Class-specific attribute reducts in rough set theory

The concept of attribute reducts plays a fundamental role in rough set analysis. There are at least two possibilities to define an attribute reduct. A classification-based or global attribute reduct is a minimal subset of condition attributes that preserves the positive region of the decision classification, namely, the positive regions of all decision classes, in a decision table. A class-specific, class-dependent, or local attribute reduct is a minimal subset of condition attributes that preserves the positive region of a particular decision class. While a classification-based reduct may not work equally well for every decision class, a class-specific attribute reduct is optimally tailored to a particular decision class. However, studies in rough set theory are dominated by classification-based reducts; there is very limited investigation on class-specific reducts. An objective of this paper is to draw attention to class-specific reducts. We systematically compare the two types of reducts and investigate their relationships with respect to both individual reducts and families of all reducts. It is possible to derive a class-specific reduct from a classification-based reduct and to derive a classification-based reduct from a family of class-specific reducts. The families of all class-specific reducts provide a pair of lower and upper bounds of the family of all classification-based reducts. Based on a three-way classification of attributes into the pair-wise disjoint sets of core, marginal, and nonuseful attributes, we examine relationships between the corresponding classes of classification-based and class-specific attributes. The union of the sets of class-specific core attributes is the set of classification-based core attributes. It is only possible to obtain an upper bound for the set of classification-based marginal attributes and a lower bound for the set of classification-based nonuseful attributes from the family of the class-specific corresponding sets of attributes.