This paper focuses on rough set theory which provides mathematical foundations of set-theoretical approximation for concepts, as well as reasoning about data. Also presented in this paper is the concept of relative reducts which is one of the most important notions for rule generation based on rough set theory. In this paper, from the viewpoint of approximation, the authors introduce an evaluation criterion for relative reducts using roughness of partitions that are constructed from relative reducts. The proposed criterion evaluates each relative reduct by the average of coverage of decision rules based on the relative reduct, which also corresponds to evaluate the roughness of partition constructed from the relative reduct,
TopIntroduction
In rough set theory (Pawlak, 1982; Pawlak, 1991), set-theoretical approximation of concepts and reasoning about data are the two main topics. In the former, lower and upper approximations of concepts and their evaluations are the main topics. Accuracy, quality of approximation, and quality of partition are well-known criteria in evaluation of approximations; these criteria are based on the correctness of the approximation. However, the roughness of the approximation is not explicitly treated in these criteria. In reasoning about data, the relative reduct is one of the most important concepts for rule generation based on rough set theory, and many methods for exhaustive or heuristic calculation of relative reducts have been proposed (Bao, 2004; Guan, 1998; Heder, 2008; Hu, 2008; Hu, 2003; Ślęzak, 2002;, Pawlak, 1991; Skowron & Rauszer, 1992; Xu, 2008; Xu, 2007; Zhang, 2003). As an evaluation criterion for relative reducts, the cardinality of a relative reduct, i.e., the number of attributes in the relative reduct, is typical and is widely used (for example, in (Heder, 2008; Hu, 2008; Hu, 2003; Xu, 2008; Zhang, 2003)). In addition, other kinds of criteria related to evaluation of partitions are also considered with respect to the following evaluation functions: a normalized decision function generated from a relative reduct B (Ślęzak, 2000), the information entropy H(B) of B (Ślęzak, 2002), and the number of decision rules induced from B (Wróblewski, 2001).
In this paper, we consider evaluating relative reducts based on the roughness of partitions constructed from them. The outline of relative reduct evaluation we propose is:
“Good” relative reducts = relative reducts that provide partitions with approximations as rough and correct as possible.
In this sense, we think that evaluation of relative reducts is strictly concerned with evaluation of roughness of approximation.