Structural expansion of fuzzy relations and its role in interactive system modeling

This study is concerned with a concept of structural expansion of fuzzy relations. Roughly speaking, when dealing with a finite family of highly dimensional fuzzy relations R_i, i = 1,2,...c, the intent is to provide a way of expressing them as Cartesian products of some fuzzy sets and ldquoreducedrdquo fuzzy relations of lower dimensionality. More descriptively, this expansion is regarded as a vehicle to isolate fuzzy sets which are easy to interpret and retain some relational ldquoremainderrdquo which is far less interpretable yet still effective in terms of processing. The expansion process can be performed stepwise by moving toward isolation of more fuzzy sets. We formulate the problem as a certain optimization task where the fuzzy sets obtained in the expansion process minimize an underlying performance index. Two main strategies of structural expansion are discussed. We distinguish between expansion relying on the use of individual variables and collections of variables. While the concept of structural expansion is of a substantial level of generality, in this study, we assume that fuzzy relations are formed through fuzzy clustering (Fuzzy C-Means, FCM to be brief) so that prototypes of the clusters can be directly used in the optimization procedure. Further refinement of fuzzy sets obtained as the result of the expansion is carried out through some logic transformation realized by means of uninorms. The links of structural expansion with rule-based computing with the related issue of interpretability and effectiveness of processing are discussed as well.