Probability Applications

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