Estimation and Prediction in the Spatial Linear Model

Abstract

Often in environmental monitoring studies interesting ecological factors will be observed at several locations repeatedly over time. Generally these space-time data are subject to a sequential spatial data analysis. In geostatistics, spatial data describing an environmental phenomenon like the pH value in precipitation at several locations are regarded as a realisation from a stochastic process. Component models are used to interpret the spatial variation of the process. Decomposing the spatial process into single components is based on the theory of linear models. Trend surface analysis is seen to be the geostatistical method for best linear unbiased estimation (BLUE) of the trend component, whereas universal kriging is equivalent to best linear unbiased prediction (BLUP) of the realisation of the spatial process. Furthermore trend surface analysis and universal kriging are shown to agree with the estimation of fixed effects and prediction of fixed and random effects in mixed linear models. Since estimation and prediction for spatial data result in different interpolations the differences are explained also graphically by example. The example uses acid-precipitation monitoring data. The extension of these spatial methods for application to space-time problems by combination with dynamic linear models is treated in the discussion.