Fuzzy Logic and Probability Applications

by Ross, Timothy J.; Booker, Jane M.; Parkinson, W. Jerry; Zadeh, Lotfi A.; Suppes, Patrick

• ISBN13: 9780898715255
• ISBN10: 0898715253
• Format: Hardcover
• Copyright: 2002-11-01
• Publisher: Society for Industrial & Applied

Summary

Probabilists and fuzzy enthusiasts tend to disagree about which philosophy is best and they rarely work together. As a result, textbooks usually suggest only one of these methods for problem solving, but not both. This book is an exception. The authors, investigators from both fields, have combined their talents to provide a practical guide showing that both fuzzy logic and probability have their place in the world of problem solving. They work together with mutual benefit for both disciplines, providing scientists and engineers with examples of and insight into the best tool for solving problems involving uncertainty. Fuzzy Logic and Probability Applications: Bridging the Gap makes an honest effort to show both the shortcomings and benefits of each technique, and even demonstrates useful combinations of the two. It provides clear descriptions of both fuzzy logic and probability, as well as the theoretical background, examples.

Table of Contents

Foreword	xv
Lotfi A. Zadeh
Foreword	xix
Patrick Suppes
Preface	xxi
Acknowledgments	xxiii
Part I. Fundamentals	1	(124)
Jane M. Booker
Chapters 1--6	1	(1)
Suggested reading	2	(1)
References	2	(1)
Introduction	3	(26)
Timothy J. Ross
Jane M. Booker
W. Jerry Parkinson
Some history and initial thoughts	3	(3)
The great debate	6	(14)
The debate literature	6	(1)
The issues and controversy	7	(13)
Fuzzy logic and probability: The best of both worlds	20	(2)
Organization of the book	22	(7)
References	24	(5)
Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems	29	(26)
Timothy J. Ross
W. Jerry Parkinson
Introduction	29	(5)
Fuzzy sets	31	(1)
Fuzzy set operations	32	(2)
Fuzzy relations	34	(2)
Operations on fuzzy relations	34	(1)
Fuzzy Cartesian product and composition	35	(1)
Fuzzy and classical logic	36	(19)
Classical logic	36	(5)
Fuzzy logic	41	(2)
Approximate reasoning	43	(1)
Fuzzy systems	43	(3)
An example numerical simulation	46	(6)
Summary	52	(1)
References	53	(2)
Probability Theory	55	(18)
Nozer D. Singpurwalla
Jane M. Booker
Thomas R. Bement
The calculus of probability	55	(2)
Popular views of probability	57	(6)
The interpretation of probability	57	(1)
The classical theory of probability	57	(1)
The a priori theory of probability	58	(1)
The relative frequency theory	58	(2)
The personalistic or subjective theory	60	(1)
Choosing an interpretation of probability	61	(1)
The use of expert testimonies in personalistic/subjective probability	62	(1)
Concepts for probability theory	63	(5)
Concepts of a random variable and sample space	63	(1)
Probability distribution functions	64	(2)
Conditional probability and dependence	66	(1)
Comparing distributions	66	(1)
Representing data, information, and uncertainties as distributions	67	(1)
Information, data, and knowledge	68	(5)
References	69	(4)
Bayesian Methods	73	(14)
Kimberly F. Sellers
Jane M. Booker
Introduction	73	(3)
Probability theory of Bayesian methods	76	(3)
The incorporation of actual information (expansion of H)	77	(1)
The likelihood principle	78	(1)
Distribution function formulation of Bayes' theorem	78	(1)
Issues with Bayes' theory	79	(2)
Criticisms and interpretations of Bayes' theorem	79	(1)
Uses and interpretations	80	(1)
Bayesian updating: Implementation of Bayes' theorem in practical applications	81	(2)
Binomial/beta example	81	(1)
Exponential/gamma example	82	(1)
Normal/normal example	82	(1)
Bayesian networks	83	(2)
Relationship with fuzzy sets	85	(2)
References	85	(2)
Considerations for Using Fuzzy Set Theory and Probability Theory	87	(18)
Timothy J. Ross
Kimberly F. Sellers
Jane M. Booker
Vagueness, imprecision, and chance: Fuzziness versus probability	87	(2)
A historical perspective on vagueness	88	(1)
Imprecision	89	(1)
Chance versus vagueness	89	(1)
Many-valued logic	90	(1)
Axiomatic structure of probability and fuzzy logics	91	(4)
