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A graph-based estimator of the number of clusters

Published online by Cambridge University Press:  19 June 2007

Gérard Biau ,
Benoît Cadre  and
Bruno Pelletier
Gérard Biau
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier, UMR CNRS 5149, Équipe de Probabilités et Statistique, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France; biau@math.univ-montp2.fr; cadre@math.univ-montp2.fr; pelletier@math.univ-montp2.fr
Benoît Cadre
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier, UMR CNRS 5149, Équipe de Probabilités et Statistique, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France; biau@math.univ-montp2.fr; cadre@math.univ-montp2.fr; pelletier@math.univ-montp2.fr
Bruno Pelletier
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier, UMR CNRS 5149, Équipe de Probabilités et Statistique, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France; biau@math.univ-montp2.fr; cadre@math.univ-montp2.fr; pelletier@math.univ-montp2.fr
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Abstract

Assessing the number of clusters of a statistical population is one of the essential issues of unsupervised learning. Given n independent observations X1,...,Xn drawn from an unknown multivariate probability density f, we propose a new approach to estimate the number of connected components, or clusters, of the t-level set $\mathcal L(t)=\{x:f(x) \geq t\}$. The basic idea is to form a rough skeleton of the set $\mathcal L(t)$ using any preliminary estimator of f, and to count the number of connected components of the resulting graph. Under mild analytic conditions on f, and using tools from differential geometry, we establish the consistency of our method.

Keywords

Cluster analysisconnected componentlevel setgraphtubular neighborhood.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 272 - 280
Copyright
© EDP Sciences, SMAI, 2007

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