Abstract
Maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on Gaussian processes. But calculating the likelihood can be computationally infeasible for large data sets, requiring O(n3) calculations for a data set with n observations. This article proposes the method of covariance tapering to approximate the likelihood in this setting. In this approach, covariance matrixes are “tapered,” or multiplied element wise by a sparse correlation matrix. The resulting matrixes can then be manipulated using efficient sparse matrix algorithms. We propose two approximations to the Gaussian likelihood using tapering. One of these approximations simply replaces the model covariance with a tapered version, whereas 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 estimator, 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.
Highlights
Much recent work has focused on the problem of estimating the autocovariance functions of spatially correlated stochastic processes
Consider taking the direct product of the true covariance function K0(x; θ) and a “tapering function” Ktaper(x; γ), an isotropic correlation function which is identically zero outside a range described by γ
When taking an increasing number of observations from a spatial process, we either can assume that the domain of the sampling region S increases to infinity, called “increasing domain” asymptotics, or we can assume that S is bounded and that observations become increasingly dense within S, called “bounded domain” or “infill” asymptotics
Summary
Much recent work has focused on the problem of estimating the autocovariance functions of spatially correlated stochastic processes. If numerical methods such as numerical maximization or Markov chain Monte Carlo are required, estimation will involve repeated evaluations of the likelihood Techniques for overcoming this computational hurdle have been developed mainly for datasets in which the sampling locations form a regular lattice, in which case spectral methods can be used (Whittle, 1954; Guyon, 1982; Stein, 1995; Dahlhaus, 2000). For analyzing large irregularly spaced spatial datasets, it is desirable to have a method which will reduce computational expense, while producing results comparable to those that would have been given by exact likelihood-based techniques To address this need, we propose using the method of covariance tapering, in which covariance matrices are multiplied element-wise by a compactly supported correlation matrix, giving matrices which can be be manipulated using more efficient sparse matrix algorithms. The approximations we have developed will be used to facilitate fitting the model
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