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

TL;DR

This paper introduces a novel Riemannian trajectory analysis framework using the TSRVF representation, enabling invariant comparison, statistical summaries, and modeling of trajectories on various manifolds, demonstrated on bird migration, hurricanes, and videos.

Contribution

It proposes a new distance and statistical analysis method for trajectories on Riemannian manifolds that is invariant to time-warping, improving mean estimation and variability modeling.

Findings

Enhanced mean structure estimation and variance reduction in real data

Effective statistical models capturing trajectory variability

Successful application to bird migration, hurricanes, and video data

Abstract

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.