Covariance Tapering for Likelihood-Based Estimation in Large Spatial Data Sets

Maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on Gaussian processes. However, calculating the likelihood can be computationally infeasible for large datasets, requiring O (n³) observations. This article proposes the method of covariance tapering to approximate the likelihood in this setting. In this approach, covariance matrices are ``tapered, '' or multiplied element-wise by a sparse correlation matrix. The resulting matrices can then be manipulated using efficient sparse matrix algorithms. We propose two approximations to the Gaussian likelihood using tapering. One simply replaces the model covariance with a tapered version; the other is motivated by the theory of unbiased estimating equations. Focusing on the particular case of the Matérn class of covariance functions, we give conditions under which estimators maximizing the tapering approximations are, like the maximum likelihood estimator, strongly consistent. Moreover, we show in a simulation study that the tapering estimators can have sampling densities quite similar to that of the maximum likelihood estimate, even when the degree of tapering is severe. We illustrate the accuracy and computational gains of the tapering methods in an analysis of yearly total precipitation anomalies at weather stations in the United States.