Uncertain Rule-Based Fuzzy Logic Systems:

Introduction and New Directions

Jerry M. Mendel

University of Southern California, Los Angeles, CA

An expanded and richer fuzzy logic

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From the Preface

Uncertainty is the fabric that makes life interesting. For millenia human beings have developed strategies to cope with lots of uncertainties, never absolutely sure what the consequences would be, but hopeful that the bad 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. By handle, I mean to 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 have been found for type-1 FL, it is its application to rule-based systems that 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.

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Organization of the Book

The book is organized 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.

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Who Can Read the Book?

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. Close to 100 worked-out examples are included in the text, and more than 100 homework problems are also included at the end of most chapters so that the book can be used in a classroom setting as well as a technical reference.

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Ways in Which the Book Can be Used

• For the person totally unfamiliar with FL who wants a quick introduction to it, read the Supplement to Chapter 1 and Chapter 5 (Sections 5.1­5.8).
• For the person who wants an in-depth treatment of type-1 rule-based FLSs, read the Supplement to Chapter 1 and Chapters 4-6.
• For the person who is only interested in type-2 fuzzy set theory, read Chapters 3, 7-9, and Appendices A and B.
• For a person who wants to give a course on rule-based fuzzy logic systems, use Chapters 1-12 and 13 (if time permits). Chapter 14 should be of interest to people with a background in digital communications, pattern recognition, or communication networks and will suggest projects for a course.
• For a person who is a proponent of Takagi-Sugeno-Kang (TSK) fuzzy systems and wants to see what their type-2 counterparts look like, read Chapters 3, 7-9, and 13.
• For a person who is interested in forecasting of time-series and wants to get a quick overview of the benefits to modeling uncertainties on forecasting performance when using rule-based forecasters, read Chapters 4 (Section 4.2), 5 (Section 5.10), 6 (Section 6.7), 10 (Section 10.11), 11 (Section 11.5), and 12 (Section 12.5).
• For a person who is interested in knowledge mining and wants to get a quick overview of the benefits to modeling uncertainties on judgment making when using rule-based advisors, read Chapters 4 (Section 4.3), 5 (Section 5.11), and 10 (Section 10.12).

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Software is Available For Free

So that people will start using type-2 FL as soon as possible, free software is available online for implementing and designing type-1 and type-2 FLSs. It is MATLAB®-based (MATLAB is a registered trademark of The MathWorks, Inc.), and can be reached at: http://sipi.usc.edu/~mendel/software.

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How to Order the Book

To order the book, go to Prentice-Hall .

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Solution Manual is Available

Solutions Manual is available only to faculty from Prentice-Hall, ISBN 0130423173. Faculty must provide a university department mailing address and department telephone number. They should send a request for the Solution Manual to: Michelle_Vincenti@prenhall.com.

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Table of Contents (abbreviated)

Click here to access the detailed Table of Contents, or click on each chapter below to see its detailed contents)

Part 1: Preliminaries

Chapter 1: Introduction

Supplementary Material: Short Primers on Fuzzy Sets and Fuzzy Logic

Chapter 2: Sources of Uncertainty

Chapter 3: Membership Functions and Uncertainty

Chapter 4: Case Studies

Part 2: Type-1 Fuzzy Logic Systems

Chapter 5: Singleton Type-1 Fuzzy Logic Systems: No Uncertainties

Chapter 6: Non-Singleton Type-1 Fuzzy Logic Systems

Part 3: Type-2 Fuzzy Sets

Chapter 7: Operations on and Properties of Type-2 Fuzzy Sets

Chapter 8: Type-2 Relations and Compositions
Chapter 9: Centroid of a Type-2 Fuzzy Set: Type-Reduction

Part 4: Type-2 Fuzzy Logic Systems

Chapter 10: Singleton Type-2 Fuzzy Logic Systems

Chapter 11: Type-1 Non-Singleton Type-2 Fuzzy Logic Systems

Chapter 12: Type-2 Non-Singleton Type-2 Fuzzy Logic Systems

Chapter 13: TSK Fuzzy Logic Systems

Chapter 14: Epilogue

 

Appendix A: Join, Meet, and Negation Operations for Non-Interval Type-2 Fuzzy Sets
Appendix B: Properties of Type-1 and Type-2 Fuzzy Sets
Appendix C: Computation

