Affiliations: [a] Department of Computer Science, Meiji University, Kawasaki-shi, Japan 214-8571 | [b] Petra Christian University, Siwalankerto 121-131, Surabaya, Indonesia 60236. E-mail: [email protected], [email protected]
Abstract: In 1982, Pawlak proposed the concept of rough sets with practical purpose of representing indiscernibility of elements. Although rough set theory built on equivalence relation has the advantage of being easy to analyze, it may not be a widely applicable model as equivalence relations, because of their properties of symmetry and transitivity, may not provide a realistic view of relationships between elements in real world. Therefore a covering of the universe was introduced in order to represent a more realistic model. However, it is still unclear regarding what kinds of relations may use in defining the coverings. In this paper, the notion of weak fuzzy similarity relations, a generalization of fuzzy similarity relations, is used to provide a more realistic description of relationships between elements in which properties of symmetry and transitivity are no longer hold. A special type (concrete example) of weak fuzzy similarity relations called conditional probability relation is discussed. A generalized concept of rough set approximations are proposed based on α-coverings of the universe induced by conditional probability relations. Rough membership functions are also re-defined into three values, minimum, maximum and average. Their properties are also examined. In addition, by extending the concept of α-coverings of the universe, some properties and applications related to Knowledge Discovery and Data Mining (KDD) are provided. First, application of α-redundancy of objects is proposed in order to reduce decision rules in the presence of decision table. Next, an important concept of dependency of domain attributes is introduced in corresponding to the concept of fuzzy functional dependency.
Keywords: rough sets, covering of the universe, fuzzy conditional probability relation, redundancy of tuples, dependency of attributes
