Fuzzy sets and typicality theory

Abstract

A set whose elements take grades of membership in the interval [0,1] is known as a fuzzy set. In practice, assignment of grades of membership to the elements of a fuzzy set is based on statistical considerations. This paper is concerned with the problem of identifying what should be meant by a “typical” element of a fuzzy set. Because of the subjective nature of membership grades, use of the common statistical measures such as the mean or median is theoretically unjustifiable. In an effort to solve this problem, we draw heavily on the works of Kruse, Kwakernaak, Ralescu, Kandel and others. In particular, we employ the concepts of the fuzzy expected value and fuzzy random variable as the basis for a measure of what is typical of a fuzzy set, an approach which we believe shows promise in fields like pattern recognition and expert systems. We propose a flexible algorithm which incorporates subjective ideas of typicality and some simple statistical techniques to arrive at the typical element of a fuzzy set. No attempt is made to further fuzzify the element of the given fuzzy set with respect to the typical element; we believe, however, that it is an issue worth considering at a later stage. The investigation assumes that we are given a well-defined fuzzy set; i.e., the grades of membership are known. Subjective feelings of “cohesiveness” within a set (based on proximity and density, for example) are incorporated within the algorithm.