The information entropy of rough relational databases

Beaubouef, Petry and Buckles proposed the generalized rough set database analysis (GRSDA) to discuss rough relational databases. Given any rough relational database (U, A) and an attribute a is an element of A, as in rough set theory, a definition of the lower and upper approximations based on phi(a) is given. The entropy and conditional entropy of similarity relations in a rough relational database are defined. The examples show that the entropy of a similarity relation does not decrease as the similarity relation is refined. It will be proved that given any two similarity relations phi and psi, defined by a set C of conditional attributes and a decision attribute d, respectively, if d similarly depends on C in a rough relational database then the conditional entropy of phi with respect to psi is equal to the entropy of phi.