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OrderTheoretical Ranking
 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCES (JASIS
, 2000
"... Current bestmatch ranking (BMR) systems perform well but cannot handle word mismatch between a query and a document. The best known alternative ranking method, hierarchical clusteringbased ranking (HCR), seems to be more robust than BMR with respect to this problem, but it is hampered by theoretic ..."
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Current bestmatch ranking (BMR) systems perform well but cannot handle word mismatch between a query and a document. The best known alternative ranking method, hierarchical clusteringbased ranking (HCR), seems to be more robust than BMR with respect to this problem, but it is hampered by theoretical and practical limitations. We present an approach to document ranking that explicitly addresses the word mismatch problem by exploiting interdocument similarity information in a novel way. Document ranking is seen as a querydocument transformation driven by a conceptual representation of the whole document collection, into which the query is merged. Our approach is based on the theory of concept (or Galois) lattices, which, we argue, provides a powerful, wellfounded, and computationallytractable framework to model the space in which documents and query are represented and to compute such a transformation. We compared information retrieval using concept latticebased ranking (CLR) to BMR and HCR. The results showed that HCR was outperformed by CLR as well as by BMR, and suggested that, of the two best methods, BMR achieved better performance than CLR on the whole document set while CLR compared more favorably when only the first retrieved documents were used for evaluation. We also evaluated the three methods' specific ability to rank documents that did not match the query, in which case the superiority of CLR over BMR and HCR (and that of HCR over BMR) was apparent.
Inferring Dependencies from Relations: A Conceptual Clustering Approach
 COMPUTATIONAL INTELLIGENCE
, 1999
"... In this paper we consider two related types of data dependencies that can hold in a relation: conjunctive implication rules between attributevalue pairs, and functional dependencies. We present a conceptual clustering approach that can be used, with some small modifications, for inferring a cover f ..."
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In this paper we consider two related types of data dependencies that can hold in a relation: conjunctive implication rules between attributevalue pairs, and functional dependencies. We present a conceptual clustering approach that can be used, with some small modifications, for inferring a cover for both types of dependencies. The approach consists of two steps. First, a particular clustered representation of the relation, called concept (or Galois) lattice is built; then, a cover is extracted from the lattice built in the earlier step. The main emphasis of this paper is on the second step. We study the computational complexity of the proposed approach and present an experimental comparison with other methods that confirms its validity. The results of the experiments show that our algorithm for extracting implication rules from concept lattices clearly outperforms an earlier algorithm, and suggest that the overall latticebased approach to inferring functional dependencies from relations can be seen as an alternative to traditional methods.
General approach to triadic concept analysis
"... Abstract. Triadic concept analysis (TCA) is an extension of formal concept analysis (dyadic case) which takes into account modi (e.g. time instances, conditions, etc.) in addition to objects and attributes. Thus instead of 2dimensional binary tables TCA concerns with 3dimensional binary tables. In ..."
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Abstract. Triadic concept analysis (TCA) is an extension of formal concept analysis (dyadic case) which takes into account modi (e.g. time instances, conditions, etc.) in addition to objects and attributes. Thus instead of 2dimensional binary tables TCA concerns with 3dimensional binary tables. In our previous work we generalized TCA to work with grades instead of binary data; in the present paper we study TCA in even more general way. In order to cover up an analogy of isotone conceptforming operators (known from dyadic case in fuzzy setting) we developed an unifying framework in which both kinds of conceptforming operators are particular cases of more general operators. We describe the unifying framework, properties of the general conceptforming operators, show their relationship to those we used in our previous work. 1
Triadic Factor Analysis
"... Abstract. This article is an extension of work which suggests using formal concepts as optimal factors of Factor Analysis. They discussed a method for decomposing a p × q binary matrix W into the Boolean matrix product P ◦ Q of a p × n binary matrix P and a n × q binary matrix Q with n as small as p ..."
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Abstract. This article is an extension of work which suggests using formal concepts as optimal factors of Factor Analysis. They discussed a method for decomposing a p × q binary matrix W into the Boolean matrix product P ◦ Q of a p × n binary matrix P and a n × q binary matrix Q with n as small as possible. We have generalised this factorization problem to the triadic case, looking for a decomposition of a p × q × r Boolean 3dmatrix B into the Boolean 3dmatrix product P ◦ Q ◦ R for p × n, q × n and r × n binary matrices P, Q and R with n as small as possible. The motivation is given by the increasing interest in Triadic Concept Analysis due to Web 2.0 applications.
DOI 10.1007/s000120062004y c○Birkhäuser Verlag, Basel, 2006 Algebra Universalis nclosure systems and nclosure operators
, 2005
"... This paper is dedicated to Walter Taylor. Abstract. It is very well known and permeating the whole of mathematics that a closure operator on a given set gives rise to a closure system, whose constituent sets form a complete lattice under inclusion, and viceversa. Recent work of Wille on triadic con ..."
