Limitations on low rank approximations for covariance matrices of spatial data

Abstract

Evaluating the likelihood function for Gaussian models when a spatial process is observed irregularly is problematic for larger datasets due to constraints of memory and calculation. If the covariance structure can be approximated by a diagonal matrix plus a low rank matrix, then both the memory and calculations needed to evaluate the likelihood function are greatly reduced. When neighboring observations are strongly correlated, much of the variation in the observations can be captured by low frequency components, so the low rank approach might be thought to work well in this setting. Through both theory and numerical results, where the diagonal matrix is assumed to be a multiple of the identity, this paper shows that the low rank approximation sometimes performs poorly in this setting. In particular, an approximation in which observations are split into contiguous blocks and independence across blocks is assumed often provides a much better approximation to the likelihood than a low rank approximation requiring similar memory and calculations. An example with satellite-based measurements of total column ozone shows that these results are relevant to real data and that the low rank models also can be highly statistically inefficient for spatial interpolation.