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

Mainfold-valued functional data analysis (FDA) recently becomes an active area of research motivated by the raising availability of trajectories or longitudinal data observed on non-linear manifolds. The challenges of analyzing such data comes from many aspects, including infinite dimensionality and nonlinearity, as well as time domain or phase variability. In this paper, we study the amplitude part of manifold-valued functions on $\S^2$, which is invariant to random time warping or re-parameterization of the function. Utilizing the nice geometry of $\S^2$, we develop a set of efficient and accurate tools for temporal alignment of functions, geodesic and sample mean calculation. At the heart of these tools, they rely on gradient descent algorithms with carefully derived gradients. We show the advantages of these newly developed tools over its competitors with extensive simulations and real data, and demonstrate the importance of considering the amplitude part of functions instead of mixing it with phase variability in mainfold-valued FDA.