Abstract
The k-weak hierarchies, for k ≥ 2, are the cluster collections such that the intersection of any (k + 1) members equals the intersection of some k of them. Any cluster collection turns out to be a k-weak hierarchy for some integer k. Weak hierarchies play a central role in cluster analysis in several aspects: they are defined as the 2-weak hierarchies, so that they not only extend directly the well-known hierarchical structure, but they are also characterized by the rank of their closure operator which is at most 2. The main aim of this chapter is to present, in a unique framework, two distinct weak hierarchical clustering approaches. The first one is based on the idea that, since clusters must be isolated, it is natural to determine them as weak clusters defined by a positive weak isolation index. The second one determines the weak subdominant quasi-ultrametric of a given dissimilarity, and thus an optimal closed weak hierarchy by means of the bijection between quasi-ultrametrics and (indexed) closed weak hierarchies. Furthermore, we highlight the relationship between weak hierarchical clustering and formal concepts analysis, through which concept extents appear to be weak clusters of some multiway dissimilarity functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The hierarchical structure is a highly versatile structure, as attested by hierarchies in cluster analysis, ontologies in knowledge representation, decision trees in supervised classification and by tree-based data structures such as PQ-trees.
References
Bandelt, H.-J.: Four point characterization of the dissimilarity functions obtained from indexed closed weak hierarchies. In: Mathematisches Seminar. Universität Hamburg, Germany (1992)
Bandelt, H.-J., Dress, A.W.M.: Weak hierarchies associated with similarity measures: an additive clustering technique. Bull. Math. Biol. 51, 113–166 (1989)
Barbut, M., Monjardet, B.: Ordre et classification. Hachette, Paris (1970)
Batbedat, A.: Les dissimilarités médas ou arbas. Statistique et Analyse des Données 14, 1–18 (1988)
Batbedat, A.: Les isomorphismes H T S et H T E (après la bijection de Benzécri/Johnson) (première partie). Metron 46, 47–59 (1988)
Benzécri, J.-P.: L’Analyse des données: la Taxinomie. Dunod, Paris (1973)
Bertrand, P.: Set systems and dissimilarities. Eur. J. Comb. 21, 727–743 (2000)
Bertrand, P., Brucker, F.: On lower-maximal paired-ultrametrics. In: Brito, P., Bertrand, P., Cucumel, G., Carvalho, F.D. (eds.) Selected Contributions in Data Analysis and Classification, pp. 455–464. Springer, Berlin (2007)
Bertrand, P., Diday, E.: A visual representation of the compatibility between an order and a dissimilarity index: the pyramids. Comput. Stat. Q. 2, 31–42 (1985)
Bertrand, P., Janowitz, M.F.: Pyramids and weak hierarchies in the ordinal model for clustering. Discrete Appl. Math. 122, 55–81 (2002)
Birkhoff, G.: Lattice Theory. Colloquium Publications, vol. XXV, 3rd edn. American Mathematical Society, Providence (1967)
Brito, P.: Order structure of symbolic assertion objects. IEEE Trans. Knowl. Data Eng. 6(5), 830–835 (1994)
Brucker, F.: Sub-dominant theory in numerical taxonomy. Discrete Appl. Math. 154, 1085–1099 (2006)
Daniel-Vatonne, M.-C., Higuera, C.D.L.: Les termes: un modèle algébrique de repésentation et de structuration de données symboliques. Math. Inf. Sci. Hum. 122, 41–63 (1993)
Diatta, J.: A relation between the theory of formal concepts and multiway clustering. Pattern Recognit. Lett. 25, 1183–1189 (2004)
Diatta, J.: Description-meet compatible multiway dissimilarities. Discrete Appl. Math. 154, 493–507 (2006)
Diatta, J., Fichet, B.: From Apresjan hierarchies and Bandelt-Dress weak hierarchies to quasi-hierarchies. In: Diday, E., Lechevalier, Y., Schader, M., Bertrand, P., Burtschy, B. (eds.) New Approaches in Classification and Data Analysis, pp. 111–118. Springer, Berlin (1994)
Diatta, J., Fichet, B.: Quasi-ultrametrics and their 2-ball hypergraphs. Discrete Math. 192, 87–102 (1998)
Diatta, J., Ralambondrainy, H.: The conceptual weak hierarchy associated with a dissimilarity measure. Math. Soc. Sci. 44, 301–319 (2002)
Diday, E.: Une représentation visuelle des classes empiétantes: les pyramides. Tech. Rep. 291, INRIA, France (1984)
Domenach, F., Leclerc, B.: On the roles of Galois connections in classification. In: Schwaiger, O.O.M. (ed.) Explanatory Data Analysis in Empirical Research, pp. 31–40. Springer, Berlin (2002)
Durand, C., Fichet, B.: One-t-one correspondences in pyramidal representation: a unified approach. In: Bock, H.H. (ed.) Classification and Related Methods of Data Analysis, pp. 85–90. North-Holland, Amsterdam (1988)
Fichet, B.: Data analysis: geometric and algebraic structures. In: Prohorov, Y.A., Sazonov, V.V. (eds.) Proceedings of the First World Congress of the Bernoulli Society (Tachkent, 1986), vol. 2, pp. 123–132. V.N.U. Science Press, Utrecht (1987)
Ganter, B., Kuznetsov, S.O.: Pattern structures and their projections. In: Conceptual Structures: Broadening the Base. Lecture Notes in Computer Science, vol. 2120, pp. 129–142. Springer, Berlin (2001)
Johnson, S.C.: Hierarchical clustering schemes. Psychometrika 32, 241–254 (1967)
Mirkin, B., Muchnik, I.: Combinatorial optimization in clustering. In: Du, D.-Z., Pardalos, P. (eds.) Handbook of Combinatorial Optimization, vol. 2, pp. 261–329. Kluwer Academic, Dordrecht (1998)
Polaillon, G.: Interpretation and reduction of Galois lattices of complex data. In: Rizzi, A., Vichi, M., Bock, H.-H. (eds.) Advances in Data Science and Classification, pp. 433–440. Springer, Berlin (1998)
Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, pp. 445–470. Reidel, Dordrecht/Boston (1982)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Bertrand, P., Diatta, J. (2014). Weak Hierarchies: A Central Clustering Structure. In: Aleskerov, F., Goldengorin, B., Pardalos, P. (eds) Clusters, Orders, and Trees: Methods and Applications. Springer Optimization and Its Applications, vol 92. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0742-7_14
Download citation
DOI: https://doi.org/10.1007/978-1-4939-0742-7_14
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-0741-0
Online ISBN: 978-1-4939-0742-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)