Overlapping Equality Rough Neighborhoods With Application to Alzheimer’s Illness

One of the robust mathematical instruments to confront uncertainty and imperfect knowledge of data is rough‐set theory. It categorizes extracted information from given data into confirmed and potential knowledge using lower and upper approximation operators and they are calculated by equivalence classes as granules of computing. To cope with real‐life challenges and complicated issues, different types of neighborhood systems have been defined to approximate subsets of data instead of equivalences classes. In this paper, we apply the equality between original rough neighborhoods, left and right neighborhoods, and minimal left and minimal right neighborhoods, to set up fresh sorts of rough neighborhoods defined under arbitrary binary relation, namely, overlapping equality rough neighborhoods, symbolized by ‐neighborhoods. We delve into studying their core properties and derive the main characteristics relative to some kinds of binary relations, like quasiorder and partial‐order relations. Also, we point out the relationships between them as well as discuss their relationships with the existing types of rough neighborhoods. As a unique contribution, we demonstrate how one can determine whether a relation is symmetric or antisymmetric by knowing ‐neighborhoods for some cases of σ . Moreover, we demonstrate that the larger the relation, the larger ‐neighborhoods as long as the navigation from a reflexive relation to an equivalence relation. Then, we introduce novel rough‐set models induced by ‐neighborhoods and show that they retain properties of the standard paradigm of Pawlak. We prove that they decrease the area of uncertainty compared with several previous paradigms, and also, we investigate the condition under which they satisfy the monotonicity property. We provide a medical example related to Alzheimer’s to clarify the superiority of the current model over several prior models in terms of shrinking the vagueness and maximizing the accuracy, as well as we make a comparative analysis to offer a clearer understanding of where our approach excels and where further improvements could be made. Ultimately, we abstract the main contributions, refer to the merits of the proposed approach, and offer proposals for forthcoming works.