Bayesian Inference for Multivariate Spatial Models with INLA

Bayesian methods and software for spatial data analysis are well-established now in the broader scientific community generally and in the spatial data analysis community specifically. Despite the wide application of spatial models, the analysis of multivariate spatial data using the integrated nested Laplace approximation through its R package (R-INLA) has not been widely described in the existing literature. Therefore, the main objective of this article is to demonstrate that R-INLA is a convenient toolbox to analyse different types of multivariate spatial datasets. This will be illustrated by analysing three datasets which are publicly available. Furthermore, the details and the R code of these analyses are provided to exemplify how to fit models to multivariate spatial datasets with R-INLA.

Francisco Palmí-Perales (Department of Statistics and Operational Research, Faculty of Mathematics, Universitat de València) , Virgilio Gómez-Rubio (Department of Mathematics, E.T.S. Ingeniería Industrial-Albacete, Universidad de Castilla-La Mancha) , Roger S. Bivand (Department of Economics, Norwegian School of Economics) , Michela Cameletti (Department of Economics, Universitá degli studi di Bergamo) , Håvard Rue (King Abdullah University of Science and Technology)

0.1 Supplementary materials

Supplementary materials are available in addition to this article. It can be downloaded at

A. Baddeley, E. Rubak and R. Turner. Spatial point patterns: Methodology and applications with R. London: Chapman & Hall/CRC Press, 2015. URL
H. Bakka, H. Rue, G.-A. Fuglstad, A. Riebler, D. Bolin, E. Krainski, D. Simpson and F. Lindgren. Spatial modelling with R-INLA: A review. WIREs Comput Stat, 10(6): 1–24, 2018. URL
S. Banerjee, B. P. Carlin and A. E. Gelfand. Hierarchical modeling and analysis for spatial data. CRC press, 2014.
M. Blangiardo and M. Cameletti. Spatial and spatio-temporal Bayesian models with R-INLA. John Wiley & Sons, 2015.
P. J. Diggle, P. Moraga, B. Rowlingson and B. M. Taylor. Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical Paradigm. Statistical Science, 28(4): 542–563, 2013. URL
A. O. Finley, S. Banerjee and B. P. Carlin. spBayes: An R package for univariate and multivariate hierarchical point-referenced spatial models. Journal of Statistical Software, 19(4): 1, 2007. URL
A. O. Finley, S. Banerjee and A. E.Gelfand. spBayes for large univariate and multivariate point-referenced spatio-temporal data models. Journal of Statistical Software, 63(13): 1–28, 2015. URL
A. Gelman. Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Analysis, 1(3): 515–534, 2006. URL
W. R. Gilks, S. Richardson and D. J. Spiegelhalter. Markov chain monte carlo in practice. Chapman & Hall, 1995.
V. Gómez-Rubio. Bayesian inference with INLA. CRC Press, 2020.
V. Gómez-Rubio, M. Cameletti and F. Finazzi. Analysis of massive marked point patterns with stochastic partial differential equations. Spatial Statistics, 14: 179–196, 2015. URL
V. Gómez-Rubio and F. Palmí-Perales. Multivariate posterior inference for spatial models with the integrated nested Laplace approximation. Journal of the Royal Statistical Society, Series C, 68(1): 199–215, 2019. URL
V. Gómez-Rubio, F. Palmí-Perales, G. López-Abente, R. Ramis-Prieto and P. Fernández-Navarro. Bayesian joint spatio-temporal analysis of multiple diseases. SORT, 1: 51–74, 2019. URL
V. Gómez-Rubio, P. Zheng, P. Diggle, D. C. Sterratt, R. D. Peng, D. Murdoch and B. Rowlingson. spatialkernel: Non-parametric estimation of spatial segregation in a multivariate point process. 2017. URL R package version 0.4-23.
E. T. Krainski, V. Gómez-Rubio, H. Bakka, A. Lenzi, D. Castro-Camilo, D. Simpson, F. Lindgren and H. Rue. Advanced spatial modeling with stochastic partial differential equations using R and INLA. Boca Raton, FL: Chapman & Hall/CRC, 2019.
D. Lee. CARBayes: An R package for Bayesian spatial modeling with conditional autoregressive priors. Journal of Statistical Software, 55(13): 1–24, 2013. URL
