Bayesian spatially-temporally varying generalized linear model

Fits a Bayesian generalized linear model with spatially-temporally varying coefficients under fixed values of spatial-temporal process parameters and some auxiliary model parameters. The output contains posterior samples of the fixed effects, spatial-temporal random effects and, if required, finds leave-one-out predictive densities.

Usage

stvcGLMexact(
  formula,
  data = parent.frame(),
  family,
  sp_coords,
  time_coords,
  cor.fn,
  process.type,
  sptParams,
  priors,
  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. Variables in parenthesis are assigned spatially-temporally varying coefficients. See examples.
data: an optional data frame containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which stvcGLMexact is called.
family: Specifies the distribution of the response as a member of the exponential family. Supported options are 'poisson', 'binomial' and 'binary'.
sp_coords: an $n \times 2$ matrix of the observation spatial coordinates in $\mathbb{R}^2$ (e.g., easting and northing).
time_coords: an $n \times 1$ matrix of the observation temporal coordinates in $\mathcal{T} \subseteq [0, \infty)$.
cor.fn: a quoted keyword that specifies the correlation function used to model the spatial-temporal dependence structure among the observations. Supported covariance model key words are: 'gneiting-decay' (Gneiting and Guttorp 2010). See below for details.
process.type: a quoted keyword specifying the model for the spatial-temporal process. Supported keywords are 'independent' which indicates independent processes for each varying coefficients characterized by different process parameters, independent.shared implies independent processes for the varying coefficients that shares common process parameters, and multivariate implies correlated processes for the varying coefficients modeled by a multivariate Gaussian process with an inverse-Wishart prior on the correlation matrix. The input for sptParams and priors must be given accordingly.
sptParams: fixed values of spatial-temporal process parameters in usually a list of length 2. If cor.fn='gneiting-decay', then it is a list of length 2 with tags phi_s and phi_t. If process.type='independent', then phi_s and phi_t contain fixed values of the $r$ spatial-temporal processes, otherwise they will contain scalars. See examples below.
priors: (optional) a list with each tag corresponding to a hyperparameter name and containing hyperprior details. Valid tags include V.beta, nu.beta, nu.z, sigmaSq.xi and IW.scale. Values of nu.beta and nu.z must be at least 2.1. If not supplied, uses defaults.
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 by loopd.method. Default is FALSE.
loopd.method: character. Ignored if loopd=FALSE. If loopd=TRUE, valid inputs are 'exact', 'CV'. 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' is faster as it implements $K$-fold cross validation 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.method is set to either 'exact' or 'CV'.
verbose: logical. If verbose = TRUE, prints model description.
...: currently no additional argument.

Value

An object of class stvcGLMexact, 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-temporal effects (z) and the fine-scale variation term (xi). The element with tag z will again be a list of length $r$, each containing posterior samples of the spatial-temporal random effects corresponding to each varying coefficient.
loopd: If loopd=TRUE, contains leave-one-out predictive densities.
model.params: Values of the fixed parameters that includes phi_s (spatial decay), phi_t (temporal 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 spatially-temporally varying generalized linear model by sampling exactly from the joint posterior distribution utilizing the generalized conjugate multivariate distribution theory (Bradley and Clinch 2024). Suppose $\chi = (\ell_1, \ldots, \ell_n)$ denotes the $n$ spatial-temporal co-ordinates in $\mathcal{L} = \mathcal{S} \times \mathcal{T}$, the response $y$ is observed. Let $y(\ell)$ be the outcome at the co-ordinate $\ell$ endowed with a probability law from the natural exponential family, which we denote by $$ y(\ell) \sim \mathrm{EF}(x(\ell)^\top \beta + \tilde{x}(\ell)^\top z(\ell); b(\ell), \psi) $$ for some positive parameter $b(\ell) > 0$ and unit log partition function $\psi$. Here, $\tilde{x}(\ell)$ denotes covariates with spatially-temporally varying coefficients We consider the following response models based on the input supplied to the argument family.

'poisson': It considers point-referenced Poisson responses $y(\ell) \sim \mathrm{Poisson}(e^{x(\ell)^\top \beta + \tilde{x}(\ell)^\top z(\ell)})$. Here, $b(\ell) = 1$ and $\psi(t) = e^t$.
'binomial': It considers point-referenced binomial counts $y(\ell) \sim \mathrm{Binomial}(m(\ell), \pi(\ell))$ where, $m(\ell)$ denotes the total number of trials and probability of success $\pi(\ell) = \mathrm{ilogit}(x(\ell)^\top \beta + \tilde{x}(\ell)^\top z(\ell))$ at spatial-temporal co-ordinate $\ell$. Here, $b = m(\ell)$ and $\psi(t) = \log(1+e^t)$.
'binary': It considers point-referenced binary data (0 or, 1) i.e., $y(\ell) \sim \mathrm{Bernoulli}(\pi(\ell))$, where probability of success $\pi(\ell) = \mathrm{ilogit}(x(\ell)^\top \beta + \tilde{x}(\ell)^\top z(\ell))$ at spatial-temporal co-ordinate $\ell$. Here, $b(\ell) = 1$ and $\psi(t) = \log(1 + e^t)$.

The hierarchical model is given as $$ \begin{aligned} y(\ell_i) &\mid \beta, z, \xi \sim EF(x(\ell_i)^\top \beta + \tilde{x}(\ell_i)^\top z(s_i) + \xi_i - \mu_i; b_i, \psi_y), i = 1, \ldots, n\\ \xi &\mid \beta, z, \sigma^2_\xi, \alpha_\epsilon \sim \mathrm{GCM_c}(\cdots),\\ \beta &\mid \sigma^2_\beta \sim N(0, \sigma^2_\beta V_\beta), \quad \sigma^2_\beta \sim \mathrm{IG}(\nu_\beta/2, \nu_\beta/2)\\ z_j &\mid \sigma^2_{z_j} \sim N(0, \sigma^2_{z_j} R(\chi; \phi_s, \phi_t)), \quad \sigma^2_{z_j} \sim \mathrm{IG}(\nu_z/2, \nu_z/2), j = 1, \ldots, r \end{aligned} $$ where $\mu = (\mu_1, \ldots, \mu_n)^\top$ denotes the discrepancy parameter. We fix the spatial-temporal process parameters $\phi_s$ and $\phi_t$ 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 .

T. Gneiting and P. Guttorp (2010). "Continuous-parameter spatio-temporal processes." In A.E. Gelfand, P.J. Diggle, M. Fuentes, and P Guttorp, editors, Handbook of Spatial Statistics, Chapman & Hall CRC Handbooks of Modern Statistical Methods, p 427–436. Taylor and Francis.

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

data("sim_stvcPoisson")
dat <- sim_stvcPoisson[1:100, ]

# Fit a spatial-temporal varying coefficient Poisson GLM
mod1 <- stvcGLMexact(y ~ x1 + (x1), data = dat, family = "poisson",
                     sp_coords = as.matrix(dat[, c("s1", "s2")]),
                     time_coords = as.matrix(dat[, "t_coords"]),
                     cor.fn = "gneiting-decay",
                     process.type = "multivariate",
                     sptParams = list(phi_s = 1, phi_t = 1),
                     verbose = FALSE, n.samples = 100)