Construction of Fuzzy Measures over Product Spaces

Fernando Reche,Antonio Salmerón,María Morales

doi:10.3390/math8091605

Abstract

In this paper, we study the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. We propose a definition of independence of fuzzy measures and introduce different ways of constructing product measures, analyzing their properties. We derive bounds for the measure on the product space and show that it is possible to construct a single product measure when the marginal measures are capacities of order 2. We also study the combination of real functions over the marginal spaces in order to produce a joint function over the product space, compatible with the concept of marginalization, paving the way for the definition of statistical indices based on fuzzy measures.

Highlights

We have studied the problem of constructing fuzzy measures over product domains, when fuzzy measures over the marginal spaces are available
We have proposed a definition of independence of fuzzy measures and different ways of constructing product measures that are consistent with the defined concept of independence
R, we show in Theorem 2 that it is possible to construct a single product measure if the marginal measures are capacities of order 2

Summary

Introduction

Known as capacities [1], non-additive measures, or monotone measures [2]. We are interested in the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. We study the combination of real functions defined on the marginal spaces, in order to obtain a function over the product space coherent with the initial functions. The problem of combining fuzzy measures from marginal spaces in order to obtain a fuzzy measure over a product space has been approached from different perspectives, fundamentally based on the concept of conditioning [11,12,13,14,15,16,17,18]. We consider a more general setting, in which the measures to be combined are general fuzzy measures over potentially different spaces.

Preliminaries

Combining Fuzzy Measures

Maximin Product

Composition of Functions

Marginalization

Conclusions