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Amplitude Mean of Functional Data on \(\mathbb {S}^2\) and its Accurate Computation

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Manifold-valued functional data analysis (FDA) has become an active area of research motivated by the rising availability of trajectories or longitudinal data observed on nonlinear 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. We represent a smooth function on \(\mathbb {S}^2\) using a pair of components: a starting point and a transported square-root velocity curve (TSRVC). Under this representation, the space of all smooth functions on \(\mathbb {S}^2\) forms a vector bundle, and the simple \(L_2\) norm becomes a time-warping invariant metric on this vector bundle. 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 and covariance 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.

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Authors and Affiliations

  1. Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    Zhengwu Zhang

  2. Department of Neurology, University of Rochester Medical Center, New York, USA

    Bayan Saparbayeva

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  1. Zhengwu Zhang
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  2. Bayan Saparbayeva
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Correspondence to Zhengwu Zhang.

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Zhang, Z., Saparbayeva, B. Amplitude Mean of Functional Data on \(\mathbb {S}^2\) and its Accurate Computation. J Math Imaging Vis 64, 1010–1028 (2022). https://doi.org/10.1007/s10851-022-01109-8

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  • DOI: https://doi.org/10.1007/s10851-022-01109-8

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