spStack: Bayesian Geostatistics Using Predictive Stacking

Introduction

Geostatistics refers to the study of a spatially distributed variable of interest, which in theory is defined at every point over a bounded study region of interest. Statistical modelling and analysis for spatially oriented point-referenced outcomes play a crucial role in diverse scientific applications such as earth and environmental sciences, ecology, epidemiology, and economics. With the advent of Markov chain Monte Carlo (MCMC) algorithms, Bayesian hierarchical models have gained massive popularity in analyzing such point-referenced or, geostatistical data. These models involve latent spatial processes characterized by spatial process parameters, which besides lacking substantive relevance in scientific contexts, are also weakly identified and hence, impedes convergence of MCMC algorithms. Thus, even for moderately large datasets (~\(10^3\) or higher), the computation for MCMC becomes too onerous for practical use.

We introduce the R package spStack that implements Bayesian inference for a class of geostatistical models, where we obviate the issues mentioned by sampling from analytically available posterior distributions conditional upon some candidate values of the spatial process parameters and, subsequently assimilate inference from these individual posterior distributions using Bayesian predictive stacking. Besides delivering competitive predictive performance as compared to fully Bayesian inference using MCMC, our proposed algorithm is embarrassingly parallel, thus drastically improves runtime and elevating the utility of the package for a diverse group of practitioners with limited computational resources at their disposal. This package, to the best of our knowledge, is the first to implement stacking for Bayesian analysis of spatial data.

Technical details surrounding the methodology can be found in the articles Zhang, Tang, and Banerjee (2024) which discuss the case where the distribution of the point-referenced outcomes are Gaussian, and, in Pan et al. (2024) where the case of non-Gaussian outcomes is explored. The code for this package is written primarily in C/C++ with additional calls to FORTRAN routines for optimized linear algebra operations. We leverage the F77_NAME macro to interface with legacy FORTRAN functions in conjunction with efficient matrix computation libraries such as BLAS (Basic Linear Algebra Subprograms) and LAPACK (Linear Algebra Package) to implement our stacking algorithm.

The remainder of the vignette evolves as follows - the next two sections discuss Bayesian hierarchical spatial models for Gaussian and non-Gaussian outcomes, which is followed by a section providing brief details on predictive stacking and a section dedicated for illustration of functions in the package.

