoryconceptsandartificialneuralnetwork(ANN)modelsisessentialfortheir
functioning in various science and engineering disciplines. But as per the best of
ourknowledge,bookscoveringthebasicconceptsoffuzzysettheoryandANNina
detailed as well as systematic manner along with programming are scarce.
Accordingly, the authors realized the need for such a book that contains a concise
and systematic description of various concepts in fuzzy set theory and ANN along
with step-by-step C and/or MATLAB codes addressing simple solved example
problemsforabetterunderstanding.Anotherkeyfeatureofthisbookisthatitalso
provides partial attention to handle interval uncertainty.
As regards, the book is mainly divided into two parts focusing on major soft
computing techniques, viz. fuzzy set theory and ANN. The fuzzy set theory part
comprises nine chapters, and ANN comprises the rest of the chapters. As a whole,
v
vi Preface
the book consists of 14 chapters giving basic knowledge of fuzzy set theory and
ANN. Before we incorporate the details of the book, the authors assume that the
readers have prerequisite knowledge of calculus, set theory, and linear algebra. In
fuzzy set theory, a fuzzy number may be approximately represented in terms of
closed intervals using a-cut approach. As such, Chap. 1 comprises basic prelimi-
naries related to intervals, interval arithmetic and interval matrices, accompanied
withCandMATLABprograms.Then,Chap.2introducesthefundamentalsrelated
to fuzzy sets with respect to the extension of crisp (classical) sets based on its
membership function. Chapter 3 discusses various types of fuzzy numbers, and
particularly, conversions of such fuzzy numbers to intervals have been done using
a-cut approach. Chapter 4 emphasizes other properties offuzzy sets such as rela-
tions and compositions on fuzzy relations. Chapter 5 elaborates different types of
fuzzy functions along with C or MATLAB codes related to fuzzy functions.
Chapter6providestheextensionofclassicaldifferentiationandintegrationtofuzzy
differentiation and integration. Chapter 7 includes the methodology of the trans-
formationoffuzzysetstocrispsetsintermsofdefuzzificationmethods.Chapters8
and 9 discuss special importance of applications using interval computation in the
system of equations and eigenvalue problems, respectively, that may be further
extended to fuzzy number with respect to the parameter a. Chapters 7–10 include
MATLAB programs only for most of the solved problems.
Chapters 10–14 introduce a computing paradigm known as ANN. Chapter 10
emphasizes mainly on the basic foundation of ANN. Chapter 11 contains
McCulloch–Pitts model. Chapters 12 and 13 contain different learning algorithms.
Further, Chap. 14 includes delta learning rule and backpropagation methods.
Chapters 10–14 present MATLAB programming codes for most of the ANNtitles
for variety of problems. In order to emphasize the importance of 14 chapters
mentioned above, few unsolved problems have also been included at the end of
some chapters for self-validation of the topics. Moreover, corresponding bibli-
ographies are given at the end of each chapter for ease and better referencing.
Thisbookismainlywritten forundergraduateandgraduatecoursesalloverthe
world.Itcoverstopicsrelatedtothebasicsofsoftcomputing,fuzzysettheory,and
neural networks. This book may also be used by researchers, industry, faculties,
etc., for understanding the two challenging soft computing techniques, viz. fuzzy
set theory and ANN, along with programming techniques. Further, the idea for
solving the uncertain system of equations and eigenvalue problems occurring in
science and engineering disciplines has also been incorporated.
Rourkela, India Snehashish Chakraverty
Punjipathra, India Deepti Moyi Sahoo
Rourkela, India Nisha Rani Mahato
Acknowledgements
The first author greatly appreciates the support, encouragement, and patience pro-
vided by hisfamily members, in particular his wife Shewli and daughters Shreyati
andSusprihaa.Further,thebookmaynothavebeenpossiblewithouttheblessings
of his beloved parents late Sh. Birendra K. Chakraborty and Smt. Parul
Chakraborty.
The second author would like to express her sincere gratitude to her family
members, especially Sh. Daya Nidhi Sahoo, Smt. Tilottama Sahoo, Trupti Moyi
Sahoo, and Deepak Kumar Sahoo, for their love and blessings. Last but not least,
the author also wants to thank her beloved son Nishit, husband Mr. Soumyaranjan
Sahoo, father-in-law Sh. Duryodhan Sahoo, and mother-in-law Smt. Rukmini
Sahoo for showing their faith and giving liberty to do all the things.
Finally, the third author’s warmest gratitude goes to her family members for
their continuous support and motivation, especially Sh. Devendra Mahato,
Smt. Premshila, Tanuja, Devasish, and Satish.
Alsosecondandthirdauthorsgreatlyappreciatetheinspirationofthefirstauthor
for his support and inspiration which helped a lot to learn new things. This work
would not have been possible without his guidance, support, and encouragement.
Authors sincerely acknowledge the reviewers for their fruitful suggestions and
appreciations of the book proposal. Further, all the authors do appreciate the sup-
portandhelpofthewholeteamofSpringer.Finally,authorsaregreatlyindebtedto
the authors/researchers mentioned in the Bibliography section given at the end of
each chapter.
