Uncertain Rule-Based Fuzzy Logic Systems Introduction and New Directions

by Mendel, Jerry M.

• ISBN13: 9780130409690
• ISBN10: 0130409693
• Edition: 1st
• Format: Paperback
• Copyright: 2000-12-22
• Publisher: Prentice Hall

Summary

Type-2 fuzzy logic: Breakthrough techniques for modeling uncertainty Key applications: digital mobile communications, computer networking, and video traffic classification Detailed case studies: Forecasting time series and knowledge mining Contains 90+ worked examples, 110+ figures, and brief introductory primers on fuzzy logic and fuzzy sets Breakthrough fuzzy logic techniques for handling real-world uncertainty. The world is full of uncertainty that classical fuzzy logic can't model. Now, however, there's an approach to fuzzy logic that can model uncertainty: "type-2" fuzzy logic. In this book, the developer of type-2 fuzzy logic demonstrates how it overcomes the limitations of classical fuzzy logic, enabling a wide range of applications from digital mobile communications to knowledge mining. Dr. Jerry Mendel presents a bottom-up approach that begins by introducing traditional "type-1" fuzzy logic, explains how it can be modified to handle uncertainty, and, finally, adds layers of complexity to handle increasingly sophisticated applications. Coverage includes: The sources of uncertainty and the role of membership functions Type-2 fuzzy sets: operations, properties, and centroids Singleton, non-singleton, and TSK Type 2 fuzzy logic systems Comparing "type-2" and "type 1" results Extensive applications coverage: digital mobile communications, computer networking, and video traffic classification Two start-to-finish case studies: Forecasting time series and knowledge mining Carefully balanced between theory and design, the book contains over 90 worked examplesand more than 110 figures. It is ideal for engineers, scientists, computer science researchers, and mathematicians interested in AI, rule-based systems, and modeling uncertainty. Since it contains brief introductory primers on fuz

Author Biography

DR. JERRY MENDEL is Professor of Electrical Engineering and Associate Director of the Integrated Media Systems Center at the University of Southern California. He has published over 380 technical papers and seven books, and has been involved in fuzzy logic research for over 14 years.