References
Index

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Detailed Table of Contents

Preface

Part 1: Preliminaries

Chapter 1: Introduction

1.1: Rule-Based FLSs
1.2: A New Direction for FLSs
1.3: New Concepts and Their Historical Background
1.4: Fundamental Design Requirement
1.5: The Flow of Uncertainties
1.6: Existing Literature on Type-2 Fuzzy Sets
1.7: Coverage
1.8: Applicability Outside of Rule-Based FLSs
1.9: Computation

Supplementary Material: Short Primers on Fuzzy Sets and Fuzzy Logic

1.10: Primer on Fuzzy Sets

1.10.1 Crisp sets
1.10.2 From crisp sets to fuzzy sets
1.10.3 Linguistic variables
1.10.4 Membership functions
1.10.5 Some terminology
1.10.6 Set theoretic operations for crisp sets
1.10.7 Set theoretic operations for fuzzy sets
1.10.8 Crisp relations and compositions on the same product space
1.10.9 Fuzzy relations and compositions on the same product space
1.10.10 Crisp relations and compositions on different product spaces
1.10.11 Fuzzy relations and compositions on different product spaces
1.10.12 Hedges
1.10.13 Extension principle

1.11: Primer on FL

1.11.1 Crisp logic
1.11.2 From crisp logic to FL

1.12: Remarks

Exercises

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Chapter 2: Sources of Uncertainty

2.1: Uncertainties in a FLS

2.2.1 Uncertainty: General discussions
2.2.2 Uncertainty: In a FLS

2.2: Words Mean Different Things to Different People

Exercises

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Chapter 3: Membership Functions and Uncertainty

3.1: Introduction
3.2: Type-1 Membership Functions
3.3: Type-2 Membership Functions

3.3.1 The concept of a type-2 fuzzy set
3.3.2 Definition of a type-2 fuzzy set and associated concepts
3.3.3 More examples of type-2 fuzzy sets and FOUs
3.3.4 Upper and lower membership functions
3.3.5 Embedded type-2 and type-1 sets
3.3.6 Type-1 fuzzy sets represented as type-2 fuzzy sets
3.3.7 Zero and one memberships in a type-2 fuzzy set

3.4: Returning to Linguistic Labels
3.: Multivariable Membership Functions

3.5.1 Type-1 membership functions
3.5.2 Type-2 membership functions

3.6: Computation

Exercises

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Chapter 4: Case Studies

4.1: Introduction
4.2: Forecasting of Time-Series

4.2.1 Extracting rules from the data
4.2.2 Mackey­Glass chaotic time-series

4.3: Knowledge Mining Using Surveys

4.3.1 Methodology for knowledge mining
4.3.2 Survey results
4.3.3 Methodology for designing a FLA
4.3.4 How to use a FLA

Exercises

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Part 2: Type-1 Fuzzy Logic Systems

Chapter 5: Singleton Type-1 Fuzzy Logic Systems: No Uncertainties

5.1: Introduction
5.2: Rules
5.3: Fuzzy Inference Engine
5.4: Fuzzification and Its Effect on Inference

5.4.1 Fuzzifier
5.4.2 Fuzzy inference engine

5.5: Defuzzification

5.5.1 Centroid defuzzifier
5.5.2 Center-of-sums defuzzifier
5.5.3 Height defuzzifier
5.5.4 Modified height defuzzifier
5.5.5 Center-of-sets defuzzifier
5.5.6 An interesting fact

5.6: Possibilities
5.7: Fuzzy Basis Functions
5.8: FLSs Are Universal Approximators
5.9: Designing FLSs

5.9.1 One-pass methods
5.9.2 Least-squares method
5.9.3 Back-propagation (steepest descent) method
5.9.4 SVD­QR method
5.9.5 Iterative design method

5.10: Case Study: Forecasting of Time-Series

5.10.1 One-pass design
5.10.2 Back-propagation design
5.10.3 A change in the measurements

5.11: Case Study: Knowledge Mining Using Surveys

5.11.1 Averaging the responses
5.11.2 Preserving all the responses

5.12: A Final Remark
5.13: Computation

Exercises

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Chapter 6: Non-Singleton Type-1 Fuzzy Logic Systems

6.1: Introduction
6.2: Fuzzification and Its Effect on Inference

6.2.1 Fuzzifier
6.2.2 Fuzzy inference engine

6.3: Possibilities
6.4: FBFs
6.5: Non-Singleton FLSs Are Universal Approximators
6.6: Designing Non-Singleton FLSs

