Extending conceptualisation modes for generalised Formal Concept Analysis

Abstract

Formal Concept Analysis (FCA) is an exploratory data analysis technique for boolean relations based on lattice theory. Its main result is the existence of a dual order isomorphism between two set lattices induced by a binary relation between a set of objects and a set of attributes. Pairs of dually isomorphic sets of objects and attributes, called formal concepts, form a concept lattice, but actually model only a conjunctive mode of conceptualisation.In this paper we augment this formalism in two ways: first we extend FCA to consider different modes of conceptualisation by changing the basic dual isomorphism in a modal-logic motivated way. This creates the three new types of concepts and lattices of extended FCA, viz., the lattice of neighbourhood of objects, that of attributes and the lattice of unrelatedness.Second, we consider incidences with values in idempotent semirings-concretely the completed max-plus or schedule algebra R max,+ -and focus on generalising FCA to try and replicate the modes of conceptualisation mentioned above.To provide a concrete example of the use of these techniques, we analyse the performance of multi-class classifiers by conceptually analysing their confusion matrices.

Activating Generalized Fuzzy Implications from Galois Connections

“…On idempotent semifields, however, [18] have proven that the four different types of Galois connection emerge from variations of the following construction: Given a scalar product x, y = x t ⊗ R ⊗ y , and a ϕ ∈ K , let the ϕ-polars be:…”

“…Note that such fuzzy algebras can alternatively be described as fuzzy semirings [17]. An independently motivated generalisation of FCA, K -Formal Concept Analysis (KFCA), uses an idempotent semifield K -a kind of semiring with a multiplicative group structureas the range of the relation [18]. Whereas fuzzy semirings are mostly used to capture a "degree of truth", semifields are used to capture the concept of "cost" or, dually, "utility".…”

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.…”

Mining gene expression data with pattern structures in formal concept analysis

“…complex object descriptions, defining so-called similarity operators which induce a semi-lattice on data descriptions. Several attempts were made for defining such semi-lattices on sets of graphs [8,17,18,22] and logical formulas [5,7] (see also [10,37] for FCA extensions). Indeed, if one is able to order object descriptions in complex data, e.g.…”