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Statistical models and stochastic algorithms for the analysis of longitudinal Riemanian manifold valued data with multiple dynamic

Abstract : Beyond transversal studies, temporal evolution of phenomena is a field of growing interest. For the purpose of understanding a phenomenon, it appears more suitable to compare the evolution of its markers over time than to do so at a given stage. The follow-up of neurodegenerative disorders is carried out via the monitoring of cognitive scores over time. The same applies for chemotherapy monitoring: rather than tumors aspect or size, oncologists asses that a given treatment is efficient from the moment it results in a decrease of tumor volume. The study of longitudinal data is not restricted to medical applications and proves successful in various fields of application such as computer vision, automatic detection of facial emotions, social sciences, etc.Mixed effects models have proved their efficiency in the study of longitudinal data sets, especially for medical purposes. Recent works (Schiratti et al., 2015, 2017) allowed the study of complex data, such as anatomical data. The underlying idea is to model the temporal progression of a given phenomenon by continuous trajectories in a space of measurements, which is assumed to be a Riemannian manifold. Then, both a group-representative trajectory and inter-individual variability are estimated. However, these works assume an unidirectional dynamic and fail to encompass situations like multiple sclerosis or chemotherapy monitoring. Indeed, such diseases follow a chronic course, with phases of worsening, stabilization and improvement, inducing changes in the global dynamic.The thesis is devoted to the development of methodological tools and algorithms suited for the analysis of longitudinal data arising from phenomena that undergo multiple dynamics and to apply them to chemotherapy monitoring. We propose a nonlinear mixed effects model which allows to estimate a representative piecewise-geodesic trajectory of the global progression and together with spacial and temporal inter-individual variability. Particular attention is paid to estimation of the correlation between the different phases of the evolution. This model provides a generic and coherent framework for studying longitudinal manifold-valued data.Estimation is formulated as a well-defined maximum a posteriori problem which we prove to be consistent under mild assumptions. Numerically, due to the non-linearity of the proposed model, the estimation of the parameters is performed through a stochastic version of the EM algorithm, namely the Markov chain Monte-Carlo stochastic approximation EM (MCMC-SAEM). The convergence of the SAEM algorithm toward local maxima of the observed likelihood has been proved and its numerical efficiency has been demonstrated. However, despite appealing features, the limit position of this algorithm can strongly depend on its starting position. To cope with this issue, we propose a new version of the SAEM in which we do not sample from the exact distribution in the expectation phase of the procedure. We first prove the convergence of this algorithm toward local maxima of the observed likelihood. Then, with the thought of the simulated annealing, we propose an instantiation of this general procedure to favor convergence toward global maxima: the tempering-SAEM.
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Submitted on : Tuesday, November 26, 2019 - 1:35:10 PM
Last modification on : Wednesday, September 2, 2020 - 3:37:28 AM


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Juliette Chevallier. Statistical models and stochastic algorithms for the analysis of longitudinal Riemanian manifold valued data with multiple dynamic. Statistics [math.ST]. Université Paris Saclay (COmUE), 2019. English. ⟨NNT : 2019SACLX059⟩. ⟨tel-02380752⟩



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