A triadic approach to formal concept analysis

Abstract

A b s t r a c t . Formal Concept Analysis, developed during the last fifteen years, has been based on the dyadic understanding of a concept constituted by its extension and its intension. The pragmatic philosophy of Charles S. Peirce with his three universal categories, and experiences in data analysis, have suggested a triadic approach to Formal Concept • C Y which is maximal with respect to component-wise inclusion. The triadic concepts are structured by three quasiorders given by the inclusion order within each of the three components. In analogy to the dyadic case, we discuss how the ordinal structure of the triadic concepts of a triadic context can be analysed and graphically represented. A basic result is that those structures can be understood order-theoretically as "complete trilattices" up to isomorphism. Triadic C o n t e x t sFormal Concept Analysis views concepts as means of intersubjective understanding in situations of purpose-oriented action. The formalization of concepts and concept systems shall especially support the interpretation and communication of conceptual relationships in different situations. This requires that the formalization of concepts has to be transparent and simple, but also comprehensive so that all main aspects of a concept may have explicit references in the formal model.The extensional and the intensional aspect, which were made explicit first by the "Port Royal Logic" [AN62] in the 17th century, are basic for the developed dyadic approach to Formal Concept Analysis. This approach starts with the primitive notion of a formal context as a triple (G, M, I) where G is a set of (formal) objects and M is a set of (formal) attributes, while I is a relation between G and M indicating when an object g of G has a certain attribute m of M. Formal contexts are often described by a two-dimensional cross table with a cross in position (g, m) if the relationship gIm holds, i.e., if the object g has the attribute m. A formal concept of a formal context (G, M, I) is defined as

Mining Association Rules in Folksonomies

“…I. e., ≺ = ∅, and we will simply note a folksonomy as a quadruple F := (U, T, R, Y ). This structure is known in Formal Concept Analysis (Wille (1982); Ganter and Wille (1999)) as a triadic context (Lehmann and Wille (1995);Stumme (2005)). An equivalent view on folksonomy data is that of a tripartite (undirected) hypergraph G = (V, E), where V = U∪T∪R is the set of nodes, and E = {{u, t, r} | (u, t, r) ∈ Y } is the set of hyperedges.…”

“…Several such projections have already been introduced in Lehmann and Wille (1995). In Stumme (2005), we provide a more complete approach, which we here adapt slightly to the association rule mining scenario.…”

Mining Triadic Association Rules

“…In the literature, there's various works some of them date back to 1995, as those [24], [16]have focused on triadic contexts analysis, concepts and diagrams and concept lattices. They define, in this way, the theoretical basis for the triadic concept analysis (TCA).…”

“…Triadic concept analysis was originally introduced by [16] and [24]as an extension to formal concept analysis, to analyze data described by three setsK 1 …”

“…The set of triadic concepts can be ordered and form a complete trilattice [4], [16]. Indeed, for each i∈ {1, 2, 3}, the relation ( …”