Geostatistical prediction through convex combination of Archimedean copulas

Abstract

A common problem in geostatistics is to interpolate a variable at unsampled locations using available data. Kriging has been the conventional method of solving this problem by providing the weighted average of samples, which is determined by minimizing the estimation variance. Kriging variance is a function of the samples’ spatial configuration and the variable’s spatial dependence structure. The latter is described by covariance that reduces complex dependence structures of natural phenomena to a single measure, introducing substantial simplifications. Another issue about kriging is that its variance does not depend on the sample values. Therefore, applying new methods such as spatial copulas that better describe the spatial dependence structure of variables and take advantage of sample values and spatial dependence structure would be helpful. This study compares prediction through the convex combination of Archimedean copulas to kriging using seven variables of the Jura data set. The empirical marginal distribution of variables and fitted kernel density estimates based on the Gaussian, triangular, Epanechnikov and gamma functions were used to investigate the effects of margins on the results. The mixed copulas were capable of describing various types of dependencies with asymmetric upper and lower tails. However, the Gaussian copula failed to explain the spatial dependence structure of variables and had the worst results among the copula-based approaches. The application of empirical marginal distribution of variables has generally given better results than the fitted models. For variables with large ratios of nugget effect to sill, in general, the copula-based approaches showed an advantage over kriging due to better reproduction of the mean values and distributions of the variables, having lower mean squared errors and higher correlation coefficients between the predicted and observed values. On the other hand, for variables with small nugget effects, kriging has better performance regarding all criteria except for the mean value reproduction. This study suggests using a convex combination of Archimedean copulas to predict variables with significant nugget effects.