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2013 | Buch

Contextual Implications between Attributes and Some Representation Properties for Finite Lattices Kapitel lesen Erstes Kapitel lesen

Formal Concept Analysis

11th International Conference, ICFCA 2013, Dresden, Germany, May 21-24, 2013. Proceedings

herausgegeben von: Peggy Cellier, Felix Distel, Bernhard Ganter

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Computer Science

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book constitutes the refereed proceedings of the 11th International Conference on Formal Concept Analysis, ICFCA 2013, held in Dresden, Germany, in May 2013. The 15 regular papers presented in this volume were carefully reviewed and selected from 46 submissions. The papers present current research from a thriving theoretical community and a rapidly expanding range of applications in information and knowledge processing including data visualization and analysis (mining), knowledge management, as well as Web semantics, and software engineering. In addition the book contains a reprint of the first publication in english describing the seminal stem-base construction by Guigues and Duquenne; and a position paper pointing out potential future applications of FCA.

Inhaltsverzeichnis

Frontmatter

Historic Paper

Contextual Implications between Attributes and Some Representation Properties for Finite Lattices
Abstract
In the starting paper on Formal Concept Analysis (see WILLE (1982)), it is claimed that the aim of this young discipline is to ”Restructure Lattice Theory”, using an approach which ”goes back to the origin of the lattice concept in the nineteenth-century attempts to formalize logic, where a fundamental step was a reduction of a concept to its ”extent”. We propose to make the reduction less abstract by retaining in some measure ”the intent” of a concept”.
Vincent Duquenne

