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Modèle de régression pour des données non-Euclidiennes en grande dimension. Application à la classification de taxons en anatomie computationnelle.

Abstract : In this thesis, we study a regression model with distribution entries and the testing hypothesis problem for signal detection in a regression model. We aim to apply these models in hearing sensitivity measured by the transient evoked otoacoustic emissions (TEOAEs) data to improve our knowledge in the auditory investigation. In the first part, a new distribution regression model for probability distributions is introduced. This model is based on a Reproducing Kernel Hilbert Space (RKHS) regression framework, where universal kernels are built using Wasserstein distances for distributions belonging to \Omega) and \Omega is a compact subspace of the real space. We prove the universal kernel property of such kernels and use this setting to perform regressions on functions. Different regression models are first compared with the proposed one on simulated functional data. We then apply our regression model to transient evoked otoascoutic emission (TEOAE) distribution responses and real predictors of the age. This part is a joint work with Loubes, J-M., Risser, L. and Balaresque, P..In the second part, considering a regression model, we address the question of testing the nullity of the regression function. The testing procedure is available when the variance of the observations is unknown and does not depend on any prior information on the alternative. We first propose a single testing procedure based on a general symmetric kernel and an estimation of the variance of the observations. The corresponding critical values are constructed to obtain non asymptotic level \alpha tests. We then introduce an aggregation procedure to avoid the difficult choice of the kernel and of the parameters of the kernel. The multiple tests satisfy non-asymptotic properties and are adaptive in the minimax sense over several classes of regular alternatives.
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Thi Thien Trang Bui. Modèle de régression pour des données non-Euclidiennes en grande dimension. Application à la classification de taxons en anatomie computationnelle.. Mathématiques générales [math.GM]. INSA de Toulouse, 2019. Français. ⟨NNT : 2019ISAT0021⟩. ⟨tel-02497588⟩

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