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Constructing Hierarchical Set Systems
Claudine Devauchelle1, AndreasW.M. Dress2, Alexander Grossmann1, Stefan Grünewald3 and Alain Henaut1
1Laboratoire Génome et Informatique, Tour Evry2, 523 Place des Terrasses, 91034 Evry Cedex France
devauchelle@genopole.cnrs.fr, alain.henaut@wanadoo.fr
2 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22-26, 04103 Leipzig Germany
dress@mathematik.uni-bielefeld.de
3 Linnaeus Centre for Bioinformatics, BMC, Box 598, SE 75124 Uppsala, Sweden
Stefan.Grunewald@lcb.uu.se
Annals of Combinatorics 8 (4) p.441-456 December, 2004
AMS Subject Classification: 05C90, 92B10, 92D15, 62G30, 62-07
Abstract:
In this note, it is shown that by applying ranking procedures to data that allow, for any three objects a1, a2, b in a collection X of objects of interest, to make consistent decisions about which of the two objects a1 or a2 is more similar to b, a family of cluster systems A(k)(k = 0, 1, . . .) can be constructed that start with the associated Apresjan Hierarchy and keep growing until, for k = #X - 1, the full set P(X) of all subsets of X is reached. Various ideas regarding canonical modifications of the similarity values so that these cluster systems contain as many clusters as possible for small values of k (and in particular for k := 0 ) and/or are rooted at a specific element in X, possible applications, e.g. concerning (i) the comparison of distinct dissimilarity data defined on the same set X or (ii) diversity optimization, and new tasks arising in ranking statistics are also discussed.
Keywords: set systems, hierarchies, Apresjan hierarchies, similarity, dissimilarity, Farris transform, ranking procedures, ranking statistics, diversity optimization

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