Uncertain Rule-Based Fuzzy Systems

This chapter describes what this book is about. It explains four kinds of uncertainty partitions—crisp, first-order, second-order with uniform weighting, and second-order with nonuniform weighting—and that they can be respectively mathematically modeled using classical (crisp) set theory, classical (type-1) fuzzy set theory, interval type-2 fuzzy set theory, and general type-2 fuzzy set theory; provides the structure of a rule-based fuzzy system, and explains its four components—rules, fuzzifier, inference, and output processor; explains why type-2 fuzzy sets are a new direction for fuzzy systems; states and explains the fundamental design requirement of a type-2 fuzzy system; provides an impressionistic brief history of type-1 fuzzy sets and fuzzy logic; reviews the early literature (1975–1992) about type-2 fuzzy sets and systems (the literature that was heavily used when the first edition of this book was written), and some literature about applications of type-2 fuzzy set and systems; and provides a brief summary of what is covered in Chaps. 2–11, a very short statement about the applicability of the book’s coverage outside of the field of rule-based fuzzy systems, and a list of sources that are available for software that can be used to implement much of what is in this book.

This chapter formally introduces type-1 fuzzy sets and fuzzy logic. It is the backbone for Chap. 3 and provides the foundation upon which type-2 fuzzy sets and systems are built in later chapters. Its coverage includes: crisp sets, type-1 fuzzy sets and associated concepts [including a short biography of Prof. Zadeh (the father of fuzzy sets and fuzzy logic)], type-1 fuzzy set defined, linguistic variables, returning to linguistic variables from a numerical value of a membership function, set theoretic operations for crisp and type-1 fuzzy sets, crisp and fuzzy relations and compositionson the same or different product spaces , compositions of a type-1 fuzzy set with a type-1 fuzzy relation, hedges, the Extension Principle (which is about functions of fuzzy sets), α-cuts (which are a powerful way to represent a type-1 fuzzy set in terms of intervals), functions of type-1 fuzzy sets computed by using α-cuts, multivariable membership functions and Cartesian products, crisp logic, going from crisp logic to fuzzy logic, Mamdani (engineering) implications, some final remarks, and an appendix about properties/laws of type-1 fuzzy sets. 35 examples are used to illustrate this chapter’s important concepts.

This chapter explores many aspects of the type-1 fuzzy system that was introduced in Chap. 1. It provides a very comprehensive and unified description of the two major kinds of type-1 fuzzy systems that are widely used in real-world applications—Mamdani and TSK fuzzy systems. The coverage of this chapter includes rules, singleton, and non-singleton fuzzifiers, input–output formulas for the fuzzy inference engine, type-1 first- and second-order rule partitions, the effects of the two kinds of fuzzifiers on the input–output formulas, combining or not combining fired-rule output sets on the way to defuzzification, defuzzifiers (centroid, height, and center-of-sets), fuzzy basis functions which provide a mathematical description of a fuzzy system from its input to its output, remarks and insights about a type-1 fuzzy system (including layered architecture interpretations for it, universal approximation by it, continuity of it, rule explosion and some ways to control it, and rule interpretability for it). Eighteen examples are used to illustrate the important concepts and there is also a comprehensive numerical example in Sect. 3.7 that is continued in later chapters. Chap. 9 builds upon the material that is in this chapter.

This chapter focuses first on what exactly “design of a type-1 fuzzy system” means, and then provides a tabular way for making the choices that are needed in order to fully specify a type-1 fuzzy system, and introduces two approaches to design, the partially dependent approach and the totally independent approach. It then describes six design methods for designing a type-1 fuzzy system, namely: one-pass, least squares, derivative-based, SVD-QR, derivative-free and iterative. It then introduces and covers three case studies (forecasting of time series, knowledge mining using surveys, and fuzzy logic control, all of which are reexamined in Chap. 10), as well as the applications of forecasting of compressed video traffic, and rule-based classification of video traffic. Twelve examples are used to illustrate the chapter’s important concepts.

This chapter examines the kinds of uncertainties that motivate the use of type-2 fuzzy sets and systems. Its coverage includes general discussions about the occurrence, causes, and nature of uncertainty, uncertainties and sets, uncertainties in a fuzzy system, and collecting word data from a group of subjects to demonstrate that words mean different things to different people. It is demonstrated that uncertainty is a commodity that can be used to control the rule explosion that is so common in a fuzzy system.

This chapter formally introduces type-2 fuzzy sets and is the backbone for the rest of this book. It includes a lot of new terminologies. Coverage includes: the concept of a type-2 fuzzy set, definitions of general type-2 fuzzy sets and associated concepts, definitions of interval type-2 fuzzy sets and associated concepts, examples of two popular footprints of uncertainty, interval type-2 fuzzy numbers, a hierarchy of different kinds of type-2 fuzzy sets, mathematical representations of type-2 fuzzy sets including the vertical slice, wavy slice, and horizontal slice representations, which mathematical representations are most useful for optimal design applications, how to represent non-type-2 fuzzy sets as type-2 fuzzy sets, returning to linguistic labels for type-2 fuzzy sets, and multivariable MFs. 24 examples are used to illustrate the important concepts.

This chapter explains how to work with type-2 fuzzy sets (T2 FSs). Most of its topics are needed in the rest of this book. Coverage includes: set-theoretic operations (union, intersection, and complement) for general type-2 fuzzy sets (GT2 FSs) computed using the Extension Principle, set-theoretic operations for interval type-2 fuzzy sets (IT2 FSs), set-theoretic operations for GT2 FSs computed using horizontal slices, type-2 relations and compositions on the same product space and on different product spaces, compositions of a T2 FS with a type-2 relation, type-2 hedges, Extension Principle for IT2 and GT2 FSs, functions of GT2 FSs computed using \( \alpha \)-planes, Cartesian product of T2 FSs, implications, an appendix about the properties of T2 FSs and an appendix that has detailed proofs of many theorems. 27 examples are used to illustrate the chapter’s important concepts.