6.6.1 One-pass methods
6.6.2 Least-squares method
6.6.3 Back-propagation (steepest descent) method
6.6.4 SVD­QR method
6.6.5 Iterative design method

6.7: Case Study: Forecasting of Time-Series

6.7.1 One-pass design
6.7.2 Back-propagation design

6.8: A Final Remark
6.9: Computation

Exercises

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Part 3: Type-2 Fuzzy Sets

Chapter 7: Operations on and Properties of Type-2 Fuzzy Sets

7.1: Introduction
7.2: Extension Principle
7.3: Operations on General Type-2 Fuzzy Sets

7.3.1 Set theoretic operations
7.3.2 Algebraic operations on fuzzy numbers

7.4: Operations on Interval Type-2 Fuzzy Sets

7.4.1 Set theoretic operations
7.4.2 Algebraic operations on interval fuzzy numbers

7.5: Summary of Operations
7.6: Properties of Type-2 Fuzzy Sets

7.6.1 Type-1 fuzzy sets
7.6.2 Type-2 fuzzy sets

7.7: Computation

Exercises

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Chapter 8: Type-2 Relations and Compositions

8.1: Introduction
8.2: Relations in General
8.3: Relations and Compositions on the Same Product Space
8.4: Relations and Compositions on Different Product Spaces
8.5: Composition of a Set with a Relation
8.6: Cartesian Product of Fuzzy Sets
8.7: Implications

Exercises

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Chapter 9: Centroid of a Type-2 Fuzzy Set: Type-Reduction

9.1: Introduction
9.2: General Results for the Centroid
9.3: Generalized Centroid for Interval Type-2 Fuzzy Sets
9.4: Centroid of an Interval Type-2 Fuzzy Set
9.5: Type-Reduction: General Results

9.5.1 Centroid type-reduction
9.5.2 Center-of-sums type-reduction
9.5.3 Height type-reduction
9.5.4 Modified height type-reduction
9.5.5 Center-of-sets type-reduction
9.5.6 Computational complexity of type-reduction
9.5.7 Concluding example

9.6: Type-Reduction: Interval Sets

9.6.1 Centroid type-reduction
9.6.2 Center-of-sums type-reduction
9.6.3 Height type-reduction
9.6.4 Modified height type-reduction
9.6.5 Center-of-sets type-reduction
9.6.6 Concluding example

9.7: Concluding Remark
9.8: Computation

Exercises

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Part 4: Type-2 Fuzzy Logic Systems

Chapter 10: Singleton Type-2 Fuzzy Logic Systems

10.1: Introduction
10.2: Rules
10.3: Fuzzy Inference Engine
10.4: Fuzzification and Its Effect on Inference

10.4.1 Fuzzifier
10.4.2 Fuzzy inference engine

10.5: Type-Reduction
10.6: Defuzzification
10.7: Possibilities
10.8: FBFs: The Lack Thereof
10.9: Interval Type-2 FLSs

10.9.1 Upper and lower membership functions for interval type-2 FLSs
10.9.2 Fuzzy inference engine revisited
10.9.3 Type-reduction and defuzzification revisited
10.9.4 FBFs revisited 3

10.10: Designing Interval Singleton Type-2 FLSs

10.10.1 One-pass method
10.10.2 Least-squares method
10.10.3 Back-propagation (steepest descent) method
10.10.4 SVD­QR method
10.10.5 Iterative design method

10.11: Case Study: Forecasting of Time-Series
10.12: Case Study: Knowledge Mining Using Surveys
10.13: Computation

Exercises

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Chapter 11: Type-1 Non-Singleton Type-2 Fuzzy Logic Systems

11.1: Introduction
11.2: Fuzzification and Its Effect on Inference

11.2.1 Fuzzifier
11.2.2 Fuzzy inference engine

11.3: Interval Type-1 Non-Singleton Type-2 FLSs
11.4: Designing Interval Type-1 Non-Singleton Type-2 FLSs

11.4.1 One-pass method
11.4.2 Least-squares method
11.4.3 Back-propagation (steepest descent) method
11.4.4 SVD­QR method
11.4.5 Iterative design method

11.5 Case Study: Forecasting of Time-Series
11.6 Final Remark
11.7 Computation

Exercises

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Chapter 12: Type-2 Non-Singleton Type-2 Fuzzy Logic Systems

