Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New

Book Review

Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions by Jery M. Mendel, Prentice-Hall, Englewood Cliffs, NJ, 2001, pp. 555, softcover. ISBN: 0-13-040969-3

ncertainty and vagueness are factors that exist in most real-life applications and domains. Fuzzy set theory has been proposed as a means for modeling the vagueness and ambiguity in complex systems. The power of fuzzy logic began to be recognized in the late 1980s and early 1990s. Since then, it has had a significant influence on developments in the field of computational intelligence. Despite the commercial success of fuzzy logic, an ordinary (type-1) fuzzy set does not capture uncertainty in all of its manifestations, particularly when it arises from vagueness in the shape of the membership function. Such uncertainties need to be depicted by fuzzy sets that have blur boundaries. The imprecise boundaries of a type-2 fuzzy set give rise to truth/membership values that are fuzzy sets in [0], [1], instead of a crisp number. Type-2 fuzzy sets are slowly gaining popularity, and the book “Uncertain rule-based fuzzy logic systems: Introduction and new directions” is a valuable resource for professionals seeking to work with fuzzy sets in general and type2 fuzzy sets in particular. It provides a comprehensive coverage of the theory of type-2 fuzzy sets and details how type-2 fuzzy systems may be designed.

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Woei Wan Tan and Teck Wee Chua National University of Singapore, SINGAPORE

The book comprises 14 chapters and three appendices. The chapters are organized into four parts: Preliminaries, Type-1 Fuzzy Logic Systems, Type-2 Fuzzy sets, and Type-2 Fuzzy Logic Systems. Part 1 lays the groundwork for the rest of the book by motivating the need for type-2 fuzzy sets. The core theoretical concepts are also introduced. Chapter 1 begins by describing the structure of a rule-based fuzzy logic system. Next, the historical background of type-2 fuzzy logic is briefly described and some of its important characteristics are highlighted. A primer about fuzzy sets and fuzzy logic completes this chapter. Readers without prior knowledge of fuzzy logic will find the primer to be very valuable in assisting them to appreciate the contents in the rest of the book. In Chapter 2, the meaning of uncertain rule-based fuzzy logic system is explained. The author proposes that “words mean different things to different people” and goes on to show how uncertainty associated with the words used in rules translates to uncertainties in the shapes of membership functions. Chapter 3 is an important chapter where the membership functions for type-2 fuzzy sets is formally

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introduced. Key concepts such as Footprint of Uncertainty, upper and lower membership functions, embedded type2 and embedded type-1 sets are also defined. The last chapter in Part 1 describes two practical problems, forecasting of time-series and knowledge mining using surveys, which are used as case studies throughout the entire book. Part 2—Type-1 Fuzzy Logic Systems—briefly reviews the developments in type-1 fuzzy logic systems. In Chapter 5, the baseline fuzzy logic system for the case where there are no rule uncertainties and the measurement of inputs is perfect is described. Each block in an ordinary fuzzy logic system, namely the fuzzification process, the fuzzy rule base, fuzzy inference engine and defuzzification techniques (centroid, centerof-sums, height, modified height, and center-of-sets), is described. Next, the important universal approximation property of a fuzzy logic system is presented, although without proof. To provide readers with tools for selecting the parameters of the fuzzy logic system, six design methods (two one-pass methods, a least-squares method, a back-propagation method,

