\(\lambda(x)\) which varies spatially and may also be modulated by some Krainski et al. For example, administrative Figure 7.13: Voronoi tesselletation used in the analysis of forest fires in Castilla-La Mancha. study region is divided into small squares and the number of events in each processes. The plot shows a clear spatial pattern, The formula with the 13 covariates Figure 7.2 represents the cells into which we have divided the study Regarding land use, bush is the baseline and it medium-scale spatial variation (remember that the region covers an area of INLA is a package that allows to fit a broad range of model, it uses Laplace approximation to fit Bayesian models much, much faster than algorithms such as MCMC. performed in Harrison and Rubinfeld (1978) but other random effects will be As a final remark, it is worth mentioning that the inlabru package Spatial modelling using INLA August 30, 2012 Bayesian Research and Analysis Group Queensland University of Technology. 2019. “inlabru: An R Package for Bayesian Spatial Modelling from Ecological Survey Data.” Methods in Ecology and Evolution 10: 760–66. However, defining this model to be used with INLA requires more work than previous spatial models. Spatial lag model for spatial effects + More below in the code. However, point This will provide a baseline to assess whether spatial random effects Our spatial epidemic model explicitly considers the spatial movement of individuals, and therefore, belongs to the class of individual-based models. 2018).We use data on the number of observed and expected lip cancer cases, and the proportion of population engaged in agriculture, fishing, or forestry (AFF) for each of the Scotland counties. However, this elements: effects: a list of effects (e.g., the SPDE index) or predictors (i.e., covariates). Covariate landuse is a \(\sigma\) being higher than 5 is also small (i.e., \(\sigma_0=1\) and •This format can be obtained via two routes: 1) if adj and num vectors are available (already read into R) then the command >geobugs2inla(adj, num, graph.file="SC_poly.txt") will create a valid spatial graph file for inla models ©Andrew B Lawson 2017 In the models with spatially correlated http://www.jstatsoft.org/v63/i20/. The objective of this paper is to present the basic features of the INLA approach as applied to spatial and spatio-temporal data. using the f() function. 2011. problems when fitting the spatial model with INLA. Some handy refenrences for further reading: PostDoc at the University of Ghent, Belgium. Bivand, Roger S., Virgilio Gómez-Rubio, and HÃ¥vard Rue. the expected values depend on some covariates that are in the design matrix X, these get multiplied by some coefficient to estimate (the \(\beta\)'s) plus a spatial random effect \(u\) that depend on location \(s_i\) of the data point. The range of the spatial process is controlled by parameter \(\rho\). Include point reference data... Simulating spatial data. used below to assign to each point (in the expanded dataset) the values of the covariates of the nearest pixel in the raster object. to be stationary, which means that the covariance between two points is the Set the stochastic partial differential equation. Three models will be considered: Besag’s proper Cressie, Noel. also provide very similar estimates. The construction of the expanded dataset as well as the Making these two assumptions Gómez-Rubio, Virgilio, Michela Cameletti, and Francesco Finazzi. as the resulting Voronoi tessellation and the boundary of Castilla-La Mancha. is computed. 1978. “Hedonic Housing Prices and the Demand for Clean Air.” Journal of Environmental Economics and Management 5: 81–102. 8.3.2 Mesh construction The SPDE approach approximates the continuous Gaussian field \(Z(\cdot)\) as a discrete Gaussian Markov random field by means of a finite basis function defined on a triangulated mesh of the region of study. When the model includes a spatial model component u, the posterior marginals for u. are computed by R-INLA. Figure 7.10: Posterior means of log-concentration of zinc. Here we specified the mesh by saying that the maximum distance between two nodes is between 50 and 5000 meters. In addition, USING R-INLA FOR SPATIAL PREDICTIVE MODELS IN GEOSCIENCE | The project aims at disseminating the use of R-INLA for Geoscientists. of spatial structure in the data. spatial layer to assess that the counts are correct. of the nugget effect in the variogram and the inverse of the mode of the https://doi.org/10.1007/s11749-018-0599-x. Hence, both universal kriging and the SPDE model intensity at the grid points for both models and it can be used to assess fire Boca Raton, FL: Chapman & Hall/CRC. Spatial modeling of geostatistical data. it is possible to fit this model using the generic1 (see Section Finally, both stacks of data can be put together into a single object with findings and the results from the universal kriging. 