New Insights into Rough Set Theory: Transitive Neighborhoods and Approximations

Rough set theory is a methodology that defines the definite or probable membership of an element for exploring data with uncertainty and incompleteness. It classifies data sets using lower and upper approximations to model uncertainty and missing information. To contribute to this goal, this study presents a newer approach to the concept of rough sets by introducing a new type of neighborhood called j-transitive neighborhood or j-TN. Some of the basic properties of j-transitive neighborhoods are studied. Also, approximations are obtained through j-TN, and the relationships between them are investigated. It is proven that these approaches provide almost all the properties provided by the approaches given by Pawlak. This study also defines the concepts of lower and upper approximations from the topological view and compares them with some existing topological structures in the literature. In addition, the applicability of the j-TN framework is demonstrated in a medical scenario. The approach proposed here represents a new view in the design of rough set theory and its practical applications to develop the appropriate strategy to handle uncertainty while performing data analysis.