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a singular-value decomposition (SVDQR) method, and an iterative design method that combines SVD-QR method with back-propagation method) are presented. Chapter 6 parallels the previous chapter except that the inputs to the fuzzy logic system are now corrupted by additive measurement noise. The process for fuzzifying noisy inputs into fuzzy sets (non-singleton fuzzification) is detailed. At the end of both Chapters 5 and 6, the respective fuzzy logic systems are employed to solve the two case studies described in Chapter 4. These results serve as a baseline upon which the results from type-2 fuzzy logic systems can be compared. Part 3—Type-2 Fuzzy Sets— contains three chapters, each of which is devoted to one aspect of type-2 fuzzy sets. Chapter 7 is titled “Operations on and properties of type-2 fuzzy sets.” Leveraging on the Extension Principle, it presents results for union, intersection, complement, algebraic operations (addition and multiplication), minimum t-norm and product tnorm of general type-2 fuzzy sets. The remaining sections of the chapter focus on interval type-2 fuzzy sets, a type of fuzzy set that is used extensively in practice because it requires the least computation effort to implement. The chapter is also supplemented by Appendices A and B, which provide additional results on the operations of and properties of general type-2 fuzzy sets. Chapter 8 deals with the heart of a type-2 fuzzy logic system—the inference engine. The backbone of an ordinary fuzzy logic system, the sup-star composition, is extended using results from the previous chapter. The extended sup-star composition produces a type-2 set that needs to be type-reduced before defuzzification techniques can be applied to obtain a crisp output. Type-reduction is achieved using the new concept of

Readers without prior knowledge of fuzzy logic will find the primer to be very valuable in assisting them to appreciate the contents in the rest of the book. centroid of a type-2 fuzzy set, which is introduced in Chapter 9. The chapter also presents different kinds of typereduction methods and eventually demonstrates how the calculation of the centroid of specific type-2 fuzzy sets may be simplified. The final part of the book—Type-2 Fuzzy Logic Systems—consists of five chapters, three chapters dealing with Mamdani type-2 fuzzy logic systems, one on TSK type-2 fuzzy logic system and an epilogue. Chapters 10–13 closely mirror the organization adopted in Part 2. The exception being that interval type-2 fuzzy sets, instead of type-1 fuzzy sets, are employed as antecedent and/or consequent sets to account for uncertainties in the fuzzy rule base. The decision to concentrate on Interval Type-2 fuzzy logic systems is driven by the need to derive analytical results that can be implemented in practice. Chapter 10 delineates a Singleton Type-2 fuzzy logic system. As singleton fuzzification is adopted, the fuzzy logic system does not explicitly account for uncertainties in inputs. Different design methods are then presented. One-pass methods are straightforward extensions of their type-1 counterparts while the back-propagation method requires the new concept of an active branch of a lower or upper membership function. In Chapter 11, Type-1 Non-Singleton Type-2 Fuzzy Logic System is introduced, where Type-1 Non-Singleton fuzzy sets are used to model the measurement uncertainties in inputs. Chapter 12 examines Type-2 Non-Singleton Type-2 Fuzzy Logic System. In this case, the input uncertainties are modeled as type-2

fuzzy sets. This type of fuzzy logic system is particularly useful when the measurement noise is non-stationary, but it is also the most complex one. Chapter 13 covers Singleton Type-1 and Type-2 TSK fuzzy logic systems. Design techniques such as least-squares and steepest-descent methods are presented. Finally, Chapter 14 summarizes the concepts and results presented in the book. A very handy table that allows the reader to compare type-1 and type-2 singleton and nonsingleton fuzzy logic systems is also included in the chapter. Four additional applications, where type-2 fuzzy logic systems clearly outperform their type-1 counterparts, are described. A list of potential application areas completes the chapter. To encourage readers to apply the results in the book, MATLAB M-files for implementing and designing type-1 and type-2 FLSs are available for download as freeware from http://sipi.usc.edu/~mendel/ software. The software is also described in Appendix C of the book. The clear and in-depth coverage makes the book an excellent introduction to type-2 fuzzy sets theory and type-2 fuzzy logic systems. As the various parts of the book are organized in a similar manner and common application examples are used throughout the book, a reader is guided along as ideas and concepts are evolved so it is a firstrate reference for both beginners and experts in the fuzzy logic field. In conclusion, this is an outstanding book that is highly recommended for graduate students, practitioners and researchers working in the general area of computational intelligence.

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