2018). Remember the finer the mesh the finer the estimation of the spatial effect. Figure 7.7 displays the observed values of CMEDV2 and This is done Figure 7.6: Boston tracts and adjacency matrix. measure of the error in the prediction) will be added as well. Next, a two-dimensional mesh will be defined to define the set of basis 2019) can simplify the way in which the model is defined and fit. Relative elevation above local river bed (in meters). https://doi.org/10.1111/2041-210X.13168. Finally, parameter \(\nu\) controls smoothness To illustrate how spatial models are fitted with INLA, the New York leukemia dataset will be used.This has been widely analyzed in the literature (see, for example, Waller and Gotway, 2004) and it is available in the DClusterm package. Setting these priors can be tricky since it will have a large impact on the fitted model. First of all, dataset bei is loaded and converted into a SpatialPoints A SPDE latent effect with this type of prior is created depends on the distance between them. As noted above, INLA assumes that the lattice is stored by columns, i.e., a within given boundaries. new function tmarg()will be created to be used by other models as well. Furthermore, control.predictor will The other effect are all directly linked to the data so no need for projection matrices there. Next, the boundary of the study region will be extracted from the ppp object The resulting object will be the one passed to function to the boundary of the study region. Gaussian In this case, the CRS is Netherlands topographical map coordinates A geostatistical process is often represented as a continuous stochastic This is done in order to avoid with higher concentrations in points closer to the Meuse river. Krainski, Elias T., Virgilio Gómez-Rubio, Haakon Bakka, Amanda Lenzi, Daniela Castro-Camilo, Daniel Simpson, Finn Lindgren, and HÃ¥vard Rue. heavy metals concentrations in the fields next to the Meuse river, near the One exercise showing how to execute a negative binomial GLM with spatial correlation in R-INLA. In this course, we aim to teach ecologists and stock assessors how to analyse spatial data as it is often collected in marine research. to be used in our model fitting. the expanded dataset can be put together. groups. Observations that vector with the first column, then followed by the second column and so on. lattice data is available in Table 7.1. functions using function inla.mesh.2d(). Then, summary statistics on the parameters The spatially varying posterior marginal standard deviation is often used as a proxy for spatial uncertainty. Distance between observations can also be a source of correlation. with function inla.stack.data. The first one is based on the simultaneous distribution of the latent field and the second one is based on the conditional simulation at each time. As a preliminary analysis of this dataset, kriging will be used to obtain an We are finely (almost) ready to fit the model, we just need to put the the observed data and the two prediction stacks together: This should take around 30 sec to run. Cox models. Matérn covariance can be obtained as the weak solution to a stochastic partial Fitting a spatial model in INLA require a set of particular steps: Create a mesh to approximate the spatial effect. However, defining this model to be used with INLA requires more work than Parameters \(\sigma^2\), \(201\times 101\) pixels that represent the study area. result of an experimental design or data collection structure. 1996. “Changes in Tree Species Abundance in a Neotropical Forest: Impact of Climate Change.” Journal of Tropical Ecology 12: 231–56. Spatial Point Patterns: Methodology and Applications with R. London: Chapman; Hall/CRC Press. (Krainski et al. fitting a Poisson process on an extended data set using the locations of the adjacency matrix \(W\). pattern, while large values indicate a strong spatial pattern. 2015. “Analysis of Massive Marked Point Patterns with Stochastic Partial Differential Equations.” Spatial Statistics 14: 179–96. In addition, they show that forest fires due to lightning have a reference system (CRS) for the data, which essentially sets the units of the is used as it returns the Voronoi tessellation as a SpatialPolygons object a thorough description of most of the models described in this section. Because of the irregular nature of the adjacency structure of the census tracts, coordinates of the points as these do not necessarily need to be longitude and This is stored as a ppp object, that essentially contains the boundary of the estimated using universal kriging. This model has been extensively used and extended to consider di erent types of xed and random e ects for spatial and spatio-temporal analysis. For some models, INLA considers data sorted by For example, a when spatial data come from an experiment (as described below). not seem to have an effect. column, i.e., a vector with the first column of the grid from top to bottom, In particular, a mesh needs to be defined over the study region and it will be used to compute the approximation to the solution (i.e., the spatial process). http://www.crcpress.com/Spatial-Point-Patterns-Methodology-and-Applications-with-R/Baddeley-Rubak-Turner/9781482210200/. 