Inverses of Matern Covariances on Grids

TL;DR

This paper analyzes the spectral properties of Matern covariance functions on grids, revealing limitations of SPDE-based approximations at high frequencies and their impact on parameter estimation.

Contribution

It provides a theoretical and numerical investigation of the aliasing effects in Matern covariances and evaluates the accuracy of SPDE approximations.

Findings

SPDE approximation overestimates high-frequency power.

Approximation accuracy does not improve as grid spacing decreases, except in 1D exponential case.

SPDE tends to overestimate spatial range parameters.

Abstract

We conduct a study of the aliased spectral densities of Mat\'ern covariance functions on a regular grid of points, providing clarity on the properties of a popular approximation based on stochastic partial differential equations; while others have shown that it can approximate the covariance function well, we find that it assigns too much power at high frequencies and does not provide increasingly accurate approximations to the inverse as the grid spacing goes to zero, except in the one-dimensional exponential covariance case. We provide numerical results to support our theory, and in a simulation study, we investigate the implications for parameter estimation, finding that the SPDE approximation tends to overestimate spatial range parameters.