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

Abstract

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.

Control of nonlinear systems using non-stationary embedded recurrent fuzzy neural networks

“…Therefore, the term "perturbation function" was intentionally chosen for that parameter changes by the function are "small", and temporary alterations in f-l A (t, x) . The alternative forms of the non-stationary that can be formalized as follows [2,17]: (2) where aCt) is a constant for any given t. Thus, the membership function is shifted, as a whole, left (a(t)<O) or right (a(t»O). The membership function which is added the concept of non-stationary fuzzy sets can be formed as follow, and in Figure 1.…”

“…By introducing a range of membership values associated with each value of the base variable, type-2 fuzzy sets can capture the concept of uncertainty in membership functions but they do not capture the notion of variability [15]. Therefore 978-1-4673-1487-9/12/$31.00 ©2012 IEEE Garibaldi proposed the notion of non-stationary fuzzy reasoning [1][2][3][4]17]. The variability of non-stationary fuzzy reasoning is introduced into the membership function of a fuzzy system through random alterations to the parameters of these functions.…”

“…The membership functions in Type-l FLSs are type-l fuzzy sets and they cannot di rectly handle rule uncertainties [15]. To deal with this problem, type-2 FLSs have been proposed; the antecedent or consequent membership functions in type-2 FLSs are type-2 fuzzy sets which can handle rule uncertainties [15,17]. However, the computation complexity of type-reduction is a distinct drawback in type-2 fuzzy sets.…”

Using nonstationary fuzzy sets to improve the tractability of fuzzy association rules

“…Note also that: Nonstationary fuzzy sets have been employed in a range applications mostly in the medical decision making domain [15]. They have also been used in control problems [16], [17] and have been compared to systems employing type-2 fuzzy sets [18]. There is a clear relationship between nonstationary fuzzy sets and type-2 fuzzy sets.…”

Type-2 Fuzzy membership function design method through a piecewise-linear approach

“…An example of 166 this procedure can be found in Ozen et al (2004) where original 167 T1MF are blurred by an arbitrary factor as well as in Benatar,168 Aickelin, and Garibaldi (2012). A different procedure which tries 169 to combine heuristically defined MFs in order to eliminate single 170 expert dependence is presented by Pagola,171 Fernandez, Barrenechea, and Bustince (2013), and a 172 non-stationary fuzzy sets proposal through arbitrary variations 173 can be found on Musikasuwan et al (2006). 174 Data-based approaches build the FOU from the analysis of col-175 lected data.…”