Residual kriging for functional data. Application to the spatial prediction of salinity curves

Abstract

Recently several methodologies for carrying out geostatistical analysis of functional data has been proposed. All of them assume that the spatial functional process considered is stationary. However in practice we often have nonstationary functional data sets because there is spatial trend in the mean. Here we propose a methodology to extend kriging predictors for functional data to the case where the mean function is not constant through the region of interest. We consider an approach based on the classical residual kriging method used in univariate geostatistcs. We propose a three steps procedure. Initially a functional regression model is used for detrending the mean. Posteriorly we apply kriging methods for functional data to the regression residuals for doing prediction of a residual curve on a non-data location. Finally the prediction curve is obtained as the sum of the trend and the residual prediction. We apply the methodology to a salinity data set corresponding to 21 salinity curves recorded a the Ci´enaga Grande de Santa Marta estuary, located in the Caribbean coast of Colombia. A cross-validation analysis was carried out in order to establish the performance of the methodology proposed.