Phase-Amplitude Separation and Modeling of Spherical Trajectories

ABSTRACT

The problems of analysis and modeling of spherical trajectories, that is, continuous longitudinal data on ${\mathbb{S}}^2$, are important in several disciplines. These problems are challenging for two reasons: (1) nonlinear geometry of ${\mathbb{S}}^2$ and (2) the presence of phase variability in given data. This article develops a geometric framework for separating phase variability from given trajectories, leaving only the shape or the amplitude variability. The key idea is to represent each trajectory with a pair of variables, a starting point, and a transported square-root velocity curve (TSRVC), a curve in the tangent (vector) space at the starting point. The space of all such curves forms a vector bundle and the ${\mathbb{L}}^2$ norm, along with the standard Riemannian metric on ${\mathbb{S}}^2$, provides a natural, warping-invariant metric on this vector bundle. This leads to an efficient algorithm for registration of trajectories, that is, phase-amplitude separation, and computational tools, such as clustering, sample means, and principal component analysis (PCA) of the two components separately. It also helps derive simple statistical models of phase-amplitude components of spherical trajectories. This comprehensive framework is demonstrated using two datasets: a set of bird-migration trajectories and a set of hurricane paths in the Atlantic ocean. Supplementary material for this article is available online.

KEYWORDS:

• Alignment
• Manifold functional PCA
• Spherical trajectories
• Vector bundle

Acknowledgments

The authors thank two anonymous reviewers and the associated editor for their constructive comments.

Additional information

Funding

Srivastava’s research is supported by NSF 1621787 and NSF 1617397. Eric Klassen gratefully acknowledges the support of the Simons Foundation Grant 317865. Zhang’s research is partially supported by the NSF under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.