Abstract
The principles behind the interface to continuous domain spatial models in the RINLA software package for R are described. The integrated nested Laplace approximation (INLA) approach proposed by Rue, Martino, and Chopin (2009) is a computationally effective alternative to MCMC for Bayesian inference. INLA is designed for latent Gaussian models, a very wide and flexible class of models ranging from (generalized) linear mixed to spatial and spatio-temporal models. Combined with the stochastic partial differential equation approach (SPDE, Lindgren, Rue, and Lindstrom 2011), one can accommodate all kinds of geographically referenced data, including areal and geostatistical ones, as well as spatial point process data. The implementation interface covers stationary spatial models, non-stationary spatial models, and also spatio-temporal models, and is applicable in epidemiology, ecology, environmental risk assessment, as well as general geostatistics.
Highlights
Markov models in image analysis and spatial statistics have been largely confined to discrete spatial domains, such as lattices and regional adjacency graphs
As discussed in Lindgren et al (2011), one can express a large class of random field models as solutions to continuous domain stochastic partial differential equations (SPDEs), and write down explicit links between the parameters of each SPDE and the elements of precision matrices for weights in a discrete basis function representation
As shown by Whittle (1963), such models include those with Matern covariance functions, which are ubiquitous in traditional spatial statistics, but in contrast to covariance based models it is far easier to introduce non-stationarity into the SPDE models
Summary
When building and using hierarchical models with latent random fields it is important to remember that the latent fields often represent real-world phenomena that exist independently of whether they are observed in a given location or not. As discussed in the introduction, an alternative to traditional covariance based modelling is to use SPDEs, but carry out the practical computations using Gaussian Markov random field (GMRF) representations This is done by approximating the full set of spatial random functions with weighted sums of simple basis functions, which allows us to hold on to the continuous interpretation of space, while the computational algorithms only see discrete structures with Markov properties. Since τ and κ have a joint influence on the marginal variances of the resulting field, it is often more natural to construct the parameter model using the standard deviation σ and range ρ, where ρ = (8ν)1/2/κ is the distance for which the correlation functions have fallen to approximately 0.13, for all ν > 1/2. Wrapper functions for constructing such models are expected to be added in the future, as well as extensions for advection-diffusion models
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