Pages 145-152
A probabilistic model for the determination of membership function based on fuzzy data is proposed. First, an experiment on the perception of triangle area by human is carried out in order to see the relationship between subjective value, у, and objective value, χ. It is found that the subjective value can be expressed by a probability density function f_Y(у), in which mean of f_Y(у) has a distance from χ and standard deviation of f_Y(у) is a function of χ. 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 χ is classified into small, medium or large, can be calculated by the integration of f_Y(у) in the range between respective classification boundaries. Actual calculation of membership grade is carried out on objective coordinate. The f_Y(у) and the subjective classification boundaries are all mapped onto objective coordinate and a probability density function f_X(χ) and objective classification boundaries are newly introduced. Normal, beta and uniform distributions are assumed for f_X(χ). The mean of f_X(χ) is approximated by the objective value χ. The standard deviation of f_X(χ) and the objective classification boundaries 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 the membership function can be determined for every type of f_X(χ). Further, it is found that the proposed method can generate triangle and trapezoid membership functions too.
