Attribute Reduction in an Incomplete Interval-Valued Decision Information System

An incomplete interval-valued decision information system (IIVDIS) is a significant type of data decision table, which is ubiquitous in real life. Interval value is a form of knowledge representation, and it seems to be an embodiment of the uncertainty of research objects. In this paper, we focus on attribute reduction on the basis of a parameterized tolerance-based rough set model in an IIVDIS. Firstly, we give the similarity degree between information values on each attribute in an IIVDIS by considering incomplete information. Then, we present tolerance relations on the object set of an IIVDIS based on this similarity degree. Next, we define the rough approximations by means of the presented tolerance relation. Based on Kryszkiewicz's ideal, we introduce alpha-generalized decision and consider attribute reduction in an IIVDIS by means of this decision. Furthermore, we put forward the notions of alpha-information entropy, alpha-conditional information entropy and alpha-joint information entropy in an IIVDIS. And we prove that alpha-positive region reduction theorem, alpha-conditional entropy reduction theorem, alpha-dependency reduction theorem and alpha-generalized decision reduction theorem are equivalent to each other. Finally, we propose two attribute reduction methods in an IIVDIS by using entropy measurement and the rough approximations, and design the relevant algorithms. We carry out a series of numerical experiments to verify the effectiveness of the proposed algorithms. The experimental results show that proposed algorithms often choose fewer attributes and improve classification accuracies in most cases.