Evidence Theory in Incomplete Information Tables

Abstract

(the power set of the universe of discourse) to itself, we show that they are mutually dual, and that both of them are order-preserving. We then introduce the belief and plausibility functions, respectively, over U, based on the -lower and -upper approximations, respectively, in VPGRS model, and we incorporate the concepts of evidence theory and VPGRS model to examine incomplete information tables.Index Terms-Rough sets, belief functions, reflexive relations, variable precision rough set models, lower and upper approximations. I. INTRODUCTIONEvidence theory is a useful tool in knowledge representation which plays an important role in dealing with many aspects of problem solving. This includes handling incomplete information tables [1]. One of the most important concepts an intelligent system needs to understand is the concept of knowledge. It may or may not be perfect. Also, one wants to know what knowledge is needed to achieve particular goals, and how that knowledge can be obtained. So, one of the important problems along this line is to seek an appropriate approach to analyze imperfect knowledge. The problem related to imperfect knowledge or an incomplete information table has been investigated by many researchers in different areas. Our approach is to apply evidence theory which is essentially Dempster-Shafer theory [2]. This theory is a generalization of the Bayesian theory of subjective probability, also known as the theory of belief functions. Some important features of Dempster-Shafer theory are that it has the capability to cope with varying levels of precision regarding the information and allows for direct representation of uncertainty of system responses where an imperfect information can be characterized by a set or an interval. With these features, we consider the concept of variable precision rough set model [3] certain. We need a method that can handle the classification with some degree of uncertainty. In this paper, we focus on applying belief functions to representing partial knowledge of incomplete information tables [7], [8]. In what follows, we set up the notations and recall lower and upper approximations in Variable Precision of Generalized Rough Sets (VPGRS) [9]-[12]. We then define belief functions and incomplete information tables. We establish several relationships on incomplete information tables by analyzing lower and upper approximations in VPGRS models. We also connect evidence theory and VPGRS models. II. PRELIMINARIESLet U be a nonempty finite set, referred as the universe of discourse (in short, the universe). The power set of U, denoted by 2 , is the collection of all subsets of U, including the whole set U and the empty set Ø. That is,The Cartesian product × is the set of all ordered pairs of elements of U. A binary relation on U is a subset of × .For a relation ⊆ × , we often write to represent ( , ) ∈ . In case R is an equivalence relation, we say that objects x and y are equivalent.Let ⊆ × . For each ∈ , the image of x under a relation R is defined as ( ) = { ∈ | }. Notic...