  We consider the statistical analysis of trajectories on Riemannian manifolds
that are observed under arbitrary temporal evolutions. Past methods rely on
cross-sectional analysis, with the given temporal registration, and
consequently may lose the mean structure and artificially inflate observed
variances. We introduce a quantity that provides both a cost function for
temporal registration and a proper distance for comparison of trajectories.
This distance is used to define statistical summaries, such as sample means and
covariances, of synchronized trajectories and "Gaussian-type" models to capture
their variability at discrete times. It is invariant to identical time-warpings
(or temporal reparameterizations) of trajectories. This is based on a novel
mathematical representation of trajectories, termed transported square-root
vector field (TSRVF), and the $\mathbb{L}^2$ norm on the space of TSRVFs. We
illustrate this framework using three representative
manifolds---$\mathbb{S}^2$, $\mathrm {SE}(2)$ and shape space of planar
contours---involving both simulated and real data. In particular, we
demonstrate: (1) improvements in mean structures and significant reductions in
cross-sectional variances using real data sets, (2) statistical modeling for
capturing variability in aligned trajectories, and (3) evaluating random
trajectories under these models. Experimental results concern bird migration,
hurricane tracking and video surveillance.
