Affiliations: The Key Laboratory of “Embedded System and Service Computing”, Ministry of Education, China | Department of Computer Science and Technology, Tongji University, Shanghai, P.R. China | Shanghai University of Electronic Power, Shanghai, P.R. China
Note: [] Corresponding author. Duo-Qian Miao, The Key Laboratory of “Embedded System and Service Computing”, Ministry of Education, China. Tel./Fax: +86 021 6958 5800; E-mail: [email protected]
Abstract: The quantitative analysis of the degree of knowledge granularity poses theoretical challenges for the development of granular computing. Information-theoretic measures have been proposed to address this problem, which exhibit usefulness in complete information systems. However, mathematical analysis of relationships between these information-theoretic measures and knowledge granularity has not been done. In this paper, after introducing Shannon's entropy and mutual information into complete information systems, we prove, for the first time, that these information-theoretic measures decrease monotonously as partition becomes coarser under complete information systems. Moreover, we illustrate that their inverse relationships do not hold generally and present an additional condition under which the inverse relationships are valid. By generalizing Shannon's entropy to incomplete information systems, we further discuss the relationship between the generalized Shannon's entropy (termed as rough information entropy) and knowledge granularity based on covering generalized rough sets. We find that in incomplete information systems, the rough information entropy varies nonmonotonously as covering becomes coarser. An illustrative example is given to verify the above observation result.
Keywords: Granular computing, information entropy, rough sets, complete information systems, incomplete information systems