Bayesian Gaussian spatial regression models

Let \(\chi = \{s_1, \ldots, s_n\} \in \mathcal{D}\) be a be a set of \(n\) spatial locations yielding measurements \(y = (y_1, \ldots, y_n)^{ \scriptstyle \top }\) with known values of predictors at these locations collected in the \(n \times p\) full rank matrix \(X = [x(s_1), \ldots, x(s_n)]^{ \scriptstyle \top }\). A customary geostatistical model is \[\begin{equation} y_i = x(s_i)^{ \scriptstyle \top }\beta + z(s_i) + \epsilon_i, \quad i = 1, \ldots, n, \end{equation}\] where \(\beta\) is the \(p \times 1\) vector of slopes, \(z(s) \sim \mathrm{GP}(0, R(\cdot, \cdot; {\theta_{\text{sp}}}))\) is a zero-centered spatial Gaussian process on \(\mathcal{D}\) with spatial correlation function \(R(\cdot, \cdot; {\theta_{\text{sp}}})\) characterized by process parameters \({\theta_{\text{sp}}}\), \(\sigma^2\) is the spatial variance parameter (“partial sill”) and \(\epsilon_i \sim \mathcal{N}(0, \tau^2), i = 1, \ldots, n\) are i.i.d. with variance \(\tau^2\) (“nugget”) capturing measurement error. The spatial process \(z(\cdot)\) is assumed to be independent of the measurement errors \(\{\epsilon_i, i = 1, \ldots, n\}\). Let \(z = (z(s_1), \ldots, z(s_n))^{ \scriptstyle \top }\) denotes the realization of the spatial process on \(\chi\) and \(n \times n\) correlation matrix \(R(\chi; {\theta_{\text{sp}}}) = (R(s_i, s_j {\theta_{\text{sp}}}))_{1 \leq i,j \leq n}\). We build a conjugate Bayesian hierarchical spatial model, \[\begin{equation} \begin{split} y \mid z, \beta, \sigma^2 &\sim N(X\beta + z, \delta^2 \sigma^2 I_n), \\ z \mid \sigma^2 &\sim N(0, \sigma^2 R(\chi; {\theta_{\text{sp}}})), \\ \beta \mid \sigma^2 &\sim N(\mu_\beta, \sigma^2 V_\beta), \quad \sigma^2 \sim \mathrm{IG}(a_\sigma, b_\sigma), \end{split} \end{equation}\] where we fix the noise-to-spatial variance ratio \(\delta^2 = \tau^2 / \sigma^2\), the process parameters \({\theta_{\text{sp}}}\) and the hyperparameters \(\mu_\beta\), \(V_\beta\), \(a_\sigma\) and \(b_\sigma\). In this package, we use the Matern covariogram specified by spatial decay parameter \(\phi\) and smoothness parameter \(\nu\) i.e., \({\theta_{\text{sp}}}= \{\phi, \nu\}\), given by \[\begin{equation} R(s, s'; {\theta_{\text{sp}}}) = \frac{(\phi \lvert s - s' \rvert)^\nu}{2^{\nu - 1} \Gamma(\nu)} K_\nu (\phi \lvert s - s' \rvert)). \end{equation}\] We utilize a composition sampling strategy to sample the model parameters from their joint posterior distribution which can be written as \[\begin{equation} p(\sigma^2, \beta, z \mid y) = p(\sigma^2 \mid y) \times p(\beta \mid \sigma^2, y) \times p(z \mid \beta, \sigma^2, y). \end{equation}\] We proceed by first sampling \(\sigma^2\) from its marginal posterior, then given the samples of \(\sigma^2\), we sample \(\beta\) and subsequently, we sample \(z\) conditioned on the posterior samples of \(\beta\) and \(\sigma^2\) (Banerjee 2020). The function spLMexact() delivers samples from this posterior distribution. More details can be found in Zhang, Tang, and Banerjee (2024).

Bayesian non-Gaussian spatial regression models

Analyzing non-Gaussian spatial data typically requires introducing spatial dependence in generalized linear models through the link function of an exponential family distribution. Let \(y(s)\) be the outcome at location \(s \in \mathcal{D}\) endowed with a probability law from the natural exponential family, which we denote by \[\begin{equation} y(s) \sim \mathrm{EF}(x(s)^{ \scriptstyle \top }\beta + z(s); b, \psi_y) \end{equation}\] for some positive parameter \(b > 0\) and unit log partition function \(\psi_y\). Fixed effects regression and spatial dependence, e.g., \(x(s)^{{ \scriptstyle \top }}\beta + z(s)\), is introduced in the natural parameter, where \(x(s)\) is a \(p \times 1\) vector of predictors referenced with respect to \(s\), \(\beta\) is a \(p \times 1\) vector of slopes measuring the trend, \(z(s)\) is a zero-centered spatial process on \(\mathcal{D}\) specified by a scale parameter \(\sigma_z\) and a spatial correlation function \(R(\cdot, \cdot ; {\theta_{\text{sp}}})\) with \({\theta_{\text{sp}}}\) consisting of spatial-temporal decay and smoothness parameters.

Unlike in Gaussian likelihoods, inference is considerably encumbered by the inability to analytically integrate out the random effects and reduce the dimension of the parameter space. Iterative algorithms such as Markov Chain Monte Carlo (MCMC), thus attempt to sample from a very high-dimensional posterior distribution, and convergence is often hampered by high auto-correlations and weakly identified spatial process parameters \({\theta_{\text{sp}}}\).

