Uncertain Rule-Based Fuzzy Logic Systems:

Transcription

1 Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions Jerry M. Mendel University of Southern California Los Angeles, CA PH PTR Prentice Hall PTR Upper Saddle River, NJ ISBN *^-3

2 Contents PREFACE xv *i PART 1: PRELIMINARIES 1 INTRODUCTION Rule-BasedFLSs 1.2 A New Direction for FLSs New Concepts and Their Historical Background Fundamental Design Requirement The Flow of Uncertainties H 1.6 Existing Literature on Type-2 Fuzzy Sets Coverage 1.8 Applicability Outside of Rule-Based FLSs Computation 18 SUPPLEMENTARY MATERIAL: SHORT PRIMERS ON FUZZY SETS AND FUZZY LOGIC 1.10 Primer on Fuzzy Sets Crisp sets From crisp sets to fuzzy sets Linguistic variables Membership functions Some terminology Set theoretic Operations for crisp sets Set theoretic Operations for fuzzy sets Crisp relations and compositions on the same product space Fuzzy relations and compositions on the same product space Crisp relations and compositions on different product Spaces Fuzzy relations and compositions on different product Spaces Hedges 42 6

3 VI Contents Extension principle Primer on FL Crisplogic From crisp logic to FL Remarks 59 Exercises 59 2 SoURCES OF UNCERTAINTY Uncertainties in a FLS Uncertainty: General discussions Uncertainty: In a FLS Words Mean Different Things to Different People 70 Exercises 78 3 MEMBERSHIP FUNCTIONS AND UNCERTAINTY Introduction Type-1 Membership Functions Type-2 Membership Functions The concept of a type-2 fuzzy set Definition of a type-2 fiizzy set and associated concepts More examples of type-2 fuzzy sets and FOUs Upper and lower membership functions Embedded type-2 and type-1 sets Type-1 fuzzy sets represented as type-2 fuzzy sets Zero and one memberships in a type-2 fuzzy set Returning to Linguistic Labels Multivariable Membership Functions Type-1 membership functions Type-2 membership functions Computation 107 Exercises 108

4 Table of Contents vii 4 CASESTUDIES Introduction Forecasting of Time-Series Extracting rules from the data Mackey-Glass chaotic time-series Knowledge Mining Using Surveys Methodology for knowledge mining Survey results Methodology for designing a FLA HowtouseaFLA 124 Exercises 126 PART 2: TYPE-1 FUZZY LOGIC SYSTEMS 5 SINGLETON TYPE-1 FUZZY LOGIC SYSTEMS: NO UNCERTAINTIES Introduction Rules Fuzzy Inference Engine Fuzzification and Its Effect on Inference Fuzzifier Fuzzy inference engine Defuzzification Centroid defuzzifier Center-of-sums defuzzifier Height defuzzifier Modified height defuzzifier Center-of-sets defuzzifier An interesting fact Possibilities 149

5 VIII Contents 5.7 Fuzzy Basis Functions FLSs Are Universal Approximators Designing FLSs One-pass methods Least-squares method Back-propagation (steepest descent) method SVD-QR method Iterative design method Case Study: Forecasting of Time-Series One-pass design Back-propagation design A change in the measurements Case Study: Knowledge Mining Using Surveys A veraging the responses Preserving all the responses A Final Remark Computation 184 Exercises NON-SINGLETON TYPE-1 FUZZY LOGIC SYSTEMS m 6.1 Introduction Fuzzification and Its Effect on Inference Fuzzifier Fuzzy inference engine Possibilities FBFs Non-Singleton FLSs Are Universal Approximators Designing Non-Singleton FLSs One-pass methods Least-squares method Back-propagation (steepest descent) method SVD-QR method Iterative design method Case Study: Forecasting of Time-Series One-pass design Back-propagation design A Final Remark 209

6 Table of Contents IX 6.9 Computation 209 Exercises 209 Part 3: Type-2 Fuzzy Sets 7 OPERATIONS ON AND PROPERTIES OF TYPE-2 FUZZY SETS Introduction Extension Principle Operations on General Type-2 Fuzzy Sets S et theoretic Operations Algebraic Operations on fuzzy numbers 223 Operations on Interval Type-2 Fuzzy Sets Set theoretic Operations Algebraic Operations on interval fuzzy numbers 227 Summary of Operations Properties of Type-2 Fuzzy Sets Type-1 fuzzy sets Type-2 fuzzy sets 230 Computation Exercises TYPE-2 RELATIONS AND COMPOSITIONS 8.1 Introduction Relations in General Relations and Compositions on the Same Product Space Relations and Compositions on Different Product Spaces Composition of a Set with a Relation Cartesian Product of Fuzzy Sets

7 X Contents.7 Implications 246 Exercises CENTROID OF A TYPE-2 FUZZY SET: TYPE-REDUCTION Introduction General Results for the Centroid Generalized Centroid for Interval Type-2 Fuzzy Sets Centroid of an Interval Type-2 Fuzzy Set Type-Reduction: General Results Centroid type-reduction Center-of-sums type-reduction Height type-reduction Modified height type-reduction Center-of-sets type-reduction Computational complexity of type-reduction Concluding example Type-Reduction: Interval Sets Centroid type-reduction Center-of-sums type-reduction Height type-reduction Modified height type-reduction Center-of-sets type-reduction Concluding example Concluding Remark Computation 280 Exercises 281 PART 4: TYPE-2 FUZZY LOGIC SYSTEMS

