Non-Monotonic Attribute Reduction in Decision-Theoretic Rough Sets

For most attribute reduction in Pawlak rough set model (PRS), monotonicity is a basic property for the quantitative measure of an attribute set. Based on the monotonicity, a series of attribute reductions in Pawlak rough set model such as positive-region-preserved reductions and condition entropy-preserved reductions are defined and the corresponding heuristic algorithms are proposed in previous rough sets research. However, some quantitative measures of attribute set may be non-monotonic in probabilistic rough set model such as decision-theoretic rough set (DTRS), and the non-monotonic definition of the attribute reduction should be reinvestigated and the heuristic algorithm should be reconsidered. In this paper, the monotonicity of the positive region in PRS and DTRS are comparatively discussed. Theoretic analysis shows that the positive region in DTRS model may be expanded with the decrease of the attributes, which is essentially different from that in PRS model. Hereby, a new non-monotonic attribute reduction is presented for the DTRS model in this paper, and a heuristic algorithm for searching the newly defined attribute reduction is proposed, in which the positive region is allowed to be expanded instead of remaining unchanged in the process of attribute reduction. Experimental analysis is included to validate the theoretic analysis and quantify the effectiveness of the proposed attribute reduction algorithm.