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.

Abstract-In this paper a notion termed non-stationary fuzzy sets is introduced and the concept of random perturbations that can be used for generating these non-stationary fuzzy sets is also presented. A case study was carried out to investigate the relationship between the performance of non-stationary fuzzy logic systems and interval type-2 fuzzy logic systems. It can be observed that in case of centre variation, the lower-upper boundaries of outputs predicted by non-stationary systems are slightly narrower than those from the corresponding interval type-2 systems. On the other hand, in case of width variation, the lower-upper boundaries of outputs predicted by nonstationary systems are slightly wider than those from the type-2 systems. Moreover, an interesting observation is that the secondary membership function of the type-2 sets corresponding to non-stationary fuzzy sets generated using Normally distributed perturbations are non-uniform. In contrast to non-interval type-2 sets, this does not affect the inference process of the nonstationary sets. In this sense, the use of non-stationary fuzzy sets may enable approximations to be made of general type-2 fuzzy inferencing.

This naner nments the results of the recent re-a rule is evaluated the mfs are undated to vary according to search on modeUi&'the -variation in human decision making. The relationship between the uncertainty introduced to the membership functions (I&) of a fuzzy logic system (FLS) and the variation in ~S ' S decision making is explored using two separate methods. Initially uncertainty is introduced to a type-1 FLS by adding noise to its mfs and the effect on decision making is examined. Secondly an interval type-2 FLS is developed by representing the terms used in the FLS with interval type-2 fuzzy sets and the variation in decision making is studied using the FLS's interval outputs. The variations in ranking of umbilical acid-base assessments by six experts is compared to the simulation results from the developed FLSs. It is shown that there is a direct relationship between the variation in decision making and the uncertainty in the linguistic terms used, and the level of variation is proportional to the magnitude of uncertainty.

Abstract-The aim of this study was to explore relationships between the shape of the primary membership functions and the uncertainties obtained in the output sets for both non-stationary and interval type-2 fuzzy systems. The study was carried out on a fuzzy system implementing the standard XOR problem, in which either Gaussian or Triangular membership functions were employed, using a range of input values and recording the size of the output intervals obtained. It can be observed that the shape of the surfaces of the output intervals are related to the primary membership function and that the surface is divided into four roughly symmetrical parts. Furthermore, it can be observed that there are complex differences between the surfaces produced by interval type-2 systems and various kinds of non-stationary systems. Detailed differences between the output surfaces of uniformly distributed non-stationary systems are examined and the implications are discussed.

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