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The New England Journal of Statistics in Data Science


Modeling Multivariate Spatial Dependencies Using Graphical Models
Volume 1, Issue 2 (2023), pp. 283–295
Pub. online: 6 September 2023      Type: Spatial And Environmental Statistics      Open accessOpen Access
Accepted
30 May 2023
Published
6 September 2023

Abstract

Graphical models have witnessed significant growth and usage in spatial data science for modeling data referenced over a massive number of spatial-temporal coordinates. Much of this literature has focused on a single or relatively few spatially dependent outcomes. Recent attention has focused upon addressing modeling and inference for substantially large number of outcomes. While spatial factor models and multivariate basis expansions occupy a prominent place in this domain, this article elucidates a recent approach, graphical Gaussian Processes, that exploits the notion of conditional independence among a very large number of spatial processes to build scalable graphical models for fully model-based Bayesian analysis of multivariate spatial data.

References

[1] 
Apanasovich, T. V. and Genton, M. G. (2010). Cross-covariance functions for multivariate random fields based on latent dimensions. Biometrika 97(1) 15–30. https://doi.org/10.1093/biomet/asp078. MR2594414
[2] 
Apanasovich, T. V., Genton, M. G. and Sun, Y. (2012). A valid Matérn class of cross-covariance functions for multivariate random fields with any number of components. Journal of the American Statistical Association 107(497) 180–193. https://doi.org/10.1080/01621459.2011.643197. MR2949350
[3] 
Banerjee, S., Gelfand, A. E., Finley, A. O. and Sang, H. (2008). Gaussian Predictive Process Models for Large Spatial Datasets. Journal of the Royal Statistical Society, Series B 70 825–848. https://doi.org/10.1111/j.1467-9868.2008.00663.x. MR2523906
[4] 
Banerjee, S. (2017). High-Dimensional Bayesian Geostatistics. Bayesian Analysis 12 583–614. https://doi.org/10.1214/17-BA1056R. MR3654826
[5] 
Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2014) Hierarchical modeling and analysis for spatial data. CRC Press, Boca Raton, FL. MR3362184
[6] 
Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society: Series B (Methodological) 36(2) 192–225. MR0373208
[7] 
Besag, J., York, J. and Mollié, A. (1991). Bayesian image restoration, with two applications in spatial statistics. Annals of the institute of statistical mathematics 43(1) 1–20. https://doi.org/10.1007/BF00116466. MR1105822
[8] 
Carey, V., Long, L. and Gentleman, R. (2020). RBGL: An interface to the BOOST graph library. R package version 1.66.0. http://www.bioconductor.org.
[9] 
Chilés, J. and Delfiner, P. (1999) Geostatistics: Modeling Spatial Uncertainty. John Wiley: New York. https://doi.org/10.1002/9780470316993. MR1679557
[10] 
Cressie, N. (1993) Statistics for Spatial Data, Revised ed. Wiley-Interscience. https://doi.org/10.1002/9781119115151. MR1239641
[11] 
Cressie, N. and Wikle, C. K. (2015) Statistics for spatio-temporal data. John Wiley & Sons, Hoboken, NJ. MR2848400
[12] 
Cressie, N. and Zammit-Mangion, A. (2016). Multivariate spatial covariance models: a conditional approach. Biometrika 103(4) 915–935. https://academic.oup.com/biomet/article-pdf/103/4/915/8339199/asw045.pdf. https://doi.org/10.1093/biomet/asw045. MR3620448
[13] 
Datta, A., Banerjee, S., Finley, A. O., Hamm, N. A. S. and Schaap, M. (2016). Non-Separable Dynamic Nearest-Neighbor Gaussian Process Models For Large Spatio-Temporal Data With An Application To Particulate Matter Analysis. Annals of Applied Statistics 10 1286–1316. https://doi.org/10.1214/16-AOAS931. MR3553225
[14] 
Datta, A., Banerjee, S., Finley, A. O. and Gelfand, A. E. (2016). Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets. Journal of the American Statistical Association 111 800–812. https://doi.org/10.1080/01621459.2015.1044091. MR3538706
[15] 
Datta, A., Banerjee, S., Finley, A. O. and Gelfand, A. E. (2016). Hierarchical Nearest-Neighbor Gaussian Process Models For Large Geostatistical Datasets. Journal of the American Statistical Association 111(514) 800–812. https://doi.org/10.1080/01621459.2015.1044091. MR3538706
[16] 
