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Title: Sparsely observed functional data on the sphere Authors:  Alessia Caponera - EPFL (Switzerland) [presenting]
Julien Fageot - McGill University (Canada)
Matthieu Simeoni - EPFL (Switzerland)
Victor Panaretos - EPFL (Switzerland)
Abstract: Asymptotic theory for sparsely observed functional data has been largely developed when the domain is the interval $[0, 1]$, in both the i.i.d. and time-dependent settings. For instance, the covariance/autocovariance functions can be suitably estimated by local polynomials, which are well defined in a 2-dimensional planar domain. However, when the domain is the sphere, the task consists in estimating a function on a 4-dimensional non-flat surface (i.e., $\mathbb{S}^2 \times \mathbb{S}^2$) and it could be convenient to consider other methods which naturally incorporates such structure. To this purpose, we define our estimator as the minimizer of a Tikhonov regularization problem. We hence make use of the machinery of reproducing kernel Hilbert spaces and specifically spherical Sobolev spaces to give a full characterization of the solution. The main result consists of an optimal rate of convergence for the covariance function estimator which can be interpreted in both the dense and sparse regimes. Additionally, we provide the extension to the stationary time-series framework, thus considering the autocovariance functions at different lags. The findings are validated through numerical experiments.