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 issues of linguistic interpretation and universal approximation of systems using hierarchical fuzzy models. Fuzzy relational equations are used to implement the sub-models of a hierarchical structure that has two very important properties: i) it can be converted into a completely equivalent nonhierarchical model, which in turn allows the extraction of linguistic knowledge in the form of consistent fuzzy If-Then rules; and ii) it is a universal approximator. These properties are analytically derived and the proposed model is illustrated by means of an example.