Classification et mélanges de processus
Comptes Rendus. Mathématique, Tome 335 (2002) no. 2, pp. 189-193.

Nous proposons une méthode de classification basée sur l'estimation de mélanges de lois, le point nouveau étant que les unités statistiques sont décrites par des lois de probabilité. Les composantes du mélange sont des processus de Dirichlet, des processus Gamma pondérés normalisés ou des processus de Kraft utilisés en satististique non paramétrique Bayesienne. Les mélanges obtenus par des algorithmes appliqués aux marginales des composantes en dimension finie convergent vers le mélange souhaité lorsque la dimension augmente car les composantes sont orthogonales grâce à un théorème de Kakutani et leur support sont alors les classes recherchées.

We propose a clustering method based on the estimation of mixtures of probability distributions, the new point being that the statistical units are described by probability distributions. The components of the mixtures are Dirichlet processes, normalized weighted Gamma processes, and Kraft processes. Mixtures obtained by applying some algorithms to the finite dimensional distributions of the components converge to the desired mixture as the dimension increases, since the components are mutually singular due to a theorem of Kakutani. The desired clusters are then the support of these components.

Reçu le :
Révisé le :
Publié le :
DOI : 10.1016/S1631-073X(02)02432-9
Emilion, Richard 1, 2

1 UFR SEGMI, Modalx, Université Paris X, 200, avenue de la République, 92001 Nanterre, France
2 Lise-Ceremade, Université Paris IX–Dauphine, place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
@article{CRMATH_2002__335_2_189_0,
     author = {Emilion, Richard},
     title = {Classification et m\'elanges de processus},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {189--193},
     publisher = {Elsevier},
     volume = {335},
     number = {2},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02432-9},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02432-9/}
}
TY  - JOUR
AU  - Emilion, Richard
TI  - Classification et mélanges de processus
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 189
EP  - 193
VL  - 335
IS  - 2
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02432-9/
DO  - 10.1016/S1631-073X(02)02432-9
LA  - fr
ID  - CRMATH_2002__335_2_189_0
ER  - 
%0 Journal Article
%A Emilion, Richard
%T Classification et mélanges de processus
%J Comptes Rendus. Mathématique
%D 2002
%P 189-193
%V 335
%N 2
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02432-9/
%R 10.1016/S1631-073X(02)02432-9
%G fr
%F CRMATH_2002__335_2_189_0
Emilion, Richard. Classification et mélanges de processus. Comptes Rendus. Mathématique, Tome 335 (2002) no. 2, pp. 189-193. doi : 10.1016/S1631-073X(02)02432-9. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02432-9/

[1] Celeux, G.; Diebolt, J. S.A.E.M. algorithm, Stochastics Stochastics Rep., Volume 41 (1992), pp. 119-134

[2] Darwich, A. About the absolute continuity and orthogonality of two probability measures, Statist. Probab. Lett., Volume 52 (2001) no. 1, pp. 1-8

[3] Diday, E. Analysis of Symbolic Data (Bock; Diday, eds.), Springer-Verlag, 2000

[4] Diday, E.; Schroeder, A. A new approach in mixing distributions detection, RAIRO Operational Research, Volume 10 (1976) no. 6

[5] Dykstra, R.L.; Laud, P. A Bayesian nonparametric approach to reliability, Ann. Statist., Volume 9 (1981), pp. 356-367

[6] Ferguson, T.S. A Bayesian analysis of some nonparametric problems, Ann. Statist., Volume 1 (1981), pp. 209-230

[7] Ferguson, T.S. Prior distributions on spaces of probability measures, Ann. Statist., Volume 2 (1974), pp. 615-629

[8] Hewitt, E.; Stromberg, K. Real and Abstract Analysis, Springer-Verlag, 1969

[9] Kraft, C.H. A class of distribution function processes which have derivatives, J. Appl. Probab., Volume 1 (1964), pp. 385-388

[10] Lo, A.Y. Bayesian nonparametric statistical inference for Poisson point process, Z. Wahrsch. Verw. Gebiete, Volume 59 (1982), pp. 55-66

Cité par Sources :