Fits a Bayesian spatial generalized linear model with fixed values of spatial process parameters and some auxiliary model parameters. The output contains posterior samples of the fixed effects, spatial random effects and, if required, finds leave-one-out predictive densities.
Usage
spGLMexact(
formula,
data = parent.frame(),
family,
coords,
cor.fn,
priors,
spParams,
boundary = 0.5,
n.samples,
loopd = FALSE,
loopd.method = "exact",
CV.K = 10,
loopd.nMC = 500,
verbose = TRUE,
...
)Arguments
- formula
a symbolic description of the regression model to be fit. See example below.
- data
an optional data frame containing the variables in the model. If not found in
data, the variables are taken fromenvironment(formula), typically the environment from whichspGLMexactis called.- family
Specifies the distribution of the response as a member of the exponential family. Supported options are
'poisson','binomial'and'binary'.- coords
an \(n \times 2\) matrix of the observation coordinates in \(\mathbb{R}^2\) (e.g., easting and northing).
- cor.fn
a quoted keyword that specifies the correlation function used to model the spatial dependence structure among the observations. Supported covariance model key words are:
'exponential'and'matern'. See below for details.- priors
(optional) a list with each tag corresponding to a hyperparameter name and containing hyperprior details. Valid tags include
V.beta,nu.beta,nu.zandsigmaSq.xi. Values ofnu.betaandnu.zmust be at least 2.1. If not supplied, uses defaults.- spParams
fixed values of spatial process parameters.
- boundary
Specifies the boundary adjustment parameter. Must be a real number between 0 and 1. Default is 0.5.
- n.samples
number of posterior samples to be generated.
- loopd
logical. If
loopd=TRUE, returns leave-one-out predictive densities, using method as given byloopd.method. Default isFALSE.- loopd.method
character. Ignored if
loopd=FALSE. Ifloopd=TRUE, valid inputs are'exact','CV'and'PSIS'. The option'exact'corresponds to exact leave-one-out predictive densities which requires computation almost equivalent to fitting the model \(n\) times. The options'CV'and'PSIS'are faster and they implement \(K\)-fold cross validation and Pareto-smoothed importance sampling to find approximate leave-one-out predictive densities (Vehtari et al. 2017).- CV.K
An integer between 10 and 20. Considered only if
loopd.method='CV'. Default is 10 (as recommended in Vehtari et. al 2017).- loopd.nMC
Number of Monte Carlo samples to be used to evaluate leave-one-out predictive densities when
loopd.methodis set to either 'exact' or 'CV'.- verbose
logical. If
verbose = TRUE, prints model description.- ...
currently no additional argument.
Value
An object of class spGLMexact, which is a list with the
following tags -
- priors
details of the priors used, containing the values of the boundary adjustment parameter (
boundary), the variance parameter of the fine-scale variation term (simasq.xi) and others.- samples
a list of length 3, containing posterior samples of fixed effects (
beta), spatial effects (z) and the fine-scale variation term (xi).- loopd
If
loopd=TRUE, contains leave-one-out predictive densities.- model.params
Values of the fixed parameters that includes
phi(spatial decay),nu(spatial smoothness).
The return object might include additional data that can be used for subsequent prediction and/or model fit evaluation.
Details
With this function, we fit a Bayesian hierarchical spatial
generalized linear model by sampling exactly from the joint posterior
distribution utilizing the generalized conjugate multivariate distribution
theory (Bradley and Clinch 2024). Suppose \(\chi = (s_1, \ldots, s_n)\)
denotes the \(n\) spatial locations the response \(y\) is observed. Let
\(y(s)\) be the outcome at location \(s\) endowed with a probability law
from the natural exponential family, which we denote by
$$
y(s) \sim \mathrm{EF}(x(s)^\top \beta + z(s); b, \psi)
$$
for some positive parameter \(b > 0\) and unit log partition function
\(\psi\). We consider the following response models based on the input
supplied to the argument family.
'poisson'It considers point-referenced Poisson responses \(y(s) \sim \mathrm{Poisson}(e^{x(s)^\top \beta + z(s)})\). Here, \(b = 1\) and \(\psi(t) = e^t\).
'binomial'It considers point-referenced binomial counts \(y(s) \sim \mathrm{Binomial}(m(s), \pi(s))\) where, \(m(s)\) denotes the total number of trials and probability of success \(\pi(s) = \mathrm{ilogit}(x(s)^\top \beta + z(s))\) at location \(s\). Here, \(b = m(s)\) and \(\psi(t) = \log(1+e^t)\).
'binary'It considers point-referenced binary data (0 or, 1) i.e., \(y(s) \sim \mathrm{Bernoulli}(\pi(s))\), where probability of success \(\pi(s) = \mathrm{ilogit}(x(s)^\top \beta + z(s))\) at location \(s\). Here, \(b = 1\) and \(\psi(t) = \log(1 + e^t)\).