F. Lindgren, H. Rue, et al. Bayesian spatial modelling with R-INLA. Journal of Statistical Software, 63(19): 1–25, 2015. URL
F. Lindgren, H. Rue and J. Lindström. An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(4): 423–498, 2011. URL
Y. C. MacNab. Some recent work on multivariate Gaussian Markov random fields. Test, 27(3): 497–541, 2018. URL
M. Martínez-Beneito. A general modelling framework for multivarite disease mapping. Biometrika, 100(3): 539–553, 2013. URL
M. A. Martínez-Beneito, P. Botella-Rocamora and S. Banerjee. Towards a multidimensional approach to Bayesian disease mapping. Bayesian analysis, 12(1): 239, 2017. URL
M. A. Martı́nez-Beneito and P. Botella-Rocamora. Disease Mapping: From Foundations to Multidimensional Modeling. CRC Press, 2019.
J. Møller, A. R. Syversveen and R. P. Waagepetersen. Log Gaussian Cox Processes. Scandinavian Journal of Statistics, 25(3): 451–482, 1998. URL
F. Palmí-Perales, V. Gómez-Rubio, G. López-Abente, R. Ramis, J. M. Sanz-Anquela and P. Fernández-Navarro. Approximate Bayesian inference for multivariate point pattern analysis in disease mapping. Biometrical Journal, 63(3): 632–649, 2021a. URL
F. Palmí-Perales, V. Gómez-Rubio and M. A. Martínez-Beneito. Bayesian Multivariate Spatial Models for Lattice Data with INLA. Journal of Statistical Software, 98(2): 1–29, 2021b. URL
E. J. Pebesma. Multivariable geostatistics in S: The gstat package. Computers & Geosciences, 30(7): 683–691, 2004.
E. J. Pebesma and C. G. Wesseling. Gstat: A Program for Geostatistical Modelling, Prediction and Simulation. Computers & Geosciences, 24(1): 17–31, 1998.
R Core Team. R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing, 2022. URL
H. Rue and L. Held. Gaussian markov random fields. Theory and applications. New York: Chapman & Hall, 2005.
H. Rue, F. Lindgren, D. Simpson, S. Martino, E. Teixeira Krainski, H. Bakka, A. Riebler and G.-A. Fuglstad. INLA: Full bayesian analysis of latent gaussian models using integrated nested laplace approximations. 2020. R package version 20.03.17.
H. Rue, S. Martino and N. Chopin. Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. Journal of the royal statistical society: Series b (statistical methodology), 71(2): 319–392, 2009. URL
D. Simpson, J. Illian, F. Lindgren, S. H. Sørbye and H. Rue. Going off grid: Computationally efficient inference for log-Gaussian Cox processes. Biometrika, 103(1): 49–70, 2016. URL
D. Simpson, H. Rue, A. Riebler, T. G. Martins and S. H. Sørbye. Penalising model component complexity: A principled, practical approach to constructing priors. Statistical Science, 32(1): 1–28, 2017. URL
S. H. Sørbye, J. B. Illian, D. P. Simpson, D. Burslem and H. Rue. Careful prior specification avoids incautious inference for log-Gaussian Cox point processes. Journal of the Royal Statistical Society: Series C (Applied Statistics), 68(3): 543–564, 2019. URL
M. N. M. Van Lieshout and A. J. Baddeley. Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics, 26(4): 511–532, 1999. URL
R. Waagepetersen, Y. Guan, A. Jalilian and J. Mateu. Analysis of multispecies point patterns by using multivariate log-Gaussian Cox processes. Journal of the Royal Statistical Society. Series C (Applied Statistics), 65(1): 77–96, 2016. URL



Text and figures are licensed under Creative Commons Attribution CC BY 4.0. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".


For attribution, please cite this work as

Palmí-Perales, et al., "Bayesian Inference for Multivariate Spatial Models with INLA", The R Journal, 2023

BibTeX citation

  author = {Palmí-Perales, Francisco and Gómez-Rubio, Virgilio and Bivand, Roger S. and Cameletti, Michela and Rue, Håvard},
  title = {Bayesian Inference for Multivariate Spatial Models with INLA},
  journal = {The R Journal},
  year = {2023},
  note = {},
  doi = {10.32614/RJ-2023-068},
  volume = {15},
  issue = {3},
  issn = {2073-4859},
  pages = {172-190}