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Fuzzy logic and probability applications : bridging the gap

Author: Timothy J Ross; Jane M Booker; W J Parkinson
Publisher: Philadelphia, Pa. : Society for Industrial and Applied Mathematics ; Alexandria, Va. : American Statistical Assoxciation, ©2002.
Series: ASA-SIAM series on statistics and applied probability.
Edition/Format:   Print book : EnglishView all editions and formats
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Additional Physical Format: Online version:
Fuzzy logic and probability applications.
Philadelphia, Pa. : Society for Industrial and Applied Mathematics ; Alexandria, Va. : American Statistical Assoxciation, ©2002
(OCoLC)606975413
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Timothy J Ross; Jane M Booker; W J Parkinson
ISBN: 0898715253 9780898715255
OCLC Number: 50022722
Description: xxiii, 409 p. : ill. (some col.) ; 26 cm.
Contents: 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 --
2 Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems / Timothy J. Ross, W. Jerry Parkinson 29 --
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 --
2.3.4 Fuzzy systems 43 --
2.3.5 An example numerical simulation 46 --
3 Probability Theory / Nozer D. Singpurwalla, Jane M. Booker, Thomas R. Bement 55 --
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 --
4 Bayesian Methods / Kimberly F. Sellers, Jane M. Booker 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 applications 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 --
5 Considerations for Using Fuzzy Set Theory and Probability Theory / Timothy J. Ross, Kimberly F. Sellers, Jane M. Booker 87 --
5.1 Vagueness, imprecision, and chance: Fuzziness versus probability 87 --
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 --
6 Guidelines for Eliciting Expert Judgment as Probabilities or Fuzzy Logic / Mary A. Meyer, Kenneth B. Butterfield, William S. Murray, Ronald E. Smith, Jane M. Booker 105 --
6.2.2 Summary table of phases and steps 108 --
6.2.3 Phases and steps for expert elicitation 108 --
Part II. Applications / Timothy J. Ross 125 --
7 Image Enhancement: Probability Versus Fuzzy Expert Systems / Aly El-Osery, Mo Jamshidi 127 --
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 --
8 Engineering Process Control / W. Jerry Parkinson, Ronald E. Smith 145 --
8.2.1 Basic PID control 148 --
8.2.2 Basic fuzzy logic control 150 --
8.2.3 Basic probabilistic control 152 --
8.3 SISO control systems 156 --
8.3.1 A system model and a PID controller 160 --
8.4 Multi-input-multi-output control systems 167 --
8.5 The three-tank MIMO problem 174 --
8.5.1 The fuzzy and probabilistic control systems 174 --
8.5.2 The PI controller 178 --
8.5.3 Setpoint tracking: Comparison between the controllers 179 --
8.5.4 Disturbance rejection: The fuzzy and probabilistic controllers 181 --
8.5.5 Disturbance rejection: Comparison between the controllers 186 --
9 Structural Safety Analysis: A Combined Fuzzy and Probability Approach / Timothy J. Ross, Jonathan L. Lucero 193 --
9.2 An example 194 --
9.3 Typical uncertainties 194 --
9.4 Current treatment of uncertainties 196 --
9.4.1 Response surface method 197 --
9.5 Problems with current methods 200 --
9.6 The fuzzy set alternative 201 --
9.7.1 Deterministic/random uncertainties 204 --
9.7.2 Modeling uncertainties 204 --
9.7.3 Interpretation of damage 211 --
10 Aircraft Integrity and Reliability / Carlos Ferregut, Roberto A. Osegueda, Yohans Mendoza, Vladik Kreinovich, Timothy J. Ross 219 --
10.1 Case study: Aircraft structural integrity: Formulation of the problem 219 --
10.1.1 Aerospace testing: Why 219 --
10.1.2 Aerospace testing: How 219 --
10.1.3 Aerospace integrity testing is very time-consuming and expensive 220 --
10.2 Solving the problem 220 --
10.2.1 Our main idea 220 --
10.2.2 Steps necessary for implementing the main idea 220 --
10.2.3 We do not have sufficient statistical data, so we must use expert estimates 221 --
10.2.4 Soft computing 221 --
10.2.5 The choices of transformation and combination functions are very important 221 --
