Hierarchical fuzzy relational models: linguistic interpretation and universal approximation

Hierarchical fuzzy structures composed of a series of sub-models connected in cascade have been found to be effective tools for dealing with the dimensionality problem in fuzzy systems. This paper addresses both the problems of universal approximation and linguistic interpretation of complex systems using hierarchical fuzzy models. Fuzzy relational equations are used to implement the sub-models of a hierarchical structure which has two very important properties: It is a universal approximator and it can be converted into a completely equivalent non-hierarchical model which in turn allows the extraction of linguistic knowledge in the form of consistent fuzzy IF-THEN rules. These properties are analytically derived and their application is illustrated by means of an example.