Kriging Riemannian Data via Random Domain Decompositions

Abstract

Data taking value on a Riemannian manifold and observed over a complex spatial domain are becoming more frequent in applications, e.g. in environmental sciences and in geoscience. The analysis of these data needs to rely on local models to account for the non stationarity of the generating random process, the non linearity of the manifold and the complex topology of the domain. In this paper, we propose to use a random domain decomposition approach to estimate an ensemble of local models and then to aggregate the predictions of the local models through Fréchet averaging. The algorithm is introduced in complete generality and is valid for data belonging to any smooth Riemannian manifold but it is then described in details for the case of the manifold of positive definite matrices, the hypersphere and the Cholesky manifold. The predictive performance of the method are explored via simulation studies for covariance matrices and correlation matrices, where the Cholesky manifold geometry is used. Finally, the method is illustrated on an environmental dataset observed over the Chesapeake Bay (USA).

Variograms for spatial functional data with phase variation

“…Chief amongst these are universal kriging methods (Caballero et al, 2013;Menafoglio et al, 2013;Reyes et al, 2015;Menafoglio & Petris, 2016) wherein observed functions are preprocessed to better manage deviations from the stationarity assumption. Menafoglio et al (2018) generalized kriging of functional data to data on a Riemannian manifold.…”

Grassmannian kriging with applications in POD-based model order reduction

“…To do so, two types of approaches are available in the literature: (i) approaches relying on a referent tangent space and (ii) totally intrinsic approaches. The first class of methods [25,28,24] is based on the choice of a reference point on the manifold and on the transport of the kriging interpolation problem to the tangent space at that point using the geodesic logarithm map. The interpolated point is then brought back on the manifold by the geodesic exponential map.…”