This chapter introduces a computation called type-reduction that lets a type-2 fuzzy set (T2 FS) be projected into a type-1 fuzzy set. Type-reduction is often used in a type-2 fuzzy system as a first step in going from a T2 FS to a number. Coverage includes: the interval weighted average (IWA), because it is the basic building block for type-reduction; three algorithms (KM, EKM, and EIASC) for computing the IWA; centroid type-reduction for interval T2 FSs and systems; height and center-of-sets type-reduction for IT2 fuzzy systems; centroid type-reduction for general T2 FSs and systems; height and center-of-sets type-reduction for GT2 fuzzy systems; an appendix that presents (for historical reasons) the early approach to type-reduction; and, an appendix about the mathematical properties of the IWA, and about continuous algorithms for performing centroid type-reduction. Fourteen examples are used to illustrate the important concepts.

This chapter explores many aspects of the interval type-2 fuzzy system that was introduced in Chap. 1. As was done for type-1 fuzzy systems, it provides a very comprehensive and unified description of the two major kinds of interval type-2 fuzzy systems that are widely used in real-world applications—IT2 Mamdani and TSK fuzzy systems. Importantly, it also distinguishes between IT2 fuzzy systems that include type-reduction followed by defuzzification and those that bypass type-reduction and use direct defuzzification. The coverage of this chapter includes IT2 rules, three kinds of fuzzifiers (singleton, type-1 non-singleton, and IT2 non-singleton), input–output formulas for the fuzzy inference engine (also valid for GT2 fuzzy systems), the effects of the three kind of fuzzifiers on the input–output formulas (valid for IT2 fuzzy systems), IT2 first-and second-order rule partitions, combining or not combining fired-rule output sets on the way to defuzzification, type-reduction (centroid, height, and center-of-sets) + defuzzification for an IT2 Mamdani fuzzy system, type-reduction + defuzzification for four kinds of IT2 TSK fuzzy systems, novelty partitions, approximate type-reduction and defuzzification (the Wu–Mendel Uncertainty Bounds), direct defuzzification (Nie–Tan and Biglarbegian–Melek–Mendel), IT2 fuzzy basis functions which provide a mathematical description of an IT2 fuzzy system from its input to its output, remarks, and insights about an IT2 fuzzy system (including layered architecture interpretations for it, fundamental differences between type-1 and IT2 fuzzy systems, universal approximation by it, continuity of it, rule explosion and some ways to control it, and rule interpretability for it), and historical notes. Seventeen examples are used to illustrate the important concepts and there is also a comprehensive numerical example in Sects. 9.7 and 9.11.

This chapter is the IT2 version of Chap. 4. It focuses first on what exactly “design of an IT2 fuzzy system” means, and then provides a tabular way for making the choices that are needed in order to fully specify an IT2 fuzzy system. It introduces two approaches to design, the partially dependent approach and the totally independent approach, but this time for singleton, T1 non-singleton and IT2 non-singleton IT2 fuzzy systems. It then describes the extension of the six design methods that were covered for type-1 fuzzy systems in Chap. 4 to IT2 fuzzy systems, namely: IT2 WM, least squares, derivative-based, SVD-QR, derivative-free, and iterative. It continues the three Chap. 4 case studies (forecasting of time series, knowledge mining using surveys, and fuzzy logic control), and continues the Chap. 4 applications of forecasting of compressed video traffic using IT2 Mamdani and TSK fuzzy systems, and IT2 rule-based classification of video traffic. The application of equalization of time-varying nonlinear digital communication channels is also covered. Thirteen examples are used to illustrate the important concepts.

This chapter explores many aspects of the general type-2 fuzzy system that was introduced in Chap. 1. As was done for interval type-2 fuzzy systems, it provides a very comprehensive and unified description of the two major kinds of general type-2 fuzzy systems that may be used in real-world applications—GT2 Mamdani and GT2 TSK fuzzy systems. Importantly, it also distinguishes between GT2 fuzzy systems that include type-reduction followed by defuzzification and those that bypass type-reduction and use direct defuzzification.

The coverage of this chapter focuses on singleton fuzzification and the use of the horizontal-slice representation of a GT2 FS, and includes: GT2 rules, horizontal-slice formulas for firing sets and fired-rules output sets, horizontal-slice first- and second-order rule partitions, combining or not combining fired-rule output sets on the way to defuzzification, horizontal-slice type-reduction (centroid and center-of-sets) for horizontal-slice GT2 Mamdani and TSK fuzzy systems, defuzzification (this is where horizontal slices are aggregated), a summary of the computational steps for two horizontal-slice Mamdani and two horizontal-slice TSK GT2 fuzzy systems, horizontal-slice versions of the NT and BMM direct defuzzification methods, GT2 fuzzy basis functions which provide a mathematical description of a GT2 fuzzy system from its input to its output, remarks and insights about a GT2 fuzzy system, what exactly “design of a GT2 fuzzy system” means as well as a tabular way for making the choices that are needed to fully specify a GT2 fuzzy system, two approaches to design—the partially dependent approach and the totally independent approach, but only for singleton GT2 fuzzy systems—requirements that need to be met in the study of real-world applications of GT2 fuzzy systems, and a case study of GT2 fuzzy logic control. Ten examples are used to illustrate the important concepts and there is also a comprehensive numerical example in Sects. 11.9 and 11.11.