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Uncertainty measures are important tools for analyzing various data. However, there are relatively few studies on the uncertainty measures for covering rough set models. In this paper, from the viewpoint of the lower and upper approximations, we propose new uncertainty measures, the lower rough entropy and the upper rough entropy, for covering rough set models. Then, we define the concepts of the joint entropy and the conditional entropy in the covering rough set models. Some important properties of these measures are obtained, and their relationships are investigated. Furthermore, we provide a certain characterization of reducible element of a covering by means of the proposed measures, and apply the proposed rough entropy to evaluate the significance of covering granules of a covering. Finally, we apply these rough entropies to measure a dual degree between covering lower and upper approximations. The theoretical analysis and examples show that the proposed uncertainty measures for covering rough set models are reasonable and useful. [ABSTRACT FROM AUTHOR]

Covering rough set model is an important extension of Pawlak rough set model, and its structure is the foundation of covering rough set theory. This paper considers four covering approximations and studies the structures of the families of their covering upper (or lower) definable sets by means of lattice theory. We provide some conditions under which the families of covering upper (or lower) definable sets with respect to these covering approximations are lattices of sets, or distributive lattices, or geometric lattices, or Boolean lattices. Furthermore, based on these results, we give the relationship among the four covering approximations and establish the connection between matroids and covering rough sets from the viewpoint of lattice theory. [ABSTRACT FROM AUTHOR]

Covering rough set model is an important extension of Pawlak rough set model, and its structure is the foundation of covering rough set theory. This paper considers four covering approximations and studies the structures of the families of their covering upper (or lower) definable sets by means of lattice theory. We provide some conditions under which the families of covering upper (or lower) definable sets with respect to these covering approximations are lattices of sets, or distributive lattices, or geometric lattices, or Boolean lattices. Furthermore, based on these results, we give the relationship among the four covering approximations and establish the connection between matroids and covering rough sets from the viewpoint of lattice theory. [ABSTRACT FROM AUTHOR]

Approximation is an important issue in rough set theory. In this study, we consider approximation by the matroidal approach. First, we study three lattices induced by an information system. Two of the three lattices are selected as the macrostructure and microstructure for approximation, respectively. Second, based on the two lattices, we define double-matroid lattices, where the upper and lower approximations with respect to an information system are depicted. Since the two lattices are geometric, we actually present approximation by the matroidal approach. Finally, we study the connection between our double-matroid lattices and granular partition lattices. Specifically, the comparison of these two structures is presented in both micro-level and macro-level. [ABSTRACT FROM AUTHOR]