Dempster, A. P. (1972). Covariance selection. Biometrics 157–175. MR3931974
[17] 
Dey, D., Datta, A. and Banerjee, S. (2021). Graphical Gaussian Process Models for Highly Multivariate Spatial Data. Biometrika. asab061. https://doi.org/10.1093/biomet/asab061. https://academic.oup.com/biomet/advance-article-pdf/doi/10.1093/biomet/asab061/41512896/asab061.pdf.
[18] 
Finley, A. O., Datta, A., Cook, B. C., Morton, D. C., Andersen, H. E. and Banerjee, S. (2019). Efficient algorithms for Bayesian Nearest Neighbor Gaussian Processes. Journal of Computational and Graphical Statistics 28(2) 401–414. https://doi.org/10.1080/10618600.2018.1537924. MR3974889
[19] 
Friedman, J., Hastie, T. and Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3) 432–441.
[20] 
Gamerman, D. and Moreira, A. R. (2004). Multivariate spatial regression models. Journal of multivariate analysis 91(2) 262–281. https://doi.org/10.1016/j.jmva.2004.02.016. MR2087846
[21] 
Gelfand, A. E. and Banerjee, S. (2010). Multivariate Spatial Process Models. In Handbook of Spatial Statistics (A. E. Gelfand, P. J. Diggle, M. Fuentes and P. Guttorp, eds.) 495–516 Boca Raton, FL: CRC Press. https://doi.org/10.1201/9781420072884-c28. MR2730963
[22] 
Gelfand, A. E. and Banerjee, S. (2010). Multivariate Spatial Process Models. In Handbook of Spatial Statistics (A. E. Gelfand, P. J. Diggle, M. Fuentes and P. Guttorp, eds.) 217–244 Boca Raton, FL: CRC Press. https://doi.org/10.1201/9781420072884-c28. MR2730963
[23] 
Gelfand, A. E. and Vounatsou, P. (2003). Proper multivariate conditional autoregressive models for spatial data analysis. Biostatistics 4(1) 11–15.
[24] 
Genton, M. G. and Kleiber, W. (2015). Cross-covariance functions for multivariate geostatistics. Statistical Science 147–163. https://doi.org/10.1214/14-STS487. MR3353096
[25] 
Gneiting, T., Kleiber, W. and Schlather, M. (2010). Matérn cross-covariance functions for multivariate random fields. Journal of the American Statistical Association 105(491) 1167–1177. https://doi.org/10.1198/jasa.2010.tm09420. MR2752612
[26] 
Goulard, M. and Voltz, M. (1992). Linear coregionalization model: tools for estimation and choice of cross-variogram matrix. Mathematical Geology 24(3) 269–286.
[27] 
Jin, X., Banerjee, S. and Carlin, B. P. (2007). Order-free co-regionalized areal data models with application to multiple-disease mapping. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69(5) 817–838. https://doi.org/10.1111/j.1467-9868.2007.00612.x. MR2368572
[28] 
Jin, X., Carlin, B. P. and Banerjee, S. (2005). Generalized hierarchical multivariate CAR models for areal data. Biometrics 61(4) 950–961. https://doi.org/10.1111/j.1541-0420.2005.00359.x. MR2216188
[29] 
Katzfuss, M. and Guinness, J. (2021). A General Framework for Vecchia Approximations of Gaussian Processes. Statistical Science 36(1) 124–141. https://doi.org/10.1214/19-STS755. https://doi.org/10.1214/19-STS755. MR4194207
[30] 
Krock, M., Kleiber, W., Hammerling, D. and Becker, S. (2021). Modeling massive multivariate spatial data with the basis graphical lasso. arXiv e-prints 2101.
[31] 
Lauritzen, S. L. (1996). Graphical Models. Clarendon Press, Oxford, United Kingdom. MR1419991
[32] 
Le, N., Sun, L. and Zidek, J. V. (2001). Spatial prediction and temporal backcasting for environmental fields having monotone data patterns. Canadian Journal of Statistics 29(4) 529–554. https://doi.org/10.2307/3316006. MR1888503
[33] 
Le, N. D. and Zidek, J. V. (2006) Statistical analysis of environmental space-time processes. Springer Science & Business Media. MR2223933
[34] 
Le, N. D., Sun, W. and Zidek, J. V. (1997). Bayesian multivariate spatial interpolation with data missing by design. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 59(2) 501–510. https://doi.org/10.1111/1467-9868.00081. MR1440593
[35] 
Lopes, H. F., Salazar, E. and Gamerman, D. (2008). Spatial Dynamic Factor Analysis. Bayesian Analysis 3(4) 759–792. https://doi.org/10.1214/08-BA329. MR2469799
[36] 
Majumdar, A. and Gelfand, A. E. (2007). Multivariate spatial modeling for geostatistical data using convolved covariance functions. Mathematical Geology 39(2) 225–245. https://doi.org/10.1007/s11004-006-9072-6. MR2324633
[37] 
