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Journal of Data Science


Scalable Predictions for Spatial Probit Linear Mixed Models Using Nearest Neighbor Gaussian Processes
Volume 20, Issue 4 (2022): Special Issue: Large-Scale Spatial Data Science, pp. 533–544
Pub. online: 3 November 2022      Type: Statistical Data Science      Open accessOpen Access
Received
16 August 2022
Accepted
6 October 2022
Published
3 November 2022

Abstract

Spatial probit generalized linear mixed models (spGLMM) with a linear fixed effect and a spatial random effect, endowed with a Gaussian Process prior, are widely used for analysis of binary spatial data. However, the canonical Bayesian implementation of this hierarchical mixed model can involve protracted Markov Chain Monte Carlo sampling. Alternate approaches have been proposed that circumvent this by directly representing the marginal likelihood from spGLMM in terms of multivariate normal cummulative distribution functions (cdf). We present a direct and fast rendition of this latter approach for predictions from a spatial probit linear mixed model. We show that the covariance matrix of the cdf characterizing the marginal cdf of binary spatial data from spGLMM is amenable to approximation using Nearest Neighbor Gaussian Processes (NNGP). This facilitates a scalable prediction algorithm for spGLMM using NNGP that only involves sparse or small matrix computations and can be deployed in an embarrassingly parallel manner. We demonstrate the accuracy and scalability of the algorithm via numerous simulation experiments and an analysis of species presence-absence data.

Supplementary material

 Supplementary Material
This supplementary material contains discussion on why is it infeasible to directly use a Monte Carlo sampling to estimate p ( Y ) in (4), evaluation of the algorithms under consideration with respect to misclassification error, and details of the code and data used in the article.

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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
binary data generalized linear mixed models spatial, Gaussian processes

Funding
Abhirup Datta was partially supported by National Institute of Environmental Health Sciences (NIEHS) grant R01 ES033739 and by National Science Foundation (NSF) Division of Mathematical Sciences grant DMS-1915803. Sudipto Banerjee was partially supported by the National Science Foundation (NSF) from grants NSF/DMS 1916349 and NSF/IIS 1562303, and by the National Institute of Environmental Health Sciences (NIEHS) from grants R01ES030210 and 5R01ES027027.

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