Information entropy, rough entropy and knowledge granulation in incomplete information systems

Rough set theory is a relatively new mathematical tool for use in computer applications in circumstances that are characterized by vagueness and uncertainty. Rough set theory uses a table called an information system, and knowledge is defined as classifications of an information system. In this paper, we introduce the concepts of information entropy, rough entropy, knowledge granulation and granularity measure in incomplete information systems, their important properties are given, and the relationships among these concepts are established. The relationship between the information entropy E(A) and the knowledge granulation GK(A) of knowledge A can be expressed as E(A)+GK(A) = 1, the relationship between the granularity measure G(A) and the rough entropy E r(A) of knowledge A can be expressed as G(A)+E r(A) = log2|U|. The conclusions in Liang and Shi (2004 Liang, J.Y. and Shi, Z.Z. 2004. The information entropy, rough entropy and knowledge granulation in rough set theory. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12(1): 37–46. [Crossref], [Web of Science ®] , [Google Scholar]) are special instances in this paper. Furthermore, two inequalities − log2 GK(A) ≤ G(A) and E r(A) ≤ log2(|U|(1 − E(A))) about the measures GK, G, E and E r are obtained. These results will be very helpful for understanding the essence of uncertainty measurement, the significance of an attribute, constructing the heuristic function in a heuristic reduct algorithm and measuring the quality of a decision rule in incomplete information systems.