Fuzziness and incremental information of disjoint regions in double-quantitative decision-theoretic rough set model

Double-quantitative decision-theoretic rough set (Dq-DTRS), as a new model considering double quantification to reflect the distinct degrees of quantitative information, satisfies the quantitative completeness properties and exhibits much stronger fault tolerance capabilities than decision-theoretic rough set (DTRS) and graded rough set (GRS). Since the Dq-DTRS was proposed, there have been few studies on the uncertainty analysis of the model. In this paper, we investigate the uncertainty measure of the four disjoint regions in Dq-DTRS models by introducing a fuzziness formula for rough set, and then describe the changing regularities of fuzziness of disjoint regions in DqI-DTRS model and DqII-DTRS model along with the variation of two parameters alpha, beta and the grade k, respectively. In addition, three kinds of incremental information for Dq-DTRS model, namely useful incremental information, useless incremental information and error-correction incremental information are presented being formed with regard to the changes of boundary regions, and also the related assessment methods for these special types of incremental information are discussed in the form of several important theorems.