No abstract

We introduce a new class of nonstationary covariance functions for spatial modelling. Nonstationary covariance functions allow the model to adapt to spatial surfaces whose variability changes with location. The class includes a nonstationary version of the Matérn stationary covariance, in which the differentiability of the spatial surface is controlled by a parameter, freeing one from fixing the differentiability in advance. The class allows one to knit together local covariance parameters into a valid global nonstationary covariance, regardless of how the local covariance structure is estimated. We employ this new nonstationary covariance in a fully Bayesian model in which the unknown spatial process has a Gaussian process (GP) prior distribution with a nonstationary covariance function from the class. We model the nonstationary structure in a computationally efficient way that creates nearly stationary local behavior and for which stationarity is a special case. We also suggest non-Bayesian approaches to nonstationary kriging.To assess the method, we use real climate data to compare the Bayesian nonstationary GP model with a Bayesian stationary GP model, various standard spatial smoothing approaches, and nonstationary models that can adapt to function heterogeneity. The GP models outperform the competitors, but while the nonstationary GP gives qualitatively more sensible results, it shows little advantage over the stationary GP on held-out data, illustrating the difficulty in fitting complicated spatial data.

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.