HAL : hal-00722668, version 1
 Lasso and probabilistic inequalities for multivariate point processes
 (02/08/2012)
 Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive $\ell_1$-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Non-asymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activities inference, we finally lead a simulation study for multivariate Hawkes processes and compare our methodology with the {\it adaptive Lasso procedure} proposed by Zou in \cite{Zou}. We observe an excellent behavior of our procedure with respect to the problem of supports recovery. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of \cite{Zou}, our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.
 1 : Department of Mathematical Sciences University of Copenhagen 2 : Laboratoire Jean Alexandre Dieudonné (JAD) CNRS : UMR6621 – Université Nice Sophia Antipolis [UNS] 3 : CEntre de REcherches en MAthématiques de la DEcision (CEREMADE) CNRS : UMR7534 – Université Paris IX - Paris Dauphine
 Domaine : Mathématiques/StatistiquesStatistiques/Théorie
 Mots Clés : Multivariate counting process – Hawkes processes – adaptive estimation – Lasso procedure – Bernstein-type inequalities
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 hal-00722668, version 1 http://hal.archives-ouvertes.fr/hal-00722668 oai:hal.archives-ouvertes.fr:hal-00722668 Contributeur : Vincent Rivoirard <> Soumis le : Vendredi 3 Août 2012, 08:07:42 Dernière modification le : Vendredi 3 Août 2012, 08:12:04