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

Abstract

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.

Rainfall Potential and Consequences on Structural Soil Degradation of the Most Important Agricultural Region of Mexico

“…The emphasis on using t_stat_VIF stemmed from its consistent and successful utilization across various works. It was employed to derive confidence intervals for site velocities derived from the Global Navigation Satellite System (GNSS), as well as from meteorology/climatology and soil contamination studies with multivariate data [33][34][35].…”

Effective Sample Size with the Bivariate Gaussian Common Component Model

“…Therefore, based on the Vallejos and Osorio [4,6] proposals for the univariate ESS (Equation ( 5)) and bivariate ESS with the BCRM model, and considering the parameters estimated by BGCCM, the ESS bi value was calculated by means of the following expression:…”

“…Let us suppose that n is the sample size that is currently collected in a given agricultural area, for example, and that these samples are used to prepare and analyze thematic maps of spatially correlated variables. One problem that several researchers have been discussing and seeking solutions to is determining the equivalent number of independent observations in [2][3][4][5][6]. This number is called effective sample size (ESS).…”

“…Canton et al [11] made a sample reduction using the multivariate ESS; however, the method only considered a weighted ESS version, to avoid using a spatial correlation between the variables. Vallejos and Acosta [6] proposed a multivariate method to estimate ESS, dividing the attributes into groups of two and using the bivariate coregionalization model (BCRM). The BCRM proposes a parametric structure in which one spatial correlation parameter is common to two variables, and another spatial correlation parameter is exclusive to one of the variables.…”

The variance inflation factor to account for correlations in likelihood ratio tests: deformation analysis with terrestrial laser scanners

“…One of the most popular method to evaluate the AT is the initial sequence estimator, as derived in Straatsma et al (1986). This formulation was used, for example, in Wang (2022) to derive the confidence interval of GNSS-derived site velocities, for GPS positioning (Jansson and Persson 2013), or in meteorology and climatology (Dickinson 1985;Kamenkovich et al 2011), see also Vallejos and Acosta (2021) for an application to soil contamination for multivariate data. The AT formula links the variance of the mean with the variance itself and depends on the empirical ACF.…”