We consider the following three cases -

1. Point-referenced Poisson count data: Here \(b = 1\) and \(\psi_y(t) = e^t\). \[\begin{equation} \begin{split} y(s_i) &\sim \mathrm{Poisson}(\lambda(s_i)), \quad i = 1, \dots, n.\\ \lambda(s_i) & = \exp(x(s_i)^{ \scriptstyle \top }\beta + z(s_i)) \end{split} \end{equation}\]
2. Point-referenced binomial count data: Here \(b = m(s_i)\) for each \(i\) and \(\psi_y(t) = \log(1 + e^t)\). \[\begin{equation} \begin{split} y(s_i) &\sim \mathrm{Binomial}(m(s_i), \pi(s_i)), \quad i = 1, \dots, n.\\ \pi(s_i) & = \mathrm{ilogit}(x(s_i)^{ \scriptstyle \top }\beta + z(s_i)) \end{split} \end{equation}\]
3. Point-referenced binary data: Here \(b = 1\) and \(\psi_y(t) = \log(1 + e^t)\). \[\begin{equation} \begin{split} y(s_i) &\sim \mathrm{Bernoulli}(\pi(s_i)), \quad i = 1, \dots, n.\\ \pi(s_i) & = \mathrm{ilogit}(x(s_i)^{ \scriptstyle \top }\beta + z(s_i)) \end{split} \end{equation}\]

Following Bradley and Clinch (2024), we introduce a Bayesian hierarchical spatial model as \[\begin{equation} \begin{split} y(s_i) \mid \beta, z, \xi & \sim \mathrm{EF}\left(x(s_i)^{ \scriptstyle \top }\beta + z(s_i) + \xi_i - \mu_i; b_i, \psi_y\right), i = 1, \ldots, n\\ \beta \mid \sigma^2_\beta &\sim N(0, \sigma^2_\beta V_\beta), \quad \sigma^2_\beta \sim \mathrm{IG}(\nu_\beta/2, \nu_\beta/2)\\ z \mid \sigma^2_z &\sim N\left(0, \sigma^2_z R(\chi; {\theta_{\text{sp}}})\right), \quad \sigma^2_z \sim \mathrm{IG}(\nu_z/2, \nu_z/2),\\ \xi \mid \beta, z, \sigma^2_\xi, \alpha_\epsilon &\sim \mathrm{GCM_c}\left(\tilde{\mu}_\xi, H_\xi, \epsilon, \kappa_\xi; \psi_\xi\right), \end{split} \end{equation}\] where \(\mu = (\mu_1, \ldots, \mu_n)^{ \scriptstyle \top }\) denotes the discrepancy parameter. We fix the spatial process parameters \({\theta_{\text{sp}}}\), the boundary adjustment parameter \(\epsilon\) and the hyperparameters \(V_\beta\), \(\nu_\beta\), \(\nu_z\) and \(\sigma^2_\xi\). The term \(\xi\) is known as the fine-scale variation term which is given a conditional generalized conjugate multivariate distribution (\(\mathrm{GCM_c}\)) as prior. For details, see Pan et al. (2024).

Predictive stacking

Following Yao et al. (2018), we consider a set of candidate models based on a grid of values of the parameters in \(\{ {\theta_{\text{sp}}}, \delta^2 \}\) for the Gaussian case, and \(\{ {\theta_{\text{sp}}}, \epsilon \}\) for the non-Gaussian case, as will be supplied by the user. We build a set of candidate models based on the Cartesian product of the collection of values for each individual parameter as \(\mathcal{M} = \{M_1, \ldots, M_G\}\). Then, for each \(g = 1, \ldots, G\), we sample from the posterior distribution \(p(\sigma^2, \beta, z \mid y, M_g)\) under the model \(M_g\) and find leave-one-out predictive densities \(p(y_i \mid y_{-i}, M_g)\). Then we solve the optimization problem \[\begin{equation} \begin{split} \max_{w_1, \ldots, w_G}& \, \frac{1}{n} \sum_{i = 1}^n \log \sum_{g = 1}^G w_g p(y_i \mid y_{-i}, M_g) \\ \text{subject to} & \quad w_g \geq 0, \sum_{g = 1}^G w_g = 1 \end{split} \end{equation}\] to find the optimal stacking weights \(\hat{w}_1, \ldots, \hat{w}_G\). After obtaining the optimal stacking weights, posterior inference of any quantity of interest subsequently proceed from the stacked posterior, \[\begin{equation} \tilde{p}(\cdot \mid y) = \sum_{g = 1}^G \hat{w}_g p(\cdot \mid y, M_g). \end{equation}\]

