Book Review: "Uncertain rule-based fuzzy logic systems: introduction and new directions" by Jerry M. Mendel

Ordinary, type-1 fuzzy sets require that each element of a universal set be assigned a particular real number of the unit interval. In many circumstances, we may be unable to precisely identify the membership function except approximately via, e.g., lower and upper bounds of membership grades for each element of the universal set. A membership function that assigns a closed interval of real numbers in [0,1] within the identi:ed lower and upper bounds characterizes an interval-valued fuzzy set. Interval-valued fuzzy sets can be further be generalized by allowing their intervals to be fuzzy. These sets are referred to as fuzzy sets of type-2. Type-2 fuzzy set is the main concept behind the word uncertain of Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions and, as such, brings a new direction in the sense that type-2 fuzzy set provides a way to capture more about uncertainty than just a single number. In the author’s own words, type-2 fuzzy sets provide a measure of dispersion and seem to be as fundamental to the design of fuzzy systems as variance is to the mean. We may safely say that the book provides a comprehensive introduction to the theory of type-2 fuzzy sets in general and rule-based fuzzy systems in particular. The fundamentals are clearly addressed assuming a fundamental requirement: when all sources of uncertainty disappear, a type-2 fuzzy system must reduce to a comparable ordinary fuzzy system. The general treatment of the main concepts (fuzzy sets, operations, relations, composition, rule semantics, reasoning) is pursued, with emphasis on interval-valued sets using lower and upper membership functions to bring computational tractability. The organization of the book clearly re?ects the experience, maturity and authority of the author as a teacher, researcher, and a major contributor of the :eld. He begins with the motivation and groundwork of Parts 1 and 2 chapters, smoothly ?ows through the core theoretical and methodological concepts with applications and analysis at the heart of Parts 3 and 4, and ends with an epilogue. Three appendices and a list of key references complete the book. Interestingly, chapters of Parts 2 and 4 share similar organization and use a common application (forecasting time-series) example as illustration. This helps the reader to follow and to appreciate the evolution of ideas and concepts. But the use of the term implication for Larsen (product) and the Mamdani (minimum) conjunctions as the respective rules meaning could eventually be avoided since the author is clearly aware that, strictly speaking, they are not implications. More precisely, the book contains 14 chapters and 3 appendices. The chapters are organized into four parts. Part 1—(Preliminaries)—contains 4 chapters that provide background materials about uncertainty, membership functions, two case studies: forecasting of time series and knowledge discovery using surveys, respectively. Chapter 1 (Introduction) introduces and motivates the reader to