Incomplete Multigranulation Rough Set

The original rough-set model is primarily concerned with the approximations of sets described by a single equivalence relation on a given universe. With granular computing point of view, the classical rough-set theory is based on a single granulation. This correspondence paper first extends the rough-set model based on a tolerance relation to an incomplete rough-set model based on multigranulations, where set approximations are defined through using multiple tolerance relations on the universe. Then, several elementary measures are proposed for this rough-set framework, and a concept of approximation reduct is introduced to characterize the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in this rough-set model. Finally, several key algorithms are designed for finding an approximation reduct.