Towards a Generalisation of Formal Concept Analysis for Data Mining Purposes

Abstract

Abstract. In this paper we justify the need for a generalisation of Formal Concept Analysis for the purpose of data mining and begin the synthesis of such theory. For that purpose, we first review semirings and semimodules over semirings as the appropriate objects to use in abstracting the Boolean algebra and the notion of extents and intents, respectively. We later bring to bear powerful theorems developed in the field of linear algebra over idempotent semimodules to try to build a Fundamental Theorem for K-Formal Concept Analysis, where K is a type of idempotent semiring. Finally, we try to put Formal Concept Analysis in new perspective by considering it as a concrete instance of the theory developed.

Detecting Features from Confusion Matrices Using Generalized Formal Concept Analysis

“…FCA, unfortunately, cannot deal in an automatic way with non-binary incidences, but generalizations of it to cater for the notion of degree of incidence have been developed [2,3,4,5,6].…”

“…In this paper, we use K-Formal Concept Analysis (kFCA) [5,7], which enables the analysis of practical real-valued CM by embedding them into an idempotent semifield K-actually a bounded lattice-ordered group [8]-, to try and prove that a concept lattice can elicit a symbolic description of the features being used in the classification process and how they are misused by the classifier.…”

“…The Basic Theorem of K-Formal Concept Analysis asserts that the set of formal ϕ-concepts is a complete lattice B ϕ (G, M, C) R max,+ (see [5,7] for details). The parameter ϕ ∈ R is called the threshold of existence and it describes a minimum degree of confusion required for concepts to be considered members of the…”

Galois Connections Between Semimodules and Applications in Data Mining

“…K-Formal Concept Analysis was introduced in [1] as a generalisation of standard Formal Concept Analysis in the sense that incidences R ∈ K n×p represented as matrices may take values in an idempotent, reflexive semifield K and we take R(i, j) = λ to mean "object g i has attribute m j in degree λ." Adequate analogues of basic objects in Formal Concept Analysis become therefore available.…”

“…Two serious obstacles may prevent widespread adoption of K-Formal Concept Analysis as a data exploration technique complementary to the standard theory: on the one hand, the K-Formal Concept Analysis analogue of the main theorem of Formal Concept Analysis is incomplete and this may worry the user willing to be on a sound mathematical ground; on the other hand, [1] did not provide an algorithm for constructing the lattice of a K-valued formal context, which prevents its use as a data-intensive exploration procedure.…”