Kriging Riemannian Data via Random Domain Decompositions

TL;DR

This paper introduces a novel approach for Riemannian data spatial prediction using random domain decompositions and Fréchet averaging, applicable to complex manifolds and spatial domains, with demonstrated effectiveness on simulated and real environmental data.

Contribution

It proposes a general random domain decomposition method for Riemannian data, enabling local modeling and aggregation via Fréchet averaging, applicable to various manifolds and complex spatial domains.

Findings

Method improves prediction accuracy on simulated covariance matrices.

Effective on real environmental dataset from Chesapeake Bay.

Applicable to multiple Riemannian manifolds.

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 Fr\'{e}chet 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).