  There are often two important types of variation in functional data: the
horizontal (or phase) variation and the vertical (or amplitude) variation.
These two types of variation have been appropriately separated and modeled
through a domain warping method (or curve registration) based on the Fisher Rao
metric. This paper focuses on the analysis of the horizontal variation,
captured by the domain warping functions. The square-root velocity function
representation transforms the manifold of the warping functions to a Hilbert
sphere. Motivated by recent results on manifold analogs of principal component
analysis, we propose to analyze the horizontal variation via a Principal Nested
Spheres approach. Compared with earlier approaches, such as approximating
tangent plane principal component analysis, this is seen to be the most
efficient and interpretable approach to decompose the horizontal variation in
some examples.
