Fuzzy entropies for class-specific and classification-based attribute reducts in three-way probabilistic rough set models

There exist two formulations of the theory of rough sets, consisting of the conceptual formulations and the computational formulations. Class-specific and classification-based attribute reducts are two crucial notions in three-way probabilistic rough set models. In terms of conceptual formulations, the two types of attribute reducts can be defined by considering probabilistic positive or negative region preservations of a decision class and a decision classification, respectively. However, in three-way probabilistic rough set models, there are few studies on the computational formulations of the two types of attribute reducts due to the non-monotonicity of probabilistic positive and negative regions. In this paper, we examine the computational formulations of the two types of attribute reducts in three-way probabilistic rough set models based on fuzzy entropies. We construct monotonic measures based on fuzzy entropies, from which we can obtain the computational formulations of the two types of attribute reducts. On this basis, we develop algorithms for finding the two types of attribute reducts based on addition-deletion method or deletion method. Finally, the experimental results verify the monotonicity of the proposed measures with respect to the set inclusion of attributes and show that class-specific attribute reducts provide a more effective way of attribute reduction with respect to a particular decision class compared with classification-based attribute reducts.