Formal Concept Analysis Based On Rough Set And Axiomatic Fuzzy Sets

Formal concept analysis (FCA) is an effective tool for concept discovery from data, in which the relationship of concepts is embodied by concept lattice. Formal concept analysis has been widely used in information retrieval, digital library, software engineering and knowledge discovery, etc. Rough set theory is a new mathematical tool dealing with vagueness and uncertainty, has been successfully used in many areas such as knowledge discovery, pattern recognition and classification and fault diagnostication. Axiomatic fuzzy sets (AFS) theory is another method to deal with fuzzy information, which provides an effective tool to concert the information in the training examples and databases into the membership functions and their fuzzy logic operations. Recently, AFS theory has been developed further and applied to many fields such as fuzzy clustering analysis, fuzzy decision trees and concept representations et al. In this paper, some new theories and applications about FCA are discussed based on rough set and AFS theory. Main topics include:Firstly, in order to overcome shortcoming of monotone concept, AFS formal concept is proposed by combining AFS theory and concept lattice, in which the intent and extent can be determined by each other. AFS formal concept can be viewed as the generalization and development of monotone concept. Moreover, we show that the set of all AFS formal concepts forms a complete lattice under the order relation. Furthermore, we give an approach to find some AFS formal concepts whose intents (extents) approximate any element of AFS algebras by virtue of rough set theory, which overcomes the shortcoming of concept approximating by using monotone concept. Moreover, we introduce fuzzy formal concept based AFS structure by combining AFS structure and formal context, which can be used to express the uncertainty relations between the objects and the attributes, and the relation is directly determined by the origin data and facts.Secondly, with the aim of deriving the mathematical properties of formal concept and the relationships between concept lattice and AFS theory, we propose a new AFS algebra system on formal context, called E~CII algebra, by which the algebra properties of FCA can be explored and show that AFS theory is closely related to FCA. Thirdly, a new similarity model is proposed, which using irreducible attributes and objects according to structure elements to evaluate the similarity degree of the two concepts of concept lattice. We give a new method to find irreducible elements of concept lattice by using attributes classes and objects classes, rather than constructing Hasse Diagram. The proposed method combines featural and structural information into decision and has a higher correlation with human judgement, which can be viewed as the generalization of Souza and Davis's similarity model. In order further to extent the ability of the above model in real world, two extension models are discussed. A measure evaluating non-definable pairs of objects and attributes is proposed based on the proposed model and rough set. Moreover, the proposed model is also extended to fuzzy formal context.Fourthly, a new near set is established based on AFS theory, in which every object has an AFS fuzzy description with definitely semantics and distinguished among other objects at maximum extent. The proposed approach to assessing the nearness (closeness) of objects is not defined directly using any distance metric, but depend on their fuzzy descriptions. Near set based on AFS logic can be used to discover the "nearness" of objects that are possibly disjoint and, yet, qualitatively near each other. Furthermore, by combining approximation space and AFS theory, two new approximation spaces are established, which can be viewed as multi-granulations forms of approximation spaces and approximation spaces based on nearness relation.