A Concept of Multi Rough Sets Defined on Multi-contextual Information Systems

Abstract

Rough set theory, introduced by Pawlak in 1982 [1], is an important concept in constructing many applications of Data Mining and Knowledge Discovery. Rough set as a generalization of crisp set, deals with crisp granularity of objects by providing an alternative to formulate a given crisp set with imprecise boundaries. In rough set theory, a given crisp set of object is approximated into two different subsets derived from a crisp partition defined on the universal set of objects. The universal set of objects is characterized by a non-empty finite set of attributes, called data table or information system. The information system is formally represented by a pair (U, A) in which U is a universal set of objects and A is a finite set of attributes. In the real application, depending on the context, a given object may have different values of attributes. Thus, a given set of objects might be approximated based on multi-context of attributes, called multi-contextual information systems. Here, n context of attributes will provide n partitions. Clearly, a given set of object, X ⊆ U, may then be represented by n pairs of lower and upper approximations. The n pairs of lower and upper approximations are denoted as multi rough sets of X as already proposed in [2, 3]. This paper extends the concept of multi rough sets by providing more properties and examining more set operations.