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

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.