12.1: Introduction
12.2: Fuzzification and Its Effect on Inference

12.2.1 Fuzzifier
12.2.2 Fuzzy inference engine

12.3: Interval Type-2 Non-Singleton Type-2 FLSs
12.4: Designing Interval Type-2 Non-Singleton Type-2 FLSs

12.4.1 One-pass method
12.4.2 Least-squares method
12.4.3 Back-propagation (steepest descent) method
12.4.4 SVD­QR method
12.4.5 Iterative design method

12.5: Case Study: Forecasting of Time-Series

12.5.1 Six-epoch back-propagation design
12.5.2 One-epoch combined back-propagation and SVD­QR design
12.5.3 Six-epoch iterative combined back-propagation and SVD­QR design

12.6: Computation

Exercises

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Chapter 13: TSK Fuzzy Logic Systems

13.1: Introduction
13.2: Type-1 TSK FLSs

13.2.1 First-order type-1 TSK FLS
13.2.2 A connection between type-1 TSK and Mamdani FLSs
13.2.3 TSK FLSs are universal approximators
13.2.4 Designing type-1 TSK FLSs

13.3: Type-2 TSK FLSs

13.3.1 First-order type-2 TSK FLS
13.3.2 Interval type-2 TSK FLSs
13.3.3 Unnormalized interval type-2 TSK FLSs
13.3.4 Further comparisons of TSK and Mamdani FLSs
13.3.5 Designing interval type-2 TSK FLSs using a back-propagation
(steepest descent) method

13.4: Example: Forecasting of Compressed Video Traffic

13.4.1 Introduction to MPEG video traffic
13.4.2 Forecasting I frame sizes: General information
13.4.3 Forecasting I frame sizes: Using the same number of rules
13.4.4 Forecasting I frame sizes: Using the same number of design
parameters
13.4.5 Conclusion

13.5: Final Remark
13.6: Computation

Exercises

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Chapter 14: Epilogue

14.1: Introduction
14.2: Type-2 Versus Type-1 FLSs
14.3: Appropriate Applications for a Type-2 FLS
14.4: Rule-Based Classification of Video Traffic

14.4.1 Selected features
14.4.2 FOUs for the features
14.4.3 Rules
14.4.4 FOUs for the measurements
14.4.5 Design parameters in a FL RBC
14.4.6 Computational formulas for type-1 FL RBCs
14.4.7 Computational formulas for type-2 FL RBCs
14.4.8 Optimization of rule design-parameters
14.4.9 Testing the FL RBCs
14.4.10 Results and conclusions

14.5: Equalization of Time-Varying Non-linear Digital Communication
Channels

14.5.1 Preliminaries for channel equalization
14.5.2 Why a type-2 FAF is needed
14.5.3 Designing the FAFs
14.5.4 Simulations and conclusions

14.6: Overcoming CCI and ISI for Digital Communication Channels

14.6.1 Communication system with ISI and CCI
14.6.2 Designing the FAFs
14.6.3 Simulations and conclusions

14.7: Connection Admission Control for ATM Networks

14.7.1 Survey-based CAC using a type-2 FLS: Overview
14.7.2 Extracting the knowledge for CAC
14.7.3 Choosing membership functions for the linguistic labels
14.7.4 Survey processing
14.7.5 CAC decision boundaries and conclusions

14.8: Potential Application Areas for a Type-2 FLS

14.8.1 Perceptual computing
14.8.2 FL control
14.8.3 Diagnostic medicine
14.8.4 Financial applications
14.8.5 Perceptual designs of multimedia systems

Exercises

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Appendix A: Join, Meet, and Negation Operations for Non-Interval
Type-2 Fuzzy Sets

A.1: Introduction
A.2: Join Under Minimum or Product t-Norms
A.3: Meet Under Minimum t-Norm
A.4: Meet Under Product t-Norm
A.5: Negation
A.6: Computation

Exercises

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Appendix B: Properties of Type-1 and Type-2 Fuzzy Sets

B.1: Introduction
B.2: Type-1 Fuzzy Sets
B.3: Type-2 Fuzzy Sets

Exercises

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Appendix C: Computation

C.1: Type-1 FLSs
C.2: General Type-2 FLSs
C.3: Interval Type-2 FLSs

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References
Index

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