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

Luc Steinbuch,Dick J Brus,Thomas G Orton

doi:10.1007/s11004-019-09840-6

Abstract

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 (<,50 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.

Highlights

An important challenge often encountered in scientific research is spatial prediction using areal-support data, that is, data about the variable of interest that is available as areal means only
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
Brus et al (2018) summarised earlier work by Pardo-Igzquiza and Dowd (2001) showing that uncertainty in the variogram parameters can be quantified by the inverse Fisher matrix of the variogram parameters, but did not integrate this uncertainty in the kriging prediction uncertainty itself

Summary

Introduction

An important challenge often encountered in scientific research is spatial prediction using areal-support data, that is, data about the variable of interest that is available as areal means only. In ATPK, regression coefficients (in the presence of covariates) and variogram parameters (describing spatial relationships) have to be estimated, for example by a least square estimator for the regression coefficients combined with an iterative variogram fitting deconvolution algorithm (‘method of moments’) on the regression residuals (Goovaerts 2008). More recent methods such as restricted maximum likelihood (REML) in combination with universal kriging (UK) (both, and from hereon, referring to their application in the ATPK setting) consider the uncertainty in the regression coefficients (Webster and Oliver 2007). Uncertainty in the variogram model parameters might be a relevant source of uncertainty (Jansen 1998; Kitanidis 1986; Minasny et al 2011). Truong et al (2014) showed that variogram uncertainty can have a substantial impact on ATPK variances. Brus et al (2018) summarised earlier work by Pardo-Igzquiza and Dowd (2001) showing that uncertainty in the variogram parameters can be quantified by the inverse Fisher matrix of the variogram parameters, but did not integrate this uncertainty in the kriging prediction uncertainty itself

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