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Journal of Data Science


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 accessOpen Access
Received
2 August 2022
Accepted
17 October 2022
Published
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 Material
The code used to perform the maximum likelihood estimation in Section 5.2 is available at https://github.com/mike-pereira/matrix-free-mle.

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Copyright
2022 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.
Open access article under the CC BY license.

Keywords
anisotropy finite elements Gaussian process Laplace-Beltrami operator nonstationarity

Funding
The authors acknowledge the support of the Mines Paris / INRAE chair “Geolearning”.

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