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This paper is dedicated to Walter Taylor. Abstract. It is very well known and permeating the whole of mathematics that a closure operator on a given set gives rise to a closure system, whose constituent sets form a complete lattice under inclusion, and viceversa. Recent work of Wille on triadic concept analysis and subsequent work by the author on polyadic concept analysis led to the introduction of complete trilattices and complete nlattices, respectively, that generalize complete lattices and capture the ordertheoretic structure of the collection of concepts associated with polyadic formal contexts. In the present paper, polyadic closure operators and polyadic closure systems are introduced and they are shown to be in a relationship similar to the one that exists between ordinary (dyadic) closure operators and ordinary (dyadic) closure systems. Finally, the algebraic case is given some special consideration. 1. Background: Closure operators and nordered Sets This section contains a brief account of the wellknown correspondence between closure operators and closure systems and of the notion of an nordered set. Our main source for the former is [3] (see also [2]) and for the latter [10] (see also [9] for the triadic case). Given a set X, a family L of subsets of X, such that
© 2002 Kluwer Academic Publishers. Printed in the Netherlands. Polyadic Concept Analysis ⋆
, 2002
"... Abstract. The framework and the basic results of Wille on triadic concept analysis, including his Basic Theorem of Triadic Concept Analysis, are here generalized to ndimensional formal contexts. ..."
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Abstract. The framework and the basic results of Wille on triadic concept analysis, including his Basic Theorem of Triadic Concept Analysis, are here generalized to ndimensional formal contexts.
Issues in the Efficient use of a Relation Ontology for Conceptual Reasoning
"... In this paper we extend the framework of combinatorial and conceptual worlds to allow arbitrary relations. Conceptual worlds of properties provide a framework for knowledge representation, reasoning, and in particular for conceptual belief revision. However, it is desirable to consider such worlds w ..."
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In this paper we extend the framework of combinatorial and conceptual worlds to allow arbitrary relations. Conceptual worlds of properties provide a framework for knowledge representation, reasoning, and in particular for conceptual belief revision. However, it is desirable to consider such worlds where relations are not restricted to properties. Firstly, we discuss foundations of the conceptual world framework such as the notion of a combinatorial world, mappings which allow us to build conceptual worlds out of combinatorial worlds, concepts and conceptual worlds. Then we present a sketch of a relation ontology and comment on its relationship to the framework. We also present initial steps for generalizing the framework to the case of unrestricted relations. Finally, we examine efficiency issues. We demonstrate that conceptual worlds are Boolean lattices and thus can be stored and retrieved efficiently. Moreover, proposed methods of computing entailment and accessibility relations on ...
Discovering Shared Conceptualizations in Folksonomies
"... Abstract: Social bookmark tools are rapidly emerging on the Web. In such systems users are setting up lightweight conceptual structures called folksonomies. Unlike ontologies, shared conceptualisations are not formalised, but rather implicit. We present a new data mining task, the \emph{mining of al ..."
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Abstract: Social bookmark tools are rapidly emerging on the Web. In such systems users are setting up lightweight conceptual structures called folksonomies. Unlike ontologies, shared conceptualisations are not formalised, but rather implicit. We present a new data mining task, the \emph{mining of all frequent triconcepts}, together with an efficient algorithm, for discovering these implicit shared conceptualisations. Our approach extends the data mining task of discovering all closed itemsets to threedimensional data structures to allow for mining folksonomies. We provide a formal definition of the problem, and present an efficient algorithm for its solution. Finally, we show the applicability of our approach on three large realworld examples. LaTeX manuscript
MINING TRIADIC ASSOCIATION RULES
"... The objective of this research is to extract triadic association rules from a triadic formal context K: = (K1, K2, K3, Y) where K1, K2 and K3 respectively represent the sets of objects, properties (or attributes) and conditions while Y is a ternary relation between these sets. Our approach consists ..."
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The objective of this research is to extract triadic association rules from a triadic formal context K: = (K1, K2, K3, Y) where K1, K2 and K3 respectively represent the sets of objects, properties (or attributes) and conditions while Y is a ternary relation between these sets. Our approach consists to define a procedure to map a set of dyadic association rules into a set of triadic ones. The advantage of the triadic rules compared to the dyadic ones is that they are less numerous and more compact than the second ones and convey a richer semantics of data. Our approach is illustrated through an example of ternary relation representing a set of Customers who purchase theirProducts from Suppliers. The algorithms and approach proposed have been validated with experimentations on large real datasets.
FuzzyValued Triadic Implications
"... Abstract. We present a new approach for handling fuzzy triadic data in the setting of Formal Concept Analysis. The starting point is a fuzzyvalued triadic context (K1, K2, K3, Y), where K1, K2 and K3 are sets and Y is a ternary fuzzy relation between these sets. First, we generalise the methods of T ..."
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Abstract. We present a new approach for handling fuzzy triadic data in the setting of Formal Concept Analysis. The starting point is a fuzzyvalued triadic context (K1, K2, K3, Y), where K1, K2 and K3 are sets and Y is a ternary fuzzy relation between these sets. First, we generalise the methods of Triadic Concept Analysis to our setting and show how they fit other approaches to Fuzzy Triadic Concept Analysis. Afterwards, we develop the fuzzyvalued triadic implications as counterparts of the various triadic implications studied in the literature. These are of major importance for the integrity of Fuzzy and FuzzyValued Triadic Concept Analysis.