Modeling Massive Highly Multivariate Nonstationary Spatial Data with the Basis Graphical Lasso

Abstract

We propose a new modeling framework for highly multivariate spatial processes that synthesizes ideas from recent multiscale and spectral approaches with graphical models. The basis graphical lasso writes a univariate Gaussian process as a linear combination of basis functions weighted with entries of a Gaussian graphical vector whose graph is estimated from optimizing an ${\mathcal{l}}_1$ penalized likelihood. This article extends the setting to a multivariate Gaussian process where the basis functions are weighted with Gaussian graphical vectors. We motivate a model where the basis functions represent different levels of resolution and the graphical vectors for each level are assumed to be independent. Using an orthogonal basis grants linear complexity and memory usage in the number of spatial locations, the number of basis functions, and the number of realizations. An additional fusion penalty encourages a parsimonious conditional independence structure in the multilevel graphical model. We illustrate our method on a large climate ensemble from the National Center for Atmospheric Research’s Community Atmosphere Model that involves 40 spatial processes. Supplementary materials for this article are available online.

KEYWORDS:

• Climate ensemble
• Graphical model
• Multivariate Gaussian process
• Nonstationary
• Spatial basis function

Supplementary Materials

Supplementary material contains: (1) a simulation study (2) additional details regarding CAM data analysis, including estimation of the additive error variance, selection of penalty parameters, and simulations from our multivariate spatial model. Further supplementary materials are available at https://gdex.ucar.edu/dataset/371_abaker.html.

Notes

1 See the appendix of Krock et al. (2021) for classification of convexity/concavity for the terms in (5).

2 See Section 3.1 for a description of this estimation procedure.

3 See Section 3.3 for a description of this estimation procedure.

Additional information

Funding

This research was funded by grants NSF DMS-1821074 and NSF DMS-1923062.