TY - JOUR
T1 - Statistical analysis of trajectories on riemannian manifolds
T2 - Bird migration, hurricane tracking and video surveillance
AU - Su, Jingyong
AU - Kurtek, Sebastian
AU - Klassen, Eric
AU - Srivastava, Anuj
PY - 2014/3
Y1 - 2014/3
N2 - 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 L2 norm on the space of TSRVFs. We illustrate this framework using three representative manifolds-S2, 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.
AB - 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 L2 norm on the space of TSRVFs. We illustrate this framework using three representative manifolds-S2, 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.
KW - Parallel transport
KW - Rate invariant
KW - Riemannian manifold
KW - Temporal trajectory
KW - Time warping
KW - Variance reduction
UR - http://www.scopus.com/inward/record.url?scp=84898039655&partnerID=8YFLogxK
U2 - 10.1214/13-AOAS701
DO - 10.1214/13-AOAS701
M3 - Article
AN - SCOPUS:84898039655
SN - 1932-6157
VL - 8
SP - 530
EP - 552
JO - Annals of Applied Statistics
JF - Annals of Applied Statistics
IS - 1
ER -