The hierarchical model is given as $$ \begin{aligned} y(s_i) &\mid \beta, z, \xi \sim EF(x(s_i)^\top \beta + z(s_i) + \xi_i - \mu_i; b_i, \psi_y), i = 1, \ldots, n\\ \xi &\mid \beta, z, \sigma^2_\xi, \alpha_\epsilon \sim \mathrm{GCM_c}(\cdots),\\ \beta &\mid \sigma^2_\beta \sim N(0, \sigma^2_\beta V_\beta), \quad \sigma^2_\beta \sim \mathrm{IG}(\nu_\beta/2, \nu_\beta/2)\\ z &\mid \sigma^2_z \sim N(0, \sigma^2_z R(\chi; \phi, \nu)), \quad \sigma^2_z \sim \mathrm{IG}(\nu_z/2, \nu_z/2), \end{aligned} $$ where \(\mu = (\mu_1, \ldots, \mu_n)^\top\) denotes the discrepancy parameter. We fix the spatial process parameters \(\phi\) and \(\nu\) 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 as prior. For more details, see Pan et al. 2024. Default values for \(V_\beta\), \(\nu_\beta\), \(\nu_z\), \(\sigma^2_\xi\) are diagonal with each diagonal element 100, 2.1, 2.1 and 0.1 respectively.
References
Bradley JR, Clinch M (2024). "Generating Independent Replicates Directly from the Posterior Distribution for a Class of Spatial Hierarchical Models." Journal of Computational and Graphical Statistics, 0(0), 1-17. doi:10.1080/10618600.2024.2365728 .
Pan S, Zhang L, Bradley JR, Banerjee S (2024). "Bayesian Inference for Spatial-temporal Non-Gaussian Data Using Predictive Stacking." doi:10.48550/arXiv.2406.04655 .
Vehtari A, Gelman A, Gabry J (2017). "Practical Bayesian Model Evaluation Using Leave-One-out Cross-Validation and WAIC." Statistics and Computing, 27(5), 1413-1432. ISSN 0960-3174. doi:10.1007/s11222-016-9696-4 .
Author
Soumyakanti Pan span18@ucla.edu
Examples
# Example 1: Analyze spatial poisson count data
data(simPoisson)
dat <- simPoisson[1:10, ]
mod1 <- spGLMexact(y ~ x1, data = dat, family = "poisson",
coords = as.matrix(dat[, c("s1", "s2")]),
cor.fn = "matern",
spParams = list(phi = 4, nu = 0.4),
n.samples = 100, verbose = TRUE)
#> ----------------------------------------
#> Model description
#> ----------------------------------------
#> Model fit with 10 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 = 4.00, and, nu = 0.40.
#>
#> Number of posterior samples = 100.
#> ----------------------------------------
# summarize posterior samples
post_beta <- mod1$samples$beta
print(t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975)))))
#> 2.5% 50% 97.5%
#> [1,] 0.07462851 2.2074935 5.541867
#> [2,] -2.32273414 -0.3911022 1.343105
# Example 2: Analyze spatial binomial count data
data(simBinom)
dat <- simBinom[1:10, ]
mod2 <- 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 = 0.4),
n.samples = 100, verbose = TRUE)
#> ----------------------------------------
#> Model description
#> ----------------------------------------
#> Model fit with 10 observations.
#>
#> Family = binomial.
#>
#> 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 = 3.00, and, nu = 0.40.
#>
#> Number of posterior samples = 100.
#> ----------------------------------------
# summarize posterior samples
post_beta <- mod2$samples$beta
print(t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975)))))
#> 2.5% 50% 97.5%
#> [1,] -0.3347374 1.1426362 2.9358010
#> [2,] -1.9595693 -0.6214757 0.4642419
# Example 3: Analyze spatial binary data
data(simBinary)
dat <- simBinary[1:10, ]
mod3 <- spGLMexact(y ~ x1, data = dat, family = "binary",
coords = as.matrix(dat[, c("s1", "s2")]),
cor.fn = "matern",
spParams = list(phi = 4, nu = 0.4),
n.samples = 100, verbose = TRUE)
#> ----------------------------------------
#> Model description
#> ----------------------------------------
#> Model fit with 10 observations.
#>
#> Family = binary.
#>
#> 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 = 4.00, and, nu = 0.40.
#>
#> Number of posterior samples = 100.
#> ----------------------------------------
# summarize posterior samples
post_beta <- mod3$samples$beta
print(t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975)))))
#> 2.5% 50% 97.5%
#> [1,] -2.012890 1.2453228 4.693207
#> [2,] -2.096343 -0.2022237 1.335701