How do Markov approximations compare with other methods for large spatial data sets?

TL;DR

This paper compares Markov approximations, covariance tapering, and process convolution methods for large spatial data sets, showing Markov methods offer superior accuracy at similar computational costs.

Contribution

It extends Hilbert space-based Markov approximations of Gaussian Matérn fields using wavelet basis functions and compares their performance with other popular methods.

Findings

Markov methods outperform other methods in accuracy for the same computational cost.

Wavelet-based Markov approximations improve upon previous Hilbert space approaches.

Results demonstrate the efficiency of Markov methods for large spatial datasets.

Abstract

The Mat\'ern covariance function is a popular choice for modeling dependence in spatial environmental data. Standard Mat\'ern covariance models are, however, often computationally infeasible for large data sets. In this work, recent results for Markov approximations of Gaussian Mat\'{e}rn fields based on Hilbert space approximations are extended using wavelet basis functions. These Markov approximations are compared with two of the most popular methods for efficient covariance approximations; covariance tapering and the process convolution method. The results show that, for a given computational cost, the Markov methods have a substantial gain in accuracy compared with the other methods.