An anisotropic Matern spatial covariance model: REML estimation and properties.

This thesis concerns the development, estimation and investigation of a general anisotropic spatial correlation function, within model-based geostatistics, expressed as a Gaussian linear mixed model, and estimated using residual maximum likelihood (REML). The Matern correlation function is attractive because of its parameter which controls smoothness of the spatial process, and which can be estimated from the data. This function is combined with geometric anisotropy, with an extension permitting different distance metrics, forming a flexible spatial covariance model which incorporates as special cases many infiniterange spatial covariance functions in common use. Derivatives of the residual log-likelihood with respect to the four correlation-model parameters are derived, and the REML algorithm coded in Splus for testing and refinement as a precursor to its implementation into the software ASReml, with additional generality of linear mixed models. Suggestions are given regarding initial values for the estimation. A residual likelihood ratio test for anisotropy is also developed and investigated. Application to three soil-based examples reveals that anisotropy does occur in practice, and that this technique is able to fit covariance models previously unavailable or inaccessible. Simulations of isotropic and anisotropic data with and without a nugget effect reveal the following principal points. Inclusion of some closely-spaced locations greatly improves estimation, particularly of the Matern smoothness parameter, and of the nugget variance when present. The presence of geometric anisotropy does not adversely affect parameter estimation. Presence of a nugget effect introduces greater uncertainty into the parameter estimates, most dramatically for the smoothness parameter, and also increases the chance of non-convergence and decreases the power of the test for anisotropy. Estimation is more difficult with very “unsmooth” processes (Matern smoothness parameter 0.1 or 0.25) — nonconvergence is more likely and estimates are less precise and/or more biased. However it is still often possible to fit the full model including both anisotropy and nugget effect using REML with as few as 100 observations. Additional simulations involving model misspecification reveal that ignoring anisotropy when it is present can substantially increase the mean squared error of prediction, but overfitting by attempting to model anisotropy when it is absent is less damaging. Further, plug-in estimates of prediction error variance are reasonable estimates of the actual mean squared error of prediction, regardless of the model fitted, weakening the argument requiring Bayesian approaches to properly allow for uncertainty in the parameter estimates when estimating prediction error variance. The most valuable outcome of this research is the implementation of an anisotropic Matern correlation function in ASReml, including the full generality of Gaussian linear