A coherent framework for learning spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data

Juliette Chevallier (1, 2) , Vianney Debavelaere (1) , Stéphanie Allassonnière (3, 4)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 MAASAI - Modèles et algorithmes pour l’intelligence artificielle
3 CRC (UMR_S_1138 / U1138) - Centre de Recherche des Cordeliers
4 HeKA - Health data- and model- driven Knowledge Acquisition

This paper provides a coherent framework for studying longitudinal manifold-valued data. We introduce a Bayesian mixed-effects model which allows estimating both a group-representative piecewise-geodesic trajectory in the Riemannian space of shape and inter-individual variability. We prove the existence of the maximum a posteriori estimate and its asymptotic consistency under reasonable assumptions. Due to the non-linearity of the proposed model, we use a stochastic version of the Expectation-Maximization algorithm to estimate the model parameters. Our simulations show that our model is not noise-sensitive and succeeds in explaining various paths of progression.

Mots clés: Spatio-temporal analysis; EM like algorithm; MCMC methods; Nonlinear mixed-effects model; Longitudinal data; Bayesian estimation; Spatiotemporal analysis

Domaines: Mathématiques [math]; Statistiques [math.ST]; Modélisation et simulation; Applications [stat.AP]; Machine Learning [stat.ML]

Juliette Chevallier, Vianney Debavelaere, Stéphanie Allassonnière. A coherent framework for learning spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data. SIAM Journal on Imaging Sciences, 2021, 14 (1), pp.349-388. ⟨10.1137/20M1328026⟩. ⟨hal-01646298v4⟩