Interval Type-2 Fuzzy Systems

This chapter explores many aspects of the interval type-2 (IT2) fuzzy system that was introduced in Chap. 1 . As was done for type-1 (T1) fuzzy systems, it provides a very comprehensive and unified description of the two major kinds of IT2 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. Not only are derivations provided, but a lot of emphasis is also placed on understanding the potential benefits of using an IT2 fuzzy system over using a T1 fuzzy system. The coverage of this chapter includes: the basic architectures for two kinds of IT2 fuzzy systems, one that uses type-reduction and the other that does not; IT2 (IF-THEN) rules; three kinds of fuzzifiers (singleton, type-1 non-singleton, and IT2 non-singleton); (derivation of) input–output formulas for the fuzzy inference engine (also valid for GT2 fuzzy systems); the effects of the three kinds of fuzzifiers on the input–output formulas (valid for IT2 fuzzy systems); combining or not combining fired-rule output sets on the way to defuzzification for Mamdani fuzzy systems; 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; a comprehensive numerical example that illustrates all of the computations for one kind of IT2 Mamdani fuzzy system and two kinds of IT2 TSK fuzzy systems (this example is continued in Chap. 11 ); approximate type-reduction and defuzzification [the Wu-Mendel Uncertainty Bounds (WMUB)]; direct defuzzification [Nie-Tan (NT) and Biglarbegian-Melek-Mendel (BMM)]; a summary that explains how the end-points of the firing interval are used in different kinds of IT2 fuzzy systems; the continuation of the comprehensive example to illustrate all of the computations for the WMUB, NT, and BMM IT2 fuzzy systems; IT2 fuzzy basis functions, which provide a mathematical description of an IT2 fuzzy system from its input to its output; course and fine sculpting of the state space as well as novelty partitions used to explain the potential for improved performance of an IT2 fuzzy system over a T1 fuzzy system; and remarks and insights about an IT2 fuzzy system (including: unique features of an IT2 fuzzy system, layered architecture interpretations for it, functional equivalence of it to other machine learning methods, universal approximation by it, continuity of it, rule explosion and some ways to control it, interpretability and explainability for it, a top-down approach for obtaining it, and some important historical remarks). There are two appendixes: Appendix 1 derives some of the IT2 formulas using T1 fuzzy set mathematics; and Appendix 2 explains how to construct IT2 first- and second-order rule partitions for singleton and non-singleton fuzzification. Seventeen examples are used to illustrate the important concepts.