Spatial prediction for infinite-dimensional compositional data

Abstract

There is a growing interest in the analysis of geostatistical functional data. Such a fusion between geostatistical methods and functional data analysis has been shown to open promising area of research. The present paper is devoted to a kriging predictor for functional data where the functions are probability densities functions (PDFs for short), being also a special case of infinite dimensional compositional data. The predictor proposed in this paper is the analogue of the classic ordinary kriging predictor, defined in terms of scalar parameters, but considering PDFs (with support on a finite interval) instead of one-dimensional data. The methodology is applied to both simulated and real data. The statistical performance of our predictor is then evaluated through cross validation techniques using real and simulated data. From the mathematical point of view, in order to assess the properties of our predictor we need to work under the framework of Hilbert valued random fields as well as with Aitchison geometries. We present some original results being of interest for themselves, and that give a complete picture of the framework illustrated through the paper.