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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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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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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- 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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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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To order the book, go to Prentice-Hall .

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

- 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

- Chapter 5: Singleton Type-1 Fuzzy Logic Systems: No Uncertainties
- Chapter 6: Non-Singleton Type-1 Fuzzy Logic Systems

- 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

- 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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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 principle1.11: Primer on FL

1.11.1 Crisp logic

1.11.2 From crisp logic to FL1.12: Remarks

Exercises

2.1: Uncertainties in a FLS

2.2.1 Uncertainty: General discussions

2.2.2 Uncertainty: In a FLS2.2: Words Mean Different Things to Different People

Exercises

3.1: Introduction

3.2: Type-1 Membership Functions

3.3: Type-2 Membership Functions3.3.1 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 set3.4: Returning to Linguistic Labels

3.: Multivariable Membership Functions3.5.1 Type-1 membership functions

3.5.2 Type-2 membership functions3.6: Computation

Exercises

4.1: Introduction

4.2: Forecasting of Time-Series4.2.1 Extracting rules from the data

4.2.2 MackeyGlass chaotic time-series4.3: 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 FLAExercises

5.1: Introduction

5.2: Rules

5.3: Fuzzy Inference Engine

5.4: Fuzzification and Its Effect on Inference5.4.1 Fuzzifier

5.4.2 Fuzzy inference engine5.5: 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 fact5.6: Possibilities

5.7: Fuzzy Basis Functions

5.8: FLSs Are Universal Approximators

5.9: Designing FLSs5.9.1 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 method5.10: 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 measurements5.11: Case Study: Knowledge Mining Using Surveys

5.11.1 Averaging the responses

5.11.2 Preserving all the responses5.12: A Final Remark

5.13: ComputationExercises

6.1: Introduction

6.2: Fuzzification and Its Effect on Inference6.2.1 Fuzzifier

6.2.2 Fuzzy inference engine6.3: Possibilities

6.4: FBFs

6.5: Non-Singleton FLSs Are Universal Approximators

6.6: Designing Non-Singleton FLSs6.6.1 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 method6.7: Case Study: Forecasting of Time-Series

6.7.1 One-pass design

6.7.2 Back-propagation design6.8: A Final Remark

6.9: ComputationExercises

7.1: Introduction

7.2: Extension Principle

7.3: Operations on General Type-2 Fuzzy Sets7.3.1 Set theoretic operations

7.3.2 Algebraic operations on fuzzy numbers7.4: Operations on Interval Type-2 Fuzzy Sets

7.4.1 Set theoretic operations

7.4.2 Algebraic operations on interval fuzzy numbers7.5: Summary of Operations

7.6: Properties of Type-2 Fuzzy Sets7.6.1 Type-1 fuzzy sets

7.6.2 Type-2 fuzzy sets7.7: Computation

Exercises

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: ImplicationsExercises

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 Results9.5.1 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 example9.6: 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 example9.7: Concluding Remark

9.8: ComputationExercises

10.1: Introduction

10.2: Rules

10.3: Fuzzy Inference Engine

10.4: Fuzzification and Its Effect on Inference10.4.1 Fuzzifier

10.4.2 Fuzzy inference engine10.5: Type-Reduction

10.6: Defuzzification

10.7: Possibilities

10.8: FBFs: The Lack Thereof

10.9: Interval Type-2 FLSs10.9.1 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 310.10: 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 method10.11: Case Study: Forecasting of Time-Series

10.12: Case Study: Knowledge Mining Using Surveys

10.13: ComputationExercises

11.1: Introduction

11.2: Fuzzification and Its Effect on Inference11.2.1 Fuzzifier

11.2.2 Fuzzy inference engine11.3: Interval Type-1 Non-Singleton Type-2 FLSs

11.4: Designing Interval Type-1 Non-Singleton Type-2 FLSs11.4.1 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 method11.5 Case Study: Forecasting of Time-Series

11.6 Final Remark

11.7 ComputationExercises

12.1: Introduction

12.2: Fuzzification and Its Effect on Inference12.2.1 Fuzzifier

12.2.2 Fuzzy inference engine12.3: Interval Type-2 Non-Singleton Type-2 FLSs

12.4: Designing Interval Type-2 Non-Singleton Type-2 FLSs12.4.1 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 method12.5: 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 design12.6: Computation

Exercises

13.1: Introduction

13.2: Type-1 TSK FLSs13.2.1 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 FLSs13.3: 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) method13.4: 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 Conclusion13.5: Final Remark

13.6: ComputationExercises

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 Traffic14.4.1 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 conclusions14.5: Equalization of Time-Varying Non-linear Digital Communication

Channels14.5.1 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 conclusions14.6: 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 conclusions14.7: 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 conclusions14.8: 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 systemsExercises

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: ComputationExercises

B.1: Introduction

B.2: Type-1 Fuzzy Sets

B.3: Type-2 Fuzzy SetsExercises

C.1: Type-1 FLSs

C.2: General Type-2 FLSs

C.3: Interval Type-2 FLSs

Index

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