Relationship between vagueness and membership functions	94	(1)
Relationship between fuzzy set theory and Bayesian analysis	95	(1)
Early works comparing fuzzy set theory and probability theory	95	(2)
Treatment of uncertainty and imprecision: Treating membership functions as likelihoods	97	(1)
Ambiguity versus chance: Possibility versus probability	98	(4)
Conclusions	102	(3)
References	103	(2)
Guidelines for Eliciting Expert Judgment as Probabilities or Fuzzy Logic	105	(20)
Mary A. Meyer
Kenneth B. Butterfield
William S. Murray
Ronald E. Smith
Jane M. Booker
Introduction	105	(1)
Method	106	(15)
Illustration	106	(2)
Summary table of phases and steps	108	(1)
Phases and steps for expert elicitation	108	(13)
Summary	121	(4)
References	122	(3)
Part II. Applications	125	(268)
Timothy J. Ross
Chapters 7--15	125	(2)
Image Enhancement: Probability Versus Fuzzy Expert Systems	127	(18)
Aly El-Osery
Mo Jamshidi
Introduction	127	(1)
Background	128	(5)
Digital image-processing elements	128	(1)
Image formats	128	(4)
Spatial domain	132	(1)
Probability density function	132	(1)
Histogram equalization	133	(4)
Expert systems and image enhancement	137	(6)
SOI detection	138	(1)
Fuzzy expert system development	139	(1)
Example	139	(4)
Final remarks	143	(2)
References	143	(2)
Engineering Process Control	145	(48)
W. Jerry Parkinson
Ronald E. Smith
Introduction	145	(1)
Background	146	(10)
Basic PID control	148	(2)
Basic fuzzy logic control	150	(2)
Basic probabilistic control	152	(4)
SISO control systems	156	(11)
A system model and a PID controller	160	(7)
Multi-input-multi-output control systems	167	(7)
The three-tank MIMO problem	174	(14)
The fuzzy and probabilistic control systems	174	(4)
The PI controller	178	(1)
Setpoint tracking: Comparison between the controllers	179	(2)
Disturbance rejection: The fuzzy and probabilistic controllers	181	(5)
Disturbance rejection: Comparison between the controllers	186	(2)
Conclusions	188	(5)
References	190	(3)
Structural Safety Analysis: A Combined Fuzzy and Probability Approach	193	(26)
Timothy J. Ross
Jonathan L. Lucero
Introduction	193	(1)
An example	194	(1)
Typical uncertainties	194	(2)
Current treatment of uncertainties	196	(4)
Response surface method	197	(3)
Problems with current methods	200	(1)
The fuzzy set alternative	201	(2)
Examples	203	(13)
Deterministic/random uncertainties	204	(1)
Modeling uncertainties	204	(7)
Interpretation of damage	211	(5)
Summary	216	(3)
References	217	(2)
Aircraft Integrity and Reliability	219	(24)
Carlos Ferregut
Roberto A. Osegueda
Yohans Mendoza
Vladik Kreinovich
Timothy J. Ross
Case study: Aircraft structural integrity: Formulation of the problem	219	(1)
Aerospace testing: Why	219	(1)
Aerospace testing: How	219	(1)
Aerospace integrity testing is very time-consuming and expensive	220	(1)
Solving the problem	220	(2)
Our main idea	220	(1)
Steps necessary for implementing the main idea	220	(1)
We do not have sufficient statistical data, so we must use expert estimates	221	(1)
Soft computing	221	(1)
The choices of transformation and combination functions are very important	221	(1)
How can we solve the corresponding optimization problem?	222	(1)
How to determine probabilities from observed values of excess energy: Optimal way (use of simulated annealing)	222	(4)
An expression for probabilities	222	(1)
Best in what sense?	223	(1)
An optimality criterion can be nonnumeric	223	(1)
The optimality criterion must be final	224	(1)
The criterion must not change if we change the measuring unit for energy	224	(1)
Definitions and the main result	225	(1)
How to determine the probability of detection: Optimal way (use of neural networks)	226	(2)
The POD function must be smooth and monotonic	226	(1)
We must choose a family of functions, not a single function	227	(1)
Definition and the main result	227	(1)
How to combine probabilities (use of fuzzy techniques)	228	(3)
Traditional probabilistic approach: Maximum entropy	228	(1)
Traditional approach is not always sufficient	228	(1)
Main idea: Describe general combination operations	229	(1)
