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We propose nonparametric estimators for the second-order central moments ofpossibly anisotropic spherical random fields, within a functional data analysiscontext. We consider a measurement framework where each random field among anidentically distributed collection of spherical random fields is sampled at afew random directions, possibly subject to measurement error. The collection ofrandom fields could be i.i.d. or serially dependent. Though similar setups havealready been explored for random functions defined on the unit interval, thenonparametric estimators proposed in the literature often rely on localpolynomials, which do not readily extend to the (product) spherical setting. Wetherefore formulate our estimation procedure as a variational problem involvinga generalized Tikhonov regularization term. The latter favours smoothcovariance/autocovariance functions, where the smoothness is specified by meansof suitable Sobolev-like pseudo-differential operators. Using the machinery ofreproducing kernel Hilbert spaces, we establish representer theorems that fullycharacterize the form of our estimators. We determine their uniform rates ofconvergence as the number of random fields diverges, both for the dense(increasing number of spatial samples) and sparse (bounded number of spatialsamples) regimes. We moreover demonstrate the computational feasibility andpractical merits of our estimation procedure in a simulation setting, assuminga fixed number of samples per random field. Our numerical estimation procedureleverages the sparsity and second-order Kronecker structure of our setup toreduce the computational and memory requirements by approximately three ordersof magnitude compared to a naive implementation would require.

Functional Estimation of Anisotropic Covariance and Autocovariance Operators on the Sphere