Fuzzy Logic and Probability Applications: A Practical Guide (ASA-SIAM Series on Statistics and Applied Probability)

434 Pages·2002·25.45 MB·English

by  Timothy J. Ross

Fuzzy Logic and
Probability Applications
ASA-SIAM Series on
Statistics and Applied Probability
The ASA-SIAM Series on Statistics and Applied Probability is published
jointly by the American Statistical Association and the Society for Industrial and
Applied Mathematics. The series consists of a broad spectrum of books on topics
in statistics and applied probability. The purpose of the series is to provide
inexpensive, quality publications of interest to the intersecting membership of the
two societies.
Editorial Board
Robert N. Rodriguez Lisa LaVange
SAS Institute Inc., Editor-in-Chief Inspire Pharmaceuticals, Inc.
Janet P. Buckingham Gary C. McDonald
Southwest Research Institute National Institute of Statistical
Sciences
Richard K. Burdick
Arizona State University Paula Roberson
University of Arkansas
James A. Calvin for Medical Sciences
Texas A&M University
Dale L. Zimmerman
Katherine Bennett Ensor University of Iowa
Rice University
Douglas M. Hawkins
University of Minnesota
Ross, T. J., Booker, J. M., and Parkinson, W. J., eds., Fuzzy Logic and
Probability Applications: Bridging the Gap
Nelson, W., Recurrent Events Data Analysis for Product Repairs, Disease Recurrence,
and Other Applications
Mason, R. L. and Young, J. C, Multivariate Statistical Process Control with
Industrial Applications
Smith, P. L., A Primer for Sampling Solids, Liquids, and Gases: Based on the Seven
Sampling Errors of Pierre Gy
Meyer, M. A. and Booker, J. M., Eliciting and Analyzing Expert Judgment:
A Practical Guide
Latouche, G. and Ramaswami, V., Introduction to Matrix Analytic Methods in
Stochastic Modeling
Peck, R., Haugh, L., and Goodman, A., Statistical Case Studies: A Collaboration
Between Academe and Industry, Student Edition
Peck, R., Haugh, L., and Goodman, A., Statistical Case Studies: A Collaboration
Between Academe and Industry
Barlow, R., Engineering Reliability
Czitrom, V. and Spagon, P. D., Statistical Case Studies for Industrial Process
Improvement
Fuzzy Logic and
Probability Applications
Bridging the Gap
Edited by
Timothy J. Ross
University of New Mexico
Albuquerque, New Mexico
Jane M. Booker
Los Alamos National Laboratory
Los Alamos, New Mexico
W. Jerry Parkinson
Los Alamos National Laboratory
Los Alamos, New Mexico
ASA
Society for Industrial and Applied Mathematics American Statistical Association
Philadelphia, Pennsylvania Alexandria, Virginia
The correct bibliographic citation for this book is as follows: Ross, Timothy J., Jane M.
Booker, and W. Jerry Parkinson, eds., Fuzzy Logic and Probability Applications: Bridging
the Cap, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia, ASA,
Alexandria, VA, 2002.
Copyright © 2002 by the American Statistical Association and the Society for Industrial
and Applied Mathematics.
10987654321
All rights reserved. Printed in the United States of America. No part of this book may be
reproduced, stored, or transmitted in any manner without the written permission of the
publisher. For information, write to the Society for Industrial and Applied Mathematics,
3600 University City Science Center, Philadelphia, PA 19104-2688.
Library of Congress Cataloging-in-Publication Data
Fuzzy logic and probability applications : bridging the gap / edited by Timothy J. Ross,
Jane M. Booker, W. Jerry Parkinson.
p. cm. — (ASA-SIAM series on statistics and applied probability)
Includes bibliographical references and index.
ISBN 0-89871-525-3
1. Fuzzy logic. 2. Probabilities. 3. Fuzzy logic—Industrial applications. 4.
Probabilities—Industrial applications. I. Ross, Timothy J. II. Booker, Jane M. III.
Parkinson, W. J. (William Jerry), 1939- IV. Series.
QA9.64.F8942002
511.3-dc21
2002075756
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To the memory of our colleague
and his inspiration for this book,
Dr. Thomas (Tom) R. Bement,
Los Alamos, New Mexico
List of Contributors
Hrishikesh Aradhye Yohans Mendoza
SRI International Sirius Images, Inc.
Thomas R. Bement (dec.) Mary A. Meyer
Los Alamos National Laboratory Los Alamos National Laboratory
Jane M. Booker William S. Murray
Los Alamos National Laboratory Los Alamos National Laboratory
Kenneth B. Butterfield Roberto A. Osegueda
Los Alamos National Laboratory University of Texas at El Paso
Aly El-Osery W. Jerry Parkinson
New Mexico Institute of Mining and Los Alamos National Laboratory
Technology
Timothy J. Ross
Carlos Ferregut University of New Mexico
University of Texas at El Paso
Kimberly F. Sellers