Table of Contents

Preface	xvii
Part 1: Preliminaries
Introduction	3	(63)
Rule-Based FLSs	6	(2)
A New Direction for FLSs	8	(1)
New Concepts and Their Historical Background	9	(2)
Fundamental Design Requirement	11	(1)
The Flow of Uncertainties	11	(1)
Existing Literature on Type-2 Fuzzy Sets	12	(2)
Coverage	14	(4)
Applicability Outside of Rule-Based FLSs	18	(1)
Computation	18	(1)
Supplementary Material: Short Primers on Fuzzy Sets and Fuzzy Logic
Primer on Fuzzy Sets	19	(29)
Crisp sets	19	(1)
From crisp sets to fuzzy sets	20	(2)
Linguistic variables	22	(1)
Membership functions	23	(2)
Some terminology	25	(1)
Set theoretic operations for crisp sets	26	(1)
Set theoretic operations for fuzzy sets	27	(4)
Crisp relations and compositions on the same product space	31	(2)
Fuzzy relations and compositions on the same product space	33	(3)
Crisp relations and compositions on different product spaces	36	(3)
Fuzzy relations and compositions on different product spaces	39	(3)
Hedges	42	(2)
Extension principle	44	(4)
Primer on FL	48	(11)
Crisp logic	48	(5)
From crisp logic to FL	53	(6)
Remarks	59	(7)
Exercises	59	(7)
Sources of Uncertainty	66	(13)
Uncertainties in a FLS	66	(4)
Uncertainty: General discussions	66	(2)
Uncertainty: In a FLS	68	(2)
Words Mean Different Things to Different People	70	(9)
Exercises	78	(1)
Membership Functions and Uncertainty	79	(31)
Introduction	79	(1)
Type-1 Membership Functions	79	(1)
Type-2 Membership Functions	80	(22)
The concept of a type-2 fuzzy set	81	(1)
Definition of a type-2 fuzzy set and associated concepts	81	(10)
More examples of type-2 fuzzy sets and FOUs	91	(2)
Upper and lower membership functions	93	(5)
Embedded type-2 and type-1 sets	98	(4)
Type-1 fuzzy sets represented as type-2 fuzzy sets	102	(1)
Zero and one memberships in a type-2 fuzzy set	102	(1)
Returning to Linguistic Labels	102	(3)
Multivariable Membership Functions	105	(2)
Type-1 membership functions	105	(2)
Type-2 membership functions	107	(1)
Computation	107	(3)
Exercises	108	(2)
Case Studies	110	(21)
Introduction	110	(1)
Forecasting of Time-Series	110	(8)
Extracting rules from the data	112	(3)
Mackey--Glass chaotic time-series	115	(3)
Knowledge Mining Using Surveys	118	(13)
Methodology for knowledge mining	119	(2)
Survey results	121	(1)
Methodology for designing a FLA	122	(2)
How to use a FLA	124	(2)
Exercises	126	(5)
Part 2: Type-1 Fuzzy Logic Systems
Singleton Type-1 Fuzzy Logic Systems: No Uncertainties	131	(55)
Introduction	131	(1)
Rules	132	(3)
Fuzzy Inference Engine	135	(3)
Fuzzification and Its Effect on Inference	138	(4)
Fuzzifier	139	(1)
Fuzzy inference engine	139	(3)
Defuzzification	142	(7)
Centroid defuzzifier	143	(1)
Center-of-sums defuzzifier	143	(2)
Height defuzzifier	145	(2)
Modified height defuzzifier	147	(1)
Center-of-sets defuzzifier	147	(1)
An interesting fact	148	(1)
Possibilities	149	(2)
Fuzzy Basis Functions	151	(5)
FLSs Are Universal Approximators	156	(1)
Designing FLSs	157	(12)
One-pass methods	160	(2)
Least-squares method	162	(2)
Back-propagation (steepest descent) method	164	(2)
SVD-QR method	166	(2)
Iterative design method	168	(1)
Case Study: Forecasting of Time-Series	169	(7)
One-pass design	171	(1)
Back-propagation design	171	(2)
A change in the measurements	173	(3)
Case Study: Knowledge Mining Using Surveys	176	(7)
Averaging the responses	178	(5)
Preserving all the responses	183	(1)
A Final Remark	183	(1)
Computation	184	(2)
Exercises	184	(2)
Non-Singleton Type-1 Fuzzy Logic Systems	186	(27)
Introduction	186	(1)
Fuzzification and Its Effect on Inference	187	(6)
Fuzzifier	187	(1)
Fuzzy inference engine	188	(5)
Possibilities	193	(1)
FBFs	193	(2)
Non-Singleton FLSs Are Universal Approximators	195	(2)
Designing Non-Singleton FLSs	197	(6)
One-pass methods	200	(1)
Least-squares method	200	(1)
Back-propagation (steepest descent) method	200	(2)
SVD-QR method	202	(1)
Iterative design method	203	(1)
Case Study: Forecasting of Time-Series	203	(6)
One-pass design	204	(1)
Back-propagation design	205	(4)
A Final Remark	209	(1)
Computation	209	(4)
Exercises	209	(4)
Part 3: Type-2 Fuzzy Sets
Operations on and Properties of Type-2 Fuzzy Sets	213	(22)
Introduction	213	(1)
Extension Principle	214	(2)
Operations on General Type-2 Fuzzy Sets	216	(8)
Set theoretic operations	217	(6)
Algebraic operations on fuzzy numbers	223	(1)
Operations on Interval Type-2 Fuzzy Sets	224	(5)
Set theoretic operations	224	(3)
Algebraic operations on interval fuzzy numbers	227	(2)
Summary of Operations	229	(1)
Properties of Type-2 Fuzzy Sets	230	(1)
Type-1 fuzzy sets	230	(1)
Type-2 fuzzy sets	230	(1)
Computation	231	(4)
Exercises	231	(4)
Type-2 Relations and Compositions	235	(13)
Introduction	235	(1)
Relations in General	235	(3)
Relations and Compositions on the Same Product Space	238	(3)
Relations and Compositions on Different Product Spaces	241	(2)
Composition of a Set with a Relation	243	(1)
Cartesian Product of Fuzzy Sets	244	(2)
Implications	246	(2)
Exercises	247	(1)
Centroid of a Type-2 Fuzzy Set: Type-Reduction	248	(39)
Introduction	248	(1)
General Results for the Centroid	248	(8)
Generalized Centroid for Interval Type-2 Fuzzy Sets	256	(4)
Centroid of an Interval Type-2 Fuzzy Set	260	(5)
Type-Reduction: General Results	265	(12)
Centroid type-reduction	265	(2)
Center-of-sums type-reduction	267	(1)
Height type-reduction	268	(2)
Modified height type-reduction	270	(1)
Center-of-sets type-reduction	270	(2)
Computational complexity of type-reduction	272	(1)
