Mathematical foundations of functional Kriging in Hilbert spaces and Riemannian manifolds

In this chapter, we review the mathematical framework for spatial prediction (kriging) for complex data. We focus here on the approach developed within the area of Object-Oriented Spatial Statistics which grounds on the foundational idea that the atom of the geostatistical analysis is the entire data point, regardless of its complexity. This is seen as an indivisible unit rather than a collection of features, and accordingly embedded as a point within a space of objects, called feature space. We illustrate here the kriging methods when data belong to Hilbert space and Riemannian manifolds, in stationary or nonstationary settings and discuss the estimators that can be used for the mean and the covariance structure.