77 research outputs found
High-dimensional clustering
International audienceHigh-dimensional (HD) data sets are now frequent, mostly motivated by technological reasons which concern automation in variable acquisition, cheaper availability of data storage and more powerful standard computers for quick data management possibility. All fields are impacted by this general phenomenon of variable number inflation, only the definition of ``high'' being domain dependent. In marketing, this number can be of order 10e2, in microarray gene expression between 10e2 and 10e4, in text mining 10e3 or more, of order 10e6 for single nucleotide polymorphism (SNP) data, etc. Note also that sometimes much more variables can be involved, what can be typically the case with discretized curves, for instance curves coming from temporal sequences.Such a technological revolution has a huge impact in other scientific fields, as societal or also mathematical ones. In particular, high-dimensional data management brings some new challenges to statisticians since standard (low-dimensional) data analysis methods struggle to directly apply to the new (high-dimensional) data sets. The reason can be twofold, sometimes linked, involving either combinatorial difficulties or disastrously large estimate variance increase. Data analysis methods are essential for providing a synthetic view of data sets, allowing data summary and data exploratory for future decision making for instance. This need is even more acute in the high-dimensional setting since on the one hand the large number of variables suggests that a lot of information is conveyed by data but, in the other hand, such information may be hidden behind their volume
A sparse variable selection procedure in model-based clustering
Au vu de l'augmentation du nombre de jeux de données de grande dimension, la sélection de variables pour la classification non supervisée est un enjeu important. Dans le cadre de la classification par mélanges gaussiens, nous reformulons le problÚme de sélection de variables en un problÚme général de sélection de modÚle. Dans un premier temps, notre procédure consiste à construire une sous-collection de modÚles grùce à une méthode de régularisation l1. Puis, l'estimateur du maximum de vraisemblance est déterminé via un algorithme EM pour chaque modÚle. Enfin un critÚre pénalisé non asymptotique est proposé pour sélectionner à la fois le nombre de composants du mélange et l'ensemble des variables informatives pour la classification. D'un point de vue théorique, un théorÚme général de sélection de modÚles dans le cadre de l'estimation par maximum de vraisemblance avec une collection aléatoire de modÚles est établi. Il permet en particulier de justifier la forme de la pénalité de notre critÚre, forme qui dépend de la complexité de la collection de modÚles. En pratique, ce critÚre est calibré grùce à la méthode dite de l'heuristique de pente. Cette procédure est illustrée sur deux jeux de données simulées. Finalement, une extension, associée à une modélisation plus générale des variables non informatives pour la classification, est proposée
Pratique de l'heuristique de pente et le package CAPUSHE
National audienceLa mise en oeuvre des méthodes "data-driven" de calibration de critÚres pénalisés, issues de l'heuristique de pente de Birgé et Massart (2007), implique des difficultés pratiques
Multidimensional two-component Gaussian mixtures detection
International audienceLet be a -dimensional i.i.d sample from a distribution with density . The problem of detection of a two-component mixture is considered. Our aim is to decide whether is the density of a standard Gaussian random -vector () against is a two-component mixture: where are unknown parameters. Optimal separation conditions on and the dimension are established, allowing to separate both hypotheses with prescribed errors. Several testing procedures are proposed and two alternative subsets are considered
Slope Heuristics: Overview and Implementation
RR INRIA-7223, Version 1Model selection is a general paradigm which includes many statistical problems. One of the most fruitful and popular approaches to carry it out is the minimization of a penalized criterion. Birgé and Massart (2006) have proposed a promising data-driven method to calibrate such criteria whose penalties are known up to a multiplicative factor: the ``slope heuristics''. Theoretical works validate this heuristic method in some situations and several papers report a promising practical behavior in various frameworks. The purpose of this work is twofold. First, an introduction to the slope heuristics and an overview of the theoretical and practical results about it are presented. Second, we focus on the practical difficulties occurring for applying the slope heuristics. A new practical approach is carried out and compared to the standard dimension jump method. All the practical solutions discussed in this paper in different frameworks are implemented and brought together in a Matlab graphical user interface called capushe
Selective inference after convex clustering with penalization
Classical inference methods notoriously fail when applied to data-driven test
hypotheses or inference targets. Instead, dedicated methodologies are required
to obtain statistical guarantees for these selective inference problems.
Selective inference is particularly relevant post-clustering, typically when
testing a difference in mean between two clusters. In this paper, we address
convex clustering with penalization, by leveraging related selective
inference tools for regression, based on Gaussian vectors conditioned to
polyhedral sets. In the one-dimensional case, we prove a polyhedral
characterization of obtaining given clusters, than enables us to suggest a test
procedure with statistical guarantees. This characterization also allows us to
provide a computationally efficient regularization path algorithm. Then, we
extend the above test procedure and guarantees to multi-dimensional clustering
with penalization, and also to more general multi-dimensional
clusterings that aggregate one-dimensional ones. With various numerical
experiments, we validate our statistical guarantees and we demonstrate the
power of our methods to detect differences in mean between clusters. Our
methods are implemented in the R package poclin.Comment: 40 pages, 8 figure
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