The probability distribution and fuzzy system based on bounded product implication

In this paper, a method of constructing joint probability density function and fuzzy system using non-singleton fuzzifier is proposed based on a set of input-output data. Firstly, we create fuzzy relations using bounded product implication operator. With the fuzzy relations, we get the joint probability density function of binary random variable. Secondly, the marginal density functions and numerical characteristics of bounded product distribution are discussed separately. Finally, the regression function is obtained based on the probability distribution. It is actually the center of gravity fuzzy system, which further reveals the probability sense of fuzzy systems with center of gravity method. Meanwhile, the sufficient condition for the fuzzy system as an universal approximator is given.

I. Introduction

Currently, there are mainly three types of fuzzy systems: Mamdani, Takagi-Sugeno and Boolean fuzzy systems. In Mamdani and Takagi-Sugeno fuzzy systems, the Center-average defuzzifier is used and implication operators are chosen as t-norms[1], [2], [3], [4], [5], [6]. In Boolean fuzzy systems, the defuzzifier is used as Maximum defuzzifier and the implication operators are chosen as R-implication, S-implication and QL-implication[7], [8], [9], [10]. While the fuzzy system with center of gravity defuzzifier can be drawn to the form of interpolation and it is the best approximation to the original system in the least squares sense[ll], [12]. However, the available literature have not been the specific expression of this fuzzy system so far.