Some ideas about graphic representations of discrete fuzzy measures

TL;DR

This paper explores visual graph representations of discrete fuzzy measures, illustrating their properties, special families, and integrals, and providing tools for measure fitting and comparison.

Contribution

It introduces graphical visualization methods for various fuzzy measures and integrals, enhancing understanding and comparison of these mathematical concepts.

Findings

Graphical rules and properties of fuzzy measures are established.

Visualizations of special fuzzy measure families are provided.

Tools for fuzzy measure fitting and comparison are developed.

Abstract

Graphs serve as efficient tools for visualizing mathematical concepts and their interrelationships. In this paper, focusing on the discrete case with universal set with finite elements, we first introduce the rules and characteristics of graph representation of fuzzy measure and discuss graphic properties of fuzzy measure's duality, symmetry, nonadditivity and nonmodularity. Then we show the graphic presentations of some special families of fuzzy measures, such as the k-additive measure, the k-maxitive and minitive measure, k-order representative measure, k-order interactive measure and the p-symmetric fuzzy measure, as well as of three nonlinear integrals, i.e., the Choquet integral, the Sugeno integral and the pan integral. Finally, we provide visualizations for the fuzzy measure fitting procedure and tools for comparing fuzzy measures.