Bounding Uncertainty in Civil Engineering (eBook)

Preface 5
Contents 7
Motivation 11
Why Use Random Sets? 11
Histograms 11
Empirical Limitations in Data Gathering 12
Modeling 17
Imprecise Information Cannot Give Precise Conclusions 21
Describing Void Information 23
Bounding Uncertainty 24
Review of Theory of Probability and Notation 25
Probability Measures 25
Random Variable 28
Joint Probability Spaces 31
Random Vectors 33
Random Sets and Imprecise Probabilities 35
Extension of Probabilistic Information 35
Multi-valued Mapping from a Probability Space 35
Theory of Evidence 36
Inner/Outer Extension of a Probability Space 37
Random Sets 39
Formal Definition of Random Sets 39
Equivalent Representations of Random Sets 41
Probability Distributions Compatible with a Random Set 44
Consonant Random Sets 57
Conditioning 63
Imprecise Probabilities and Monotone Non-additive Measures 67
Introduction 67
Coherent Upper and Lower Previsions 72
Choquet and Alternating Choquet Capacities of Order k 79
Expectation Bounds and Choquet Integral for Real Valued Functions 83
The Generalized Bayes’ Rule 87
Credal Sets 93
Interval Valued Probabilities 93
P-Boxes 96
Convex Sets of Parametric Probability Distributions 105
Conclusions 109
Notes 110
Random Relations 112
Random Relations and Marginals 112
Stochastic Independence in the Theory of Imprecise Probabilities 118
Unknown Interaction 123
Epistemic Independence and Irrelevance 138
Strong Independence 160
Relationships between the Four Types of Independence 166
Independence When Marginals Are Random Sets 167
Random Set Independence 168
Unknown Interaction 172
Epistemic Independence 179
Strong Independence 180
Fuzzy Cartesian Product or Consonant Random Cartesian Product 183
Relationships between the Five Types of Independence 185
Correlation 186
The Entire Random Relation Is Given 188
Only the Marginals Are Given 190
Conclusions 205
Notes 207
Inclusion and Mapping of Random Sets/Relations 211
Inclusion of Random Sets 211
Weak Inclusion 211
Strong Inclusion 220
Including a Random Set in a Consonant Random Set 229
Inclusion Properties for Random Relations under the Hypotheses of Random Set Independence and Non-interactivity 235
Mappings of Sets/Relations 247
Extension Principle 247
Monotonicity of Operations on Random Relations 257
Conclusions 260
Approximate Reasoning 262
The Basic Problem 262
Combination and Updating within Set Theory 263
Statistical Combination and Updating 269
Bayesian Combining and Updating in Probability Theory 271
Limits Entailed by the Probabilistic Solution 276
Set-Valued Mapping 276
Variables Linked by a Joint Random Relation 276
Conditioning a Random Set to an Event B 277
Not Deterministic Mapping 280
Probability Kinematics and Nets of Italian Flags 281
Combination of Random Sets 284
Evidence Theory: Dempster’s Rule of Combination 284
A Critical Discussion of Dempster’s Rule: Yager’s Rule of Combination 288
Fuzzy Logic and Fuzzy Composition Rule 291
Introduction 291
Fuzzy Extension of Set Operations 292
Fuzzy Composition Rule 294
Fuzzy Approximate Reasoning 299
Introduction 299
Inference from Conditional Fuzzy Propositions 300
Pattern Recognition and Clustering 302
Fuzzy Model of Multi-objective Decision Making 306
Conclusions 313
Notes 314
References 316
Subject Index 325