Illustrations

In this section, we thoroughly illustrate our method on synthetic Gaussian as well as non-Gaussian spatial data and provide code to analyze the output of our functions. We start by loading the package.

library(spStack)

spStack.R Some synthetic spatial data are lazyloaded which includes synthetic spatial Gaussian data simGaussian, Poisson data simPoisson, binomial data simBinom and binary data simBinary. One can use the function sim_spData() to simulate spatial data. We will be applying our functions on these datasets.

data("simGaussian")
dat <- simGaussian[1:200, ] # work with first 200 rows

muBeta <- c(0, 0)
VBeta <- cbind(c(10.0, 0.0), c(0.0, 10.0))
sigmaSqIGa <- 2
sigmaSqIGb <- 2
phi0 <- 2
nu0 <- 0.5
noise_sp_ratio <- 0.8
prior_list <- list(beta.norm = list(muBeta, VBeta),
                   sigma.sq.ig = c(sigmaSqIGa, sigmaSqIGb))
nSamples <- 2000

set.seed(1729)
mod1 <- spLMexact(y ~ x1, data = dat,
                  coords = as.matrix(dat[, c("s1", "s2")]),
                  cor.fn = "matern",
                  priors = prior_list,
                  spParams = list(phi = phi0, nu = nu0),
                  noise_sp_ratio = noise_sp_ratio, n.samples = nSamples,
                  loopd = TRUE, loopd.method = "exact",
                  verbose = TRUE)
#> ----------------------------------------
#>  Model description
#> ----------------------------------------
#> Model fit with 200 observations.
#> 
#> Number of covariates 2 (including intercept).
#> 
#> Using the matern spatial correlation function.
#> 
#> Priors:
#>  beta: Gaussian
#>  mu: 0.00    0.00    
#>  cov:
#>  10.00   0.00    
#>  0.00    10.00   
#> 
#>  sigma.sq: Inverse-Gamma
#>  shape = 2.00, scale = 2.00.
#> 
#> Spatial process parameters:
#>  phi = 2.00, and, nu = 0.50.
#> Noise-to-spatial variance ratio = 0.80.
#> 
#> Number of posterior samples = 2000.
#> 
#> LOO-PD calculation method = exact.
#> ----------------------------------------

post_beta <- mod1$samples$beta
summary_beta <- t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975))))
rownames(summary_beta) <- mod1$X.names
print(summary_beta)
#>                 2.5%      50%    97.5%
#> (Intercept) 1.388249 2.129140 2.932428
#> x1          4.865843 4.954446 5.045627

mod2 <- spLMexact(y ~ x1, data = dat,
                  coords = as.matrix(dat[, c("s1", "s2")]),
                  cor.fn = "matern",
                  priors = prior_list,
                  spParams = list(phi = phi0, nu = nu0),
                  noise_sp_ratio = noise_sp_ratio, n.samples = nSamples,
                  loopd = TRUE, loopd.method = "PSIS",
                  verbose = FALSE)

loopd_exact <- mod1$loopd
loopd_psis <- mod2$loopd
loopd_df <- data.frame(exact = loopd_exact, psis = loopd_psis)

