On learning capacities of Sugeno integrals with systems of fuzzy relational equations
This work addresses a specific challenge in fuzzy systems and decision-making for researchers in that domain, representing an incremental advancement.
The paper tackles the problem of learning capacities for Sugeno integrals from training data by introducing a method based on systems of fuzzy relational equations, resulting in the ability to derive extremal capacities and approximate solutions for inconsistent cases.
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.