Weak Hierarchies: A Central Clustering Structure

Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 92)

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.

Keywords

Weak hierarchy Quasi-ultrametric 2-Ball Weak cluster Formal concept 

References

  1. 1.
    Bandelt, H.-J.: Four point characterization of the dissimilarity functions obtained from indexed closed weak hierarchies. In: Mathematisches Seminar. Universität Hamburg, Germany (1992)Google Scholar
  2. 2.
    Bandelt, H.-J., Dress, A.W.M.: Weak hierarchies associated with similarity measures: an additive clustering technique. Bull. Math. Biol. 51, 113–166 (1989)MathSciNetGoogle Scholar
  3. 3.
    Barbut, M., Monjardet, B.: Ordre et classification. Hachette, Paris (1970)MATHGoogle Scholar
  4. 4.
    Batbedat, A.: Les dissimilarités médas ou arbas. Statistique et Analyse des Données 14, 1–18 (1988)MathSciNetGoogle Scholar
  5. 5.
    Batbedat, A.: Les isomorphismes H T S et H T E (après la bijection de Benzécri/Johnson) (première partie). Metron 46, 47–59 (1988)MATHMathSciNetGoogle Scholar
  6. 6.
    Benzécri, J.-P.: L’Analyse des données: la Taxinomie. Dunod, Paris (1973)MATHGoogle Scholar
  7. 7.
    Bertrand, P.: Set systems and dissimilarities. Eur. J. Comb. 21, 727–743 (2000)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    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)MATHGoogle Scholar
  10. 10.
    Bertrand, P., Janowitz, M.F.: Pyramids and weak hierarchies in the ordinal model for clustering. Discrete Appl. Math. 122, 55–81 (2002)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Birkhoff, G.: Lattice Theory. Colloquium Publications, vol. XXV, 3rd edn. American Mathematical Society, Providence (1967)Google Scholar
  12. 12.
    Brito, P.: Order structure of symbolic assertion objects. IEEE Trans. Knowl. Data Eng. 6(5), 830–835 (1994)CrossRefGoogle Scholar
  13. 13.
    Brucker, F.: Sub-dominant theory in numerical taxonomy. Discrete Appl. Math. 154, 1085–1099 (2006)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    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)MATHGoogle Scholar
  15. 15.
    Diatta, J.: A relation between the theory of formal concepts and multiway clustering. Pattern Recognit. Lett. 25, 1183–1189 (2004)CrossRefGoogle Scholar
  16. 16.
    Diatta, J.: Description-meet compatible multiway dissimilarities. Discrete Appl. Math. 154, 493–507 (2006)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    Diatta, J., Fichet, B.: Quasi-ultrametrics and their 2-ball hypergraphs. Discrete Math. 192, 87–102 (1998)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Diatta, J., Ralambondrainy, H.: The conceptual weak hierarchy associated with a dissimilarity measure. Math. Soc. Sci. 44, 301–319 (2002)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Diday, E.: Une représentation visuelle des classes empiétantes: les pyramides. Tech. Rep. 291, INRIA, France (1984)Google Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    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)Google Scholar
  23. 23.
    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)Google Scholar
  24. 24.
    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)Google Scholar
  25. 25.
    Johnson, S.C.: Hierarchical clustering schemes. Psychometrika 32, 241–254 (1967)CrossRefGoogle Scholar
  26. 26.
    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)Google Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    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)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.CEREMADE, Université Paris DauphineParisFrance
  2. 2.LIM-EA2525, Université de la RéunionSaint-DenisFrance

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