Bayesian spatial linear model using predictive stacking

Fits Bayesian spatial linear model on a collection of candidate models constructed based on some candidate values of some model parameters specified by the user and subsequently combines inference by stacking predictive densities. See Zhang, Tang and Banerjee (2024) for more details.

Usage

spLMstack(
  formula,
  data = parent.frame(),
  coords,
  cor.fn,
  priors,
  params.list,
  n.samples,
  loopd.method,
  parallel = FALSE,
  solver = "ECOS",
  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 from environment(formula), typically the environment from which spLMstack is called.
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: a list with each tag corresponding to a parameter name and containing prior details. If not supplied, uses defaults.
params.list: a list containing candidate values of spatial process parameters for the cor.fn used, and, noise-to-spatial variance ratio.
n.samples: number of posterior samples to be generated.
loopd.method: character. Valid inputs are 'exact' and 'PSIS'. The option 'exact' corresponds to exact leave-one-out predictive densities. The option 'PSIS' is faster, as it finds approximate leave-one-out predictive densities using Pareto-smoothed importance sampling (Gelman et al. 2024).
parallel: logical. If parallel=FALSE, the parallelization plan, if set up by the user, is ignored. If parallel=TRUE, the function inherits the parallelization plan that is set by the user via the function future::plan() only. Depending on the parallel backend available, users may choose their own plan. More details are available at https://cran.R-project.org/package=future.
solver: (optional) Specifies the name of the solver that will be used to obtain optimal stacking weights for each candidate model. Default is "ECOS". Users can use other solvers supported by the CVXR-package package.
verbose: logical. If TRUE, prints model-specific optimal stacking weights.
...: currently no additional argument.

Value

An object of class spLMstack, which is a list including the following tags -

samples: a list of length equal to total number of candidate models with each entry corresponding to a list of length 3, containing posterior samples of fixed effects (beta), variance parameter (sigmaSq), spatial effects (z) for that model.
loopd: a list of length equal to total number of candidate models with each entry containing leave-one-out predictive densities under that particular model.
n.models: number of candidate models that are fit.
candidate.models: a matrix with n_model rows with each row containing details of the model parameters and its optimal weight.
stacking.weights: a numeric vector of length equal to the number of candidate models storing the optimal stacking weights.
run.time: a proc_time object with runtime details.
solver.status: solver status as returned by the optimization routine.

The return object might include additional data that is useful for subsequent prediction, model fit evaluation and other utilities.

Details

Instead of assigning a prior on the process parameters $\phi$ and $\nu$, noise-to-spatial variance ratio $\delta^2$, we consider a set of candidate models based on some candidate values of these parameters supplied by the user. Suppose the set of candidate models is $\mathcal{M} = \{M_1, \ldots, M_G\}$. 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 $$ \begin{aligned} \max_{w_1, \ldots, w_G}& \, \frac{1}{n} \sum_{i = 1}^n \log \sum_{g = 1}^G w_g p(y_i \mid y_{-i}, M_g) \\ \text{subject to} & \quad w_g \geq 0, \sum_{g = 1}^G w_g = 1 \end{aligned} $$ to find the optimal stacking weights $\hat{w}_1, \ldots, \hat{w}_G$.

References

Vehtari A, Simpson D, Gelman A, Yao Y, Gabry J (2024). "Pareto Smoothed Importance Sampling." Journal of Machine Learning Research, 25(72), 1-58. URL https://jmlr.org/papers/v25/19-556.html.

Zhang L, Tang W, Banerjee S (2024). "Bayesian Geostatistics Using Predictive Stacking."
doi:10.48550/arXiv.2304.12414 .

Author

Soumyakanti Pan span18@ucla.edu,
Sudipto Banerjee sudipto@ucla.edu

Examples

# load data and work with first 100 rows
data(simGaussian)
dat <- simGaussian[1:100, ]

# setup prior list
muBeta <- c(0, 0)
VBeta <- cbind(c(1.0, 0.0), c(0.0, 1.0))
sigmaSqIGa <- 2
sigmaSqIGb <- 2
prior_list <- list(beta.norm = list(muBeta, VBeta),
                   sigma.sq.ig = c(sigmaSqIGa, sigmaSqIGb))

mod1 <- 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),
                                     nu = c(0.5, 1),
                                     noise_sp_ratio = c(1)),
                  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|               1| 0.291  |
#> | Model 2 |  3.0|  0.5|               1| 0.709  |
#> | Model 3 |  1.5|  1.0|               1| 0.000  |
#> | Model 4 |  3.0|  1.0|               1| 0.000  |
#> +---------+-----+-----+----------------+--------+
#> 

post_samps <- stackedSampler(mod1)
post_beta <- post_samps$beta
print(t(apply(post_beta, 1, function(x) quantile(x, c(0.025, 0.5, 0.975)))))
#>                  2.5%      50%    97.5%
#> (Intercept) 0.9400005 1.854580 2.684189
#> x1          4.7319053 4.914079 5.081339

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])

library(ggplot2)
plot1 <- 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) +
  geom_abline(slope = 1, intercept = 0,
              color = "red", linetype = "solid") +
  xlab("True z") + ylab("Stacked posterior of z") +
  theme_bw() +
  theme(panel.background = element_blank(),
        aspect.ratio = 1)