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

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

FCAIR 2012 Formal Concept Analysis Meets Information Retrieval Workshop co-located with the 35th European Conference on Information Retrieval (ECIR 2013) March 24, 2013, Moscow, Russia

International audienceFormal Concept Analysis (FCA) is a mathematically well-founded theory aimed at data analysis and classifiation. The area came into being in the early 1980s and has since then spawned over 10000 scientific publications and a variety of practically deployed tools. FCA allows one to build from a data table with objects in rows and attributes in columns a taxonomic data structure called concept lattice, which can be used for many purposes, especially for Knowledge Discovery and Information Retrieval. The Formal Concept Analysis Meets Information Retrieval (FCAIR) workshop collocated with the 35th European Conference on Information Retrieval (ECIR 2013) was intended, on the one hand, to attract researchers from FCA community to a broad discussion of FCA-based research on information retrieval, and, on the other hand, to promote ideas, models, and methods of FCA in the community of Information Retrieval