Fuzzy rough approximations for set-valued data

Rough set theory is one of important tools of soft computing, and rough approximations are the essential elements in rough set models. However, the existing fuzzy rough set model for set-valued data, which is directly constructed based on a kind of similarity relation, fail to explicitly define fuzzy rough approximations. To solve this issue, in this paper, we propose two types of fuzzy rough approximations, and define two corresponding relative positive region reducts. Furthermore, two discernibility matrices and two discernibility functions are introduced to acquire these new proposed reducts, and the relationships among the new reducts and the existing reducts are also be provided. Theoretical analyses demonstrate that the new types of reducts have less redundancy and are more diverse (no lower number of reducts) than those obtained by means of the existing matrices, and experimental results illustrate the new reducts found by our methods outperform those obtained by existing method.