ファジィ線形回帰分析の三つの定式化
1986 年 22 巻 10 号 p. 1051-1057
詳細
1986 年 22 巻 10 号 p. 1051-1057
Since fuzzy data can be regarded as distribution of possibility, three different formulations of fuzzy linear regression analysis are proposed by possibilistic linear systems. Fuzzy data are given by expert knowledge and it becomes recently important to deal with fuzzy data.
Three formulations called Min Problem, Max Problem and Conjunction Problem have been defined based on the following relations:
[Yi]h⊆[Yi]h, [Yi]h⊆[Yi]h, [Yi]h∩[Yi]h≠φwhere [Yi]h is h-level set of fuzzy output data and [Yi]h, [Yi]h and [Yi]h are h-level sets of estimated fuzzy outputs in Min, Max and Conjunction Problems, respectively. The estimated fuzzy output Yi in Min Problem includes fuzzy output data Yi, Yi in Max Problem is included in Yi and the intersection of Yi in Conjunction Problem and Yi is not null in h-level sets, respectively. Three problems can be reduced to linear programming problems. Thus, the merit of these formulations is to be able to obtain easily fuzzy parameters in possibilistic linear systems by solving the linear programming problem and to add another constraint conditions which might be obtained from expert knowledge of fuzzy parameters.
In this paper, the mutual relations of three formulations are discussed to clarify the properties of fuzzy data analysis. There is a solution in the Min Problem and the Conjunction Problem, but it is not assured that there exists a solution in Max Problem. If there is a solution in the Max Problem, we have[Yi]h⊇[Yi]⊇⊇[Yi]h⊇[Yi]hThese properties are illustrated in numerical examples. This approach for dealing with fuzzy data can be regarded as a fuzzy interval analysis.
