Model-Based Geostatistics from a Bayesian Perspective: Investigating Area-to-Point Kriging with Small Data Sets

摘要

Area-to-point kriging (ATPK) is a geostatistical method for creating high-resolution raster maps using data of the variable of interest with a much lower resolution. The data set of areal means is often considerably smaller (<50documentclass[12pt]{minimal}usepackage{amsmath}usepackage{wasysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathrsfs}usepackage{upgreek}setlength{oddsidemargin}{-69pt}egin{document}$$<,50 $$end{document} observations) than data sets conventionally dealt with in geostatistical analyses. In contemporary ATPK methods, uncertainty in the variogram parameters is not accounted for in the prediction; this issue can be overcome by applying ATPK in a Bayesian framework. Commonly in Bayesian statistics, posterior distributions of model parameters and posterior predictive distributions are approximated by Markov chain Monte Carlo sampling from the posterior, which can be computationally expensive. Therefore, a partly analytical solution is implemented in this paper, in order to (i) explore the impact of the prior distribution on predictions and prediction variances, (ii) investigate whether certain aspects of uncertainty can be disregarded, simplifying the necessary computations, and (iii) test the impact of various model misspecifications. Several approaches using simulated data, aggregated real-world point data, and a case study on aggregated crop yields in Burkina Faso are compared. The prior distribution is found to have minimal impact on the disaggregated predictions. In most cases with known short-range behaviour, an approach that disregards uncertainty in the variogram distance parameter gives a reasonable assessment of prediction uncertainty. However, some severe effects of model misspecification in terms of overly conservative or optimistic prediction uncertainties are found, highlighting the importance of model choice or integration into ATPK.