Multivariate cross-validation and measures of accuracy and precision

Cross-validation and performance measures are standard components in the evaluation of a geostatistical model. These are well established in the univariate case, but measures for multivariate geostatistical modeling have not received as much attention. In the case of a single target variable, the univariate approaches remain valid, but in the fully multivariate case where a vector of variables needs to be estimated the evaluation needs to be based on all estimates simultaneously. An extension of cross-validation and associated performance measures to the fully multivariate case is presented and discussed for the case of regionalized compositions. The method is demonstrated by validating geostatistical models for two case studies: a sample drawn from a geochemical survey data set estimated with cokriging, and an application of direct sampling multiple point simulation.