On learning capacities of Sugeno integrals with systems of fuzzy relational equations

TL;DR

This paper presents a method for learning capacities underlying Sugeno integrals from training data by solving systems of fuzzy relational equations, providing extremal solutions and approximations under certain conditions.

Contribution

It introduces a novel approach to derive extremal and approximate capacities from fuzzy relational systems, extending existing methods with new consistency and reduction techniques.

Findings

Directly obtain extremal capacities from consistent systems

Reduced systems identify q-maxitive and q-minitive capacities

Approximate capacities can be derived when systems are inconsistent

Abstract

In this article, we introduce a method for learning a capacity underlying a Sugeno integral according to training data based on systems of fuzzy relational equations. To the training data, we associate two systems of equations: a $\max-\min$ system and a $\min-\max$ system. By solving these two systems (in the case that they are consistent) using Sanchez's results, we show that we can directly obtain the extremal capacities representing the training data. By reducing the $\max-\min$ (resp. $\min-\max$) system of equations to subsets of criteria of cardinality less than or equal to $q$ (resp. of cardinality greater than or equal to $n-q$), where $n$ is the number of criteria, we give a sufficient condition for deducing, from its potential greatest solution (resp. its potential lowest solution), a $q$-maxitive (resp. $q$-minitive) capacity. Finally, if these two reduced systems of equations are inconsistent, we show how to obtain the greatest approximate $q$-maxitive capacity and the lowest approximate $q$-minitive capacity, using recent results to handle the inconsistency of systems of fuzzy relational equations.