Distance-based double-quantitative rough fuzzy sets with logic operations

Based on various requirements, many generalized rough set models have been developed to alleviate the limitations of generic Pawlak rough set theory and tackle different categories of information systems. One of the limitations is that rough set models based on equivalence relation are only applicable to discrete data information systems, and not suitable for dealing with real-valued continuous data without any prior processing. Another limitation is that “classical” rough sets do not consider the quantitative information about the degree of overlap between equivalence classes and the basic set, so they cannot cope well with the quantification problems. In this paper, we propose a framework of distance-based double-quantitative rough fuzzy set (Db-Dq-RFS) with logic operation by forming a distance-based fuzzy similarity relation in an information system with continuous data to simultaneously solve the two limitations. It is presented how to construct the distance-based fuzzy similarity relation in a normalized information system, and how to use this fuzzy similarity relation to generate distance-based single-quantitative rough fuzzy set (Db-Sq-RFS) models and the Db-Dq-RFS models with logic operation. The proposed Db-Dq-RFS models can overcome certain limitations of the classical rough set model. Following further studies to discuss the decision rules with parameters variation in the four kinds of Db-Dq-RFS models, we present an illustrative example to interpret the proposed developments and to verify the effect of parameters variation on decision rules. To illustrate the effectiveness of the parameters variation on decision rules, experimental evaluation is performed using five datasets coming from the University of California-Irvine (UCI) repository.