The Association between Non-Stationary and Interval Type-2 Fuzzy Sets: A Case Study

Abstract

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.

On Relationships Between Primary Membership Functions and Output Uncertainties in Interval Type-2 and Non-Stationary Fuzzy Sets

“…The concept of non-stationary fuzzy sets have been formalised from the notion of 'non-deterministic fuzzy reasoning' and non-stationary fuzzy sets have previously been introduced to allow the modelling of variation in the membership value associated with a given value of the base variable of a fuzzy set by the use of random alterations to the parameters of the generating function(s) [9], [10]. The class of a nonstationary fuzzy set is determined both by which kind of non-stationarity is used (variation in location, variation in slope or noise variation) and by the form of perturbation function used to deviate the primary membership function.…”

“…Again, given that any physical notion of time is relative, any arbitrary point in time might be chosen as zero. The interested reader is particularly referred to [9], [10] for more details on nonstationary fuzzy sets and systems.…”

“…In this work, Garibaldi proposed the notion of 'non-deterministic fuzzy reasoning' in which variability is introduced into the membership functions of a fuzzy system through the use of random alterations to the parameters of the generating function(s). Later, Garibaldi and Musikasuwan [9], [10] extended and formalised this notion through the introduction 'nonstationary fuzzy sets '. Further details about non-stationary fuzzy sets and systems are provided in Section II.…”

Nonstationary Fuzzy Systems for Modelling and Control in Cyber Physical Systems under Uncertainty

“…Nonstationary sets for representing type-2 sets, can be produced by two different approach mainly. One of them is change in width, another one is change in centre [15]. The example set created by change in centre is given by fig.…”

Type-1 OWA operator based non-stationary fuzzy decision support systems for breast cancer treatments

“…However, it has often been demonstrated, particularly in breast cancer decision making, that groups of experts exhibit both intra-expert variability and inter-expert variability [6], [7]. Intra-expert variability is exhibited when an individual expert's decisions, given the same problem (i.e.…”