Uncertainty Modeling and Analysis in Engineering and the Sciences

Bilal M. Ayyub, George J. Klir9781584886440, 1584886447

Engineers and scientists often need to solve complex problems with incomplete information resources, necessitating a proper treatment of uncertainty and a reliance on expert opinions. Uncertainty Modeling and Analysis in Engineering and the Sciences prepares current and future analysts and practitioners to understand the fundamentals of knowledge and ignorance, how to model and analyze uncertainty, and how to select appropriate analytical tools for particular problems. This volume covers primary components of ignorance and their impact on practice and decision making. It provides an overview of the current state of uncertainty modeling and analysis, and reviews emerging theories while emphasizing practical applications in science and engineering. The book introduces fundamental concepts of classical, fuzzy, and rough sets, probability, Bayesian methods, interval analysis, fuzzy arithmetic, interval probabilities, evidence theory, open-world models, sequences, and possibility theory. The authors present these methods to meet the needs of practitioners in many fields, emphasizing the practical use, limitations, advantages, and disadvantages of the methods.

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Table of contents :
Uncertainty Modeling and Analysis in Engineering and the Sciences……Page 1
Preface……Page 4
STRUCTURE, FORMAT, AND MAIN FEATURES……Page 7
Acknowledgments……Page 10
About the Authors……Page 11
Books by the Authors……Page 13
Contents……Page 15
Appendix A: Historical Perspectives on Knowledgeed……Page 358
Bibliography……Page 365
1.1 DATA ABUNDANCE AND UNCERTAINTY……Page 21
1.2.1 SYSTEMS DEFINITIONS AND MODELING……Page 23
1.2.2 REALISM AND CONSTRUCTIVISM IN SYSTEMS THINKING……Page 25
1.2.3 TAXONOMY OF SYSTEMS……Page 26
1.2.3.1 Epistemological Categories of Systems……Page 27
1.2.3.2 Source (or Experimental Frame) Systems……Page 28
1.2.3.4 Generative Systems……Page 31
1.2.3.5 Structure Systems……Page 33
1.2.3.6 Metasystems……Page 34
Source Systems……Page 35
Data Systems……Page 38
Generative Systems……Page 39
Metasystems……Page 42
1.2.4 DISCIPLINARY ROOTS OF SYSTEMS SCIENCE……Page 45
1.2.5 SYSTEMS KNOWLEDGE, METHODOLOGY, AND METAMETHODOLOGY……Page 47
1.2.6 COMPLEXITY AND SIMPLIFICATION OF SYSTEMS……Page 49
1.2.7 COMPUTATIONAL COMPLEXITY AND LIMITATIONS……Page 52
1.3.1 TERMINOLOGY AND DEFINITIONS……Page 57
1.3.2 ABSOLUTE REALITY AND ABSOLUTE KNOWLEDGE……Page 59
1.3.3 KNOWLEDGE, INFORMATION, AND OPINIONS……Page 60
1.3.4 REASONING, SCIENCE, AND UNCERTAINTY……Page 63
1.3.5 COGNITION AND COGNITIVE SCIENCE……Page 66
1.4.1 KNOWLEDGE AND IGNORANCE……Page 68
1.4.2.1 Ignorance Classification……Page 70
1.4.2.2 Ignorance Hierarchy……Page 72
1.4.3.2 Aleatory and Epistemic Uncertainties……Page 76
1.4.3.3.1 Abstraction for System Modeling……Page 79
1.4.3.3.2 Ignorance and Uncertainty in Abstracted System Aspects……Page 80
1.4.3.3.4 Ignorance due to Unknown System Aspects……Page 82
1.5 FROM DATA TO KNOWLEDGE FOR DECISION MAKING……Page 83
