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Methods and algorithms to learn spatio-temporal changes from longitudinal manifold-valued observations

Abstract : We propose a generic Bayesian mixed-effects model to estimate the temporal progression of a biological phenomenon from manifold-valued observations obtained at multiple time points for an individual or group of individuals. The progression is modeled by continuous trajectories in the space of measurements, which is assumed to be a Riemannian manifold. The group-average trajectory is defined by the fixed effects of the model. To define the individual trajectories, we introduced the notion of « parallel variations » of a curve on a Riemannian manifold. For each individual, the individual trajectory is constructed by considering a parallel variation of the average trajectory and reparametrizing this parallel in time. The subject specific spatiotemporal transformations, namely parallel variation and time reparametrization, are defined by the individual random effects and allow to quantify the changes in direction and pace at which the trajectories are followed. The framework of Riemannian geometry allows the model to be used with any kind of measurements with smooth constraints. A stochastic version of the Expectation-Maximization algorithm, the Monte Carlo Markov Chains Stochastic Approximation EM algorithm (MCMC-SAEM), is used to produce produce maximum a posteriori estimates of the parameters. The use of the MCMC-SAEM together with a numerical scheme for the approximation of parallel transport is discussed. In addition to this, the method is validated on synthetic data and in high-dimensional settings. We also provide experimental results obtained on health data.
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Jean-Baptiste Schiratti. Methods and algorithms to learn spatio-temporal changes from longitudinal manifold-valued observations. Statistics [math.ST]. Université Paris Saclay (COmUE), 2017. English. ⟨NNT : 2017SACLX009⟩. ⟨tel-01512319⟩

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