Algebraic Formulations and Geometric Interpretations of Decision-Theoretic Rough Sets

Decision-theoretic rough sets (DTRS) are a probabilistic generalization of rough sets based on Bayesian decision theory. Existing studies on DTRS mainly focus on algebraic approaches. They investigate the formal properties of cost functions of three actions (i.e., assigning an object to the positive, boundary, or negative regions) and the procedure for determining a pair of thresholds by minimizing the overall cost of a rough-set based three-way classification. The objective of this paper is to propose a new direction of research towards the visualization of DTRS. As a complement and an alternative to algebraic approaches, we examine geometric interpretations of DTRS. The geometric approaches are intuitively appealing, easy-to-grasp, and easy-to-use. By looking at visual representations of the various costs, the thresholds, and the geometric relationships between the costs and thresholds, we gain new insights into, and a deeper understanding of, DTRS. Geometric approaches can help practitioners use and apply quickly and effectively DTRS. Combining algebraic approaches and geometric approaches is instrumental in pursuing future research on DTRS.