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Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions

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Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions


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  • A self-contained pedagogical approach—Not a handbook.
  • An expanded rule-based fuzzy logic—Type-2 fuzzy logic-is able to handle uncertainties because it can model them and minimize their effects; and, if all uncertainties disappear, type-2 fuzzy logic reduces to type-1 fuzzy logic, in much the same way that if randomness disappears, then probability reduces to determinism.
    • Shows students how to use fuzzy logic in new ways and how to effectively solve problems that are awash in uncertainties.

  • A bottom-up approach—Begins with type-1 fuzzy logic systems and explains how they are modified to the simplest kind of type-2 fuzzy logic system; then add layers of complexity to that system.
  • Two short primers on fuzzy sets and fuzzy logic—Amply illustrated with examples).
    • Allows students unfamiliar with fuzzy logic to read and grasp the entire book.

  • Two case studies—Carried through the entire book—forecasting of time-series and knowledge-mining using surveys.
    • Features applications that are easily understood by all readers.

  • Four major applications—Classification of video data working directly with compressed data; equalization of time-varying nonlinear digital communication channels; overcoming co-channel interference and inter-symbol interference for time-varying nonlinear digital communication channels; and connection admission control for asynchronous transfer mode networks.
    • Each demonstrates the superiority of type-2 fuzzy logic designs.

  • A balance between theory and design—Explains and illustrates design methods for every kind of fuzzy logic system. Using the time-series forecasting case study, the book demonstrates the advantages for using type-2 fuzzy logic systems over type-1 fuzzy logic systems.
  • Many worked-out examples (more than 90) and figures (more than 110).
    • Clearly illustrates the book's important points.

  • Homework problems—At the end of most chapters.
  • Readily available supporting software—Developed at USC by two of the author's students and available on-line as freeware. Includes a collection of more than 30 MATLAB m-files that have been organized into three folders: type-1 fuzzy logic systems, general type-2 fuzzy logic systems, and interval type-2 fuzzy logic systems. A “Computation” section at the end of most chapters lets students know exactly which m-files are associated with the contents of the chapter; a short word description is given for the m-file; and Appendix C of the text lists all of the m-files so that students can see the forest from the trees.
    • Enables students to immediately be able to use the book's contents.


  • Copyright 2001
  • Dimensions: 7" x 9-1/4"
  • Pages: 576
  • Edition: 1st
  • Book
  • ISBN-10: 0-13-040969-3
  • ISBN-13: 978-0-13-040969-0

  • 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 examples and 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 fuzzy logic and fuzzy sets, it's accessible to virtually anyone with an undergraduate B.S. degree—including computing professionals designing and implementing rule-based systems.


Online software includes more than 30 companion MATLAB m-files for implementing a wide variety of type-1 and type-2 fuzzy logic systems.

Sample Content

Downloadable Sample Chapter

Click here for a sample chapter for this book: 0130409693.pdf

Table of Contents

(NOTE: Each chapter concludes with Exercises.)


1. Introduction.

Rule-Based FLSs. A New Direction for FLSs. New Concepts and Their Historical Background. Fundamental Design Requirement. The Flow of Uncertainties. Existing Literature on Type-2 Fuzzy Sets. Coverage. Applicability Outside of Rule-Based FLSs. Computation.

Supplementary Material: Short Primers on Fuzzy Sets and Fuzzy Logic.

Primer on Fuzzy Sets. Primer on FL. Remarks.

2. Sources of Uncertainty.

Uncertainties in a FLS. Words Mean Different Things to Different People.

3. Membership Functions and Uncertainty.

Introduction. Type-1 Membership Functions. Type-2 Membership Functions. Returning to Linguistic Labels. Multivariable Membership Functions. Computation.

4. Case Studies.

Introduction. Forecasting of Time-Series. Knowledge Mining Using Surveys.


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

Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Defuzzification. Possibilities. Fuzzy Basis Functions. FLSs Are Universal Approximators. Designing FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. A Final Remark. Computation.

6. Non-Singleton Type-1 Fuzzy Logic Systems.

Introduction. Fuzzification and Its Effect on Inference. Possibilities. FBFs. Non-Singleton FLSs Are Universal Approximators. Designing Non-Singleton FLSs. Case Study: Forecasting of Time-Series. A Final Remark. Computation.


7. Operations on and Properties of Type-2 Fuzzy Sets.

Introduction. Extension Principle. Operations on General Type-2 Fuzzy Sets. Operations on Interval Type-2 Fuzzy Sets. Summary of Operations. Properties of Type-2 Fuzzy Sets. Computation.

8. Type-2 Relations and Compositions.

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. Implications.

9. 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. Type-Reduction: Interval Sets. Concluding Remark. Computation.


10. Singleton Type-2 Fuzzy Logic Systems.

Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Type-Reduction. Defuzzification. Possibilities. FBFs: The Lack Thereof. Interval Type-2 FLSs. Designing Interval Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. Computation.

11. Type-1 Non-Singleton Type-2 Fuzzy Logic Systems.

Introduction. Fuzzification and Its Effect on Inference. Interval Type-1 Non-Singleton Type-2 FLSs. Designing Interval Type-1 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Final Remark. Computation.

12. Type-2 Non-Singleton Type-2 Fuzzy Logic Systems.

Introduction. Fuzzification and Its Effect on Inference. Interval Type-2 Non-Singleton Type-2 FLSs. Designing Interval Type-2 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Computation.

13. TSK Fuzzy Logic Systems.

Introduction. Type-1 TSK FLSs. Type-2 TSK FLSs. Example: Forecasting of Compressed Video Traffic. Final Remark. Computation.

14. Epilogue.

Introduction. Type-2 Versus Type-1 FLSs. Appropriate Applications for a Type-2 FLS. Rule-Based Classification of Video Traffic. Equalization of Time-Varying Non-linear Digital Communication Channels. Overcoming CCI and ISI for Digital Communication Channels. Connection Admission Control for ATM Networks. Potential Application Areas for a Type-2 FLS.

A. Join, Meet, and Negation Operations For Non-Interval Type-2 Fuzzy Sets.

Introduction. Join Under Minimum or Product t-Norms. Meet Under Minimum t-Norm. Meet Under Product t-Norm. Negation. Computation.

B. Properties of Type-1 and Type-2 Fuzzy Sets.

Introduction. Type-1 Fuzzy Sets. Type-2 Fuzzy Sets.

C. Computation.

Type-1 FLSs. General Type-2 FLSs. Interval Type-2 FLSs.




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. 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 were 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.

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

Here are some specific ways that this 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 5.1-5.8).
  • 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).

So that people will start using type-2 FL as soon as possible, I have made free software 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.), was developed by my former PhD students Nilesh Karnik and Qilian Liang, and can be reached at: http://sipi.usc.edu/~mendel/software. A computation section, which directs the reader to very specific M-files, appears at the end of most chapters of this book. Appendix C summarizes all of the M-files so that the reader can see the forest from the trees.


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