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Manifold-valued functional data analysis (FDA) recently becomes an activearea of research motivated by the raising availability of trajectories orlongitudinal data observed on non-linear manifolds. The challenges of analyzingsuch data come from many aspects, including infinite dimensionality andnonlinearity, as well as time-domain or phase variability. In this paper, westudy the amplitude part of manifold-valued functions on $\mathbb{S}^2$, whichis invariant to random time warping or re-parameterization. Utilizing the nicegeometry of $\mathbb{S}^2$, we develop a set of efficient and accurate toolsfor temporal alignment of functions, geodesic computing, and sample meancalculation. At the heart of these tools, they rely on gradient descentalgorithms with carefully derived gradients. We show the advantages of thesenewly developed tools over its competitors with extensive simulations and realdata and demonstrate the importance of considering the amplitude part offunctions instead of mixing it with phase variability in manifold-valued FDA.

Amplitude Mean of Functional Data on $\mathbb{S}^2$