2019 Volume 31 Issue 2 Pages 672-689
Due to the development and maturation of recent computer and measurement technologies, opportunities to handle multidimensional data are increasing. Many research reports focus on multidimensional data processing. They aim to construct the framework of statistical analysis of multidimensional data. Construction of a methodology of statistical analysis focusing on multidimensional data processing is important. In this paper, we consider statistical method for multi-dimensional data. We define fuzzy interval data and introduce an approximate method for Bayesian estimation by using Zadeh’s probability concept of fuzzy events. In this paper, the data whose boundaries are vague are called fuzzy interval data. Here, probability variables are observed as fuzzy interval data. We formulate the approximate Bayesian estimation using the concept of the probability of fuzzy events. However, the method treating the membership functions of fuzzy interval data precisely causes the complexity of calculation. To solve this problem, we introduce the method using the middle point of membership functions as the representative points. Then, we can settle such problems. Now, we suppose that the fuzzy interval data are obtained from the multi-dimensional normal population. When a prior distribution of population parameter is a multi-dimensional normal distribution, we can show that the posterior distribution forms the multi-dimensional normal distribution approximately by using our proposed method. As a result, even if we obtain fuzzy interval data, we can formulate an approximate multi-dimensional Bayesian estimation which is not so far different from the conventional Bayesian estimation. Finally, we provide the numerical examples to illustrate our proposed model. In realistic situations, it is not limited to always being able to determine the shape of membership functions as the ideal left-right symmetrical type. In examples, the practicality of our proposed method is studied in condition in which the left-right symmetrical type of trapezoidal membership function is not perfectly satisfied. Consequently, practicality is shown.