Mardia, K. (1988). Multi-dimensional multivariate Gaussian Markov random fields with application to image processing. Journal of Multivariate Analysis 24(2) 265–284. https://doi.org/10.1016/0047-259X(88)90040-1. MR0926357
[38] 
Musgrove, D. (2016). Spatial Models for Large Spatial and Spatiotemporal Data.
[39] 
Peruzzi, M., Banerjee, S. and Finley, A. O. (2022). Highly Scalable Bayesian Geostatistical Modeling via Meshed Gaussian Processes on Partitioned Domains. Journal of the American Statistical Association 117(538) 969–982. https://doi.org/10.1080/01621459.2020.1833889. MR4436326
[40] 
Ren, Q. and Banerjee, S. (2013). Hierarchical factor models for large spatially misaligned datasets: A low-rank predictive process approach. Biometrics 69 19–30. https://doi.org/10.1111/j.1541-0420.2012.01832.x. MR3058048
[41] 
Rua, H. and Held, L. (2005) Gaussian Markov Random Fields: Theory and Applications. Monographs on statistics and applied probability. Chapman and Hall/CRC Press, Boca Raton, FL. http://opac.inria.fr/record=b1119989. https://doi.org/10.1201/9780203492024. MR2130347
[42] 
Saha, A. and Datta, A. (2018). BRISC: bootstrap for rapid inference on spatial covariances. Stat 7(1) 184. https://doi.org/10.1002/sta4.184. MR3805084
[43] 
Salvaña, M. L. O. and Genton, M. G. (2020). Nonstationary cross-covariance functions for multivariate spatio-temporal random fields. Spatial Statistics 100411. https://doi.org/10.1016/j.spasta.2020.100411. MR4109598
[44] 
Schmidt, A. M. and Gelfand, A. E. (2003). A Bayesian coregionalization approach for multivariate pollutant data. Journal of Geophysical Research: Atmospheres 108(D24).
[45] 
Speed, T. P., Kiiveri, H. T. et al. (1986). Gaussian Markov distributions over finite graphs. The Annals of Statistics 14(1) 138–150. https://doi.org/10.1214/aos/1176349846. MR0829559
[46] 
Sun, W., Le, N. D., Zidek, J. V. and Burnett, R. (1998). Assessment of a Bayesian multivariate interpolation approach for health impact studies. Environmetrics: The official journal of the International Environmetrics Society 9(5) 565–586.
[47] 
Taylor-Rodriguez, D., Finley, A. O., Datta, A., Babcock, C., Andersen, H. E., Cook, B. D., Morton, D. C. and Banerjee, S. (2019). Spatial factor models for high-dimensional and large spatial data: An application in forest variable mapping. Statistica Sinica 29(3) 1155–1180. MR3932513
[48] 
Ver Hoef, J. M. and Barry, R. P. (1998). Constructing and fitting models for cokriging and multivariable spatial prediction. Journal of Statistical Planning and Inference 69(2) 275–294. https://doi.org/10.1016/S0378-3758(97)00162-6. MR1631328
[49] 
Ver Hoef, J. M., Cressie, N. and Barry, R. P. (2004). Flexible spatial models for kriging and cokriging using moving averages and the Fast Fourier Transform (FFT). Journal of Computational and Graphical Statistics 13(2) 265–282. https://doi.org/10.1198/1061860043498. MR2063985
[50] 
Wackernagel, H. (2003) Multivariate Geostatistics, 3 ed. Springer-Verlag, Berlin.
[51] 
Xu, P. q. F., Guo, J. and He, X. (2011). An improved iterative proportional scaling procedure for Gaussian graphical models. Journal of Computational and Graphical Statistics 20(2) 417–431. https://doi.org/10.1198/jcgs.2010.09044. MR2847802
[52] 
Zapata, J., Oh, S. Y. and Petersen, A. (2021). Partial separability and functional graphical models for multivariate Gaussian processes. Biometrika. asab046. https://academic.oup.com/biomet/advance-article-pdf/doi/10.1093/biomet/asab046/41856973/asab046.pdf. https://doi.org/10.1093/biomet/asab046. MR4472841
[53] 
Zhang, L. and Banerjee, S. Spatial factor modeling: A Bayesian matrix-normal approach for misaligned data. Biometrics 78(2) 560–573. https://doi.org/10.1111/biom.13452. https://onlinelibrary.wiley.com/doi/pdf/10.1111/biom.13452.
[54] 
Zhu, H., Strawn, N. and Dunson, D. B. (2016). Bayesian Graphical Models for Multivariate Functional Data. Journal of Machine Learning Research 17(204) 1–27. MR3580357

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© 2023 New England Statistical Society
Open access article under the CC BY license.

Keywords
Bayesian inference Covariance selection Graphical models Graphical Gaussian Process Multivariate dependencies Spatial process models

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