Concept Lattice Construction Research Articles

Interval-Valued Fuzzy Galois Connections: Algebraic Requirements and Concept Lattice Construction

Fuzzy formal concept analysis is concernedwith formal contexts expressing scalar-valued fuzzy relationships between objects and their properties. Existing fuzzy approaches assume that the relationship between given and given property is matter of degree in scale L (generally [0,1]). However, the extent to which object o has property a may be sometimes hard to assess precisely. Then it is convenient to use sub-interval from the scale L rather than precise value. Such formal contexts naturally lead to interval-valued fuzzy formal concepts. The aim of the paper is twofold. We provide sound minimal set of algebraic requirements for interval-valued implications in order to fulfill the fuzzy closure properties of the resulting Galois connection. Secondly, new approach based on generalization of Godel implication is proposed for building the complete lattice of all interval-valued fuzzy formal concepts.

The Theory about CD-CAT Based on FCA and Its Application

Cognitive diagnosis (CD) plays an important role in intelligent tutoring system. Computerized adaptive testing (CAT) is adaptive, fair, and efficient, which is suitable to large-scale examination. Traditional cognitive diagnostic test needs quite large number of items, the efficient and tailored CAT could be a remedy for it, so the CAT with cognitive diagnosis (CD-CAT) is prospective. It is more benefit to the students who live in the developing area without rich source of teaching and distance education is adopted there. Although many researchers pay attention to CD, some flaws exist because it is still in its infancy (Leighton at el.2007) and only a few researches of CD-CAT are gotten. In Tatsuoka’s Q-matrix theory, it is wrong that the rows/columns could form a Boolean lattice. Based on Rule Space Model (RSM) and the Attribute Hierarchy Method (AHM), Formal Concept Analysis (FCA) is applied into CD-CAT. Concept lattices are served as the model of CD. The technology of item bank construction, item selection strategies in CD-CAT and estimation method are considered to design an initiatory and systemic CD-CAT, which diagnoses examinees on-line and offers remedial measure for examinees in time. The algorithms of constructing concept lattice for CAT, diagnosing examinees and offering the best remedial measure to examinees are discussed theoretically in detail. The result of Monte Carlo study shows that examinees’ knowledge states are well diagnosed and the precision in examinees’ abilities estimation is satisfied.

Distributed algorithm for constructing concept lattice based on index

The presented concept lattice parallel/distributed algorithm needs to search plenty of non-related concepts when dealing with a large scale data,which reduces the performance of the algorithm.A distributed concept lattice construction algorithm based on index named LCBI was put forward.When inserting a new concept,it quickly found all the greatest correlative concepts of the new concept using index,then found out cross-sub-concepts of child nodes of all greatest correlative concepts using parallel and top-down search,which decreased the search area.Theoretical analysis and experimental results show that LCBI outperforms the other distributed algorithms when dealing with dense context.

An incremental algorithm to construct a lattice of set intersections

An incremental algorithm to construct a lattice from a collection of sets is derived, refined, analyzed, and related to a similar previously published algorithm for constructing concept lattices. The lattice constructed by the algorithm is the one obtained by closing the collection of sets with respect to set intersection. The analysis explains the empirical efficiency of the related concept lattice construction algorithm that had been observed in previous studies. The derivation highlights the effectiveness of a correctness-by-construction approach to algorithm development.

An algorithm of constructing concept lattices for CAT with cognitive diagnosis

Concept lattices have proved useful in many fields such as machine learning, knowledge discovery, in which building concept lattices is extensively studied. A modification of the Godin’s algorithms is proposed. Considering attribute hierarchies, an algorithm are presented for computerized adaptive testing (CAT) with cognitive diagnosis (CD). Both of algorithms are studied experimentally, and the algorithm for CD–CAT is studied theoretically. Algorithmic complexity of the algorithms for CD–CAT is studied theoretically (time complexity is studied in the best case and the worst case). With eight attributes, the test results are generated for both algorithms. With 10 attributes, quadratic regression is given for CPU time for the algorithm of CD–CAT.

Algorithme de construction d'un treillis des concepts formels et de détermination des générateurs minimaux

The extremely large number of association rules that can be drawn from ―even reasonably sized datasets―, bootstrapped the development of more acute techniques or methods to reduce the size of the reported rule sets. In this context, the battery of results provided by the Formal Concept Analysis (FCA) allowed to define "irreducible" nuclei of association rule subset better known as generic basis. However, a thorough overview of the literature shows that all the algorithms dedicated neglected an essential component: the relation of order, or the extraction of the minimal generators. In this paper, we introduce the GenAll algorithm to build a formal concept lattice, in which each formal concept is "decorated" by its minimal generators. The GenAll algorithm aims to extract generic bases of association rules. The main novelty in this algorithm is the use of refinement process to compute immediate successor lists to simultaneously determine the set of formal concepts, their underlying partial order and the set of minimal generators associated with each formal concept. Carried out experiments showed that the GenAll algorithm is especially efficient for dense extraction contexts compared to that of Nourine et al. Response times obtained from the GenAll algorithm largely outperform those of Nourine et al. Le nombre prohibitif de règles d'association qui peuvent être dérivées même pour des bases de données de taille raisonnable est à l'origine du développement de techniques pour réduire la taille de l'ensemble de ces règles. Dans ce contexte, les résultats obtenus par l'Analyse de Concepts Formels (AFC) a permis de définir un sous-ensemble de règles appelé base générique. Cependant, un survol de la littérature montre que tous les algorithmes qui leur sont dédiés ont négligé une composante essentielle : soit la relation d'ordre sous-jacente ou c'est l'extraction des générateurs minimaux qui manque à l'appel. Dans ce papier, nous proposons un algorithme, appelé GenAll, pour construire un treillis de concepts formels, dans lequel chaque concept formel est "décoré" par ses générateurs minimaux. L'objectif de cet algorithme est d'extraire toute la connaissance nécessaire pour extraire les bases génériques des règles d'association. La principale originalité de GenAll est l'emploi d'un processus de raffinement des listes des successeurs immédiats pour déterminer, d'une manière simultanée, l'ensemble des concepts formels, leur ordre partiel sous-jacent et l'ensemble de générateurs minimaux associés à chacun des concepts formels. Les résultats expérimentaux ont prouvé que l'algorithme proposé est particulièrement efficace pour des contextes d'extraction denses comparé à Nourine et al. Le temps de réponse de l'algorithme GenAll surpasse celui de l'algorithme de Nourine et al.

Subdirect product construction of concept lattices

The idea of constructing with subdirect products was exhibited in [l] and worked out in [2] for complete lattices. The developed construction method was used in [4] to elaborate the fusion of (data) contexts via a specific subdirect product of their concept lattices. In this paper we shall give a direct approach to the (generalized) fusion of contexts and the corresponding subdirect product construction of concept lattices; this approach may conversely be used to obtain the results of [2]. We shall start with a description of the complete sublattices of a concept lattice within its underlying context. This leads to a characterization of contexts corresponding to the complete subdirect products of concept lattices. We are able to construct a subdirect product if we know the images of a generating subset under all projections to its factors. The main result will be that this construction can be imitated in the language of contexts. Finally, the construction method will be demonstrated by two examples.