Uncertainty measures for general fuzzy relations

Abstract

Relation is one of the important concepts in rough set models. Some uncertainty measures for relations were proposed in recent years by generalizing information entropy. However, the existing theories considered only special relations and do not address mixed-type data well. They always transformed each attribute into a binary relation with the same structure regardless of attribute types. As a result, the structure information can be lost for other types of attributes. In this paper, a new theory about uncertainty measures for general fuzzy relations is proposed. The proposed theory can overcome the weakness of the existing theories. We first introduce a novel entropy to compute the uncertainty information of fuzzy binary relations and then present the concepts of joint fuzzy entropy, conditional fuzzy entropy and fuzzy mutual information. The basic properties of these uncertainty measures are studied. Finally, we apply the proposed theory to perform attribute reduction of heterogeneous data sets. The experimental results show that our proposed method is feasible and effective in some cases. The proposed theory can provide a fundamental framework that integrates all kinds of methods for the uncertainty information of special relations.