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.

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