An expanded double-quantitative model regarding probabilities and grades and its hierarchical double-quantitative attribute reduction

Probabilities and grades serve as relative and absolute measures, respectively. They are used to establish the decision-theoretic rough set (DTRS) and graded rough set (GRS) two basic quantitative models. The double-quantification of probabilities and grades exhibits systematicness and completeness in view of the two-dimensional feature of the approximate space; however, double-quantitative construction becomes a problem, and double-quantitative reduction is rarely reported. Thus, this paper mainly constructs an expanded double-quantitative model by logically integrating probabilities and grades; it further studies relevant double-quantitative reduction by hierarchically preserving specific regions. (1) First, a novel model is established via the logic integration and expansion requirement, and its regional system and granular hierarchy are studied via granular computing. Thus, regional semantics is extracted via basic semantics granules. Regional calculation is realized by two algorithms, and the algorithm regarding calculation granules exhibits optimization according to algorithm analyses. (2) Second, three types of model-regional preservation reducts and their hierarchy are discussed in the two-category case. Thus, SRP-Reduct, CRP-Reduct, and APP-Reduct are studied by exploring four-region preservation properties, constructing two-region classification regions, and preserving four original approximations, respectively. Furthermore, a relevant reduction hierarchy is thoroughly achieved. (3) Moreover, the model and its reduction are illustrated by two examples of decision tables. The constructional model conducts double-quantification regarding probabilities and grades; thus, it exhibits double-quantitative semantics and benignly expands DTRS-Model, GRS-Model, and Pawlak-Model. Furthermore, its hierarchical reduction reflects some double-quantitative reduction essence; thus, its reduction expands qualitative Pawlak-Reduction while guides quantitative DTRS-Reduction and GRS-Reduction. (C) 2014 Elsevier Inc. All rights reserved.