Geostatistics for Large Datasets on Riemannian Manifolds: A Matrix-Free Approach

Volume 20, Issue 4 (2022): Special Issue: Large-Scale Spatial Data Science, pp. 512–532

Pub. online: 3 November 2022
Type: Statistical Data Science
Open Access

Received

2 August 2022

2 August 2022

Accepted

17 October 2022

17 October 2022

Published

3 November 2022

3 November 2022

#### Abstract

Large or very large spatial (and spatio-temporal) datasets have become common place in many environmental and climate studies. These data are often collected in non-Euclidean spaces (such as the planet Earth) and they often present nonstationary anisotropies. This paper proposes a generic approach to model Gaussian Random Fields (GRFs) on compact Riemannian manifolds that bridges the gap between existing works on nonstationary GRFs and random fields on manifolds. This approach can be applied to any smooth compact manifolds, and in particular to any compact surface. By defining a Riemannian metric that accounts for the preferential directions of correlation, our approach yields an interpretation of the nonstationary geometric anisotropies as resulting from local deformations of the domain. We provide scalable algorithms for the estimation of the parameters and for optimal prediction by kriging and simulation able to tackle very large grids. Stationary and nonstationary illustrations are provided.

#### Supplementary material

Supplementary MaterialThe code used to perform the maximum likelihood estimation in Section 5.2 is available at https://github.com/mike-pereira/matrix-free-mle.

#### References

Lang A, Pereira M (2021). Galerkin–Chebyshev approximation of Gaussian random fields on compact riemannian manifolds. arXiv preprint: https://arxiv.org/abs/2107.02667.

Pereira M, Desassis N, Magneron C, Palmer N (2020). A matrix-free approach to geostatistical filtering. arXiv preprint: https://arxiv.org/abs/2004.02799.