ABSTRACT
The problems of analysis and modeling of spherical trajectories, that is, continuous longitudinal data on , are important in several disciplines. These problems are challenging for two reasons: (1) nonlinear geometry of
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
norm, along with the standard Riemannian metric on
, 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.
Acknowledgments
The authors thank two anonymous reviewers and the associated editor for their constructive comments.