Concluding example	273	(4)
Type-Reduction: Interval Sets	277	(2)
Centroid type-reduction	277	(1)
Center-of-sums type-reduction	278	(1)
Height type-reduction	278	(1)
Modified height type-reduction	278	(1)
Center-of-sets type-reduction	278	(1)
Concluding example	279	(1)
Concluding Remark	279	(1)
Computation	280	(7)
Exercises	281	(6)
Part 4: Type-2 Fuzzy Logic Systems
Singleton Type-2 Fuzzy Logic Systems	287	(66)
Introduction	287	(1)
Rules	288	(1)
Fuzzy Inference Engine	289	(2)
Fuzzification and Its Effect on Inference	291	(2)
Fuzzifier	291	(1)
Fuzzy inference engine	292	(1)
Type-Reduction	293	(4)
Defuzzification	297	(1)
Possibilities	298	(2)
FBFs: The Lack Thereof	300	(2)
Interval Type-2 FLSs	302	(19)
Upper and lower membership functions for interval type-2 FLSs	302	(2)
Fuzzy inference engine revisited	304	(4)
Type-reduction and defuzzification revisited	308	(10)
FBFs revisited	318	(3)
Designing Interval Singleton Type-2 FLSs	321	(13)
One-pass method	323	(2)
Least-squares method	325	(1)
Back-propagation (steepest descent) method	326	(6)
SVD-QR method	332	(1)
Iterative design method	333	(1)
Case Study: Forecasting of Time-Series	334	(4)
Case Study: Knowledge Mining Using Surveys	338	(12)
Computation	350	(3)
Exercises	350	(3)
Type-1 Non-Singleton Type-2 Fuzzy Logic Systems	353	(29)
Introduction	353	(1)
Fuzzification and Its Effect on Inference	354	(2)
Fuzzifier	354	(1)
Fuzzy inference engine	355	(1)
Interval Type-1 Non-Singleton Type-2 FLSs	356	(13)
Designing Interval Type-1 Non-Singleton Type-2 FLSs	369	(7)
One-pass method	372	(1)
Least-squares method	372	(1)
Back-propagation (steepest descent) method	373	(3)
SVD-QR method	376	(1)
Iterative design method	376	(1)
Case Study: Forecasting of Time-Series	376	(4)
Final Remark	380	(1)
Computation	380	(2)
Exercises	380	(2)
Type-2 Non-Singleton Type-2 Fuzzy Logic Systems	382	(39)
Introduction	382	(1)
Fuzzification and Its Effect on Inference	383	(2)
Fuzzifier	383	(1)
Fuzzy inference engine	383	(2)
Interval Type-2 Non-Singleton Type-2 FLSs	385	(16)
Designing Interval Type-2 Non-Singleton Type-2 FLSs	401	(8)
One-pass method	405	(1)
Least-squares method	405	(1)
Back-propagation (steepest descent) method	406	(2)
SVD-QR method	408	(1)
Iterative design method	408	(1)
Case Study: Forecasting of Time-Series	409	(8)
Six-epoch back-propagation design	409	(4)
One-epoch combined back-propagation and SVD-QR design	413	(2)
Six-epoch iterative combined back-propagation and SVD-QR design	415	(2)
Computation	417	(4)
Exercises	420	(1)
TSK Fuzzy Logic Systems	421	(33)
Introduction	421	(1)
Type-1 TSK FLSs	422	(7)
First-order type-1 TSK FLS	422	(1)
A connection between type-1 TSK and Mamdani FLSs	423	(1)
TSK FLSs are universal approximators	424	(1)
Designing type-1 TSK FLSs	424	(5)
Type-2 TSK FLSs	429	(12)
First-order type-2 TSK FLS	429	(1)
Interval type-2 TSK FLSs	430	(4)
Unnormalized interval type-2 TSK FLSs	434	(1)
Further comparisons of TSK and Mamdani FLSs	435	(2)
Designing interval type-2 TSK FLSs using a back-propagation (steepest descent) method	437	(4)
Example: Forecasting of Compressed Video Traffic	441	(10)
Introduction to MPEG video traffic	442	(2)
Forecasting I frame sizes: General information	444	(3)
Forecasting I frame sizes: Using the same number of rules	447	(1)
Forecasting I frame sizes: Using the same number of design parameters	448	(3)
Conclusion	451	(1)
Final Remark	451	(1)
Computation	451	(3)
Exercises	452	(2)
Epilogue	454	(48)
Introduction	454	(2)
Type-2 Versus Type-1 FLSs	456	(1)
Appropriate Applications for a Type-2 FLS	457	(1)
Rule-Based Classification of Video Traffic	458	(11)
Selected features	459	(2)
FOUs for the features	461	(1)
Rules	461	(1)
FOUs for the measurements	462	(1)
Design parameters in a FL RBC	462	(1)
Computational formulas for type-1 FL RBCs	463	(1)
Computational formulas for type-2 FL RBCs	464	(2)
Optimization of rule design-parameters	466	(1)
Testing the FL RBCs	467	(1)
Results and conclusions	468	(1)
Equalization of Time-Varying Non-linear Digital Communication Channels	469	(11)
Preliminaries for channel equalization	470	(5)
Why a type-2 FAF is needed	475	(1)
Designing the FAFs	476	(1)
Simulations and conclusions	476	(4)
Overcoming CCI and ISI for Digital Communication Channels	480	(9)
Communication system with ISI and CCI	481	(4)
Designing the FAFs	485	(1)
Simulations and conclusions	486	(3)
Connection Admission Control for ATM Networks	489	(8)
Survey-based CAC using a type-2 FLS: Overview	491	(1)
Extracting the knowledge for CAC	491	(1)
Choosing membership functions for the linguistic labels	492	(1)
Survey processing	492	(3)
CAC decision boundaries and conclusions	495	(2)
Potential Application Areas for a Type-2 FLS	497	(5)
Perceptual computing	498	(1)
FL control	499	(1)
Diagnostic medicine	499	(1)
Financial applications	499	(1)
Perceptual designs of multimedia systems	500	(1)
Exercises	500	(2)
A Join, Meet, and Negation Operations for Non-Interval Type-2 Fuzzy Sets	502	(15)
A.1 Introduction	502	(1)
A.2 Join Under Minimum or Product t-Norms	503	(1)
A.3 Meet Under Minimum t-Norm	504	(5)
A.4 Meet Under Product t-Norm	509	(3)
A.5 Negation	512	(2)
A.6 Computation	514	(3)
Exercises	515	(2)
B Properties of Type-1 and Type-2 Fuzzy Sets	517	(9)
B.1 Introduction	517	(1)
B.2 Type-1 Fuzzy Sets	517	(3)
B.3 Type-2 Fuzzy Sets	520	(6)
Exercises	525	(1)
C Computation	526	(4)
C.1 Type-1 FLSs	526	(1)
C.2 General Type-2 FLSs	527	(1)
C.3 Interval Type-2 FLSs	528	(2)
References	530	(17)
Index	547