Snehashish Chakraverty
Deepti Moyi Sahoo
Nisha Rani Mahato
vii
Contents
Part I Fuzzy Set Theory
1 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Preliminaries of Intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Addition and Subtraction of Intervals . . . . . . . . . . . . . 5
1.1.2 Multiplication of Intervals. . . . . . . . . . . . . . . . . . . . . . 7
1.1.3 Division and Inverse of Intervals. . . . . . . . . . . . . . . . . 8
1.2 Interval Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Arithmetic Operations on Interval Matrices . . . . . . . . . 18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Fuzzy Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Preliminaries of Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Generating Membership Function . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Support, Height, Core, and Boundary of Fuzzy Set. . . . . . . . . . 31
2.4 Intersection, Union, and Complement. . . . . . . . . . . . . . . . . . . . 35
2.5 Other Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6 Extension Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Fuzzy Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1 Preliminaries of Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Types of Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.1 Triangular Fuzzy Number (TFN). . . . . . . . . . . . . . . . . 55
3.2.2 Trapezoidal Fuzzy Number (TrFN) . . . . . . . . . . . . . . . 58
3.2.3 Gaussian Fuzzy Number (GFN) . . . . . . . . . . . . . . . . . 61
3.3 Conversion of Fuzzy Number to Interval Form Using a-Cut . . . 65
3.3.1 Triangular Fuzzy Number a-Cut Decomposition . . . . . 65
3.3.2 Trapezoidal Fuzzy Number a-Cut Decomposition . . . . 66
3.3.3 Gaussian Fuzzy Number a-Cut Decomposition . . . . . . 66
ix
x Contents
3.4 Fuzzy Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.1 Arithmetic Operations of Fuzzy Numbers . . . . . . . . . . 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Fuzzy Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1 Preliminaries of Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Intersection, Union, and Complement. . . . . . . . . . . . . . . . . . . . 72
4.3 First, Second, and Total Projection . . . . . . . . . . . . . . . . . . . . . 78
4.4 Composition of Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Fuzzy Equivalence Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . 89
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Fuzzy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1 Preliminaries of Crisp Function . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Preliminaries of Fuzzy Function . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Types of Fuzzy Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1 Type I Fuzzy Function . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.2 Type II Fuzzy Function . . . . . . . . . . . . . . . . . . . . . . . 99
5.3.3 Type III Fuzzy Function. . . . . . . . . . . . . . . . . . . . . . . 101
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 Fuzzy Differentiation and Integration . . . . . . . . . . . . . . . . . . . . . . . 105
6.1 Fuzzy Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.1 Fuzzifying Function at Crisp Point . . . . . . . . . . . . . . . 106
6.1.2 Crisp Function at Fuzzy Point. . . . . . . . . . . . . . . . . . . 108
6.2 Fuzzy Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2.1 Fuzzifying Function Over Crisp Limits . . . . . . . . . . . . 110
6.2.2 Crisp Function Over Fuzzy Limits. . . . . . . . . . . . . . . . 112
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7 Defuzzification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.1 Defuzzification Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1.1 Aggregation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1.2 Max-Membership Method. . . . . . . . . . . . . . . . . . . . . . 121
7.1.3 Centroid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.1.4 Weighted-Average Method . . . . . . . . . . . . . . . . . . . . . 126
7.1.5 Mean–Max Method . . . . . . . . . . . . . . . . . . . . . . . . . . 126
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8 Interval System of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . 129
8.1 System of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Interval System of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.2.1 Methods for Solving ISLEs. . . . . . . . . . . . . . . . . . . . . 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Contents xi
9 Interval Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.1 Eigenvalue Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.1.1 Standard Eigenvalue Problems . . . . . . . . . . . . . . . . . . 142
9.1.2 Generalized Eigenvalue Problems . . . . . . . . . . . . . . . . 143
9.2 Interval Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2.1 Vertex Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Part II Artificial Neural Network
10 Artificial Neural Network Terminologies. . . . . . . . . . . . . . . . . . . . . 153
10.1 Artificial Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.2 Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.2.1 Single Layer Neural Network . . . . . . . . . . . . . . . . . . . 154
10.2.2 Multilayer Neural Network . . . . . . . . . . . . . . . . . . . . . 154
10.2.3 Competitive Layer Neural Network. . . . . . . . . . . . . . . 155
10.3 Different Training Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.3.1 Supervised Training . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.3.2 Unsupervised Training . . . . . . . . . . . . . . . . . . . . . . . . 156
10.4 Activation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.4.1 Some Common Activation Function . . . . . . . . . . . . . . 158
10.4.2 Weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.4.3 Bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.4.4 Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.5 Computing Net Input Using Matrix Multiplication Method . . . . 160
10.6 Matlab Programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.6.1 Write a Matlab Program to Generate the Following
Activation Functions That Are Used in Neural
Networks: (a) Identity Function, (b) Binary Step
Function with Threshold Value 1 and 0, (c) Binary
Sigmoid Function, and (d) Bipolar Sigmoid
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.6.2 Write a Program in Matlab if the Net Input to an
OutputNeuronIs0.38,CalculateItsOutputWhenthe
Activation Function Is (a) Binary Sigmoid Function
and (b) Bipolar Sigmoid Function . . . . . . . . . . . . . . . . 164
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11 McCulloch–Pitts Neural Network Model. . . . . . . . . . . . . . . . . . . . . 167
11.1 McCulloch–Pitts Neural Network Model . . . . . . . . . . . . . . . . . 167
11.2 McCulloch–Pitts Neuron Architecture . . . . . . . . . . . . . . . . . . . 167
11.3 Matlab Programs for McCulloch–Pitts Neuron Model. . . . . . . . 168
11.3.1 Write a Matlab Program to Generate the Output of
Logic AND Function by McCulloch–Pitts Neuron
Model. The Threshold on Unit Is 2. . . . . . . . . . . . . . . 168