Hierarchical sparse Cholesky decomposition with applications to high-dimensional spatio-temporal filtering

TL;DR

This paper introduces a hierarchical Vecchia approximation for sparse Cholesky decomposition, enabling scalable high-dimensional spatio-temporal filtering with improved efficiency and accuracy.

Contribution

It proposes a novel hierarchical approximation that ensures sparsity in Cholesky factors, facilitating scalable filtering in high-dimensional spatial statistics.

Findings

Outperforms alternative methods in numerical comparisons.

Enables efficient high-dimensional spatio-temporal filtering.

Extends to non-linear, non-Gaussian data using Laplace approximation.

Abstract

Spatial statistics often involves Cholesky decomposition of covariance matrices. To ensure scalability to high dimensions, several recent approximations have assumed a sparse Cholesky factor of the precision matrix. We propose a hierarchical Vecchia approximation, whose conditional-independence assumptions imply sparsity in the Cholesky factors of both the precision and the covariance matrix. This remarkable property is crucial for applications to high-dimensional spatio-temporal filtering. We present a fast and simple algorithm to compute our hierarchical Vecchia approximation, and we provide extensions to non-linear data assimilation with non-Gaussian data based on the Laplace approximation. In several numerical comparisons, including a filtering analysis of satellite data, our methods strongly outperformed alternative approaches.