Coming up with Bayesian models for spatial data is easy, but performing inference with them can be challenging. Writing fast inference code for a complex spatial model with realistically‐sized datasets from scratch is time‐consuming, and if changes are made to the model, there is little guarantee that the code performs well. The key advantages of R‐INLA are the ease with which complex models can be created and modified, without the need to write complex code, and the speed at which inference can be done even for spatial problems with hundreds of thousands of observations. R‐INLA handles latent Gaussian models, where fixed effects, structured and unstructured Gaussian random effects are combined linearly in a linear predictor, and the elements of the linear predictor are observed through one or more likelihoods. The structured random effects can be both standard areal model such as the Besag and the BYM models, and geostatistical models from a subset of the Matérn Gaussian random fields. In this review, we discuss the large success of spatial modeling with R‐INLA and the types of spatial models that can be fitted, we give an overview of recent developments for areal models, and we give an overview of the stochastic partial differential equation (SPDE) approach and some of the ways it can be extended beyond the assumptions of isotropy and separability. In particular, we describe how slight changes to the SPDE approach leads to straight‐forward approaches for nonstationary spatial models and nonseparable space–time models. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical Models > Bayesian Models Data: Types and Structure > Massive Data

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The classical tools in spatial statistics are stationary models, like the Matérn field. However, in some applications there are boundaries, holes, or physical barriers in the study area, e.g. a coastline, and stationary models will inappropriately smooth over these features, requiring the use of a non-stationary model.We propose a new model, the Barrier model, which is different from the established methods as it is not based on the shortest distance around the physical barrier, nor on boundary conditions. The Barrier model is based on viewing the Matérn correlation, not as a correlation function on the shortest distance between two points, but as a collection of paths through a Simultaneous Autoregressive (SAR) model. We then manipulate these local dependencies to cut off paths that are crossing the physical barriers. To make the new SAR well behaved, we formulate it as a stochastic partial differential equation (SPDE) that can be discretised to represent the Gaussian field, with a sparse precision matrix that is automatically positive definite.The main advantage with the Barrier model is that the computational cost is the same as for the stationary model. The model is easy to use, and can deal with both sparse data and very complex barriers, as shown in an application in the Finnish Archipelago Sea. Additionally, the Barrier model is better at reconstructing the modified Horseshoe test function than the standard models used in R-INLA.

This study evaluates the dynamics of soil organic carbon (SOC) under perennial crops across the globe. It quantifies the effect of change from annual to perennial crops and the subsequent temporal changes in SOC stocks during the perennial crop cycle. It also presents an empirical model to estimate changes in the SOC content under crops as a function of time, land use, and site characteristics. We used a harmonized global dataset containing paired‐comparison empirical values of SOC and different types of perennial crops (perennial grasses, palms, and woody plants) with different end uses: bioenergy, food, other bio‐products, and short rotation coppice. Salient outcomes include: a 20‐year period encompassing a change from annual to perennial crops led to an average 20% increase in SOC at 0–30 cm (6.0 ± 4.6 Mg/ha gain) and a total 10% increase over the 0–100 cm soil profile (5.7 ± 10.9 Mg/ha). A change from natural pasture to perennial crop decreased SOC stocks by 1% over 0–30 cm (−2.5 ± 4.2 Mg/ha) and 10% over 0–100 cm (−13.6 ± 8.9 Mg/ha). The effect of a land use change from forest to perennial crops did not show significant impacts, probably due to the limited number of plots; but the data indicated that while a 2% increase in SOC was observed at 0–30 cm (16.81 ± 55.1 Mg/ha), a decrease in 24% was observed at 30–100 cm (−40.1 ± 16.8 Mg/ha). Perennial crops generally accumulate SOC through time, especially woody crops; and temperature was the main driver explaining differences in SOC dynamics, followed by crop age, soil bulk density, clay content, and depth. We present empirical evidence showing that the FAO perennialization strategy is reasonable, underscoring the role of perennial crops as a useful component of climate change mitigation strategies.

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We investigate earthquake-induced landslides using a geostatistical model featuring a latent spatial effect (LSE). The LSE represents the spatially structured residuals in the data, which remain after adjusting for covariate effects. To determine whether the LSE captures the residual signal from a given trigger, we test the LSE in reproducing the pattern of seismic shaking from the distribution of seismically induced landslides, without prior knowledge of the earthquake being included in the model. We assessed the landslide intensity, that is, the expected number of landslides per mapping unit, for the area in which landslides triggered by the Wenchuan and Lushan earthquakes overlap. We examined this area to test our method on landslide inventories located in near and far fields of the earthquake. We generated three models for both earthquakes: (i) seismic parameters only (proxy for the trigger); (ii) the LSE only; and (iii) both seismic parameters and the LSE. The three configurations share the same morphometric covariates. This allowed us to study the LSE pattern and assess whether it approximated the seismic effects. Our results show that the LSE reproduced the shaking patterns for both earthquakes. In addition, the models including the LSE perform better than conventional models featuring seismic parameters only. Due to computational limitations we carried out a detailed analysis for a relatively small area (2,112 km 2 ), using a data set with higher spatial resolution. Results were consistent with those of a subsequent analysis for a larger area (14,648 km 2 ) using coarser-resolution data.

The INLA package provides a tool for computationally efficient Bayesian modeling and inference for various widely used models, more formally the class of latent Gaussian models. It is a non-sampling based framework which provides approximate results for Bayesian inference, using sparse matrices. The swift uptake of this framework for Bayesian modeling is rooted in the computational efficiency of the approach and catalyzed by the demand presented by the big data era. In this paper, we present new developments within the INLA package with the aim to provide a computationally efficient mechanism for the Bayesian inference of relevant challenging situations.

With modern high-dimensional data, complex statistical models are necessary, requiring computationally feasible inference schemes. We introduce Max-and-Smooth, an approximate Bayesian inference scheme for a flexible class of latent Gaussian models (LGMs) where one or more of the likelihood parameters are modeled by latent additive Gaussian processes. Our proposed inference scheme is a two-step approach. In the first step (Max), the likelihood function is approximated by a Gaussian density with mean and covariance equal to either (a) the maximum likelihood estimate and the inverse observed information, respectively, or (b) the mean and covariance of the normalized likelihood function. In the second step (Smooth), the latent parameters and hyperparameters are inferred and smoothed with the approximated likelihood function. The proposed method ensures that the uncertainty from the first step is correctly propagated to the second step. Because the prior density for the latent parameters is assumed to be Gaussian and the approximated likelihood function is Gaussian, the approximate posterior density of the latent parameters (conditional on the hyperparameters) is also Gaussian, thus facilitating efficient posterior inference in high dimensions. Furthermore, the approximate marginal posterior distribution of the hyperparameters is tractable, and as a result, the hyperparameters can be sampled independently of the latent parameters. We show that the computational cost of Max-and-Smooth is close to being insensitive to the number of independent data replicates, and that it scales well with increased dimension of the latent parameter vector provided that its Gaussian prior density is specified with a sparse precision matrix. In the case of a large number of independent data replicates, sparse precision matrices, and high-dimensional latent vectors, the speedup is substantial in comparison to an MCMC scheme that infers the posterior density from the exact likelihood function. The accuracy of the Gaussian approximation to the likelihood function increases with the number of data replicates per latent model parameter. The proposed inference scheme is demonstrated on one spatially referenced real dataset and on simulated data mimicking spatial, temporal, and spatio-temporal inference problems. Our results show that Max-and-Smooth is accurate and fast.