Information-theoretic measures associated with rough set approximations

In rough set theory, some information-theoretic measures of uncertainty and granularity have been proposed. A common feature of these measures is that they are only dependent on the partitions and the cardinality of a universe, which means that they are independent of the lower and upper approximations of rough sets. This seems somewhat unreasonable since the basic idea of rough set theory is to describe incomplete or inexact concepts by the lower and upper approximations. In light of this, we develop a new pair of information-theoretic entropy and co-entropy functions associated to partitions and approximations in this paper. Such functions are used to measure the uncertainty and granularity of an approximation space. After introducing the novel notions of entropy and co-entropy, we then examine their properties. In particular, we disclose the relationship of co-entropies between different universes. The theoretical development is accompanied by illustrative numerical examples.