10.2.6 How can we solve the corresponding optimization problem? 222 --
10.3 How to determine probabilities from observed values of excess energy: Optimal way (use of simulated annealing) 222 --
10.3.1 An expression for probabilities 222 --
10.3.2 Best in what sense? 223 --
10.3.3 An optimality criterion can be nonnumeric 223 --
10.3.4 The optimality criterion must be final 224 --
10.3.5 The criterion must not change if we change the measuring unit for energy 224 --
10.3.6 Definitions and the main result 225 --
10.4 How to determine the probability of detection: Optimal way (use of neural networks) 226 --
10.4.1 The POD function must be smooth and monotonic 226 --
10.4.2 We must choose a family of functions, not a single function 227 --
10.4.3 Definition and the main result 227 --
10.5 How to combine probabilities (use of fuzzy techniques) 228 --
10.5.1 Traditional probabilistic approach: Maximum entropy 228 --
10.5.2 Traditional approach is not always sufficient 228 --
10.5.3 Main idea: Describe general combination operations 229 --
10.5.4 The notions of t-norms and t-conorms 230 --
10.6 Preliminary results 231 --
10.7 Alternative approach to fusing probabilities: Fuzzy rules 231 --
10.7.1 Main problems with the above approach 231 --
10.7.2 The use of fuzzy rules 231 --
10.7.3 Expert rules for fault detection 232 --
10.7.4 The problem with this rule base and how we solve it 233 --
10.7.5 Experimental results 234 --
10.8 Applications to aircraft reliability 235 --
10.8.1 Reliability: General problem 235 --
10.8.2 Traditional approach to reliability 235 --
10.8.3 Traditional approach is not always sufficient: A problem 235 --
10.8.4 Proposed approach to fusing probabilities: Main idea 236 --
10.8.5 Resulting solution 237 --
Appendix Proofs 237 --
Proof of Theorem 10.1 237 --
Proof of Theorem 10.2 238 --
11 Auto Reliability Project / Jane M. Booker, Thomas R. Bement 243 --
11.1 Description of the reliability problem 243 --
11.2 Implementing the probability approach 246 --
11.2.1 Logic and reliability models 246 --
11.2.2 Expert elicitation 247 --
11.2.3 Updating methods 249 --
11.2.4 Calculating reliabilities 250 --
11.2.5 Results 252 --
11.2.6 Documentation 253 --
11.3 Auto performance using fuzzy approach 253 --
11.3.1 Aspects of the fuzzy and probability approaches 253 --
11.3.2 A fuzzy/probability hybrid approach 254 --
11.3.3 A fuzzy automotive example 256 --
11.4 Comments on approaches for system performance/reliability 259 --
12 Control Charts for Statistical Process Control / W. Jerry Parkinson, Timothy J. Ross 263 --
12.2.1 Example 1: Measured variable control chart 265 --
12.2.2 Example 2: Special-cause events 269 --
12.2.3 Example 3: Traditional SPC 272 --
12.3 Fuzzy techniques for measurement data: A case study 275 --
12.3.2 The fuzzy system 276 --
12.3.3 Plant simulation 278 --
12.3.4 Example 4: Establishing fuzzy membership values 281 --
12.3.5 Example 5: Fuzzy control 284 --
12.3.6 Example 6: A campaign 286 --
12.4 SPC techniques for measurement data requiring correlation and using regression 288 --
12.4.1 Example 7: Regression on Example 4 290 --
12.4.2 Example 8: Regression on Example 5 291 --
12.4.3 Example 9: Regression on Example 6 293 --
12.4.4 Comments on regression 294 --
12.5 SPC techniques for attribute data 294 --
12.5.1 Example 10: Log cabin revisited 294 --
12.5.2 Example 11: The log cabin with a special-cause problem 296 --
12.6 Fuzzy techniques for attribute data 296 --
12.6.1 Example 12: Fuzzy attribute data 299 --
12.6.2 The fuzzy system used to downgrade parts 300 --
12.7 Statistical techniques for multinomial attribute data 312 --
13 Fault Tree Logic Models / Jonathan L. Lucero, Timothy J. Ross 325 --
13.4 Part A: General methodology 327 --
13.4.1 Fault trees 327 --
13.4.2 General logic methodology 328 --
13.4.3 Mathematical operators 331 --
13.5 Part B: General methodology extended with fuzzy logic 341.
Series Title: ASA-SIAM series on statistics and applied probability.
Responsibility: edited by Timothy J. Ross, Jane M. Booker, W. Jerry Parkinson.
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