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Set Systems and Dissimilarities

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Date
2000
Dewey
Probabilités et mathématiques appliquées
Sujet
Cluster Analysis
Journal issue
European Journal of Combinatorics
Volume
21
Number
6
Publication date
08-2000
Article pages
727-743
Publisher
Elsevier
DOI
http://dx.doi.org/10.1006/eujc.1999.0379
URI
https://basepub.dauphine.fr/handle/123456789/1830
Collections
  • CEREMADE : Publications
Metadata
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Author
Bertrand, Patrice
Type
Article accepté pour publication ou publié
Abstract (EN)
Different one–one correspondences exist between classes of indexed clustering structures (such as indexed hierarchies) and classes of dissimilarities (such as ultrametrics). Following the line developed in previous works (i.e., Johnson (1967), Diday (1984, 1986), Bertrand and Diday (1991), Batbedat (1988), Bandelt and Dress (1994)), we associate each pseudo-dissimilarityδ defined on a finite set E with the set system of all maximal linked subsets of E. This provides a one–one correspondence that maps the set of pseudo-dissimilarities onto the collection of all the valued set systems (S, f) that satisfy two conditions. One of these conditions was introduced by Batbedat (1988) and is related to the characterization of 2-conformity in hypergraphs (see also Bandelt and Dress (1994)). We introduce the other condition which requires the index f to be a weak index. Our approach includes a characterization of the valued set systems (S, f) such that S contains (respectively is contained in) the set of all finite nonempty intersections of maximal linked subsets with respect to the pseudo-dissimilarity induced by the pair (S, f).

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