Multi-Resolution Spatial Methods on Spheres

Abstract

We consider estimating a (smooth) function on a sphere from noisy data (irregularly) sampled on the sphere. We first develop a new class of multi-resolution basis functions in the thin-plate-spline (TPS) function space on the sphere. These basis functions are ordered in terms of their degrees of smoothness, which provide a dimension reduction representation for the underlying function, and enable estimation of the model parameters using least squares. Theoretically, we show that the proposed method achieves the same convergence rate as the TPS method by using a small number of basis functions relative to the sample size. In addition, we extend the model by adding a spatial stochastic process, resulting in a spatial mixed-effects model. We apply the conditional Akaike information criterion to select the number of basis functions and to determine whether the spatial process should be included. A simulation experiment and an application to global data of sea surface temperature observed from a satellite are performed to show the effectiveness of the proposed method.