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, for example, in environmental sciences and in geoscience. The analysis of these data needs to rely on local models to account for the nonstationarity of the generating random process, the nonlinearity of the manifold, and the complex topology of the domain. In this article, 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 performances 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). Supplementary materials for this article are available online.

Keywords:

• Bagging
• Divide-and-conquer
• Fréchet mean
• Object-oriented data analysis
• Spatial analysis

Supplementary Materials

Online supplementary materials include R codes and datasets to replicate the simulation studies and the illustrative example of the Chesapeake data case study.

Acknowledgments

The authors wish to thank Ilaria Sartori and Luca Torriani for supporting them in writing in C++ parts of the code used to fit the local models, thus speeding up considerably our original R code and making possible extensive simulation studies. These codes are available in the R package Manifoldgstat, available on GitHub (github.com/LucaTorriani/KrigingManifoldData).

Notes

¹ The code was also successfully tested on more recent versions of the software R (v. 4.0.1) and of the packages: mgcv (v. 1.8.31), geoR (v. 1.8.1), sp (v. 1.4.2), igraph (v. 1.2.5), spatstat (v. 1.64.1), RTriangle (v. 1.6.0.10), fields (v. 10.3), MASS (v. 7.3.51.6), purrr (v. 0.3.4), proxy (v. 0.4.24), maptools (v. 1.0.2), rgeos (v. 0.5.3) on Windows 10 operating system.