Nonstationary Fuzzy Sets

Abstract

Abstract-In this paper, the notion termed a "nonstationary fuzzy set" is introduced, and the concept of a perturbation function that is used for generating nonstationary fuzzy sets is presented. Definitions of the basic set operators (the union, the intersection, and the complement) for nonstationary fuzzy sets are given, together with proofs of selected properties of these operators. Two case studies were carried out in order to illustrate the use of nonstationary fuzzy sets in a nonstationary fuzzy inference, and to provide an initial insight into the relationships between nonstationary and interval type-2 fuzzy sets.Index Terms-Nonstationary fuzzy sets, perturbation functions, type-2 fuzzy sets.

Constrained type-2 fuzzy sets

“…One different form of non-standard fuzzy set which is different from both type-1 and type-2 fuzzy sets is the nonstationary fuzzy set, introduced by Garibaldi [7]. A nonstationary fuzzy set consists of a standard type-1 fuzzy set that is perturbed (moves by small amounts) over time.…”

“…A nonstationary fuzzy set consists of a standard type-1 fuzzy set that is perturbed (moves by small amounts) over time. At any instant in time, it is equivalent to a type-1 fuzzy set, but the generating function is non-stationary (there are different forms of non-stationarity, see [7] for more details). Each instant in time at which the non-standard fuzzy set is invoked is termed an instantiation.…”

Fuzzy Multicriteria Decision-Making: A Literature Review

“…These include type-2 fuzzy sets and type-n fuzzy sets that incorporate uncertainty about the membership function in their definition [3]; nonstationary fuzzy sets that introduce into the membership functions a connection that expresses a slight variation in the membership function [8]; Intuitionistic fuzzy sets, introduced by Atanassov [9], extend fuzzy sets by an additional degree, which is called the degree of uncertainty; fuzzy multisets based on multisets that allow repeated elements in the set [10]; hesitant fuzzy sets (HFS) that have been recently introduced in [11] and provide a very interesting extension of fuzzy sets. HFS are especially useful when a set of values are possible to define the membership function of an element [12].…”

An Investigation Into the Distribution of Membership Grades for Non-Stationary Fuzzy Sets

“…To incorporate this variability into decision making in the context of fuzzy expert system, they proposed the notion of 'nondeterministic fuzzy reasoning' in which variability is introduced into the membership function of a fuzzy system through random alterations to the parameters of the functions. Garibaldi et al (Garibaldi et al, 2008) formalized this notion and called it a nonstationary fuzzy sets which is a set (collection) of type-1 fuzzy sets, obtained by instantiations of the underlying fuzzy membership function. The variations in the nonstationary fuzzy sets were generated as a result of perturbations in the underlying membership function.…”

“…Various approaches to perturb the underlying membership function viz. variation in location, variation in width, noise variation, were discussed in (Garibaldi et al, 2008). They discussed two case studies which revealed that the distribution of membership grades was following some pattern dependent on choice of underlying membership function and also the point of interest.…”