Geostatistical analysis of spatially dependent functional data: Universal Kriging in a Hilbert space

When dealing with high-dimensional georeferenced data, the need of spatial predictions often results in both theoretical and practical issues. In this work, we address this problem by proposing an extension of some geostatistical techniques to nonstationary functional random fields. A new theoretical framework is established in order to perform Universal Kriging of spatially dependent functional data belonging to a Hilbert space; moreover, estimators for the spatial mean and the spatial covariance structure are derived. Thanks to the generality of the proposed approach, not only the pointwise but also the differential information brought by the data can be exploited by embedding the analysis in a proper Hilbert space, possibly other than L2, such as a Sobolev space. Nevertheless, the non-stationary approach allows to model the spatial mean through a georeferenced external drift or to include cluster-varying spatial mean models. Proper algorithms to deal with real data are proposed and their performance is tested through an extensive simulation study. Finally, the proposed method is applied to daily mean temperature curves recorded in 35 meteorological stations located in Canada’s Maritime Provinces