Phase-Amplitude Separation and Modeling of Spherical Trajectories

This paper studies the problem of separating phase-amplitude components in sample paths of a spherical process (longitudinal data on a unit two-sphere). Such separation is essential for efficient modeling and statistical analysis of spherical longitudinal data in a manner that is invariant to any phase variability. The key idea is to represent each path or trajectory with a pair of variables, a starting point and a Transported Square-Root Velocity Curve (TSRVC). A TSRVC is a curve in the tangent (vector) space at the starting point and has some important invariance properties under the L2 norm. The space of all such curves forms a vector bundle and the L2 norm, along with the standard Riemannian metric on S2, provides a natural metric on this vector bundle. This invariant representation allows for separating phase and amplitude components in given data, using a template-based idea. Furthermore, the metric property is useful in deriving computational procedures for clustering, mean computation, principal component analysis (PCA), and modeling. This comprehensive framework is demonstrated using two datasets: a set of bird-migration trajectories and a set of hurricane paths in the Atlantic ocean.