Learning Rule Parameters of Possibilistic Rule-Based System

In this paper, we introduce a learning paradigm of the rule parameters of a possibilistic rule-based system, given training data. For a rule-based system composed of n if-then parallel possibilistic rules, we introduce an equation system denoted (Sigma(n)), which is analogous to the Farreny-Prade equation system. The unknown part of the system (Sigma(n)) is a vector composed of the rule parameters, whose values must be determined according to training data. We establish necessary and sufficient conditions for the system (Sigma(n)) to be consistent. If this is the case, we show that the set of solutions of the system is a Cartesian product of subintervals of [0, 1] whose bounds are computed. Then, we deduce that there are a unique maximal solution and, as it is well known by Sanchez's work on the solving of min -max fuzzy relational equations, a unique minimal one. These results are proved by relating the solutions of (Sigma(n)) to those of the equation system given by the first n - 1 possibilistic rules equipped with a second member which is constructed from that of (Sigma(n)). Finally, our results are illustrated by an example.