Accurate similarity analysis and computing of Gaussian membership functions for FNN simplification

This paper provides a complete solution for the problem how to accurately compute the similarity between fuzzy sets with Gaussian membership functions, which is a fundamental issue for the identification and simplification of FNNs. It is shown that there are three different types of similarities between a pair of Gaussian membership functions dependent on the relative positioning between the given pair of membership functions, and the accurate and detailed computing formulas are given in each type. A simulation example is given to compare the proposed accurate similarity analysis method with the existing approximation approaches and to show how much more accuracy can be obtained than the approximation one in terms of absolute percentage approximation error.

I. Introduction

Similarity measure and analysis between fuzzy sets has attracted substantial research for the last four decades and a huge number of different types of similarity measures and analysis methods have been proposed. Further and more important, similarity analysis and computing for fuzzy sets has played a critical role in the learning and system identification of fuzzy systems or fuzzy neural networks (FNNs) [1]–[11] as well as in other applications such as pattern recognition and classification [12] [13] and fuzzy clustering [14].