Robustness Problems in the Analysis of Spatial Data

Abstract

Kriging is a widely used method of spatial prediction, particularly in earth and environmental sciences. It is based on a function which describes the spatial dependence, the so called variogram. Estimation and fitting of the variogram, as well as variogram model selection, are crucial stages of spatial prediction, because the variogram determines the kriging weights. These three steps must be carried out carefully, otherwise kriging can produce noninformative maps. The classical variogram estimator proposed by Matheron is not robust against outliers in the data, nor is it enough to make simple modifications such as the ones proposed by Cressie and Hawkins in order to achieve robustness. The use of a variogram estimator based on a highly robust estimator of scale is proposed. The robustness properties of these three variogram estimators are analyzed by means of the influence function and the classical breakdown point. The latter is extended to a spatial breakdown point, which depends on the construction of the most unfavorable configurations of perturbation. The effect of linear trend in the data and location outliers on variogram estimation is also discussed. Variogram model selection is addressed via nonparametric estimation of the derivative of the variogram. Variogram estimates at different spatial lags are correlated, because the same observation is used for different lags. The correlation structure of variogram estimators has been analyzed for Gaussian data, and then extended to elliptically contoured distributions. Its use for variogram fitting by generalized least squares is presented. Results show that our techniques improve the estimation and the fit significantly. Two new Splus functions for highly robust variogram estimation and variogram fitting by generalized least squares, as well as a Matlab code for variogram model selection via nonparametric derivative estimation, are available on the Web at http://www.math.mit.edu/~genton/].