8 Table of Contents xi 10 SINGLETON TYPE-2 FUZZY LOGIC SYSTEMS Introduction Rules Fuzzy Inference Engine Fuzzification and Its Effect on Inference Fuzzifier Fuzzy inference engine Type-Reduction Defuzzification Possibilities FBFs: The Lack Thereof Interval Type-2 FLSs Upper and lower membership functions for interval type-2 FLSs Fuzzy inference engine revisited Type-reduction and defuzzification revisited FBFs revisited Designing Interval Singleton Type-2 FLSs One-pass method Least-squares method Back-propagation (steepest descent) method SVD-QR method Iterative design method Case Study: Forecasting of Time-Series Case Study: Knowledge Mining Using Surveys Computation 350 Exercises TYPE-1 NON-SINGLETON TYPE-2 FUZZY LOGIC SYSTEMS Intr o ducti on Fuzzification and Its Effect on Inference Fuzzifier Fuzzy inference engine Interval Type-1 Non-Singleton Type-2 FLSs 356

9 XII Contents 11.4 Designing Interval Type-1 Non-Singleton Type-2 FLSs One-pass method Least-squares method Back-propagation (steepest descent) method SVD-QR method Iterative design method Case Study: Forecasting of Time-Series Final Reraark Computation 380 Exercises TYPE-2 NON-SINGLETON TYPE-2 FUZZY LOGIC SYSTEMS Introduction Fuzzification and Its Effect on Inference Fuzzifier Fuzzy inference engine Interval Type-2 Non-Singleton Type-2 FLSs Designing Interval Type-2 Non-Singleton Type-2 FLSs One-pass method Least-squares method Back-propagation (steepest descent) method SVD-QR method Iterative design method Case Study: Forecasting of Time-Series Six-epoch back-propagation design One-epoch combined back-propagation and SVD-QR design Six-epoch iterative combined back-propagation and SVD-QR design Computation 417 Exercises TSK FUZZY LOGIC SYSTEMS Introduction Type-1 TSK FLSs First-order type-1 TSK FLS 422

10 Table of Contents xiii A connection between type-1 TSK and Mamdani FLSs TSK FLSs are universal approximators Designing type-1 TSK FLSs Type-2 TSK FLSs First-order type-2 TSK FLS Interval type-2 TSK FLSs Unnormalized interval type-2 TSK FLSs Further comparisons of TSK and Mamdani FLSs Designing interval type-2 TSK FLSs using a back-propagation (steepest descent) method Example: Forecasting of Compressed Video Traffic Introduction to MPEG video traffic Forecasting I frame sizes: General information Forecastinglframe sizes: Usingthe samenumber of rules Forecasting I frame sizes: Using the same number of design Parameters Conclusion Final Remark Computation 451 Exercises E PILOGUE Introduction Type-2 Versus Type-1 FLSs Appropriate Applications for a Type-2 FLS Rule-Based Classification of Video Traffic Selected features FOUs for the features Rules FOUs for the measurements Design parameters in a FL RBC Computational formulas for type-1 FL RBCs Computational formulas for type-2 FL RBCs Optimization ofrule design-parameters Testing the FL RBCs Results and conclusions Equalization of Time-Varying Non-linear Digital Communication Channels Preliminaries for Channel equalization Why a type-2 FAF is needed Designing the FAFs 476

11 xiv Contents Simulations and conclusions Overcoming CCI and ISI for Digital Communication Channels Communication System with ISI and CCI Designing the FAFs Simulations and conclusions Connection Admission Control for ATM Networks Survey-based CAC using a type-2 FLS: Overview Extracting the knowledge for CAC Choosing membership functions for the linguistic labeis Survey processing CAC decision boundaries and conclusions Potential Application Areas for a Type-2 FLS Perceptual Computing FL control Diagnostic medicine Financial applications Perceptual designs ofmultimedia Systems 500 Exercises 500 A «JOIN, MEET, AND NEGATION OP ERATIONS FOR NON-lNTERVAL TYPE-2 FUZZY SETS 502 A.l Introduction 502 A.2 Join Under Minimum or Product t-norms 503 A.3 Meet Under Minimum t-norm 504 A.4 Meet Under Product t-norm 509 A.5 Negation 512 A.6 Computation 514 Exercises 515 B PROPERTIES OF TYPE-1 AND TYPE-2 FUZZY SETS 517 B.l Introduction 517

12 Table of Contents xv B.2 Type-1 Fuzzy Sets 517 B.3 Type-2 Fuzzy Sets 520 Exercises 525 c c OMPUTATION 526 C.l Type-1 FLSs 526 C.2 General Type-2 FLSs 527 C.3 Interval Type-2 FLSs 528 R I EFERENCES 530 NDEX 547