Two basic double-quantitative rough set models of precision and grade and their investigation using granular computing

The precision and grade of the approximate space are two fundamental quantitative indexes that measure the relative and absolute quantitative information, respectively. The double quantification of the precision and grade is a relatively new subject, and its effective implementation remains an open problem. This paper approaches the double quantification problem using basic rough set models. The Cartesian product is a natural operator for combining the two indexes given their completeness and complementary natures, and we construct two new models using this strategy. The fundamental items (i.e., the complete system, quantitative semantics and optimal computing) of the model regions are studied using granular computing. First, the model regions (MR granules) and basic model regions (BMR granules) are defined in the traditional fashion using logical double-quantitative semantics; basic semantics (BS) is provided for the double-semantic description, and the semantic extraction of the MR and BMR granules is realized within the BS framework. Computing granules (BMRC granules) are then proposed for the basic model regions to optimize the computation, and a two-dimensional plane and granular hierarchical structure are provided. Two basic algorithms for computing the MR and BMR granules are proposed and analyzed, and the BMRC-granules algorithm generally exhibits superior performance in terms of the temporal and spatial complexity. We also explore the properties of the approximation operators and the notions of attribute approximate dependence and reduction. Finally, we provide an example application from the medical field. The two models provide a basic double quantification of the precision and grade and have concrete double-quantitative semantics; they also represent a quantitatively complete expansion of the Pawlak model.

► The completeness and complementarity of precision and grade are deeply analyzed.
► Two double-quantitative rough set models of precision and grade are constructed.
► The proposed models with quantitative completeness expand the Pawlak model.
► Model regions with double-quantitative semantics develop the traditional regions.
► The BMRC-Granules algorithm optimally computes model regions.