Using Laplace Approximations To Estimate the Variogram Robustly by (Restricted) Maximum Likelihood
Hans R. Künsch1, Andreas Papritz2, Cornelia Schwierz1,2, Werner A. Stahel1
1Seminar für Statistik, ETH Zurich, Zurich, Switzerland; 2Institute of Terrestrial Ecosystems, ETH Zurich, Zurich, Switzerland

Environmental data are often tainted by outliers and require robust geostatistical techniques for prediction. Former studies on robust geostatistics mostly focused on robust estimation of the sample variogram and on robust ordinary kriging. A robust kriging method for data with trend and a robustified method for (restricted) maximum likelihood estimation, (RE)ML, of the variogram are currently lacking. We develop a new method for robust (RE)ML estimation of the variogram of a Gaussian random field, possibly contaminated by independent errors from long-tailed distributions. Besides robust estimates of fixed effects and covariance parameters, the method also provides robustified kriging predictions. We approximate the (RE)ML function by Laplace approximation.

In a first approach, we estimated the trend and variogram parameters by maximizing the approximated (RE)ML function directly. However, simulation studies revealed that the outliers had bounded influence only for re-descending ψ-functions. Furthermore, simple approaches borrowed from robust M-estimation of scale parameters for independent data failed to achieve Fisher consistency at the normal distribution for the variogram parameters.

In a second approach, we therefore derived the score equations of the Laplace approximation of the (RE)ML function and estimate the parameters now by solving those. This approach gives us the opportunity to robustify the estimation procedure further such that the influence functions are bounded also for monotone ψ-functions. Furthermore, we hope to derive asymptotic bias correction factors for Fisher consistency.

Apart from presenting our modelling framework, we shall present selected simulation results by which we explored the properties of the new method. This will be complemented by an analysis of data on heavy metals in the soils around a metal smelter in Switzerland where removal and displacement of contaminated soil has led to unpredictable irregularities in the spatial distribution of the metals. We shall compare the results of the robust with a customary geostatistical analysis.

Keywords: Robust restricted maximum likelihood estimation; Robust Kriging; Contaminated Gaussian random fields; Laplace approximation

Biography: Andreas Papritz is a researcher and lecturer at the Department of Environmental Sciences of ETH Zurich. His research interest are geostatistics and environmental statistics.