Galois Connections Between Semimodules and Applications in Data Mining

Abstract

Abstract. In [1] a generalisation of Formal Concept Analysis was introduced with data mining applications in mind, K-Formal Concept Analysis, where incidences take values in certain kinds of semirings, instead of the standard Boolean carrier set. A fundamental result was missing there, namely the second half of the equivalent of the main theorem of Formal Concept Analysis. In this continuation we introduce the structural lattice of such generalised contexts, providing a limited equivalent to the main theorem of K-Formal Concept Analysis which allows to interpret the standard version as a privileged case in yet another direction. We motivate our results by providing instances of their use to analyse the confusion matrices of multiple-input multiple-output classifiers.

Extending conceptualisation modes for generalised Formal Concept Analysis

“…Yet another path towards generalising Formal Concept Analysis was explored in K-Formal Concept Analysis [33,35], an analogue where incidences take values in idempotent, reflexive semifields-closely related to residuated lattices-being structures in which the boolean semiring is embedded.…”

Detecting Features from Confusion Matrices Using Generalized Formal Concept Analysis

“…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…”

Towards Galois Connections over Positive Semifields

“…But Standard FCA can also be understood in the context of the linear algebra of boolean spaces with sets substituted for characteristic functions, and other extensions, e.g. the K-FCA [2,3,4], FCA in a fuzzy setting [5], etc., can also be considered in the light of linear algebra over a certain subclass of semirings. In this paper, we will understand a semiring [6] to be an algebra S = S, ⊕, ⊗, , e for which -the additive structure, S, ⊕, , is a commutative monoid, -the multiplicative structure, S\{ }, ⊗, e , is a monoid, -multiplication distributes over addition from right and left -and the zero element is multiplicatively-absorbing i.e.…”

“…One of the most useful extensions to FCA uses K-valued formal contexts where K is a complete idempotent semifield: this is the basis of K-Formal Concept Analysis [2,3,4]. An idempotent semiring is one whose addition is idempotent, u⊕u = u while semifields are semirings whose multiplicative structure is a group.…”