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

Abstract

Likelihood-based methods such as maximum likelihood, REML, and Bayesian methods are attractive approaches to estimating covariance parameters in spatial models based on Gaussian processes. Finding such estimates can be computationally infeasible for large datasets, however, requiring O(n 3) calculations for each evaluation of the likelihood based on n observations. We propose 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 compactly supported correlation matrix. This produces matrices which can be be manipulated using more efficient sparse matrix algorithms. We present two approximations to the Gaussian likelihood using tapering. The first tapers the model covariance matrix only, whereas the second tapers both the model and sample covariance matrices. Tapering the model covariance matrix can be viewed as changing the underlying model to one in which the spatial covariance function is the direct product of the original covariance function and the tapering function. Focusing on the particular case of the Matérn class of covariance functions, we give conditions under which tapered and untapered covariance functions give equivalent (mutually absolutely continuous) measures for Gaussian processes on bounded domains. This allows us to evaluate the behavior of estimators maximizing our approximations to the likelihood under a bounded domain asymptotic framework. We give conditions under which estimators maximizing our approximations converge almost surely and quantify their efficiency using using the robust information criterion of Heyde (1997). We present results from a simulation study showing concordance between our asymptotic results and what we observe for moderate but increasing sample sizes. Finally, we discuss a potential application of these methods to a large spatial estimation problem, that of making statistical inference about the climatological (long-run mean) temperature difference between two sets of output from a computer model of global climate, run under two different land use scenarios.

Gaussian processes for computer experiments

“…This becomes problematic when n goes beyond 10 3 −10 4 , and there is currently a large body of literature proposing alternative computation procedures for larger values of n [17,30,31,55].…”

“…Classical methods of the literature are dedicated to this problem, as inducing points [26,43], low rank approximations [54], Gaussian Markov Random Fields [48], compactly supported covariance functions and covariance tapering [22,31,53]. These methods suffer from either the loss of interpolation properties or either difficulties to capture small or large scale dependencies.…”

Review on statistical methods for large spatial Gaussian data

“…Another approach to remark is to reduce dimension of covariance matrix by tapering method (see e.g., Furrer et al, 2006;Kaufman et al, 2008), which essentially set elements zero in a covariance matrix when a distance between two sites is located far in way to keep the resulting tapered matrix positive definite. Specifically, using tapering function K taper to the original covariance matrix Σ, we can obtain the tapered covariance matrix by Σ(θ) • K(γ), where the dot notation • refers to Schur product and K(γ) is a tapering matrix with parameter γ.…”

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

“…One approach is to use covariance tapering [25,26], in which the correct covariance matrix is tapered using an appropriately chosen compactly supported radial basis function which results in a sparse approximation of the covariance matrix that can be solved using sparse matrix algorithms. Another approach is to choose classes of covariance functions for which kriging can be done exactly using a multiresolution spatial process [27][28][29].…”