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The generation of random ultrametric matrices representing dendrograms

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Abstract

Many methods and algorithms to generate random trees of many kinds have been proposed in the literature. No procedure exists however for the generation of dendrograms with randomized fusion levels. Randomized dendrograms can be obtained by randomizing the associated cophenetic matrix. Two algorithms are described. The first one generates completely random dendrograms, i.e., trees with a random topology, random fusion level values, and random assignment of the labels. The second algorithm uses a double-permutation procedure to randomize a given dendrogram; it proceeds by randomization of the fixed fusion levels, instead of using random fusion level values. A proof is presented that the double-permutation procedure is a Uniform Random Generation Algorithmsensu Furnas (1984), and a complete example is given.

Résumé

On retrouve dans la littérature plusieurs méthodes et algorithmes destinés à générer des arbres aléatoires de toutes sortes. Il n'existe cependant aucune procédure permettant la génération de dendrogrammes comportant des niveaux de fusion aléatoires. De tels dendrogrammes peuvent être obtenus à partir des matrices cophénétiques associées. Nous décrivons deux algorithmes pour ce faire. Le premier permet de générer des dendrogrammes complètement aléatoires, c'est-à-dire des arbres possédant une topologie aléatoire, des niveaux de fusion aléatoires ainsi que des feuilles étiquetées de façon aléatoire. Le deuxième algo- rithme utilise une procédure à double permutation afin de randomiser un dendrogramme donné; on proc de dans ce cas à la permutation des véritables niveaux de fusion au lieu de génèrer des niveaux aléatoires. Nous présentons la preuve démontrant que la procédure à double permutation représente un Algorithme de Génération Aléatoire Uniformesensu Furnas (1984). Un exemple complet est également fourni.

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Authors and Affiliations

  1. Département de Sciences biologiques, Université de Montréal, Succursale A, C.P. 6128, H3C 3J7, Montréal, Québec, Canada

    François-Joseph Lapointe & Pierre Legendre

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  1. François-Joseph Lapointe
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  2. Pierre Legendre
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Additional information

This work was supported by NSERC Grant No. A7738 to P. Legendre and by a NSERC scholarship to F.-J. Lapointe.

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Lapointe, FJ., Legendre, P. The generation of random ultrametric matrices representing dendrograms. Journal of Classification 8, 177–200 (1991). https://doi.org/10.1007/BF02616238

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