A Bayesian mixed-effects model to learn trajectories of changes from repeated manifold-valued observations

International audienceWe propose a generic Bayesian mixed-effects model to estimate the temporal progression of a biological phenomenon from observations obtained at multiple time points for a group of individuals. The progression is modeled by continuous trajectories in the space of measurements. Individual trajectories of progression result from spatiotemporal transformations of an average trajectory. These transformations allow to quantify the changes in direction and pace at which the trajectories are followed. The framework of Rieman-nian geometry allows the model to be used with any kind of measurements with smooth constraints. A stochastic version of the Expectation-Maximization algorithm is used to produce produce maximum a posteriori estimates of the parameters. We evaluate our method using series of neuropsychological test scores from patients with mild cognitive impairments later diagnosed with Alzheimer's disease, and simulated evolutions of symmetric positive definite matrices. The data-driven model of the impairment of cognitive functions shows the variability in the ordering and timing of the decline of these functions in the population. We show also that the estimated spatiotemporal transformations effectively put into correspondence significant events in the progression of individuals