Type-Reduction: Uncertainty Measures

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. It does this outside of the context of a rule-based fuzzy system (Chap. 9 does this in the context of an IT2 fuzzy system, and Chap. 11 does it in the context of a GT2 fuzzy system) and leads to very useful uncertainty measures for T2 FSs—the centroid and variance. The coverage of this chapter includes: the interval weighted average (IWA), because it is the basic building block for type-reduction; the original KM algorithms for computing the IWA; centroid type-reduction for IT2 FSs (as a measure of uncertainty for IT2 FSs) and eight properties of the centroid; the variance of an IT2 FS (as another measure of uncertainty for IT2 FSs); centroid type-reduction for GT2 FSs (as a measure of uncertainty for GT2 FSs) and 10 properties of the centroid; and the variance of GT2 FSs (as another measure of uncertainty for GT2 FSs). There are three appendixes: Appendix 1 presents (for historical reasons) the early wavy-slice approach to type-reduction; Appendix 2 describes other (more efficient) type-reduction algorithms, organized by whether or not they require sorting; and Appendix 3 is about the mathematical properties of the IWA and also about continuous algorithms for performing centroid type-reduction. Twelve examples are used to illustrate the important concepts.