Learning spatio-temporal trajectories from manifold-valued longitudinal data
Résumé
We propose a Bayesian mixed-effects model to learn typical scenarios of changes
from longitudinal manifold-valued data, namely repeated measurements of the
same objects or individuals at several points in time. The model allows to estimate
a group-average trajectory in the space of measurements. Random variations of
this trajectory result from spatiotemporal transformations, which allow changes in
the direction of the trajectory and in the pace at which trajectories are followed.
The use of the tools of Riemannian geometry allows to derive a generic algorithm
for any kind of data with smooth constraints, which lie therefore on a Riemannian
manifold. Stochastic approximations of the Expectation-Maximization algorithm
is used to estimate the model parameters in this highly non-linear setting. The
method is used to estimate a data-driven model of the progressive impairments of
cognitive functions during the onset of Alzheimer’s disease. Experimental results
show that the model correctly put into correspondence the age at which each in-
dividual was diagnosed with the disease, thus validating the fact that it effectively
estimated a normative scenario of disease progression. Random effects provide
unique insights into the variations in the ordering and timing of the succession of
cognitive impairments across different individuals.
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