A data reduction method in formal fuzzy contexts
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ORIGINAL ARTICLE
A data reduction method in formal fuzzy contexts Kewen Li1 • Ming-Wen Shao1 • Wei-Zhi Wu2,3
Received: 11 June 2015 / Accepted: 16 December 2015 Ó Springer-Verlag Berlin Heidelberg 2016
Abstract As a basic operation in data mining and knowledge discovery, data reduction can reduce the volume of the data and simplify the representation of knowledge. In this paper we propose a method of attribute reduction in a formal fuzzy context based on the notion of ‘‘one-sided fuzzy concept’’. According to the importance of attributes, we classify the attributes into three types: core attributes, relatively necessary attributes and unnecessary attributes, which are also referred to the attribute characteristics. We propose judgment theorems and the corresponding algorithms for computing the three types of attribute sets. Moreover, a straightforward attribute reduction method by virtue of attribute characteristics is formulated. We show that the computation of formal concepts on the reduced data set is made more efficient, and yet produces the same lattice structure and conceptual hierarchy as the ones derived from the original formal context.
& Ming-Wen Shao [email protected] Kewen Li [email protected] Wei-Zhi Wu [email protected] 1
College of Computer and Communication Engineering, China University of Petroleum, 266580 Qingdao, Shandong, China
2
School of Mathematics, Physics and Information Science, Zhejiang Ocean University, 316022 Zhoushan, Zhejiang, China
3
Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province, 316022 Zhoushan, China
Keywords Attribute characteristics Attribute reduction Concept lattices Irreducible elements
1 Introduction Formal concept analysis (FCA), proposed by Wille [11, 42], is a branch of the lattice theory motivated by the need for a clear mathematical representation of the notions of concept and conceptual hierarchy, where the mathematical notion of ‘‘concept’’ has its origin in formal logic. FCA is formulated on the basis of a formal context, which is a binary relation between a set of objects and a set of attributes with the truth values 1 and 0 indicating whether an object has (1) or does not have (0) a particular attribute. However, in many real-life problems, truth value is not binary. Truth is to a certain degree, these are intermediate values between absolutely true (1) and absolutely false (0) [4]. Classical logic thus becomes an inappropriate construct to capture such a reasoning process. Fuzzy logic, on the other hand, allows for arguments with partial truth under vagueness. At present, there are two major types of fuzzy extension of FCA, namely fuzzy numerical method and fuzzy logic method. From the perspective of fuzzy numerical value, Jaoua and Elloumi [13] extended the binary relation into a real-valued binary relation and introduced the corresponding Galois lattices. On the other hand, Krajcˇi [15] and Yahia et al. [44], independently proposed the ‘‘one-sided fuzzy concept’’, where each fuzzy concept (X, B) has the following
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