The effective sample size for multivariate spatial processes with an application to soil contamination

Effective sample size accounts for the equivalent number of independent observations contained in a sample of correlated data. This notion has been widely studied in the context of univariate spatial variables. In that case, the effective sample size determines the reduction in the sample size due to the existing spatial correlation. In this paper, we generalize the methodology for multivariate spatial variables to provide a common effective sample size when all variables have been measured at the same locations. Together with the definition, we provide examples to investigate what an effective sample size looks like. An application for a soil contamination data set is considered. To reduce the dimensions of the process, clustering techniques are applied to obtain three bivariate vectors that are modeled using coregionalization models. Because the sample size of the data set is moderate and the locations are very unevenly distributed in the study area, the spatial analysis is challenging and interesting. We find that due to the presence of spatial autocorrelation, the sample size can be reduced by 38.53%, avoiding the duplication of information.