Kriging for Hilbert-space valued random fields: The operatorial point of view

We develop a comprehensive framework for linear spatial prediction in Hilbert spaces. We explore the problem of Best Linear Unbiased (BLU) prediction in Hilbert spaces through an original point of view, based on a new Operatorial definition of Kriging. We ground our developments on the theory of Gaussian processes in function spaces and on the associated notion of measurable linear transformation. We prove that our new setting allows (a) to derive an explicit solution to the problem of Operatorial Ordinary Kriging, and (b) to establish the relation of our novel predictor with the key concept of conditional expectation of a Gaussian measure. Our new theory is posed as a unifying theory for Kriging, which is shown to include the Kriging predictors proposed in the literature on Functional Data through the notion of finite-dimensional approximations. Our original viewpoint to Kriging offers new relevant insights for the geostatistical analysis of either finite- or infinite-dimensional georeferenced dataset.