Rough sets under incomplete information: Concept formation and three-way decision perspectives

Abstract

A research direction in generalizing Pawlak rough sets is to consider incomplete information. One of the common practices is to replace an equivalence relation on a set of objects by various similarity relations to account for the incomplete information. Typically, one defines similarity relations based on some intuitive understanding of incomplete information. What lacks in the existing studies is a satisfactory formal formulation on a sound semantics foundation. In this paper, we present a formal model for interpreting and constructing concept-based granular rough sets under incomplete information, with a direct reference to one of the main purposes of rough set theory for data analysis and concept forming. Our formulation is based on a conjunctive concept space. For an object, we identify the most specific conjunctive concept to which the object is an instance. By using the ideas of possible world semantics of incomplete information suggested by Lipski, for the conjunctive concept, we derive a pair of extensions, namely, the set of objects that are known to be instances and the set of objects that are possible instances. We show that these two extensions are the neighborhoods induced by two similarity relations, which provides a formal and sound semantics of similarity relations. Finally, we use the proposed model to probe into the ideas of concept-based granular rough sets and three-way decision under incomplete information. Our investigation on the semantics reveals some difficulties with existing methods and calls for more studies on semantics related issues. The result will be useful in guiding future research and preventing misuse of models of incomplete information.