A granulated fuzzy rough set and its measures

The theory of rough sets is an efficient mathematical tool for dealing and reasoning with uncertainty information systems. The measures of traditional rough sets are applicable to discrete-valued information systems, but not suitable to real-valued data sets. In this paper, by introducing a distance matrix to granulate these real-valued data, a granulated fuzzy rough set model is proposed, which combines fuzziness and roughness into a rough set theoretical framework. By constructing a fuzzy similar relation with a distance matrix form, real-valued data sets can be deal with. We also define some operations on the fuzzy relations and fuzzy granules. Furthermore, two kinds of measures of fuzzy granules are proposed, which are information entropy measure and information granularity measure. These measures are calculated by a novel representation with a fuzzy granule matrix. As a result, uniform representations of fuzzy rough sets and their information measures are formed in this work.