A Weighted Matrix Visualization for Fuzzy Measures and Integrals

Fuzzy integrals are useful general purpose aggregation operators, but they can be difficult to understand and visualize in practice. The interaction between an exponentially increasing number of variables–2ⁿ fuzzy measure variables for n inputs–makes it hard to understand what exactly is going on in a high dimensional space. We propose a new visualization scheme based on a weighted indicator matrix to better understand the inner workings of an arbitrary fuzzy measure. We provide ways of viewing the Shapley and interaction indices, as well as an optional data coverage histogram. This approach can give insight into which sources are the most relevant in the overall aggregation and decision making process, and it provides a way to visually compare fuzzy measures and subsequently integrals.

I. Introduction

Fuzzy measures and fuzzy integrals are powerful tools for performing generalized non-linear aggregation. Their expressiveness, however, has led to some mystification regarding the actual operation of their inner workings in practice. At the heart of the matter is the issue of understanding the interaction and relationships between the 2ⁿ variables that arise from n inputs. On a purely computational level, the fuzzy integral and measure is well understood, but it remains difficult to intuitively grasp what occurs in high dimensional spaces.