## The $$k$$-weak hierarchical representations: An extension of the indexed closed weak hierarchies.(English)Zbl 1012.62067

Summary: Several approaches have been proposed for the purpose of proving that different classes of dissimilarities (e.g., ultrametrics) can be represented by certain types of stratified clusterings which are easily visualized (e.g., indexed hierarchies). These approaches differ in the choice of the clusters that are used to represent a dissimilarity coefficient. More precisely, the clusters may be defined as the maximal linked subsets, also called $$M_{\text L}$$-sets; equally they may be defined as a particular type of 2-balls.
In this paper, we first introduce the notion of a $$k$$-ball, thereby extending the notion of a 2-ball. For an arbitrary dissimilarity coefficient, we establish some properties of $$k$$-balls that pinpoint the connection between them and the $$M_{\text L}$$-sets. We also introduce the (2,$$k$$)-point condition ($$k\geqslant 1$$) which is an extension of the Bandelt four-point condition [H.-J. Bandelt, Four-point characterization of the dissimilarity functions obtained from indexed closed weak hierarchies. Math. Seminar, Univ. Hamburg/Germany (1992)].
For $$k\geqslant 2$$, we prove that the dissimilarities satisfying the (2,$$k$$)-point condition are in one–one correspondence with a class of stratified clusterings, called $$k$$-weak hierarchical representations, whose main characteristic is that the intersection of ($$k+1$$) arbitrary clusters may be reduced to the intersection of some $$k$$ of these clusters.

### MSC:

 62H30 Classification and discrimination; cluster analysis (statistical aspects) 91C20 Clustering in the social and behavioral sciences
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### References:

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