Model-based geostatistics: some issues in modelling and model diagnostics

Abstract

Spatial modelling is examined in a model-based geostatistical context using the Gaussian linear mixed model in a likelihood framework. Complex spatial models developed provide practitioners with a practical and best-practice guide for spatial analysis. Adequate modelling theory and matrix algebra are provided to ground the methods demonstrated. A multivariate model over two time points and three-dimensional space is developed which is novel to the field of soil science. Soil organic carbon measurements at three soil depths and two time points from a cropping field with four soil classes are used. The spatial process is assessed for second-order stationarity and anisotropic correlation. Univariate spatial modelling is used to inform bivariate spatial modelling of pre- and post-harvest soil organic carbon at each soil depth. Bivariate modelling is extended to the multivariate level, where both time points and the three soil depths are incorporated in a single model to pool maximum information. A common correlation structure is tested and is supported for the response variable at each of the six time-depth combinations. Separable correlation structures are used for computational efficiency. The difficulty of estimating nugget effects suggests a sub-optimal sampling design. Preferred fitted models are all isotropic. Equations for predictions and the variance of prediction errors are extended from well-known results and maps of predicted values and variance of prediction errors are produced and show close correspondence with observed values. Finally, univariate models for spatially referenced seed counts from small sampling plots are examined within a Gaussian framework using Box-Cox transformations. The discrete nature of the data, small sample size and computational problems hamper model fitting. Anisotropy is examined using a variogram envelope diagnostic technique. ASReml-R software is shown to be a powerful analytical tool for spatial processes