Highly Scalable Bayesian Geostatistical Modeling via Meshed Gaussian Processes on Partitioned Domains

Abstract

We introduce a class of scalable Bayesian hierarchical models for the analysis of massive geostatistical datasets. The underlying idea combines ideas on high-dimensional geostatistics by partitioning the spatial domain and modeling the regions in the partition using a sparsity-inducing directed acyclic graph (DAG). We extend the model over the DAG to a well-defined spatial process, which we call the meshed Gaussian process (MGP). A major contribution is the development of an MGPs on tessellated domains, accompanied by a Gibbs sampler for the efficient recovery of spatial random effects. In particular, the cubic MGP (Q-MGP) can harness high-performance computing resources by executing all large-scale operations in parallel within the Gibbs sampler, improving mixing and computing time compared to sequential updating schemes. Unlike some existing models for large spatial data, a Q-MGP facilitates massive caching of expensive matrix operations, making it particularly apt in dealing with spatiotemporal remote-sensing data. We compare Q-MGPs with large synthetic and real world data against state-of-the-art methods. We also illustrate using Normalized Difference Vegetation Index data from the Serengeti park region to recover latent multivariate spatiotemporal random effects at millions of locations. The source code is available at github.com/mkln/meshgp. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian
• Domain partitioning
• Graphical models
• Large n
• Sparsity
• Spatial

Supplementary Materials

The online supplement includes additional theoretical and computational details on Meshed Gaussian Processes, along with discussions on tessellation designs; the choice of the reference set and partition sizes; an application to multivariate outcomes; and and a comparison with other state-of-the-art scalable methods for large spatial data.

Funding

Banerjee was supported by the NSF grants DMS-1513654, IIS-1562303, and DMS-1916349; and by the National Institute of Health grants NIEHS-R01ES027027 and NIEHS-R01ES030210. Finley and Peruzzi were supported by National Science Foundation (NSF) EF-1253225 and DMS-1916395, and National Aeronautics and Space Administration’s Carbon Monitoring System project. Peruzzi was supported in part by 1R01ES028804 of the National Institute of Environmental Health Sciences of the National Institutes of Health and European Union project 856506.