Approximate Bayesian Regression Analysis based on Fuzzy Interval Data

Abstract

In this paper, we refer to the interval data whose boundary of interval is vague as fuzzy interval data. We propose the method of approximate Bayesian regression analysis based on data processing using fuzzy interval data. Here, we formulate the approximate Bayesian regression analysis using the concept of the probability of fuzzy events defined by Zadeh. However, the method with direct usage of the membership functions of fuzzy interval data, that is to say, treating the membership functions precisely causes the complexity of calculation. But our method treating the middle points of membership functions as the representative points can solve such problems. Here, we suppose that the fuzzy interval data are obtained from the k-dimensional normal population. When a prior distribution of regression coefficient β is a n-dimensional normal distribution, we can show that the posterior distribution of β forms the n-dimensional normal distribution approximately by using our proposed method. As a result, even if we obtain fuzzy interval data. we can show the Bayesian regression analysis which is not so far different from the conventional Bayesian regression analysis. Moreover, in realistic situations, we cannot always treat the ideal symmetrical membership functions of fuzzy interval data. So we performed the computer simulations under realistic circumstances which do not satisfy completely the condition of the symmetry of trapezoidal membership functions. And we examined the practicability of our method. As a result, we could proof the practicability.