Double-quantitative decision-theoretic rough set

The probabilistic rough set (PRS) and the graded rough set (GRS) are two quantification models that measure relative and absolute quantitative information between the equivalence class and a basic concept, respectively. As a special PRS model, the decision-theoretic rough set (DTRS) mainly utilizes the conditional probability to express relative quantification. However, it ignores absolute quantitative information of the overlap between equivalence class and the basic set, and it cannot reflect the distinctive degrees of information and extremely narrow their applications in real life. In order to overcome these defects, this paper proposes a framework of double-quantitative decision-theoretic rough set (Dq-DTRS) based on Bayesian decision procedure and GRS. Two kinds of Dq-DTRS model are constructed, which essentially indicate the relative and absolute quantification. After further studies to discuss decision rules and the inner relationship between these two models, we introduce an illustrative case study about the medical diagnosis to interpret and express the theories, which is valuable for applying these theories to deal with practical issues.