Dependence space models to construct concept lattices

Abstract

Concept lattice is a kind of efficient mathematical tools for data analysis and knowledge discovery. In this paper, dependence space models are constructed to obtain a concept lattice. For a formal context, a congruence relation is first defined based on a sufficiency operator on the power set of the attribute set. A dependence space is generated. By using the dependence space, we introduce a closure operator. Related properties of the operator are discussed. It is also proved that any closed element of the closure operator is the maximum element in the attribute granule generated by the element. Further, the closed element is just the intension of some formal concept. Similarly, we define a congruence relation on the basis of a sufficiency operator, and introduce a dependence space on the power set of the object set. A closure operator is proposed for the subset of objects. The corresponding closed element of the closure operator is the maximum element in each object granule. All maximum elements generate the set of all extensions of the concept lattice. An example is used to show the validity of the approach to obtain all intensions. Finally, we introduce a dominance congruence relation, which is used to produce a closure operator. The dependence of attributes is defined based on the closure operator. The related decision rules are discussed.