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

Fuzzy sets (FSs) may have complex, non-normal, or non-convex membership functions that occur, for example, in the output of a fuzzy logic system or when automatically generating FSs from data. Measuring the distance between such non-standard FSs can be challenging as there is no clear correct method of comparison and only limited research currently exists that systematically compares existing distance measures (DMs) for these FSs. 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 FSs further enables us to rank them, learning if one represents a higher output or higher ratings than another. This paper builds on 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 FSs with both normal/non-normal and convex/non-convex membership functions. In real-world applications, where data-driven, non-convex, non-normal FSs are the norm, the proposed approaches for measuring the distance enables us to systematically reason about the real-world objects captured by the FSs.