Statistical Summarization, Modeling, and Evaluation of Temporally Registered Trajectories on Riemannian Manifolds

• Activity Number: 225
• Type: Topic Contributed
• Date/Time: Monday, August 5, 2013 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Jingyong Su*+
• Keywords: Riemannian manifold ; time warping ; Karcher mean

Abstract

We study statistical analysis of trajectories that take values on nonlinear Riemannian manifolds and are observed under arbitrary temporal evolutions. The past methods, such as those used in analysis of shape curves, are not applicable because they do not account for the variability in observation times. Based on cross-sectional analysis with fixed registration, they lose the mean structure and artificially inflate the observed variance in the given data. We introduce a quantity that provides both a cost function for temporal registration and a proper distance for comparison of trajectories. This distance, in turn, is used to define statistical summaries, such as the sample means and covariances, of given trajectories and "Gaussian-type" models to capture their variability. An essential property of this distance is that it is invariant to identical time-warpings (or temporal re-parameterizations) 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 will 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 will demonstrate: (1) improvements in mean structures and significant reductions in cross-sectional variances using real datasets, (2) statistical modeling for capturing variability in aligned trajectories, and (3) computing p-values of arbitrary trajectories under these models. Experimental results demonstrate that our framework greatly improves the mean trajectory and reduces the observed variance of these trajectories.