Probabilistic Estimation of Membership Function Based on Fuzzy Data

Abstract

A probabilistic model for the determination of membership function based on fuzzy data is proposed. First, an experiment on the perception of tiangle area by human is carried out in order to see the relationship between subjective value , y, and objective value , x. It is found that the subjective value can be expressed by a probability density function fγ(y), in which mean of fγ(y) has a distance from x and standard deviation of fγ(y) is a function of x. By introducing subjective classification boundaries, the process of subjective classification is expressed by a probabilistic model. In the model, membership grade, which is same as the probability that an objective data x is classified into small, medium of large, can be calculated by the integration of fγ(y) in the range between respective classification boundaries.Actual calculation of membership grade is carried out on objective coordinate. The fγ(y) and the subjective classification boundaries are all mapped onto objective coordinate and a probability density function fx(x) and objective classification boundaries are newly introduced. Normal, beta and uniform distributions are assumed for fx(x). The mean of fx(x) is approximated by the objective vaue x. The standard deviation of fx(x) and the objective classification boundaried are determined from the fuzzy data by the likelihood analysis.The proposed method is applied to the fuzzy data obtained by the experiments on the perception of triangle area. It is concluded tha membership function can be determined for every type of fx(x). Further, it is found that the proposed method can generate triangle and trapezoid membership functions too.