Florent Domenach

Closure systems, implicational systems, overhanging relations and the case of hierarchical classification

Abstract Moore families and closure operators, especially those appearing in hierarchical classification, are considered here from the point of view of their related implicational systems and overhanging relations. The last ones, newly introduced here, generalize the “nesting relations” defined by Adams [J. Classification 3 (1986) 299] in the case of classification trees. Here we characterize overhanging relations by three axioms, and prove that they are in a one-to-one correspondence with Moore families. We study which properties of implicational systems and overhanging relations are general, and which are specific to hierarchical structures. We also characterize canonical implication bases of hierarchies and obtain a similar result for overhangings.

Consensus of Classification Systems, with Adams’ Results Revisited

The problem of aggregating a profile of closure systems into a consensus closure system has interesting applications in classification. We first present an overview of the results obtained by a lattice approach. Then, we develop a more refined approach based on overhangings and implications that appears to be a generalization of Adams’ consensus tree algorithm. Adams’ uniqueness result is explained and generalized.

On the Roles of Galois Connections in Classification

Galois connections (or residuated mapping) are of growing interest in various domains related with or relevant from Classification. Among their many uses, we select some topics related with modelization and aggregation of dissimilarities and conceptual classification. We partially revist them in a common frame provided by a recent study about Galois connections between closure spaces.

Using pattern structures for analyzing ontology-based annotations of biomedical data

Annotating data with concepts of an ontology is a common practice in the biomedical domain. Resulting annotations, i.e., data-concept relationships, are useful for data integration whereas the reference ontology can guide the analysis of integrated data. Then the analysis of annotations can provide relevant knowledge units to consider for extracting and understanding possible cor- relations between data. Formal Concept Analysis (FCA) which builds from a binary context a concept lattice can be used for such a knowledge discovery task. However annotated biomedical data are usually not binary and a scaling procedure for using FCA is required as a prepro- cessing, leading to problems of expressivity, ranging from loss of information to the generation of a large num- ber of additional binary attributes. By contrast, pattern structures o er a general FCA-based framework for buil- ding a concept lattice from complex data, e.g., a set of objects with partially ordered descriptions. In this pa- per, we show how to instantiate this general framework when descriptions are ordered by an ontology. We illus- trate our approach with the analysis of annotations of drug related documents, and we show the capabilities of the approach for knowledge discovery.

Biclosed Binary Relations and Galois Connections

Given two closure spaces (E,ϕ) and (E′,ϕ′), a relation R⊒E×E′ is said biclosed if every row of its matrix representation corresponds to a closed subset of E′, and every column to a closed subset of E. An isomorphism between, on the one hand, the set of all biclosed relations and, on the other hand, the set of all Galois connections between the two lattices of closed sets is established. Several computational applications are derived from this result.

DASACT: A decision aiding software for axiomatic consensus theory

There have been various attempts, solutions, and approaches towards constructing an appropriate consensus tree based on a given set of phylogenetic trees. However, for practitioners, it is not always clear, for a given data set, which of these would create the most relevant consensus tree. In this paper, we introduce an open-source software called DASACT (Decision Aiding Software for Axiomatic Consensus Theory) created to assist practitioners on choosing the most appropriate consensus function. It is based on an exhaustive evaluation of axiomatic properties and consensus functions, which define the knowledge space as a concept lattice. Using a selection of axiomatic properties provided by the user, it is able to aid the user in choosing the most suitable function. DASACT is freely available at http://www.cs.unic.ac.cy/florent/software.htm.

Implications of Axiomatic Consensus Properties

Since Arrow’s celebrated impossibility theorem, axiomatic consensus theory has been extensively studied. Here we are interested in implications between axiomatic properties and consensus functions on a profile of hierarchies. Such implications are systematically investigated using Formal Concept Analysis. All possible consensus functions are automatically generated on a set of hierarchies derived from a fixed set of taxa. The list of implications is presented and discussed.

The structure of the overhanging relations associated with some types of closure systems

Over many different kinds of cryptomorphisms equivalent with closure systems (and so with closure operators), we focus here on implication relations and the related overhanging relations, as introduced and axiomatized in a previous paper (Domenach and Leclerc, Math Soc Sci 47(3):349–366, 2004). In relation with data analysis motivations, we particularize the axioms on overhanging relations in order to account for some types of closure systems, such as nested or distributive ones. We also examine the lattice structure of overhanging relations, which is isomorphic to the lattice of closure systems, and derived structures for particular sets of overhangings.RésuméParmi bien des types de structures liés par cryptomorphisme aux systèmes de fermeture (et donc aux fermetures), nous considérons ici plus particulièrement les relations d’implication et les relations d’emboitement qui leur sont liées, telles qu’introduites et caractérisées axiomatiquement dans un article précédent (Domenach and Leclerc, Math Soc Sci 47(3):349–366, 2004). En relation avec des motivations issues de l’analyse des données, nous établissons des systèmes d’axiomes particuliers pour les relations d’emboitement associées à des systèmes de fermeture particuliers, comme les familles de parties totalement ordonnées ou distributives. Nous abordons aussi l’étude du treillis des relations d’emboitement, qui est isomorphe à celui des systèmes de fermeture, ainsi que des structures qui en résultent sur les ensembles particuliers d’emboitements.

Similarity Measures of Concept Lattices

Concept lattices fulfil one of the aims of classification by providing a description by attributes of each class of objects. We introduce here two new similarity/dissimilarity measures: a similarity measure between concepts (elements) of a lattice and a dissimilarity measure between concept lattices defined on the same set of objects and attributes. Both measures are based on the overhanging relation previously introduced by the author, which are a cryptomorphism of lattices.

Similarity Measures on Concept Lattices

This paper falls within the framework of Formal Concept Analysis which provides classes (the extents) of objects sharing similar characters (the intents), a description by attributes being associated to each class. In a recent paper by the first author, a new similarity measure between two concepts in a concept lattice was introduced, allowing for a normalization depending on the size of the lattice.In this paper, we compare this similarity measure with existing measures, either based on cardinality of sets or originating from ontology design and based on the graph structure of the lattice. A statistical comparison with existing methods is carried out, and the output of the measure is tested for consistency.