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Zbl 0592.62050
Leclerc, B.
La comparaison des hiérarchies: Indices et métriques. (Comparison of hierarchies: Indices and metrics). (French)
[J] Math. Sci. Hum. 92, 5-40 (1985). ISSN 0987-6936

Summary: We are concerned with description indices on hierarchies (or n-trees) on a finite set X. This study follows a previous one, on the properties of these hierarchies, especially on the median semi-lattice structure of their set ${\cal H}\sb X$ [the author, ibid. 89, 5-34 (1985; Zbl 0563.68060)]. \par First, we recall some possible properties (isotony, symmetries, extremal properties, valuation properties) and uses (shape, or consensus, or information indices, metrics induced by valuations) of such indices, and we give a brief survey on indices already presented in the literature (and on some other ones), tentatively classified. \par Then, we tackle axiomatic construction of indices. Linear indices are presented, given by cluster values, weighted by coefficients. Cluster values are related to an internal context, while coefficients depend on positions of clusters in hierarchies. Some conditions on weights are necessary and sufficient in order to have valuations, independently of cluster values. \par We study two families of valuations. The linear indices introduced in the last section are invariant by deletion of irrelevant clusters. They are not valuations, but "significance" indices, and have interesting properties. They may be used, for instance, as criterion in problems like optimal removal of clusters.

MSC 2000:: *62H30 Statistical classification, etc.
05C05 Trees

Keywords: comparison of hierarchies; hierarchical classification; description indices on hierarchies; n-trees; axiomatic construction; cluster values

Citations: Zbl 0563.68060

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