Measuring the Directional or Non-directional Distance Between Type-1 and Type-2 Fuzzy Sets With Complex Membership Functions

Abstract

Fuzzy sets may have complex, non-normal or nonconvex membership functions that occur, for example, in the output of a fuzzy logic system or when automatically generating fuzzy sets from data. Measuring the distance between such nonstandard fuzzy sets can be challenging as there is no clear correct method of comparison and limited research currently exists that systematically compares existing distance measures for these fuzzy sets. It is useful to know the distance between these sets, which can tell us how much the results of a system change when the inputs differ, or the amount of disagreement between individual's perceptions or opinions on different concepts. In addition, understanding the direction of difference between such fuzzy sets further enables us to rank them, learning if one represents a higher output or higher ratings than another. This paper picks up previous functions of measuring directional distance and, for the first time, presents methods of measuring the directional distance between any type-1 and type-2 fuzzy sets with both normal/nonnormal and convex/non-convex membership functions. In realworld applications where data-driven, non-convex, non-normal fuzzy sets are the norm, the proposed approaches for measuring the distance enables us to systematically reason about the realworld objects captured by the fuzzy sets.