Improved rough approximations based on variable J-containment neighborhoods

Abstract

Classic generalized rough set model in neighborhood systems provides a more general framework for depicting approximations, while it may meet the non-reflexive situations. Some scholars put forward different neighborhoods, such as adhesion neighborhoods (briefly, $$P_{j}$$ P j -neighborhoods), containment neighborhoods (briefly, $$C_{j}$$ C j -neighborhoods), and $$E_{j}$$ E j -neighborhoods. However, not all of them are reflexive. Moreover, the granularity of $$P_{j}$$ P j -neighborhoods and $$C_{j}$$ C j -neighborhoods are too fine, and that of $$E_{j}$$ E j -neighborhoods too coarse. To solve the problem, we aim to design a novel construction approach of neighborhoods, called variable j-containment neighborhoods (briefly, $$V_{j}^{\beta }$$ V j β -neighborhoods), which satisfies the reflexivity and the granularity so flexible that the neighborhood space can adjust the granularity to meet the needs of problems. We generalize three kinds of rough approximations in $$V_{j}^{\beta }$$ V j β -neighborhood spaces and discuss their properties. What’s more, we analyze the topology structures relying on $$V_{j}^{\beta }$$ V j β -neighborhood spaces and compare our proposed approach with the existing approaches. By selecting the appropriate parameter $$\beta$$ β , our neighborhood system is more flexible in adjusting the granularity to fit problem requirements. And illustrative examples demonstrate the advantages of the proposed rough set model to attribute reduction in incomplete information systems.