Functional estimation of anisotropic covariance and autocovariance operators on the sphere

We propose nonparametric estimators for the second-order cen-tral moments of possibly anisotropic spherical random fields, within a func-tional data analysis context. We consider a measurement framework where each random field among an identically distributed collection of spherical random fields is sampled at a few random directions, possibly subject to measurement error. The collection of random fields could be i.i.d. or se-rially dependent. Though similar setups have already been explored for random functions defined on the unit interval, the nonparametric estima-tors proposed in the literature often rely on local polynomials, which do not readily extend to the (product) spherical setting. We therefore formulate our estimation procedure as a variational problem involving a generalized Tikhonov regularization term. The latter favours smooth covariance/auto-covariance functions, where the smoothness is specified by means of suit-able Sobolev-like pseudo-differential operators. Using the machinery of re-producing kernel Hilbert spaces, we establish representer theorems that fully characterize the form of our estimators. We determine their uniform rates of convergence as the number of random fields diverges, both for the dense (increasing number of spatial samples) and sparse (bounded number of spatial samples) regimes. We moreover demonstrate the computational feasibility and practical merits of our estimation procedure in a simulation setting, assuming a fixed number of samples per random field. Our numeri-cal estimation procedure leverages the sparsity and second-order Kronecker structure of our setup to reduce the computational and memory require-ments by approximately three orders of magnitude compared to a naive implementation would require.