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

TL;DR

This paper develops new geometric tools for analyzing the amplitude component of manifold-valued functional data on the sphere, enabling accurate temporal alignment, geodesic computation, and mean estimation, with demonstrated advantages over existing methods.

Contribution

It introduces efficient algorithms for amplitude analysis of manifold-valued functions on 2, leveraging sphere geometry and gradient descent, advancing functional data analysis on non-linear manifolds.

Findings

Tools outperform competitors in simulations and real data

Amplitude analysis improves understanding of manifold-valued functions

Gradient-based algorithms are effective for geometric computations

Abstract

Manifold-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 come 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 $\mathbb{S}^2$, which is invariant to random time warping or re-parameterization. Utilizing the nice geometry of $\mathbb{S}^2$, we develop a set of efficient and accurate tools for temporal alignment of functions, geodesic computing, 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 manifold-valued FDA.