Latent Gaussian modeling and INLA: A review with focus on space-time applications

Opitz, Thomas

Journal de la société française de statistique, Tome 158 (2017) no. 3, pp. 62-85.

Les modèles bayésiens hiérarchiques structurés par un processus gaussien latent sont largement utilisés dans la pratique statistique pour caractériser des comportements stochastiques complexes et des structures hiérarchiques dans les données en grande dimension, souvent spatiales ou spatio-temporelles. Si des méthodes d’inférence bayésienne de type MCMC, basées sur la simulation de la loi a posteriori, sont souvent entravées par une covergence lente et des instabilités numériques, l’approche inférentielle par INLA (« Integrated Nested Laplace Approximation ») utilise des approximations analytiques, souvent très précises et relativement rapides, afin de calculer des quantités liées aux lois a posteriori d’intérêt. Cette technique s’appuie fortement sur des structures de dépendance de type Gauss–Markov afin d’éviter des difficultés numériques dans les calculs matriciels en grande dimension. En mettant l’accent sur les applications spatio-temporelles, nous discutons ici les principales notions théoriques, les classes de modèles accessibles et les outils d’inférence dans le contexte d’INLA. Certains champs Markoviens Gaussiens, obtenus comme solution approximative d’une équation différentielle partielle stochastique, sont la base de la modélisation spatio-temporelle. Pour illustrer l’utilisation pratique du logiciel R-INLA et la syntaxe de ses commandes principales, un scénario de simulation-réestimation est présenté en détail, basé sur des données simulées, spatio-temporelles et non gaussiennes, avec une structure de dépendance autorégressive dans le temps.

Bayesian hierarchical models with latent Gaussian layers have proven very flexible in capturing complex stochastic behavior and hierarchical structures in high-dimensional spatial and spatio-temporal data. Whereas simulation-based Bayesian inference through Markov Chain Monte Carlo may be hampered by slow convergence and numerical instabilities, the inferential framework of Integrated Nested Laplace Approximation (INLA) is capable to provide accurate and relatively fast analytical approximations to posterior quantities of interest. It heavily relies on the use of Gauss–Markov dependence structures to avoid the numerical bottleneck of high-dimensional nonsparse matrix computations. With a view towards space-time applications, we here review the principal theoretical concepts, model classes and inference tools within the INLA framework. Important elements to construct space-time models are certain spatial Matérn-like Gauss–Markov random fields, obtained as approximate solutions to a stochastic partial differential equation. Efficient implementation of statistical inference tools for a large variety of models is available through the INLA package of the R software. To showcase the practical use of R-INLA and to illustrate its principal commands and syntax, a comprehensive simulation experiment is presented using simulated non Gaussian space-time count data with a first-order autoregressive dependence structure in time.

Keywords: Integrated Nested Laplace Approximation, R-INLA, spatio-temporal statistics
Mot clés : Integrated Nested Laplace Approximation, R-INLA, statistique spatio-temporelle

[1] Blangiardo, Marta; Cameletti, Michela Spatial and Spatio-temporal Bayesian Models with R-INLA, John Wiley & Sons, 2015 | MR

[2] Blangiardo, Marta; Cameletti, Michela; Baio, Gianluca; Rue, Håvard Spatial and spatio-temporal models with R-INLA, Spatial and spatio-temporal epidemiology, Volume 7 (2013), pp. 39-55

[3] Bisanzio, Donal; Giacobini, Mario; Bertolotti, Luigi; Mosca, Andrea; Balbo, Luca; Kitron, Uriel; Vazquez-Prokopec, Gonzalo M. Spatio-temporal patterns of distribution of West Nile virus vectors in eastern Piedmont Region, Italy, Parasites & Vectors, Volume 4 (2011) | DOI

[4] Bolin, David; Lindgren, Finn Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping, The Annals of Applied Statistics (2011), pp. 523-550 | MR | Zbl

[5] Bakka, Haakon; Vanhatalo, Jarno; Illian, Janine; Simpson, Daniel; Rue, Håvard Accounting for physical barriers in species distribution modeling with non-stationary spatial random effects, arXiv preprint arXiv:1608.03787 (2016)

[6] Cosandey-Godin, Aurelie; Krainski, Elias Teixeira; Worm, Boris; Flemming, Joanna Mills Applying Bayesian spatiotemporal models to fisheries bycatch in the Canadian Arctic, Canadian Journal of Fisheries and Aquatic Sciences, Volume 72 (2014) no. 999, pp. 1-12

[7] Cameletti, Michela; Lindgren, Finn; Simpson, Daniel; Rue, Håvard Spatio-temporal modeling of particulate matter concentration through the SPDE approach, AStA Advances in Statistical Analysis, Volume 97 (2013) no. 2, pp. 109-131 | MR | Zbl

[8] Ferkingstad, Egil; Rue, Håvard Improving the INLA approach for approximate Bayesian inference for latent Gaussian models, Electronic Journal of Statistics, Volume 9 (2015) no. 2, pp. 2706-2731 | MR | Zbl

[9] Fong, Youyi; Rue, Håvard; Wakefield, Jon Bayesian inference for generalized linear mixed models, Biostatistics, Volume 11 (2010) no. 3, pp. 397-412 | Zbl

