Rough Approaching Of Structured Rough Set Approximations

Rough set theory, a mathematical tool for dealing with fuzziness, imprecision, uncertainty, incompleteness, and so on, is widely applied in many fields, such as artificial intelligence, machine learning, decision analysis, etc. Lower and upper approximations are two basic concepts in rough set theory. It is well-known that lower and upper approximations in Pawlak rough set and structured lower and upper approximations in Bryniarski structured rough set both aim to depict the unknown knowledge with known knowledge, a partition generated by an equivalence relation. However, lower and upper approximations in Pawlak rough set describe a set of objects whose equivalence classes and the approximated set possessing some kinds of relationships, and structured lower and upper approximations in Bryniarski rough set show structured information about the relationships between equivalence classes and the approximated set.In order to weaken the inclusion relation between the equivalence classes and the approximated set in lower approximations, and strengthen the nonempty meet between the equivalence classes and the approximated set, structured probabilistic rough approximations are first proposed based on probability rough set approximations, and related properties are also discussed. According to properties of lower and upper limitations of set sequences, rough approaching of probabilistic rough set approximations and structured probabilistic rough set approximations are then investigated when the approximated set and thresholds change.Furthermore, properties of inclusion- degree-based rough set approximations are discussed. And inclusion-degree-based structured rough set approximations are then introduced. After discussing their properties, monotonicity of the lower and upper approximations and the lower and upper structured approximations based on inclusion degree are studied, respectively.Rough approaching of inclusion-degree-based rough set approximations and inclusion-degree-based structured rough set approximations are depicted, respectively,Finally, subsethood measure is introduced and related properties are discussed. Meanwhile,subsethood-measure-basedrough set approximationsand subsethood–measure –basedstructured rough set approximations are introduced. By discussing their properties, rough approaching of subsethood- measure- based rough set approximations and subsethoodmeasure-based structured rough set approximations are depicted, respectively, as the approximated set and related thresholds changed.Three different kinds of rough set approximations and corresponding structured rough set approximations characterize structured information by using different measures between equivalent classes and the approximated set.