Statistical Properties of Covariance Tapers

Abstract

Compactly supported autocovariance functions reduce computations needed for estimation and prediction under Gaussian process models, which are commonly used to model spatial and spatial-temporal data. A critical issue in using such models is the loss in statistical efficiency caused when the true autocovariance function is not compactly supported. Theoretical results indicate the value of specifying the local behavior of the process correctly. One way to obtain a compactly supported autocovariance function that has similar local behavior to an autocovariance function K of interest is to multiply K by some smooth compactly supported autocovariance function, which is called covariance tapering. This work extends previous theoretical results showing that covariance tapering has some asymptotic optimality properties as the number of observations in a fixed and bounded domain increases. However, numerical experiments show that for purposes of parameter estimation, covariance tapering often does not work as well as the simple alternative of breaking the observations into blocks and ignoring dependence across blocks. When covariance tapering is used for spatial prediction, predictions near the boundary of the observation domain are affected most. This article proposes an approach to modifying the taper to ameliorate this edge effect. In addition, a justification for a specific approach to carrying out conditional simulations based on tapered covariances is given. Supplementary materials for this article are available online.

Key Words:

• Equivalence of Gaussian measures
• Fixed-domain asymptotics
• Kriging
• Sparse matrices
• Unbiased estimating equations

ACKNOWLEDGMENTS

This research was supported by U.S. Department of Energy grant no. DE-SC0002557. The author thanks an anonymous reviewer for suggesting a simulation study along the lines described in Section 3.2.