The notions of t-norms and t-conorms	230	(1)
Preliminary results	231	(1)
Alternative approach to fusing probabilities: Fuzzy rules	231	(4)
Main problems with the above approach	231	(1)
The use of fuzzy rules	231	(1)
Expert rules for fault detection	232	(1)
The problem with this rule base and how we solve it	233	(1)
Experimental results	234	(1)
Applications to aircraft reliability	235	(2)
Reliability: General problem	235	(1)
Traditional approach to reliability	235	(1)
Traditional approach is not always sufficient: A problem	235	(1)
Proposed approach to fusing probabilities: Main idea	236	(1)
Resulting solution	237	(1)
Closing thoughts	237	(6)
Appendix: Proofs	237	(1)
Proof of Theorem 10.1	237	(1)
Proof of Theorem 10.2	238	(2)
References	240	(3)
Auto Reliability Project	243	(20)
Jane M. Booker
Thomas R. Bement
Description of the reliability problem	243	(3)
Implementing the probability approach	246	(7)
Logic and reliability models	246	(1)
Expert elicitation	247	(2)
Updating methods	249	(1)
Calculating reliabilities	250	(2)
Results	252	(1)
Documentation	253	(1)
Auto performance using fuzzy approach	253	(6)
Aspects of the fuzzy and probability approaches	253	(1)
A fuzzy/probability hybrid approach	254	(2)
A fuzzy automotive example	256	(3)
Comments on approaches for system performance/reliability	259	(4)
References	260	(3)
Control Charts for Statistical Process Control	263	(62)
W. Jerry Parkinson
Timothy J. Ross
Introduction	263	(1)
Background	264	(11)
Example 1: Measured variable control chart	265	(4)
Example 2: Special-cause events	269	(3)
Example 3: Traditional SPC	272	(3)
Fuzzy techniques for measurement data: A case study	275	(13)
Background	275	(1)
The fuzzy system	276	(2)
Plant simulation	278	(3)
Example 4: Establishing fuzzy membership values	281	(3)
Example 5: Fuzzy control	284	(2)
Example 6: A campaign	286	(2)
SPC techniques for measurement data requiring correlation and using regression	288	(6)
Example 7: Regression on Example 4	290	(1)
Example 8: Regression on Example 5	291	(2)
Example 9: Regression on Example 6	293	(1)
Comments on regression	294	(1)
SPC techniques for attribute data	294	(2)
Example 10: Log cabin revisited	294	(2)
Example 11: The log cabin with a special-cause problem	296	(1)
Fuzzy techniques for attribute data	296	(16)
Example 12: Fuzzy attribute data	299	(1)
The fuzzy system used to downgrade parts	300	(12)
Statistical techniques for multinomial attribute data	312	(10)
Discussion and conclusions	322	(3)
References	323	(2)
Fault Tree Logic Models	325	(22)
Jonathan L. Lucero
Timothy J. Ross
Introduction	325	(1)
Objective	325	(2)
Chapter overview	327	(1)
Part A: General methodology	327	(14)
Fault trees	327	(1)
General logic methodology	328	(3)
Mathematical operators	331	(4)
Examples	335	(3)
Summary	338	(3)
Part B: General methodology extended with fuzzy logic	341	(4)
Membership functions	341	(1)
Dependence	341	(3)
Summary	344	(1)
Conclusions	345	(2)
References	345	(2)
Uncertainty Distributions Using Fuzzy Logic	347	(18)
Ronald E. Smith
W. Jerry Parkinson
Thomas R. Bement
Introduction	347	(2)
Example 1: The use of fuzzy expertise to develop uncertainty distributions for cutting tool wear	349	(3)
Demonstration of the technique	350	(2)
Example 2: Development of uncertainty distributions using fuzzy and probabilistic techniques	352	(11)
Description and illustration of the fuzzy setup of the method	353	(3)
Fuzzy development of uncertainty distributions	356	(3)
Probabilistic characterization of the development of uncertainty distributions	359	(2)
Extended example	361	(1)
Concluding remarks	362	(1)
Summary	363	(2)
References	364	(1)
Signal Validation Using Bayesian Belief Networks and Fuzzy Logic	365	(28)
Hrishikesh Aradhye
A. Sharif Heger
Introduction	365	(2)
Bayesian belief networks	367	(14)
BBNs for SFDIA	373	(1)
Construction of nodes, links, and associated functions	373	(1)
A single-variable, single-sensor system	374	(1)
Incorporation of redundancies in the network structure	375	(6)
Fuzzy logic	381	(8)
Using fuzzy logic for fault detection	382	(7)
Summary	389	(4)
References	390	(3)
Index	393