# zbMATH — the first resource for mathematics

Inversions en classification hiérarchique: Application à la construction adaptative d’indices d’agregation. (English) Zbl 0538.62052
We say that there is an inversion in a hierarchy if the height of a cluster merged later in the hierarchy is lower than that of clusters merged earlier. We give: 1) Necessary and sufficient conditions on the parameters of the G. C. Lance and W. T. Williams formula (1966), generalized by M. Jambu (1978) which insure the existence or non existence of inversion in a hierarchy; one of those new results generalizes the conditions given by P. Ducimetière (1971), G. Milligan (1979) and V. Batagelj (1981).
2) A necessary and sufficient condition on the same parameters which insures the ”neighbourhood reducibility” property of M. Bruynooghe (1978) and thus permits the use of an accelerated algorithm for hierarchical clustering. By using those results and taking account of the a priori knowledge of the user we show that it is possible to obtain adaptive ultrametrics. This work arises towards learning hierarchical clustering from examples [see the author and J. V. Moreau, Learning hierarchical clustering from examples. (1984)].

##### MSC:
 62H30 Classification and discrimination; cluster analysis (statistical aspects)
Full Text:
##### References:
 [1] V. Batagelj . - Note on ultrametric hierarchical clustering algorithms . Psychometrica , Vol. 46 n^\circ 3 ( 1981 ). MR 655020 [2] E. Diday , J. Lemaire , J. Poujet , F. Testu . - Elements d’analyse des données , Dunod , ( 1982 ). Zbl 0504.62002 · Zbl 0504.62002 [3] P. Dumicetiere . - Les méthodes de classification numérique . Revue de Statistiques Appliquées , Volume 18 , n^\circ 4 , p. 5 - 25 ( 1970 ). Numdam | MR 331655 · numdam:RSA_1970__18_4_5_0 [4] M. Bruynooghe . - Classification ascendante hiérarchique de grands ensembles de données : un algorithme rapide fondé sur la construction des voisinages réductibles , Cahiers d’analyse des Données , Vol. III , n^\circ 1 , ( 1978 ). [5] M. Jambu . - Classification automatique pour l’analyse des données , Dunod . ( 1978 ). MR 634958 | Zbl 0419.62057 · Zbl 0419.62057 [6] G.C. Lance , W.T. Williams . - A general theory of classification sorting , Computer Journal 9.10 and Computer Journal 10.3 ( 1967 ). [7] G. Milligan . - U ltrametric hierarchical clustering algorithms , Psychometrica , 44 , 3 . ( 1979 ). MR 547260 | Zbl 0422.62057 · Zbl 0422.62057 · doi:10.1007/BF02294699
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.