Excerpts

Preface Uncertainty is the fabric that makes life interesting. For millenia human beings have developed strategies to cope with a plethora of uncertainties, never absolutely sure what the consequences would be, but hopeful that the deleterious effects of those uncertainties could be minimized. This book presents a complete methodology for accomplishing this within the framework of fuzzy logic (FL). This is not the original FL, but is an expanded and richer FL, one that contains the original FL within it. The original FL, founded by Lotfi Zadeh, has been around for more than 35 years, as of the year 2000, and yet it is unable to handle uncertainties. Byhandle, I meanto model and minimize the effect of. That the original FL--type-1 FL--cannot do this sounds paradoxical because the word fuzzy has the connotation of uncertainty. The expanded FL--type-2 FL--is able to handle uncertainties because it can model them and minimize their effects. And, if all uncertainties disappear, type-2 FL reduces to type-1 FL, in much the same way that if randomness disappears, probability reduces to determinism. Although many applications were found for type-1 FL, it is its application torule-based systemsthat has most significantly demonstrated its importance as a powerful design methodology. Such rule-based fuzzy logic systems (FLSs), both type-1 and type-2, are what this book is about. In it I show how to use FL in new ways and how to effectively solve problems that are awash in uncertainties. FL has already been applied in numerous fields, in many of which uncertainties are present (e.g., signal processing, digital communications, computer and communication networks, diagnostic medicine, operations research, financial investing, control, etc.). Hence, the results in this book can immediately be used in all of these fields. To demonstrate the performance advantages for type-2 FLSs over their type-1 counterparts, when uncertainties are present, I describe and provide results for the following applications in this book: forecasting of time series, knowledge-mining using surveys, classification of video data working directly with compressed data, equalization of time-varying nonlinear digital communication channels, overcoming co-channel interference and intersymbol interference for time-varying nonlinear digital communication channels, and connection admission control for asynchronous transfer mode networks. No control applications have been included, because to date type-2 FL has not yet been applied to them; hence, this book is not about FL control, although its methodologies may someday be applicable to it. I have organized this book into four parts. Part 1--Preliminaries-- contains four chapters that provide background materials about uncertainty, membership functions, and two case studies (forecasting of time-series and knowledge mining using surveys) that are carried throughout the book. Part 2--Type-1 Fuzzy Logic Systems--contains two chapters that are included to provide the underlying basis for the new type-2 FLSs, so that we can compare type-2 results for our case studies with type-1 results. Part 3--Type-2 Fuzzy Sets--contains three chapters, each of which focuses on a different aspect of such sets. Part 4--Type-2 Fuzzy Logic Systems--which is the heart of the book, contains five chapters, four having to do with different architectures for a FLS and how to handle different kinds of uncertainties within them, and one having to do primarily with four specific applications of type-2 FLSs. This book can be read by anyone who has an undergraduate BS degree and should be of great interest to computer scientists and engineers who already use or want to use rule-based systems and are concerned with how to handle uncertainties about such systems. I have inc