Regular Contributions

Mathematical Morphology Operators over Concept Lattices
Abstract
Although mathematical morphology and formal concept analysis are two lattice-based data analysis theories, they are still developed in two disconnected research communities. The aim of this paper is to contribute to fill this gap, beyond the classical relationship between the Galois connections defined by the derivation operators and the adjunctions underlying the algebraic mathematical morphology framework. In particular we define mathematical morphology operators over concept lattices, based on distances, valuations, or neighborhood relations in concept lattices. Their properties are also discussed. These operators provide new tools for reasoning over concept lattices.
Jamal Atif, Isabelle Bloch, Felix Distel, Céline Hudelot
Dismantlable Lattices in the Mirror
Abstract
We investigate properties which hold for both the lattice of a binary relation and for its ’mirror lattice’, which is the lattice of the complement relation.
We first prove that the relations whose lattice is dismantlable corres-pond to the class of chordal bipartite graphs; we provide algorithmic tools to find a doubly irreducible element in such a lattice.
We go on to show that a lattice is dismantlable and its mirror lattice is also dismantlable if and only if both these lattices are planar.
Anne Berry, Alain Sigayret
Towards an Error-Tolerant Construction of $\mathcal{EL}^\bot$ -Ontologies from Data Using Formal Concept Analysis
Abstract
In the work of Baader and Distel, a method has been proposed to axiomatize all general concept inclusions (GCIs) expressible in the description logic \(\mathcal{EL}^\bot\) and valid in a given interpretation \(\mathcal{I}\). This provides us with an effective method to learn \(\mathcal{EL}^\bot\)-ontologies from interpretations. In this work, we want to extend this approach in the direction of handling errors, which might be present in the data-set. We shall do so by not only considering valid GCIs but also those whose confidence is above a given threshold c. We shall give the necessary definitions and show some first results on the axiomatization of all GCIs with confidence at least c. Finally, we shall provide some experimental evidence based on real-world data that supports our approach.
Daniel Borchmann
Using Pattern Structures for Analyzing Ontology-Based Annotations of Biomedical Data
Abstract
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 correlations 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 preprocessing, leading to problems of expressivity, ranging from loss of information to the generation of a large number of additional binary attributes. By contrast, pattern structures offer a general FCA-based framework for building a concept lattice from complex data, e.g., a set of objects with partially ordered descriptions. In this paper, we show how to instantiate this general framework when descriptions are ordered by an ontology. We illustrate our approach with the analysis of annotations of drug related documents, and we show the capabilities of the approach for knowledge discovery.
Adrien Coulet, Florent Domenach, Mehdi Kaytoue, Amedeo Napoli
Formal Concept Analysis via Atomic Priming
Abstract
Formal Concept Analysis (FCA) looks to decompose a matrix of objects-attributes into a set of sparse matrices capturing the underlying structure of a formal context. We propose a Rank Reduction (RR) method to prime approximate FCAs, namely RRFCA. While many existing FCA algorithms are complete, lectic ordering of the lattice may not minimize search/decomposition time. Initially, RRFCA decompositions are not unique or complete; however, a set of good closures with high support is learned quickly, and then, made complete. RRFCA has its novelty in that we propose a new multiplicative two-stage method. First, we describe the theoretical foundations underpinning our RR approach. Second, we provide a representative exemplar, showing how RRFCA can be implemented. Further experiments demonstrate that RRFCA methods are efficient, scalable and yield time-savings. We demonstrate the resulting methods lend themselves to parallelization.
Ruairí de Fréin
Applications of Ordinal Factor Analysis
Abstract
Ordinal factorisation is a factor analytical tool based on Formal Concept Analysis. It groups the so-called Boolean factors, given by suitable formal concepts, into well-structured families that can be interpreted as many-valued factors. In this paper we put the ordinal factorisation to work by testing it on well-documented medical data. We also compare the results with those obtained by established data reduction methods.
Cynthia Vera Glodeanu, Bernhard Ganter
Tri-ordinal Factor Analysis
Abstract
We present a new factor analytical technique for three-way data based on Triadic Concept Analysis. Previous works in this direction focused on reducing the complexity of triadic data by using triadic concepts. In this paper we propose a method of grouping these concepts into well-structured families such that the factors can be interpreted as many-valued ones. Thereby we lift the already existing theory of ordinal factorisations to the triadic setting.
Cynthia Vera Glodeanu
Formal $\mathcal{F}$ -contexts and Their Induced Implication Rule Systems
Abstract
Formal concept analysis (FCA) provides an approach to restructuring important lattice structures such as complete lattices, distributive lattices and algebraic lattices. In this paper, we focus on the theoretical aspect of FCA and study the representation of algebraic domains by a special type of formal contexts. We first propose the notion of consistent \(\mathcal{F}\)-context and investigate the detailed properties. Then we study the induced implication rule systems of the consistent \(\mathcal{F}\)-contexts and propose the notion of formal implication rule systems as the axiomatization. The results show that \(\mathcal{F}\)-concepts inherent in consistent \(\mathcal{F}\)-contexts can be characterized equivalently by closed subsets derived from the formal implication rule systems. Furthermore, we study the order-theoretical properties of \(\mathcal{F}\)-concepts hierarchy (respectively, closed subsets family) of consistent \(\mathcal{F}\)-contexts (respectively, formal implication rule system). It is shown that both \(\mathcal{F}\)-contexts and formal implication rule systems can serve as appropriate tools to concretely represent algebraic domains.
