Formal Concept Analysis and Rough Set Theory in Clustering

Abstract

This paper is concerned with the fundamental role of two mathematical theories in some clustering problems. Formal concept analysis provides the algebraic structure and properties of possible concepts from a given context, and rough set theory provides a mathematical tool to deal with imprecise and incomplete data. Based on these theories, we developed models and algorithms for solving three clustering problems: conceptual clustering, approximate conceptual clustering, and text clustering. 1 Formal Concept Analysis and Rough Set Theory A theory of concept lattices has been studied under the name formal concept analysis (FCA) by Wille and his colleagues [1, 11]. Considers a context as a triple (O,D,R) where O be a set of objects, D be a set of primitive descriptors and R be a binary relation between O and D, i.e., R ⊆ O×D and (o, d) ∈ R is understood as the fact that object o has the descriptor d. For any object subset X ⊆ O, the largest tuple common to all objects in X is denoted by λ(X). For any tuple S ∈ T , the set of all objects satisfying S is denoted by ρ(S). A tuple S is closed if λ(ρ(S)) = S. Formally, a concept C in the classical view is a pair (X, S), X ⊆ O and S ⊆ T , satisfying ρ(S) = X and λ(X) = S. X and S are called extent and intent of C, respectively. Concept (X2, S2) is a subconcept of concept (X1, S1) if X2 ⊆ X1 which is equivalent to S2 ⊇ S1, and (X1, S1) is then a superconcept of (X2, S2). It was shown that λ and ρ define a Galois connection between the power sets ℘(O) and ℘(D), i.e., they are two order-reversing one-to-one operators. As a consequence, the following properties hold which will be exploited in the learning process: if S1 ⊆ S2 then ρ(S1) ⊇ ρ(S2) and λρ(S1) ⊆ λρ(S2) if X1 ⊆ X2 then λ(X1) ⊇ λ(X2) and ρλ(X1) ⊆ ρλ(X2) S ⊆ λρ(S), X ⊆ ρλ(X) ρλρ = ρ, λρλ = λ, λρ(λρ(S)) = λρ(S) ρ( ⋃ j Sj) = ⋂ j ρ(Sj), λ( ⋃ j Xj) = ⋂ j λ(Xj) The basic theorem in formal concept analysis [11] states that the set of all possible concepts from a context (O,D,R) is a complete lattice L, called Galois lattice, in which infimum and supremum can be described as follows: ∧