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SS13 - Analyse statistique des données fonctionnelles / SS13 - Statistical Analysis of Functional Data
Org: J. Ramsay (McGill) et/and H. Cardot (INRA Castanet-Tolosan)
- C. ABRAHAM, Agro Montpellier, 2 place P. Viala, 34060 Montpellier Cedex 1
Classification of curves: the choice of the metric
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We emphasize the importance of the choice of the metric in curves
classification with a theoretical example. It is shown that the
consistency of the moving window rule can depend strongly on the
metric. Sufficient conditions on the metric are given to ensure
consistency for this rule.
In the second part of the talk, we provide a Bayesian rule for which
the metric is automatically fitted by the data. This rule is derived
from the predictive classification method. The curves are modeled by
Gaussian processes.
- A. BOUDOU, Université Paul Sabatier, Toulouse
ACP du transformé multiplicatif d'un processus stationnaire
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Étant donnés deux processus stationnaires indépendants Xn
et Yn , avec n entier relatif, respectivement multi et
unidimensionnel, il est facile de constater que le processus
transformé multiplicatif simple Tn = Yn Xn (resp.tensoriel Un = Yn ÄXn ) est aussi stationnaire.
On peut ainsi modéliser une transformation multiplicative simple ou
tensorielle de la série p-dimensionnelle Xn par le
facteur Yn . Par exemple, dans le cas simple, ce facteur peut
être un changement d'unité de mesure entre Xn et Tn, une perturbation de Xn ou encore une inconnue dans
l'équation Tn = Yn Xn .
Nous nous proposons d'étudier ici les rapports pouvant exister entre
les ACP dans le domaine des fréquences de Xn et de Tn.
Nous commençons par exprimer la mesure aléatoire associée au
processus Tn en fonction des mesures aléatoires
respectivement associées à Xn et à Yn : à
une isométrie près, il s'agit du produit de convolution de deux
mesures aléatoires tel qu'il a été défini récemment par les
auteurs.
Ensuite, nous obtenons le résultat stipulant que la mesure (resp.la densité) spectrale du transformé Tn est le produit de
convolution des mesures (resp. des densités) spectrales respectives
de Xn et Yn .
Ainsi, alors que les ACP classiques de Xn et Tn sont
quasi-identiques, les résultats précédents montrent que les ACP
dans le domaines des fréquences de Xn et Tn sont
totalement différentes et leur lien est explicité par les formules
de convolution.
Enfin, on peut montrer que ces propriétés peuvent être
étendues, d'une part à l'étude de Xn et Un et,
d'autre part, au cas où l'ensemble indiciel est un groupe abélien
localement compact.
Travail en collaboration avec Yves Romain, Univ. P. Sabatier,
Toulouse.
- SOPHIE DABO-NIANG, Laboratoire de statistique du CREST, timbre J340, 3 avenue
Pierre Larousse, 92245 Malakoff Cedex, France
Kernel regression estimation in Banach space: application to
genetic data
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We study a nonparametric regression kernel estimate when the response
variable is in a Banach space and the explanatory variable takes
values in a semi-metric space. The present framework includes the
classical finite dimensional case, but also spaces of infinite
dimensions, particularly functional spaces. This problem was widely
studied, when the variables take values in finite dimensional spaces,
and there are many references on this topic, contrary to the infinite
dimensional observation case.
Many phenomena, in various areas (e.g. weather, medicine, ...), are
continuous in time and may or must be represented by curves.
Recently, the statistics of the functional data have met a growing
interest.
We establish some asymptotic results and give upper bounds of the
p-mean and (pointwise and integrated) almost sure estimation errors,
under general conditions. As an example, we study the case when the
explanatory variable is the Wiener process. We end by an application
to genetic data.
- FREDERIC FERRATY, Université Toulouse Le Mirail, Département Math. &
Info., 5 allée Machado, 31058 Toulouse Cedex 1
Nonparametric Methods for Functional Data: Regression,
Discrimination and Classification
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Functional aspects have become more and more popular in modern
statistics, so much so that the designation of Functional
Statistics or Functional Data emerged recently. Typically,
Functional Data occur as soon as we observe one curve per subject (or
unit). In the practice, many objects can be viewed as functional data
(spectrometric curves, vocals registration, time series,
spatio-temporal processes, ...).
We propose here to present some recent nonparametric statistical
methods taking into account the functional features of such data. The
starting point is the development of a nonparametric regression method
when we consider a scalar response and a functional explanatory
variable. Replacing the scalar response with a categorical variable,
we derive a curve discrimination method.
On the other hand, recent advances allow us to achieve density
estimation of a functional random variable. One of the main interest
of such a method is the possibility to define and estimate modal
curves, median curves, ... . These notions have been used to build
an unsupervised classification method for functional data.
Finally, practical datasets concerning food industry, vocal
recognization or satellite wave altimeter forms will illustrate our
purpose.
- ALDO GOIA, Università del Piemonte Orientale "A. Avogadro", Novara,
Italy
A Functional Nonparametric Regression Model for Time Series
Prediction
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The purpose is to illustrate a nonparametric model for time series
prediction: the originality of this model consists in using a
continuous set of past values as a predictor. This time series problem
is presented in the general framework of regression estimation from
dependent samples with regressor valued in some in finite dimensional
semi-normed vectorial space. Under suitable a-mixing
conditions, we give asymptotics for a kernel type nonparametric
predictor, linking the rates of convergence with the fractal dimension
of the functional process.
- A. MAS, Université Montpellier, 2 place Eugène Bataillon, 34095
Montpellier Cedex 5
Confidence sets and prediction in the ARH(1) model
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The Hilbertian autoregressive model (ARH(1) for short) generalizes the
classical AR(1) model to functional data. We focus on a class of
predictors. This class is determined by the pseudo inversion method
used to estimate the autocorrelation operator. We are led to solving
a problem both connected with selection of dimension and
ill-posedness. We finally prove a CLT for the statistical predictor
with a non standard normalization.
- CRISTIAN PREDA, Faculté de Médecine, Université de Lille 2, France
Regression models for functional data by reproducing kernel
Hilbert space methods
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Non-linear regression models based on RKHS approach are presented in
the context of functional data. The results of an application on
medical data are compared with those given by the linear models.
- JIM RAMSAY, McGill
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