Galois Connections between Semimodules and Applications in Data Mining

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

Towards a generalisation of formal concept analysis for data mining purposes

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

K-Formal Concept Analysis as linear algebra over idempotent semifields

We report on progress in characterizing K-valued FCA in algebraic terms, where K is an idempotent semifield. In this data mining-inspired approach, incidences are matrices and sets of objects and attributes are vectors. The algebraization allows us to write matrix-calculus formulae describing the polars and the fixpoint equations for extents and intents. Adopting also the point of view of the theory of linear operators between vector spaces we explore the similarities and differences of the idempotent semimodules of extents and intents with the subspaces related to a linear operator in standard algebra. This allows us to shed some light into Formal Concept Analysis from the point of view of the theory of linear operators over idempotent semimodules. In the opposite direction, we state the importance of FCA-related concepts for dual order homomorphisms of linear spaces over idempotent semifields, specially congruences, the lattices of extents, intents and formal concepts

Activating Generalized Fuzzy Implications from Galois Connections

This paper deals with the relation between fuzzy implications and Galois connections, trying to raise the awareness that the fuzzy implications are indispensable to generalise Formal Concept Analysis. The concrete goal of the paper is to make evident that Galois connections, which are at the heart of some of the generalizations of Formal Concept Analysis, can be interpreted as fuzzy incidents. Thus knowledge processing, discovery, exploration and visualization as well as data mining are new research areas for fuzzy implications as they are areas where Formal Concept Analysis has a niche.F.J. Valverde-Albacete—was partially supported by EU FP7 project LiMoSINe, (contract 288024). C. Peláez-Moreno—was partially supported by the Spanish Government-CICYT project 2011-268007/TEC.Publicad

Four-fold Formal Concept Analysis based on Complete Idempotent Semifields

Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is K¯¯¯¯-four-fold Formal Concept Analysis (K¯¯¯¯-4FCA) where K¯¯¯¯ is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in dually-ordered pairs—e.g., the complete max-plus and min-plus semirings—the basic construction shows dual-order-, row–column- and Galois-connection-induced dualities that appear simultaneously a number of times to provide the full spectrum of variability. Our results lead to a fundamental theorem of K¯¯¯¯-four-fold Formal Concept Analysis that properly defines quadrilattices as 4-tuples of (order-dually) isomorphic lattices of vectors and discuss its relevance vis-à-vis previous formal conceptual analyses and some affordances of their results

The Singular Value Decomposition over Completed Idempotent Semifields

In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus semifield, the tropical algebra, and the max-times algebra—are useful in a range of applications, e.g., morphological processing. We further the task of eliciting the relation between i-SVD and the extension of Formal Concept Analysis to complete idempotent semifields (K-FCA) started in a prior work. We find out that for a matrix with entries considered in a complete idempotent semifield, the Galois connection at the heart of K-FCA provides two basis of left- and right-singular vectors to choose from, for reconstructing the matrix. These are join-dense or meet-dense sets of object or attribute concepts of the concept lattice created by the connection, and they are almost surely not pairwise orthogonal. We conclude with an attempt analogue of the fundamental theorem of linear algebra that gathers all results and discuss it in the wider setting of matrix factorization.This research was funded by the Spanish Government-MinECo project TEC2017-84395-P and the Dept. of Research and Innovation of Madrid Regional Authority project EMPATIA-CM (Y2018/TCS-5046)

Towards Galois Connections over Positive Semifields

In this paper we try to extend the Galois connection construction of K-Formal Concept Analysis to handle semifields which are not idempotent. Important examples of such algebras are the extended non-negative reals and the extended non-negative rationals, but we provide a construction that suggests that such semifields are much more abundant than suspected. This would broaden enormously the scope and applications of K-Formal Concept Analysis.CPM & FVA have been partially supported by the Spanish Government-MinECo projects TEC2014-53390-P and TEC2014-61729-EX

Extending Conceptualisation Modes for Generalised Formal Concept Analysis

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 View the MathML source—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.Spanish Government-Comisión Interministerial de Ciencia y Tecnología project 2008–06382/TEC and 2008–02473/TEC and the regional project (Comunidad Autónoma de Madrid – UC3M) CCG08-UC3M/TIC-4457Publicad

Supporting scientific knowledge discovery with extended, generalized Formal Concept Analysis

In this paper we fuse together the Landscapes of Knowledge of Wille's and Exploratory Data Analysis by leveraging Formal Concept Analysis (FCA) to support data-induced scientific enquiry and discovery. We use extended FCA first by allowing K-valued entries in the incidence to accommodate other, non-binary types of data, and second with different modes of creating formal concepts to accommodate diverse conceptualizing phenomena. With these extensions we demonstrate the versatility of the Landscapes of Knowledge metaphor to help in creating new scientific and engineering knowledge by providing several successful use cases of our techniques that support scientific hypothesis-making and discovery in a range of domains: semiring theory, perceptual studies, natural language semantics, and gene expression data analysis. While doing so, we also capture the affordances that justify the use of FCA and its extensions in scientific discovery.FJVA and AP were partially supported by EUFP7 project LiMo- SINe (contract288024) for this research. CPM was partially supported by the Spanish Ministry of Economics and Competitiveness projects TEC2014-61729-EXP and TEC2014-53390-P

Gene Expressio Array Exploration Using K-Formal Concept Analysis

Proceeding of: 9th International Conference, ICFCA 2011, Nicosia, Cyprus, May 2-6, 2011.DNA micro-arrays are a mechanism for eliciting gene expression values, the concentration of the transcription products of a set of genes, under different chemical conditions. The phenomena of interest—up-regulation, down-regulation and co-regulation—are hypothesized to stem from the functional relationships among transcription products. In [1,2,3] a generalisation of Formal Concept Analysis was developed with data mining applications in mind, K-Formal Concept Analysis, where incidences take values in certain kinds of semirings, instead of the usual Boolean carrier set. In this paper, we use (Rmin+)- and (Rmax+) to analyse gene expression data for Arabidopsis thaliana. We introduce the mechanism to render the data in the appropriate algebra and profit by the wealth of different Galois Connections available in Generalized Formal Concept Analysis to carry different analysis for up- and down-regulated genes.Spanish Government-Comision Interministerial de Ciencia y Tecnología projects 2008-06382/TEC and 2008-02473/TEC and the regional projects S-505/TIC/0223 (DGUI-CM) and CCG08-UC3M/TIC- 4457 (Comunidad Aut onoma de Madrid - UC3M)