EXERCISE PROBLEMS……Page 85
2.2 IDENTIFICATION AND CLASSIFICATION OF THEORIES……Page 86
2.3.1 A UNIVERSE AND ITS ELEMENTS……Page 89
2.3.2 CLASSICAL (CRISP) SETS AND EVENTS……Page 90
2.3.3 PROPERTIES OF SETS AND SUBSETS……Page 91
2.3.5 SAMPLE SPACE AND EVENTS……Page 93
2.3.7 VENN–EULER DIAGRAMS……Page 94
2.3.8 BASIC OPERATIONS ON SETS……Page 95
2.3.9 POWER SETS……Page 97
2.4 FUZZY SETS AND OPERATIONS……Page 98
2.4.1 MEMBERSHIP FUNCTION……Page 99
2.4.2 alpha-CUT REPRESENTATION OF SETS……Page 103
2.4.4 OPERATIONS ON FUZZY SETS……Page 104
2.4.5 CARDINALITY OF FUZZY SETS……Page 116
2.4.6 FUZZY SUBSETS……Page 117
2.4.7 FUZZY INTERVALS, NUMBERS, AND ARITHMETIC……Page 118
2.4.8 FUZZY RELATIONS……Page 126
2.4.9 FUZZIFIED AND FUZZY FUNCTIONS……Page 131
2.5 GENERALIZED MEASURES……Page 133
2.6.1 ROUGH SET DEFINITIONS……Page 134
2.6.2 ROUGH SET OPERATIONS……Page 136
2.6.3 MEMBERSHIP FUNCTIONS FOR ROUGH SETS……Page 137
2.6.4 ROUGH FUNCTIONS……Page 138
EXERCISE PROBLEMS……Page 140
3.2 KNOWLEDGE, SYSTEMS, UNCERTAINTY, AND INFORMATION……Page 145
3.3 MEASURE THEORY AND CLASSICAL MEASURES……Page 147
3.4.1 DEFINITION OF MONOTONE MEASURES……Page 149
3.4.2 CLASSIFYING MONOTONE MEASURES……Page 150
3.5.1 BELIEF MEASURES……Page 154
3.5.3 INTERPRETATION OF BELIEF AND PLAUSIBILITY MEASURES……Page 155
3.5.4 MÖBIUS REPRESENTATION AS A BASIC ASSIGNMENT……Page 156
3.5.5.1 Dempster’s Rule of Combination……Page 157
3.5.5.3 Inagaki’s Rule of Combination……Page 158
3.5.5.4 Mixed or Averaging Rule of Combination……Page 159
3.6.1 CLASSICAL POSSIBILITY THEORY……Page 165
3.6.2 THEORY OF GRADED POSSIBILITIES……Page 166
3.7.2 CLASSICAL DEFINITIONS OF PROBABILITY……Page 169
3.7.4 FAILURE RATES……Page 171
3.7.5 CENTRAL TENDENCY MEASURES……Page 173
3.7.6 DISPERSION (OR VARIABILITY)……Page 174
3.7.7 PERCENTILE VALUES……Page 175
3.7.8 STATISTICAL UNCERTAINTY……Page 176
3.7.9 BAYESIAN PROBABILITIES……Page 178
3.8 IMPRECISE PROBABILITIES……Page 184
3.8.1 INTERVAL PROBABILITIES……Page 185
3.8.2 INTERVAL CUMULATIVE DISTRIBUTION FUNCTIONS……Page 189
3.8.3 DEPENDENCE MODELING AND MEASURES……Page 192
3.8.3.1 Perfect Independence……Page 193
3.8.3.2 Mutual Exclusion……Page 194
3.8.5 PERFECT DEPENDENCE……Page 195
3.8.7.1 Correlation Based on Probability Theory……Page 196
3.8.7.2 Statistical Correlation……Page 200
3.8.8 CORRELATION BETWEEN EVENTS……Page 202
3.8.9 UNKNOWN DEPENDENCE BETWEEN EVENTS……Page 203
3.8.12 PROBABILITY BOUNDS……Page 204
3.9 FUZZY MEASURES AND FUZZY INTEGRALS……Page 211
EXERCISE PROBLEMS……Page 215
4.2 UNCERTAINTY MEASURES: DEFINITION AND TYPES……Page 220
4.3.1 HARTLEY MEASURE……Page 222
4.3.2 HARTLEY-LIKE MEASURE……Page 227
4.3.3 EVIDENCE NONSPECIFICITY……Page 228
4.3.4 NONSPECIFICITY OF GRADED POSSIBILITY……Page 229
4.3.5 NONSPECIFICITY OF FUZZY SETS OR U-UNCERTAINTY……Page 230
4.4 ENTROPY-LIKE MEASURES……Page 232
4.4.1 SHANNON ENTROPY PROBABILITY DISTRIBUTIONS……Page 233
4.4.3.1 Measure of Dissonance……Page 236
4.4.3.2 Measure of Confusion……Page 237
4.4.4 AGGREGATE AND DISAGGREGATE UNCERTAINTY IN EVIDENCE THEORY……Page 238
4.5 FUZZINESS MEASURE……Page 241
4.6.2 PERCENTILES FOR COMBINING OPINIONS……Page 243