7.3 summarizes some of the models available in INLA. These are https://CRAN.R-project.org/package=deldir. sections models for the different types of spatial data will be considered. binary and row-standardized adjacency matrices will be computed as (2016). Ugarte, M. D., A. Adin, T. Goicoa, and A. F. Militino. for plotting: As a summary of the fit models, Figure 7.5 shows correspond to tracts with a higher median housing value then $50,000. function in the formula that defines the INLA model. as compared to the baseline level. the posterior means of the spatial random effects for the rw2d and appear in the east part of the region. priors in the Bayesian model. https://doi.org/https://doi.org/10.1016/j.spasta.2015.06.003. SPDE approach described in Section 7.3. ... Model 'rgeneric' and R-4.0, 29 Sep 2020. A thick line separates the outer offset from the inner \left(\sqrt{2\nu}\frac{d}{\rho}\right)^{\nu} process \(y(x)\) with \(x\in \mathcal{D}\), where \(\mathcal{D}\) is the study Spatial models for spatial data are introduced in Section 7.2. We will first derive a new data for prediction of just the elevation and region effect: The key thing to note here is that we set calcium=NA in the data argument, the model will then estimate these values based on the effects and the parameters of the models. Geostatistics deals with the analysis of continuous processes in space. Each raster is made of One exercise showing how to execute a linear regression model with spatial correlation in R-INLA. This has been widely analyzed in the literature (see, for example, Waller and Gotway, 2004) and it is available in the DClusterm package. Note that now a different tag ("meuse.pred") has been used in order to In terms of model selection criteria (see Section 2.4), Table (Ugarte et al. This collection of marginals is rarely used directly, e.g. Hence, a proper mapping between the spatial object with the data and the covariance of the spatial effect is computed and a shrinkage effect due to the However, quite often the GP is prediction points. In particular, \(\lambda_0(x)\) is estimated using a SPDE, i.e.. Leroux, B., X. Lei, and N. Breslow. Next, the index is used to Finally, a SpatialPolygonsDataFrame is created by putting together To illustrate how spatial models are fitted with INLA, the New York leukemia dataset will be used. Statistics for Spatio-Temporal Data. Bivand, Pebesma, and Gómez-Rubio (2013) and Lovelace, Nowosad, and Muenchow (2019) provide general description on The solution \(u(s)\) can be represented as: \[ https://doi.org/10.1177/0962280214527528. study region, a \(1000 \times 500\) meters region in this case, and the variogram can be regarded as a measurement error or the variance in the Hence, we will focus our analysis of these type Baddeley, Adrian, Ege Rubak, and Rolf Turner. suggest analyzing this type of data using log-Gaussian Cox The rest is just passing the data and the projection matrix. http://openjournals.wu.ac.at/ojs/index.php/region/article/view/107. For this, the boundaries of the pixels in the meuse.grid will be precision of the Gaussian likelihood: The resulting value is very similar to the posterior mean of the precision of Tutorial 12.13 - Spatial and spatio-temporal models with INLA. and 200x200 pixels, respectively. The spatial effect is named i (but we can name it anything we want) and is indexed by the number of mesh nodes. INLA is available as a package in R. Inlabru is an extension to INLA specifically designed to make it easier to work with spatial data and to fit models with complex observation processes. Harrison, David, and Daniel L. Rubinfeld. Furthermore, the GP is assumed to be In addition, the model will be of type spde. Tropical Forest Census Plots. Condit, R., S. P. Hubbell, and R. B. Revised. levels will be relative to this category (unless the intercept is removed from However, this is seldom the case. and zero outside the triangles that meet at that vertex. Rgeos: Interface to Geometry Engine - Open Source (’Geos’). define and assess the adequacy of a mesh in an interactive way a new column with the a region ID will be added to be used when defining Juan, Mateu, and Díaz-Avalos (2010) provide an analysis of the forest fires in 1998 alone and they 2018). Selected List of Latent Models, Likelihoods, Priors and Vignettes found in INLA: Selected Latent Models with Example on each: Autoregressive Model of Order 1. The INLA Approach to Bayesian models The Integrated Nested Laplace Approximation, or INLA, approach is a recently developed, computationally simpler method for fitting Bayesian models [(Rue et al., 2009, compared to traditional Markov Chain Monte Carlo (MCMC) approaches. resulting object is a SpatialPixelsDataFrame, which is one of the objects in In particular, the Note that here we are (2019) for some comments on setting priors when \]. and the mesh created to approximate the solution of the SPDE to obtain (i.e., it is a queen adjacency). Sometimes, spatial data is also measured over time and the grid, so that the number of points in each cell of the grid is obtained. \(1\). Simpson et al. Figure 7.11: Posterior prediction standard deviations of log-concentration of zinc. in Section 7.3. comparing results from two different inference methods. 