Lankun Guo, Qingguo Li, Petko Valtchev, Robert Godin
User-Friendly Fuzzy FCA
Abstract
Fuzzy FCA is developed and well understood from the algebraic point of view, but the meaning is still not clear to a (potential) real user. In the area of linear residuated lattices we try to explain the meaning of the concept forming operators and their results in a fuzzy FCA. In the area of non-linear residuated lattices we discuss meaningful criteria for using them in fuzzy FCA and we propose examples with a clear meaning for a user.
Juraj Macko
Proper Mergings of Stars and Chains Are Counted by Sums of Antidiagonals in Certain Convolution Arrays
Abstract
A proper merging of two disjoint quasi-ordered sets P and Q is a quasi-order on the union of P and Q such that the restriction to P or Q yields the original quasi-order again and such that no elements of P and Q are identified. In this article, we determine the number of proper mergings in the case where P is a star (i.e. an antichain with a smallest element adjoined), and Q is a chain. We show that the lattice of proper mergings of an m-antichain and an n-chain, previously investigated by the author, is a quotient lattice of the lattice of proper mergings of an m-star and an n-chain, and we determine the number of proper mergings of an m-star and an n-chain by counting the number of congruence classes and by determining their cardinalities. Additionally, we compute the number of Galois connections between certain modified Boolean lattices and chains.
Henri Mühle
Modeling Ceteris Paribus Preferences in Formal Concept Analysis
Abstract
We present a context-based semantics for parameterized ceteris paribus preferences over attributes subsets. Such preferences are only required to hold when the alternatives being compared agree on a specified subset of attributes. We show that ceteris paribus preferences valid in a preference context correspond to implications of a special formal context derived from the original preference context. We prove that the problem of checking the semantic consequence relation for parameterized ceteris paribus preferences is coNP-complete. We then discuss the relation between parameterized and classical, i.e., non-parameterized, ceteris paribus preferences, which are only required to hold “all other things being equal”. We show that a non-parameterized preference is a special case of a parameterized preference, while any parameterized preference can be represented by an exponentially large set of non-parameterized preferences.
Sergei Obiedkov
Concept-Forming Operators on Multilattices
Abstract
Adjoint pairs or adjoint triples defined on lattices have proven to be a useful tool when working in fuzzy formal concept analysis. This paper shows that adjoint pairs and triples can play as well an important role within the framework of multilattices, especially in order to form the Galois connections needed to build concept multilattices.
Jesús Medina-Moreno, Manuel Ojeda-Aciego, Jorge Ruiz-Calviño
Using FCA to Analyse How Students Learn to Program
Abstract
In computer science and mathematics education, universities often observe high failure rates among students if they are taught in a traditional, lecture-centric manner. Interactive engagement teaching methods are more successful but in order to develop suitable teaching materials, lecturers must be aware of potential conceptual difficulties of a domain in advance, for example, by analysing the data of student-submitted work from previous sessions. In computer science education, the data collected from computer-based assessment tools provides a possible source for analysing conceptual difficulties students encounter. The data can be analysed with data mining techniques and in particular with FCA as discussed in this paper.
Uta Priss
Soundness and Completeness of Relational Concept Analysis
Abstract
Relational Concept Analysis (rca) is an extension of Formal Concept Analysis (fca) to the processing of relational datasets, i.e., made of (objects × properties) contexts and (objects × objects) relations. rca constructs a set of fixpoint concept lattices by iteratively expanding the lattices of the initial contexts. To that end, at each iteration a scaling mechanism translates the inter-object links into relational attributes that reflect the available conceptual structures. The output of a rca task has so far only been described operationally. We propose here an analytic characterization thereof, i.e., a completeness and consistence result connecting fixpoint extents to particular relational structures in the input data.
Mohamed Rouane-Hacene, Marianne Huchard, Amedeo Napoli, Petko Valtchev
Contextual Uniformities
Abstract
Uniform spaces are topological spaces with additional structure. In this paper, we describe the uniform structure of a uniform space as a multi-context and we discuss some properties of a uniform space in the language of Formal Concept Analysis. In the second part of the paper, we define a uniform structure between two sets, G and M, called contextual uniformity and we discuss some topological aspects related to this construction.
Christian Săcărea

Position Paper

Fitting Pattern Structures to Knowledge Discovery in Big Data
Abstract
Pattern structures, an extension of FCA to data with complex descriptions, propose an alternative to conceptual scaling (binarization) by giving direct way to knowledge discovery in complex data such as logical formulas, graphs, strings, tuples of numerical intervals, etc. Whereas the approach to classification with pattern structures based on preceding generation of classifiers can lead to double exponent complexity, the combination of lazy evaluation with projection approximations of initial data, randomization and parallelization, results in reduction of algorithmic complexity to low degree polynomial, and thus is feasible for big data.
Sergei O. Kuznetsov
Backmatter
Metadaten
Titel
Formal Concept Analysis
herausgegeben von
Peggy Cellier
Felix Distel
Bernhard Ganter
Copyright-Jahr
2013
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-38317-5
Print ISBN
978-3-642-38316-8
DOI
https://doi.org/10.1007/978-3-642-38317-5

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