library(ggplot2)
plot1 <- ggplot(data = loopd_df, aes(x = exact)) +
  geom_point(aes(y = psis), size = 0.5, alpha = 0.5) +
  geom_abline(slope = 1, intercept = 0, color = "red") +
  xlab("Exact") + ylab("PSIS") + theme_bw() +
  theme(panel.background = element_blank(), aspect.ratio = 1)
plot1

mod3 <- spLMstack(y ~ x1, data = dat,
                  coords = as.matrix(dat[, c("s1", "s2")]),
                  cor.fn = "matern",
                  priors = prior_list,
                  params.list = list(phi = c(1.5, 3, 5),
                                     nu = c(0.5, 1, 1.5),
                                     noise_sp_ratio = c(0.5, 1.5)),
                  n.samples = 1000, loopd.method = "exact",
                  parallel = FALSE, solver = "ECOS", verbose = TRUE)
#> 
#> STACKING WEIGHTS:
#> 
#>            | phi | nu  | noise_sp_ratio | weight |
#> +----------+-----+-----+----------------+--------+
#> | Model 1  |  1.5|  0.5|             0.5| 0.000  |
#> | Model 2  |  3.0|  0.5|             0.5| 0.000  |
#> | Model 3  |  5.0|  0.5|             0.5| 0.000  |
#> | Model 4  |  1.5|  1.0|             0.5| 0.284  |
#> | Model 5  |  3.0|  1.0|             0.5| 0.000  |
#> | Model 6  |  5.0|  1.0|             0.5| 0.707  |
#> | Model 7  |  1.5|  1.5|             0.5| 0.000  |
#> | Model 8  |  3.0|  1.5|             0.5| 0.000  |
#> | Model 9  |  5.0|  1.5|             0.5| 0.000  |
#> | Model 10 |  1.5|  0.5|             1.5| 0.000  |
#> | Model 11 |  3.0|  0.5|             1.5| 0.009  |
#> | Model 12 |  5.0|  0.5|             1.5| 0.000  |
#> | Model 13 |  1.5|  1.0|             1.5| 0.000  |
#> | Model 14 |  3.0|  1.0|             1.5| 0.000  |
#> | Model 15 |  5.0|  1.0|             1.5| 0.000  |
#> | Model 16 |  1.5|  1.5|             1.5| 0.000  |
#> | Model 17 |  3.0|  1.5|             1.5| 0.000  |
#> | Model 18 |  5.0|  1.5|             1.5| 0.000  |
#> +----------+-----+-----+----------------+--------+

print(mod3$solver.status)
#> [1] "optimal"
print(mod3$run.time)
#>    user  system elapsed 
#>   2.033   0.386   2.077

post_samps <- stackedSampler(mod3)

post_beta <- post_samps$beta
summary_beta <- t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975))))
rownames(summary_beta) <- mod3$X.names
print(summary_beta)
#>                  2.5%      50%    97.5%
#> (Intercept) 0.8620702 2.173778 2.989719
#> x1          4.8678581 4.953678 5.030822

post_z <- post_samps$z
post_z_summ <- t(apply(post_z, 1, function(x) quantile(x, c(0.025, 0.5, 0.975))))
z_combn <- data.frame(z = dat$z_true, zL = post_z_summ[, 1],
                      zM = post_z_summ[, 2], zU = post_z_summ[, 3])

plotz <- ggplot(data = z_combn, aes(x = z)) +
  geom_point(aes(y = zM), size = 0.25, color = "darkblue", alpha = 0.5) +
  geom_errorbar(aes(ymin = zL, ymax = zU), width = 0.05, alpha = 0.15, 
                color = "skyblue") + 
  geom_abline(slope = 1, intercept = 0, color = "red") +
  xlab("True z") + ylab("Stacked posterior of z") + theme_bw() +
  theme(panel.background = element_blank(), aspect.ratio = 1)
plotz