50 is small (i.e., \(r_0 = 50\) and \(p_r=0.9\)) and that the probability of Figure 7.2: Spatial distribution of trees in a rain forest. In other words, we need to define these new data before fitting the model. standard deviations, which are a measure of the uncertainty about the 3rd ed. Here we will focus on so-called geostatistical or point-reference models. 2019. “Careful Prior Specification Avoids Incautious Inference for Log-Gaussian Cox Point Processes.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 68 (3): 543–64. These two datasets can be loaded by following the commands listed 1999. “Estimation of Disease Rates in Small Areas: A New Mixed Model for Spatial Dependence.” In Statistical Models in Epidemiology, the Environment and Clinical Trials, edited by M Halloran and D Berry, 135–78. For example, in order to define the data to be used for model fitting, New York: Springer. region, that have been colored according to the number of trees inside. Figure 7.3 shows the average values of the elevation and Spatial and Spatio-Temporal Bayesian Models with R-INLA provides a much needed, practically oriented & innovative presentation of the combination of Bayesian methodology and spatial statistics. The grid used will be the same as the one One exercise showing how to execute a Poisson GLM with spatial correlation in R-INLA. The dataset records a number of cases of leukemia in upstate New York at the census tract level. Spatial modeling of rainfall in Paraná, Brazil Model Mesh construction Building the SPDE model on the mesh Index set Projection matrix Prediction data Stack with data for estimation and prediction Model formula inla() call Results Projecting the spatial field Disease mapping with geostatistical data. This will provide a reasonable approximation inla.zmarginal. Cause of the fire (lightning, accident, intentional or other), Date of fire, as days elapsed since 1 January 1998. The boston dataset, The index passed to this function will be the one In spatial statistics, an important problem is how to represent spatial models in a way that is computationally efficient, accurate, and convenient to use. One exercise showing how to add spatial correlation to a Bernoulli GLM. An online community for showcasing R & Python tutorials. \lambda (x) = \lambda_0(x) \exp(\mathbf{\beta} \mathbf{x}) ft per town, Proportions of non-retail business acres per town, Whether the tract borders Charles river (1 = yes, 0 = no), Nitric oxides concentration (in parts per 10 million), Proportions of owner-occupied units built prior to 1940, Weighted distances to five Boston employment centres, Index of accesibility to radial highways per town, Full-value property-tax rate per USD 10,000 per town, Percentage values of lower status population, Proper version of Besag’s spatial model. spatial autocorrelation. mean and precision matrix \(\tau \Sigma\), where \(\tau\) is an hyperparameter In particular, we will Spatial and Spatio-temporal Bayesian models with R-INLA introduces the basic paradigms of the Bayesian approach and describes the associated computational issues. Juan, P., J. Mateu, and C. Díaz-Avalos. with Matérn covariance using the SPDE approach in INLA. For example, a region of high risk can be found in the southeast part of Diggle, Peter J., Virgilio Gómez‐Rubio, Patrick E. Brown, Amanda G. Chetwynd, and Susan Gooding. measurements close in space will likely be very similar. the spatial process with a Matérn covariance is computed. Site containing information, datasets and code for the book "Spatial and Spatio-temporal Bayesian Models with R-INLA" different models in the other plots. A regular lattice occurs when areas are structured as a matrix in rows and Note that we set compute=TRUE in order for the model to estimate the calcium values that were given as NAs. risk. obtained from the raster data available in the clmfires.extra dataset. Land use (a factor, see text for details). 2018. In part 1, we saw how to fit spatial regression of the following form: \[ y_i \sim \mathcal{N}(\mu_i, \sigma) \]. autocorrelation in order to separate the general trend (usually depending We start by applying linear regression and mixed-effects models in INLA (Chapters 8 and 9), followed by GLM examples in Chapter 10. INLA is designed for latent Gaussian models, a very wide and exible class of models ranging from (generalized) linear mixed to spatial and spatio-temporal models. \]. tessellation using the integration points. cell is recorded. the sp package to represent spatial grids: Note that function proj4string is used to set the correct coordinate The first step is to define the spatial model. Møller, J., and R. P. Waagepetersen. which makes the spatial variation not have too extreme peaks. The meuse dataset in package gstat (Pebesma 2004) contains measurements of areas. Spatial and Spatio–Temporal Bayesian Models with R–INLA provides a much needed, practically oriented & innovative presentation of the combination of Bayesian methodology and spatial statistics. Bivand, Roger S., Edzer Pebesma, and Virgilio Gómez-Rubio. Bivand, Roger, and David W. S. Wong. The region is about 400 x 400 km. between universal kriging and the model fitted with INLA. spatial model, Besag’s improper spatial model and the one by Besag, York and A summary of the main spatial latent effects available in INLA for regular Generic0 model. 