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We consider the statistical analysis of trajectories on Riemannian manifoldsthat are observed under arbitrary temporal evolutions. Past methods rely oncross-sectional analysis, with the given temporal registration, andconsequently may lose the mean structure and artificially inflate observedvariances. We introduce a quantity that provides both a cost function fortemporal registration and a proper distance for comparison of trajectories.This distance is used to define statistical summaries, such as sample means andcovariances, of synchronized trajectories and "Gaussian-type" models to capturetheir variability at discrete times. It is invariant to identical time-warpings(or temporal reparameterizations) of trajectories. This is based on a novelmathematical representation of trajectories, termed transported square-rootvector field (TSRVF), and the $\mathbb{L}^2$ norm on the space of TSRVFs. Weillustrate this framework using three representativemanifolds---$\mathbb{S}^2$, $\mathrm {SE}(2)$ and shape space of planarcontours---involving both simulated and real data. In particular, wedemonstrate: (1) improvements in mean structures and significant reductions incross-sectional variances using real data sets, (2) statistical modeling forcapturing variability in aligned trajectories, and (3) evaluating randomtrajectories under these models. Experimental results concern bird migration,hurricane tracking and video surveillance.

Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance