Probabilistic rough set over two universes and rough entropy

In this paper, we discuss the properties of the probabilistic rough set over two universes in detail. We present the parameter dependence or the continuous of the lower and upper approximations on parameters for probabilistic rough set over two universes. We also investigate some properties of the uncertainty measure, i.e., the rough degree and the precision, for probabilistic rough set over two universes. Meanwhile, we point out the limitation of the uncertainty measure for the traditional method and then define the general Shannon entropy of covering-based on universe. Then we discuss the uncertainty measure of the knowledge granularity and rough entropy for probabilistic rough set over two universes by the proposed concept. Finally, the validity of the methods and conclusions is tested by a numerical example. (C) 2012 Elsevier Inc. All rights reserved.