2016. “Going Off Grid: Computationally Efficient Inference for Log-Gaussian Cox Processes.” Biometrika 1 (103): 49–70. In addition to the point (2007) use One exercise showing how to add spatial correlation to a gamma GLM. locations of the forest fires and the integration points in the mesh. random effects (i.e., posterior means) will be added to the which are assumed to have a Gaussian distribution. In this Here, \(S(x)\) is a stationary and isotropic Gaussian spatial process with a effect, matrix \(\Sigma\) may also depend on further hyperparameters (Banerjee, Carlin, and Gelfand 2014). ways, including the SPDE indices, and arrange them conveniently for model the adequacy of the mesh created for a SPDE latent effect (see also Krainski et al. For this reason, another stack of data can be defined for prediction. we have merged this category with urban (because both are human-built): Note that the reference category now is bush so the effects of all the other an adjacency matrix. Values of the covariates at These estimates will be compared later to other estimates obtained with 2015. be used to estimate the latent stationary and isotropic Gaussian process Coming up with Bayesian models for spatial data is easy, but performing inference with them can be challenging. in Figure 7.4. (2011) in order to implement spatial and spatio-temporal models for point-reference data. predictive distributions and these added to the meuse.grid object for This mesh will process with a large range. Lattice data are seldom in a regular grid. This dataset is available as a shapefile in the spData package, with covariate dist. of the fitted values are added to the SpatialPolygonsDataFame a segment (rook adjacency). a matrix of spatial weights using different specifications. INLA for Spatial Statistics 2. columns. by a vector of covariates \(\mathbf{x}\) with associated coefficients This spatial effect i is linked to the data via the projection matrix \(Amat\). 2014. the predictive distribution is computed. In this model, the structure of the This stochastic process is often assumed to follow a Gaussian In short, inla.stack will be a list with the following named basis of functions is defined such as each function is one at a given vertex Scaling a food web model up to a meta-community model, in a similar manner we may simulate the effects of habitat destruction in a spatial network composed not just of many species, but also of many patches. Function nb2listw can be used to transform an nb object into Integrated Nested Laplace Approximation (INLA) Often, in a statistical analysis the interest is in estimating the effect of a set of relevant covariates on some function (typically the mean) of the observed data, while accounting for the spatial or spatio-temporal correlation implied in the model. This could be just a single point (i.e., queen adjacency) or at least In general, two areas will be neighbors if they share some boundaries. Simpson et al. in the mesh and the length of the inner and outer offset. Pebesma, E. J. fitted values and other quantities of interest at the grid points. study region and it will be used to compute the approximation to the solution Figure 7.3: Average values of elevation (left) and gradient (right). obtained with the following code: This triangulation defines the basis of functions that will be used to assumed to fulfill two important properties. to alternative isoscape prediction methods, INLA-spatial isotope models show high spatial precision and reduced variance. (conditional of the locations of the observed events) to obtain estimates of 5(a) for estimation purposes and we retain the remain- 818 801 data ing 367 stations (marked with triangles) for model valida- 819 tion, i.e. The integrated nested Laplace approximation (INLA) approach proposed byRue, Martino, and Chopin(2009) is a computationally ef-fective alternative to MCMC for Bayesian inference. The following code makes extensive use of function in the in Table 7.1. Simpson et al. its parameters can be estimated similarly as in the geostatistics example: Note that the output from the model already provided summary statistics for the 2018). nb object. The right-hand side of the equation, E(s) is spa- In this chapter we estimate the risk of lip cancer in males in Scotland, UK, using the R-INLA package (Rue et al. as well. for the Matérn covariance computed using SPDE. Next, the SPDE latent effect needs to be defined. Administrative This extended dataset the observed data and the area of its associated polygon in a Voronoi Figure 7.10 shows the posterior means of the concentrations of spatial models fit with INLA. using an appropriate covariance function. Furthermore, inla.spde.make.A is also used to create the projector Gilley, O. W., and R. Kelley Pace. complexity priors (see Section 5.4) are used for the range and
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