[10] Gelman, Andrew; Hwang, Jessica; Vehtari, Aki Understanding predictive information criteria for Bayesian models, Statistics and Computing, Volume 24 (2014) no. 6, pp. 997-1016 | MR | Zbl

[11] Gabriel, Edith; Opitz, Thomas; Bonneu, Florent Detecting and modeling multi-scale space-time structures: the case of wildfire occurrences (2016) Submitted to Journal de la Société Française de Statistique (Special Issue on Space-Time Statistics) | Numdam | MR | Zbl

[12] Gómez-Rubio, Virgilio; Bivand, Roger; Rue, Håvard A new latent class to fit spatial econometrics models with Integrated Nested Laplace Approximations, Procedia Environmental Sciences, Volume 27 (2015), pp. 116-118

[13] Gómez-Rubio, Virgilio; Cameletti, Michela; Finazzi, Francesco Analysis of massive marked point patterns with stochastic partial differential equations, Spatial Statistics, Volume 14 (2015), pp. 179-196 | MR

[14] Hu, Xiangping; Simpson, Daniel; Lindgren, Finn; Rue, Håvard Multivariate Gaussian random fields using systems of stochastic partial differential equations, arXiv preprint arXiv:1307.1379 (2013)

[15] Held, Leonhard; Schrödle, Birgit; Rue, Håvard Posterior and cross-validatory predictive checks: a comparison of MCMC and INLA, Statistical modelling and regression structures, Springer, 2010, pp. 91-110 | MR

[16] Ingebrigtsen, Rikke; Lindgren, Finn; Steinsland, Ingelin Spatial models with explanatory variables in the dependence structure, Spatial Statistics, Volume 8 (2014), pp. 20-38 | MR

[17] Illian, Janine B; Sørbye, Sigrunn H; Rue, Håvard A toolbox for fitting complex spatial point process models using integrated nested Laplace approximation (INLA), The Annals of Applied Statistics, Volume 6 (2012) no. 4, pp. 1499-1530 | MR | Zbl

[18] Lindgren, Finn; Rue, Håvard Bayesian spatial modelling with R-INLA, Journal of Statistical Software, Volume 63 (2015) no. 19

[19] Lindgren, Finn; Rue, Håvard; Lindström, Johan 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), Volume 73 (2011) no. 4, pp. 423-498 | MR | Zbl

[20] Martino, Sara; Akerkar, Rupali; Rue, Havard Approximate Bayesian inference for survival models, Scandinavian Journal of Statistics, Volume 38 (2011) no. 3, pp. 514-528 | MR | Zbl

[21] Martins, Thiago G; Simpson, Daniel; Lindgren, Finn; Rue, Håvard Bayesian computing with INLA: new features, Computational Statistics & Data Analysis, Volume 67 (2013), pp. 68-83 | MR | Zbl

[22] Rue, Havard; Held, Leonhard Gaussian Markov random fields: theory and applications, CRC Press, 2005 | MR | Zbl

[23] Rue, Håvard; Martino, Sara; Chopin, Nicolas Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations, Journal of the Royal Statistical Society: Series B, Volume 71 (2009) no. 2, pp. 319-392 | MR | Zbl

[24] Rue, Håvard; Riebler, Andrea; Sørbye, Sigrunn H; Illian, Janine B; Simpson, Daniel P; Lindgren, Finn K Bayesian computing with INLA: a review, Annual Review of Statistics and Its Application, Volume 1 (2016)

[25] Rue, Havard Marginal variances for Gaussian Markov random fields (2005) (Technical report Norwegian Institute of Science and of Technology, Trondheim)

[26] Schrödle, Birgit; Held, Leonhard Spatio-temporal disease mapping using INLA, Environmetrics, Volume 22 (2011) no. 6, pp. 725-734 | MR

[27] Schrödle, Birgit; Held, Leonhard; Rue, Håvard Assessing the impact of a movement network on the spatiotemporal spread of infectious diseases, Biometrics, Volume 68 (2012) no. 3, pp. 736-744 | MR | Zbl

[28] Simpson, Daniel P; Rue, Håvard; Martins, Thiago G; Riebler, Andrea; Sørbye, Sigrunn H Penalising model component complexity: A principled, practical approach to constructing priors, arXiv preprint arXiv:1403.4630 (2014) | MR | Zbl

[29] Serra, Laura; Saez, Marc; Mateu, Jorge; Varga, Diego; Juan, Pablo; Díaz-Ávalos, Carlos; Rue, Håvard Spatio-temporal log-Gaussian Cox processes for modelling wildfire occurrence: the case of Catalonia, 1994–2008, Environmental and Ecological Statistics, Volume 21 (2014) no. 3, pp. 531-563 | MR

[30] Saumard, Adrien; Wellner, Jon A Log-concavity and strong log-concavity: a review, Statistics surveys, Volume 8 (2014), pp. 45-114 | DOI | MR | Zbl

[31] Tierney, Luke; Kadane, Joseph B Accurate approximations for posterior moments and marginal densities, Journal of the American Statistical Association, Volume 81 (1986) no. 393, pp. 82-86 | MR | Zbl

[32] Watanabe, Sumio Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory, The Journal of Machine Learning Research, Volume 11 (2010), pp. 3571-3594 | MR | Zbl