postmedian_z <- apply(post_z, 1, median)
dat$z_hat <- postmedian_z
plot_z <- surfaceplot2(dat, coords_name = c("s1", "s2"),
                       var1_name = "z_true", var2_name = "z_hat")
library(ggpubr)
ggarrange(plotlist = plot_z, common.legend = TRUE, legend = "right")

data("simPoisson")
dat <- simPoisson[1:200, ] # work with first 200 observations

ggplot(dat, aes(x = s1, y = s2)) +
  geom_point(aes(color = y), alpha = 0.75) +
  scale_color_distiller(palette = "RdYlGn", direction = -1, 
                        label = function(x) sprintf("%.0f", x)) +
  guides(alpha = 'none') + theme_bw() +
  theme(axis.ticks = element_line(linewidth = 0.25),
        panel.background = element_blank(), panel.grid = element_blank(),
        legend.title = element_text(size = 10, hjust = 0.25),
        legend.box.just = "center", aspect.ratio = 1)

mod1 <- spGLMexact(y ~ x1, data = dat, family = "poisson",
                   coords = as.matrix(dat[, c("s1", "s2")]), cor.fn = "matern",
                   spParams = list(phi = phi0, nu = nu0),
                   boundary = 0.5,
                   n.samples = 1000, verbose = TRUE)
#> ----------------------------------------
#>  Model description
#> ----------------------------------------
#> Model fit with 200 observations.
#> 
#> Family = poisson.
#> 
#> Number of covariates 2 (including intercept).
#> 
#> Using the matern spatial correlation function.
#> 
#> Priors:
#>  beta: Gaussian
#>  mu: 0.00    0.00    
#>  cov:
#>  100.00  0.00    
#>  0.00    100.00  
#> 
#>  sigmaSq.beta ~ IG(nu.beta/2, nu.beta/2)
#>  sigmaSq.z ~ IG(nu.z/2, nu.z/2)
#>  nu.beta = 2.10, nu.z = 2.10.
#>  sigmaSq.xi = 0.10.
#>  Boundary adjustment parameter = 0.50.
#> 
#> Spatial process parameters:
#>  phi = 2.00, and, nu = 0.50.
#> 
#> Number of posterior samples = 1000.
#> ----------------------------------------

post_beta <- mod1$samples$beta
summary_beta <- t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975))))
rownames(summary_beta) <- mod1$X.names
print(summary_beta)
#>                   2.5%        50%      97.5%
#> (Intercept) -0.1302807  1.9026041  4.3045317
#> x1          -0.6664152 -0.5545303 -0.4543145

mod2 <- spGLMstack(y ~ x1, data = dat, family = "poisson",
                   coords = as.matrix(dat[, c("s1", "s2")]), cor.fn = "matern",
                   params.list = list(phi = c(3, 7, 10), nu = c(0.5, 1.5),
                                      boundary = c(0.5, 0.6)),
                   n.samples = 1000,
                   loopd.controls = list(method = "CV", CV.K = 10, nMC = 1000),
                   parallel = TRUE, solver = "ECOS", verbose = TRUE)
#> 
#> STACKING WEIGHTS:
#> 
#>            | phi | nu  | boundary | weight |
#> +----------+-----+-----+----------+--------+
#> | Model 1  |    3|  0.5|       0.5| 0.000  |
#> | Model 2  |    7|  0.5|       0.5| 0.000  |
#> | Model 3  |   10|  0.5|       0.5| 0.000  |
#> | Model 4  |    3|  1.5|       0.5| 0.000  |
#> | Model 5  |    7|  1.5|       0.5| 0.000  |
#> | Model 6  |   10|  1.5|       0.5| 0.000  |
#> | Model 7  |    3|  0.5|       0.6| 0.000  |
#> | Model 8  |    7|  0.5|       0.6| 0.000  |
#> | Model 9  |   10|  0.5|       0.6| 0.000  |
#> | Model 10 |    3|  1.5|       0.6| 0.175  |
#> | Model 11 |    7|  1.5|       0.6| 0.825  |
#> | Model 12 |   10|  1.5|       0.6| 0.000  |
#> +----------+-----+-----+----------+--------+