Mo Jamshidi Carnegie Mellon University
University of New Mexico
Nozer D. Singpurwalla
A. Sharif Heger George Washington University
Los Alamos National Laboratory
Ronald E. Smith
Vladik Kreinovich Los Alamos National Laboratory
University of Texas at El Paso
Jonathan L. Lucero
University of New Mexico
Contents
Foreword xv
Lotfi A. Zadeh
Foreword xix
Patrick Suppes
Preface xxi
Acknowledgments xxiii
Part I. Fundamentals 1
Jane M. Booker
1.1 Chapters 1-6 1
1.2 Suggested reading 2
References 2
1 Introduction 3
Timothy J. Ross, Jane M. Booker, and W. Jerry Parkinson
1.1 Some history and initial thoughts 3
1.2 The great debate 6
1.2.1 The debate literature 6
1.2.2 The issues and controversy 7
1.3 Fuzzy logic and probability: The best of both worlds 20
1.4 Organization of the book 22
References 24
2 Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems 29
Timothy J. Ross and W. Jerry Parkinson
2.1 Introduction 29
2.1.1 Fuzzy sets 31
2.1.2 Fuzzy set operations 32
2.2 Fuzzy relations 34
2.2.1 Operations on fuzzy relations 34
2.2.2 Fuzzy Cartesian product and composition 35
2.3 Fuzzy and classical logic 36
2.3.1 Classical logic 36
2.3.2 Fuzzy logic 41
2.3.3 Approximate reasoning 43
vii
viii Contents
2.3.4 Fuzzy systems 43
2.3.5 An example numerical simulation 46
2.3.6 Summary 52
References 53
3 Probability Theory 55
Nozer D. Singpunvalla, Jane M. Booker, and Thomas R. Bement
3.1 The calculus of probability 55
3.2 Popular views of probability 57
3.2.1 The interpretation of probability 57
3.2.2 The classical theory of probability 57
3.2.3 The a priori theory of probability 58
3.2.4 The relative frequency theory 58
3.2.5 The personalistic or subjective theory 60
3.2.6 Choosing an interpretation of probability 61
3.2.7 The use of expert testimonies in personalistic/subjective
probability 62
3.3 Concepts for probability theory 63
3.3.1 Concepts of a random variable and sample space 63
3.3.2 Probability distribution functions 64
3.3.3 Conditional probability and dependence 66
3.3.4 Comparing distributions 66
3.3.5 Representing data, information, and uncertainties as
distributions 67
3.4 Information, data, and knowledge 68
References 69
4 Bayesian Methods 73
Kimberly F. Sellers and Jane M. Booker
4.1 Introduction 73
4.2 Probability theory of Bayesian methods 76
4.2.1 The incorporation of actual information (expansion of H) 77
4.2.2 The likelihood principle 78
4.2.3 Distribution function formulation of Bayes'theorem . .. 78
4.3 Issues with Bayes' theory 79
4.3.1 Criticisms and interpretations of Bayes'theorem 79
4.3.2 Uses and interpretations 80
4.4 Bayesian updating: Implementation of Bayes' theorem in practical ap-
plications 81
4.4.1 Binomial/beta example 81
4.4.2 Exponential/gamma example 82
4.4.3 Normal/normal example 82
4.5 Bayesian networks 83
4.6 Relationship with fuzzy sets 85
References 85
5 Considerations for Using Fuzzy Set Theory and Probability Theory 87
Timothy J. Ross, Kimberly F. Sellers, and Jane M. Booker
5.1 Vagueness, imprecision, and chance: Fuzziness versus probability . . . 87
Contents ix
5.1.1 A historical perspective on vagueness 88
5.1.2 Imprecision 89
5.2 Chance versus vagueness 89
5.3 Many-valued logic 90
5.4 Axiomatic structure of probability and fuzzy logics 91
5.4.1 Relationship between vagueness and membership functions 94
5.4.2 Relationship between fuzzy set theory and Bayesian analysis 95
5.5 Early works comparing fuzzy set theory and probability theory . . .. 95
5.6 Treatment of uncertainty and imprecision: Treating membership
functions as likelihoods 97
5.7 Ambiguity versus chance: Possibility versus probability 98
5.8 Conclusions 102
References 103
6 Guidelines for Eliciting Expert Judgment as Probabilities or Fuzzy Logic 105
Mary A. Meyer, Kenneth B. Butterfield, William S. Murray, Ronald E. Smith,
and Jane M. Booker
6.1 Introduction 105
6.2 Method 106
6.2.1 Illustration 106
6.2.2 Summary table of phases and steps 108
6.2.3 Phases and steps for expert elicitation 108
6.3 Summary 121
References 122
Part II. Applications 125
Timothy J. Ross
II. 1 Chapters 7-15 125
7 Image Enhancement: Probability Versus Fuzzy Expert Systems 127
Aly El-Osery and Mo Jamshidi
7.1 Introduction 127
7.2 Background 128
7.2.1 Digital image-processing elements 128
7.2.2 Image formats 128
7.2.3 Spatial domain 132
7.2.4 Probability density function 132
7.3 Histogram equalization 133
7.4 Expert systems and image enhancement 137
7.4.1 SOI detection 138
7.4.2 Fuzzy expert system development 139
7.4.3 Example 139
7.5 Final remarks 143
References 143
8 Engineering Process Control 145
W. Jerry Parkinson and Ronald E. Smith
8.1 Introduction 145