PICAR: An Efficient Extendable Approach for Fitting Hierarchical Spatial Models

Abstract

Hierarchical spatial models are very flexible and popular for a vast array of applications in areas such as ecology, social science, public health, and atmospheric science. It is common to carry out Bayesian inference for these models via Markov chain Monte Carlo (MCMC). Each iteration of the MCMC algorithm is computationally expensive due to costly matrix operations. In addition, the MCMC algorithm needs to be run for more iterations because the strong cross-correlations among the spatial latent variables result in slow mixing Markov chains. To address these computational challenges, we propose a projection-based intrinsic conditional autoregression (PICAR) approach, which is a discretized and dimension-reduced representation of the underlying spatial random field using empirical basis functions on a triangular mesh. Our approach exhibits fast mixing as well as a considerable reduction in computational cost per iteration. PICAR is computationally efficient and scales well to high dimensions. It is also automated and easy to implement for a wide array of user-specified hierarchical spatial models. We show, via simulation studies, that our approach performs well in terms of parameter inference and prediction. We provide several examples to illustrate the applicability of our method, including (i) a high-dimensional cloud cover dataset that showcases its computational efficiency, (ii) a spatially varying coefficient model that demonstrates the ease of implementation of PICAR in the probabilistic programming languages stan and nimble, and (iii) a watershed survey example that illustrates how PICAR applies to models that are not amenable to efficient inference via existing methods.

Keywords:

• Basis representation
• Gaussian random field
• Markov chain Monte Carlo
• Non-Gaussian spatial data
• Ordinal spatial data
• Spatially varying coefficients

Acknowledgments

We are grateful to Ephraim Hanks, Erin Schliep, Yawen Guan, Jaewoo Park, and Klaus Keller for helpful discussions. We would like to thank Perry de Valpine and Chris Paciorek for their assistance with nimble, particularly with the massive cloud mask dataset. This study was also co-supported by the Penn State Center for Climate Risk Management. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Department of Energy, the National Science Foundation, or other funding entities. Any errors and opinions are those of the authors. We are not aware of any real or perceived conflicts of interest for any authors.

Supplementary Material

Supplemental Descriptions, Tables, and Figures: Descriptions of various hierarchical spatial models, a parallelized method to generate Moran’s basis functions, three other simulation studies, and a real-world application. Additional figures and tables for the simulation studies, the dwarf mistletoe study, and the Maryland Stream Waders application (.pdf file).

Code for the PICAR approach in nimble and stan: Code to implement PICAR for various hierarchical spatial models can be accessed at the following github repository: https://github.com/benee55/PICAR_code. (github respository link)

Dwarf mistletoe data set: Dataset used in the dwarf mistletoe application. (.csv file).

Maryland Stream Waders data set: Dataset used in the Maryland Stream Waders application (Section 5.2). (.csv file)

Additional information

Funding

The research was partially supported by the National Institute of General Medical Sciences of the National Institutes of Health under Award Number R01GM123007. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. This work was partially supported by the U.S. Department of Energy, Office of Science, Biological and Environmental Research Program, Earth and Environmental Systems Modeling, MultiSector Dynamics, Contract No. DE-SC0016162 and by the National Science Foundation through the Network for Sustainable Climate Risk Management (SCRiM) under NSF cooperative agreement GEO-1240507.