print(mod2$solver.status)
#> [1] "optimal"
print(mod2$run.time)
#>    user  system elapsed 
#>  12.283   2.035  12.247

post_samps <- stackedSampler(mod2)

post_beta <- post_samps$beta
summary_beta <- t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975))))
rownames(summary_beta) <- mod3$X.names
print(summary_beta)
#>                   2.5%        50%      97.5%
#> (Intercept) -0.8032363  2.0460414  3.8253354
#> x1          -0.6275769 -0.5471968 -0.4529202

post_z <- post_samps$z
post_z_summ <- t(apply(post_z, 1, function(x) quantile(x, c(0.025, 0.5, 0.975))))
z_combn <- data.frame(z = dat$z_true, zL = post_z_summ[, 1],
                      zM = post_z_summ[, 2], zU = post_z_summ[, 3])

plotz <- ggplot(data = z_combn, aes(x = z)) +
  geom_point(aes(y = zM), size = 0.25, color = "darkblue", alpha = 0.5) +
  geom_errorbar(aes(ymin = zL, ymax = zU), width = 0.05, alpha = 0.15,
                color = "skyblue") +
  geom_abline(slope = 1, intercept = 0, color = "red") +
  xlab("True z") + ylab("Stacked posterior of z") + theme_bw() +
  theme(panel.background = element_blank(), aspect.ratio = 1)
plotz

postmedian_z <- apply(post_z, 1, median)
dat$z_hat <- postmedian_z
plot_z <- surfaceplot2(dat, coords_name = c("s1", "s2"),
                       var1_name = "z_true", var2_name = "z_hat")
library(ggpubr)
ggarrange(plotlist = plot_z, common.legend = TRUE, legend = "right")

data("simBinom")
dat <- simBinom[1:200, ] # work with first 200 rows

mod1 <- spGLMexact(cbind(y, n_trials) ~ x1, data = dat, family = "binomial",
                   coords = as.matrix(dat[, c("s1", "s2")]), cor.fn = "matern",
                   spParams = list(phi = 3, nu = 0.5),
                   boundary = 0.5, n.samples = 1000, verbose = FALSE)

post_beta <- mod1$samples$beta
summary_beta <- t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975))))
rownames(summary_beta) <- mod1$X.names
print(summary_beta)
#>                   2.5%        50%      97.5%
#> (Intercept) -1.2039026  0.7229478  2.5872269
#> x1          -0.5815266 -0.4016808 -0.2365151

data("simBinary")
dat <- simBinary[1:200, ]

mod1 <- spGLMexact(y ~ x1, data = dat, family = "binary",
                   coords = as.matrix(dat[, c("s1", "s2")]), cor.fn = "matern",
                   spParams = list(phi = 4, nu = 0.4),
                   boundary = 0.5, n.samples = 1000, verbose = FALSE)

post_beta <- mod1$samples$beta
summary_beta <- t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975))))
rownames(summary_beta) <- mod1$X.names
print(summary_beta)
#>                   2.5%        50%      97.5%
#> (Intercept) -1.1291296  0.3061828 1.70795510
#> x1          -0.6507851 -0.3088350 0.04471555

Conclusion

We have devised and demonstrated Bayesian predictive stacking to be an effective tool for estimating spatial regression models and yielding robust predictions for Gaussian as well as non-Gaussian spatial data. We develop and exploit analytically accessible distribution theory pertaining to Bayesian analysis of linear mixed model and generalized linear mixed models that enables us to directly sample from the posterior distributions. The focus of this package is on effectively combining inference across different closed-form posterior distributions by circumventing inference on weakly identified parameters. Future developments and investigations will consider zero-inflated non-Gaussian data and adapting to variants of Gaussian process models that scale inference to massive datasets by circumventing the Cholesky decomposition of dense covariance matrices.