Intelligent decision systems often need to deal with vague and uncertain data. Several approaches are commonly used to address this problem, such as statistical methods, machine learning, and fuzzy set. Overlapping with but different from the fuzzy set theory, rough set theory is a relatively new mathematical approach to vague data analysis. A rough set is basically an approximation representation of the given data. The representation is expressed in two subsets defined on the data set: the upper and lower approximations. The main difference between the rough set theory and other approaches is that it does not rely on preliminary information about the data such as membership probabilities of the data items required for fuzzy set. One of the open questions in rough set is to decide if a subset of a covering approximation space is definable. In this paper, we answer this question by investigating the approximation operator and conclude the relation of the inner definable, outer definable, and definable subsets of a covering approximation space under certain conditions. keyword: Rough set, covering, approximation space, definable subset.
Abstract. Attribute reduction, as an important preprocessing step for knowledge acquiring in data mining, is one of the key issues in rough set theory. It can only deal with attributes of a specific type in the information system by using a specific binary relation. However, there may be attributes of multiple different types in information systems in real-life applications. A composite relation is proposed to process attributes of multiple different types simultaneously in composite information systems. In order to solve the time-consuming problem of traditional heuristic attribute reduction algorithms, a novel attribute reduction algorithm based on structure discernibility matrix was proposed in this paper. The proposed algorithms can choose the same attribute reduction as its previous version, but it can be used to accelerate a heuristic process of attribute reduction by avoiding the process of intersection and adopting the forward greedy attribute reduction approach. The theoretical analysis and experimental results with UCI data sets show that the proposed algorithm can accelerate the heuristic process of attribute reduction.
In this paper, we first introduce the notion of ππ-convergence in posets as an unified form of O-convergence and O2-convergence. Then, by studying the fundamental properties of ππ-topology which is determined by ππ-convergence according to the standard topological approach, an equivalent characterization to the ππ-convergence being topological is established. Finally, the lim-infπ-convergence in posets is further investigated, and a sufficient and necessary condition for lim-infπ-convergence to be topological is obtained.
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