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 (2025) for more details.

Usage

spLMstack(
  formula,
  data = parent.frame(),
  coords,
  cor.fn,
  priors = "flat",
  candidate.models,
  n.samples,
  loopd.method,
  parallel = FALSE,
  solver = NULL,
  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: either "flat" (default), which assigns the prior $p(\beta, \sigma^2) \propto 1/\sigma^2$, or a list with tags beta.norm (a list containing $\mu_\beta$ and $V_\beta$) and/or sigma.sq.ig (a vector containing $a_\sigma$ and $b_\sigma$). A component not supplied in the list receives its flat prior, $p(\beta) \propto 1$ or $p(\sigma^2) \propto 1/\sigma^2$.
candidate.models: an object of class candidateModels containing a list of candidate models for stacking. See candidateModels() for details.
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 order is c("CLARABEL", "ECOS", "SCS"). 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 4, containing posterior samples of fixed effects (beta), measurement error variance (sigmaSq), spatial variance (sigmaSq.z), and 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.
model.params: a list with one element per candidate model, each a named list of its parameters: phi, nu (NA for the exponential correlation function) and noise_sp_ratio (noise-to-spatial variance ratio).
stacking.summary: a matrix with one row per candidate model, containing its parameters and its optimal stacking weight, for display.
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.
diagnostics: a list of diagnostics. Element numerical is a data frame with one row per candidate model and columns min.pivot (the smallest relative Cholesky pivot of the $n \times n$ factorizations; values below 1e-8 indicate a nearly singular covariance matrix), min.cor and max.cor (the correlations of the two farthest-apart and of the two closest locations; values of min.cor above 0.95 suggest an effective range far exceeding the extent of the data, values of max.cor below 0.05 nearly uncorrelated locations), obtained from quantities the fits compute anyway. If loopd.method = 'PSIS', element pareto is a list with the Pareto $k$ diagnostic values of each candidate model (k), the threshold above which they are unreliable (threshold) and the number of values above it for each model (n.high). Element solver describes the optimization for the stacking weights: the solver used (used) and its status (status), the installed and requested solvers, the search order, the attempts with their status, and whether the fallback loo::stacking_weights() was used. If verbose = TRUE, a "Diagnostics" section is printed if there is an issue: numerical flags of candidate models with stacking weight above 0.05 (extreme candidates with negligible weight are expected in a stacking grid and are only counted), Pareto $k$ values above the threshold for any model, and solver problems (a requested solver not installed, an inaccurate solution, or the fallback).

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 (2025). "Bayesian Geostatistics Using Predictive Stacking." Journal of the American Statistical Association, In press. doi:10.1080/01621459.2025.2566449 .

Author

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

Examples

set.seed(1234)
data(simSpatial)
dat <- simSpatial[1:100, ]

cand.mod <- candidateModels(list(phi = c(2, 4, 6),
                                 nu = c(1, 1.5, 1.75),
                                 noise_sp_ratio = c(0.1, 0.25)),
                            "cartesian")

mod1 <- spLMstack(y_gauss ~ x1 + x2, data = dat,
                  coords = as.matrix(dat[, c("s1", "s2")]),
                  cor.fn = "matern",
                  candidate.models = cand.mod,
                  n.samples = 1000, loopd.method = "exact",
                  parallel = FALSE, verbose = TRUE)
#> 
#> STACKING WEIGHTS:
#> 
#>            | phi | nu   | noise_sp_ratio | weight |
#> +----------+-----+------+----------------+--------+
#> | Model 1  |    2|  1.00|            0.10| 0.000  |
#> | Model 2  |    4|  1.00|            0.10| 0.000  |
#> | Model 3  |    6|  1.00|            0.10| 0.000  |
#> | Model 4  |    2|  1.50|            0.10| 0.000  |
#> | Model 5  |    4|  1.50|            0.10| 0.000  |
#> | Model 6  |    6|  1.50|            0.10| 0.000  |
#> | Model 7  |    2|  1.75|            0.10| 0.000  |
#> | Model 8  |    4|  1.75|            0.10| 0.189  |
#> | Model 9  |    6|  1.75|            0.10| 0.000  |
#> | Model 10 |    2|  1.00|            0.25| 0.000  |
#> | Model 11 |    4|  1.00|            0.25| 0.000  |
#> | Model 12 |    6|  1.00|            0.25| 0.000  |
#> | Model 13 |    2|  1.50|            0.25| 0.000  |
#> | Model 14 |    4|  1.50|            0.25| 0.000  |
#> | Model 15 |    6|  1.50|            0.25| 0.000  |
#> | Model 16 |    2|  1.75|            0.25| 0.000  |
#> | Model 17 |    4|  1.75|            0.25| 0.000  |
#> | Model 18 |    6|  1.75|            0.25| 0.811  |
#> +----------+-----+------+----------------+--------+
#> 

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)  0.9676843  2.022228  3.0711551
#> x1           4.8147586  4.912469  4.9982113
#> x2          -1.1026691 -1.010358 -0.9012183

# compare the posterior medians of the spatial effects with the truth
cor(apply(post_samps$z, 1, median), dat$z_true)
#> [1] 0.9654177