The Research On Power Formal Concept Analysis And Fuzzy Concept Lattices

Formal concept analysis(FCA) was initiated by Wille in 1982, which has de-veloped as an e(?)cient order-theoretical tool of data analysis. The notion of formalconcept originates from the philosophical understanding that a concept is consti-tuted by its extension and intention. Through the formalization of the subconcept-superconcept relation between concepts, a conceptual hierarchical structure, namelyformal concept lattice can be constructed. In the philosophical sense, formal con-cepts can be understood as the mathematical representation of basic units ofthought formed within some special social or cultural environment. From theviewpoint of data mining and knowledge discovery, formal concepts can be viewedas a formalization of basic knowledge units hidden in a specified contextual back-ground. Particularly, FCA has a considerable potential capability of extractingbasic knowledge units from data in dealing with scientific problems. In the pastthree decades, FCA has developed rapidly on the theoretical side and has been ap-plied in many fields such as knowledge discovery, information retrieval and softwareengineering.The basis of FCA is the notion of context which is essentially a binary rela-tion between two classical sets. However, the data encountered in the real worldis usually organized in a more complex pattern. Moreover, the available conceptscomputing algorithms are challenged in many applications where the data scaleis huge. It is obvious that the classical FCA is not qualified in such applications.Therefore, it is necessary to improve or adjust the classical framework of FCA tomeet the requirements in such applications. In this paper, based on the applicationof FCA in research communities identification, we accommodate additional struc-ture represented by binary relations on objects and attributes into the classicalsetting of FCA. By lifting the original incidence relation onto the power level, wedevelop the fundamental theory of power concept analysis. Moreover, we studyseveral special types of lifting and investigate the relationship between the originalconcept lattice and the power concept lattice.In the study of fuzzy concept lattices, much e(?)ort has been made to establishthe basic theory of classical FCA in the fuzzy setting. As in the classical setting,notions of fuzzy Galois connection, fuzzy closure operator and fuzzy closure systemplay a key role in the framework of fuzzy concept analysis. Many scholars havestudied the closure systems on fuzzy powersets and extended the classical corre- spondence theory between closure systems and closure operators into the fuzzysetting. In this paper, we give a concise equivalent characterization of the notionof L-closure system proposed by Bˇelohl′avek on fuzzy powerset. In addition, wepropose the notion of L-closure L-system on L-power set and show that thereexists one-to-one correspondence between L-closure L-systems and L-closuresystems.The notion of fuzzy partially ordered set (fuzzy poset) has been proposed fordi-erent purposes and much e-ort has been made to study closure operators, clo-sure systems and Galois connections on fuzzy posets. We observe that the studyof closure systems on fuzzy posets is an imperative problem. In this paper, we pro-pose the notions of LK-closure system and LK-Galois connection, and investigatetheir relationship with LK-closure operators. Furthermore, we develop anothernotion of closure system, namely LK-closure L-system on fuzzy posets. We alsostudy the interrelation among LK-closure L-systems, LK-closure systems andLK-closure operators. All the results can be viewed as the fuzzification of thecorresponding results in the classical setting.To explore the ability of FCA to deal with incomplete or imprecise informa-tion, di-erent approaches to generalize the classical concept lattices into the fuzzysetting have been developed in the literature. Based on the multi-adjoint conceptlattices introduced by Medina et al, we extend the generalized concept latticesdeveloped by Krajˇci and propose a more general framework of concept analysis,namely MC-concept lattices. We also establish the basic theory of classical FCAin this new framework. Moreover, by comparing the MC-concept lattices withthe generalized concept lattices and multi-adjoint concept lattices, we show thatMC-concept lattices provides a more -exible framework of FCA for data analysisin the real world.