Phase-Amplitude Separation and Modeling of Spherical Trajectories

TL;DR

This paper introduces a novel invariant framework for separating phase and amplitude in spherical trajectories, enabling improved modeling and analysis of spherical longitudinal data.

Contribution

It proposes a new representation using starting points and TSRVCs, along with a natural metric, facilitating phase-amplitude separation and statistical analysis on spherical trajectories.

Findings

Effective separation of phase and amplitude in spherical data

Application to bird migration and hurricane path datasets

Enhanced clustering and PCA methods for spherical trajectories

Abstract

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.