On a set-function representation of the data constellation with strong commensurability

Abstract

The input values of the Choquet integral models satisfy the strong commensurability. Using the property, we propose the set-function representation of the data constellation. As a set-function shows the magnitude relationship among input values, the average set function of the functions keep the property. To apply the inverse M ̈obius transformation, we can use fuzzy measure representations of the set-functions and analyze the function using fuzzy measure tools such as the Shapley values. Numerical examples are the evaluation values of an AHP model and the values of FCR method (Fuzzy-Set Concurrent Rating Method).