Connections between two-universe rough sets and formal concepts
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ORIGINAL ARTICLE
Connections between two‑universe rough sets and formal concepts Ming‑Wen Shao1 · Li Guo2 · Chang‑Zhong Wang3 Received: 18 August 2017 / Accepted: 26 February 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract Rough sets and formal concept analysis are two complementary tools during the process of data analysis. Two-universe rough set model is one of generalization of the classical rough sets. In this paper, the connections between two-universe rough sets and formal concepts are discussed. We investigate the relations between two-universe rough sets and the object (attribute) oriented formal concepts. We also establish connections between revised two-universe rough sets and the object (attribute) oriented formal concepts. Meanwhile, relations between algebraic characterizations of two-universe rough sets and formal concepts are revealed. Some properties of two-universe rough sets are examined. Keywords Generalized approximation space · Two-universe rough sets · Object (Attribute)Oriented formal concepts · Concept lattices
1 Introduction The theory of rough sets (RS), proposed by Pawlak [22, 23], is an extension of the classical set theory. RS is a tool useful for dealing with imprecision, vagueness and uncertainty of information. In recent years, RS has been successfully applied to various disciplines such as data analysis, data mining, knowledge discovery, pattern recognition, decision analysis and medical diagnosis [2, 4, 10, 11, 16, 24, 27, 35, 36, 47]. The basic operators in RS are lower and upper approximations. In order to deal with complex data, several extensions to the classical RS have been developed. For instance,
* Li Guo [email protected] Ming‑Wen Shao [email protected] Chang‑Zhong Wang [email protected] 1
College of Computer and Communication Engineering, China University of Petroleum, Qingdao 266580, Shandong, People’s Republic of China
2
College of Information Science and Technology, Shihezi University, Shihezi 832000, Xinjiang, People’s Republic of China
3
Department of Mathematics, Bohai University, Jinzhou 121000, People’s Republic of China
covering rough sets [48, 49, 55], fuzzy rough sets [5, 45] and intuitionistic fuzzy rough sets [56] are proposed respectively in the last two decades. An interesting extension is two-universe rough set model proposed by Yao et al. [50, 51]. Subsequently, Wu et al. proposed two-universe fuzzy rough set model in [46]. Note that the approximations of a set on an universe are sets on the other universe in twouniverse rough set model. It seems unreasonable. Then Pei and Xu revised the model and gave a new generalized rough set model on two universes [25, 26]. The theory of formal concept analysis (FCA), originally proposed by Wille [7, 43] in 1982, is a model for the study of formal concepts and conceptual hierarchies. Currently, FCA has been successfully applied to information retrieval, knowledge discovery, data mining, machine learning, software engineering and other disciplines [1, 6, 9, 13,
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