4.6.3 WEIGHTED COMBINATIONS OF OPINIONS……Page 244
EXERCISE PROBLEMS……Page 247
5.1 INTRODUCTION……Page 250
5.2 CONSTRUCTION OF KNOWLEDGE……Page 251
5.3 MINIMUM UNCERTAINTY PRINCIPLE……Page 252
5.4 MAXIMUM UNCERTAINTY PRINCIPLE……Page 253
5.5 UNCERTAINTY INVARIANCE PRINCIPLE……Page 263
5.6 METHODS FOR OPEN-WORLD ANALYSIS……Page 265
5.6.1.1 Laplace Model……Page 266
5.6.1.2 Add-c Model……Page 267
5.6.1.3 Witten–Bell Model……Page 268
5.6.2 TRANSFERABLE BELIEF MODEL……Page 269
5.6.3 OPEN-WORLD ASSUMPTION MATHEMATICAL FRAMEWORK……Page 271
5.6.4 EVIDENTIAL REASONING MECHANISM, BELIEF REVISION, AND DIAGNOSTICS……Page 272
EXERCISE PROBLEMS……Page 273
6.1 INTRODUCTION……Page 275
6.2.1 ANALYTIC PROBABILISTIC METHODS……Page 276
6.2.1.1 Probability Distributions for Dependent Random Variables……Page 277
6.2.1.2 Mathematical Expectations……Page 281
6.2.1.3.1 Single Random Variable X……Page 282
6.2.1.3.2 Random vector X……Page 283
6.2.2 SIMULATION METHODS……Page 284
6.2.3 VERTEX METHOD FOR FUNCTIONS OF FUZZY VARIABLES……Page 287
6.3.1 A FUNDAMENTAL INPUT–OUTPUT SYSTEM……Page 291
6.3.2 INTERVAL PARAMETERS……Page 292
6.3.3 A POWER AS AN INTERVAL AND A SET OF INTERVALS……Page 293
6.3.3.1 A Consonant or Nested Set of Intervals……Page 294
6.3.3.2 A Consistent Set of Intervals……Page 296
6.3.3.3 An Arbitrary Set of Intervals……Page 298
6.3.4.1 Consonant or Nested Sets of Intervals……Page 300
6.3.4.2 Consistent Sets of Intervals……Page 302
6.3.5.1 Power Intervals and Lognormally Distributed Parameter……Page 304
6.3.5.3 A Set of Power Intervals and an Uncertain Lognormally Distributed Parameter……Page 306
EXERCISE PROBLEMS……Page 307
7.1 INTRODUCTION……Page 312
7.3 CLASSIFICATION OF ISSUES, STUDY LEVELS, EXPERTS, AND PROCESS OUTCOMES……Page 313
7.5 NEED IDENTIFICATION FOR EXPERT OPINION ELICITATION……Page 317
7.6 SELECTION OF STUDY LEVEL AND STUDY LEADER……Page 318
7.7.2 IDENTIFICATION AND SELECTION OF EXPERTS……Page 319
7.7.3 ITEMS NEEDED BY EXPERTS AND REVIEWERS BEFORE THE EXPERT OPINION ELICITATION MEETING……Page 321
7.8 IDENTIFICATION, SELECTION, AND DEVELOPMENT OF TECHNICAL ISSUES……Page 322
7.9.2 TRAINING OF EXPERTS……Page 323
7.9.4 AGGREGATION AND PRESENTATION OF RESULTS……Page 324
7.10 DOCUMENTATION AND COMMUNICATION……Page 325
Personal Flotation Device Face Plane Angle (FPA)……Page 326
Personal Flotation Device Turning Time (TT) from Facedown……Page 329
EXERCISE PROBLEMS……Page 333
8.1 INTRODUCTION……Page 335
8.2.2 POINT AND GLOBAL VISUALIZATION……Page 338
8.2.3 USE OF COLORS……Page 340
8.2.4 FINANCIAL VISUALIZATION……Page 341
8.2.5.2 Ignoricons and Uncerticons……Page 342
8.3.1 DEFINITION OF PRIMARY SELECTION CRITERIA AND WEIGHT FACTORS……Page 344
8.3.2 DEFINITION AND SELECTION SUBCRITERIA AND WEIGHT FACTORS……Page 347
8.3.4 DESIGN OF VISUALIZATION METHOD……Page 349
8.3.5 ASSESSMENT METHODS AND EXPERIMENTAL PROTOCOL……Page 351
8.3.6 CANDIDATE SHAPES FOR IGNORICONS AND UNCERTICONS……Page 353
8.4.1 INTELLIGENT AGENTS……Page 354
8.4.2 INFORMATION UNCERTAINTY AGENT……Page 355
8.4.3 PROCESSING INFORMATION FOR SYMBOLOGY SELECTION……Page 356
EXERCISE PROBLEMS……Page 357