A permutation approach to the analysis of spatio-temporal geochemical data in the presence of heteroscedasticity

This paper proposes a novel nonparametric approach to model and reveal differences in the geochemical properties of the soil, when these are described by space–time measurements collected in a spatial region naturally divided into two parts. The investigation is motivated by a real study on a space–time geochemical data set, consisting of measurements of potassium chloride pH, water pH, and percentage of organic carbon collected during the growing season in the agricultural and forest areas of a site near Brno (Czech Republic). These data are here modeled as spatially distributed functions of time. A permutation approach is introduced to test for the effect of covariates in a spatial functional regression model with heteroscedastic residuals. In this context, the proposedmethod accounts for the heterogeneous spatial structure of the data by grounding on a permutation scheme for estimated residuals of the functional model. Here, a weighted least squares model is fitted to the observations, leading to asymptotically exchangeable and, thus, permutable residuals. An extensive simulation study shows that the proposed testing procedure outperforms the competitor approaches that neglect the spatial structure, both in terms of power and size. The results of modeling and testing on the case study are shown and discussed.