Spatial and Spatiotemporal SPDE-Based GLMMs with TMB • sdmTMB

Spatial and spatiotemporal GLMMs with TMB

sdmTMB is an R package that fits spatial and spatiotemporal GLMMs (Generalized Linear Mixed Effects Models) using Template Model Builder (TMB), R-INLA, and Gaussian Markov random fields. One common application is for species distribution models (SDMs). See the documentation site and a preprint:

Anderson, S.C., E.J. Ward, P.A. English, L.A.K. Barnett, J.T. Thorson. 2024. sdmTMB: an R package for fast, flexible, and user-friendly generalized linear mixed effects models with spatial and spatiotemporal random fields. bioRxiv 2022.03.24.485545; doi: https://doi.org/10.1101/2022.03.24.485545

Installation

sdmTMB can be installed from CRAN:

install.packages("sdmTMB", dependencies = TRUE)

Assuming you have a C++ compiler installed, the development version is recommended and can be installed:

# install.packages("pak")
pak::pkg_install("pbs-assess/sdmTMB", dependencies = TRUE)

There are some extra utilities in the sdmTMBextra package.

Importantly, it is recommended to use an optimized BLAS library, which will result in major speed improvements for TMB (and other) models in R (e.g., often 8-fold speed increases for sdmTMB models). Suggested installation instructions for Mac users, Linux users, Windows users, and Windows users without admin privileges. To check that you’ve successfully linked the optimized BLAS, start a new session and run:

m <- 1e4; n <- 1e3; k <- 3e2
X <- matrix(rnorm(m*k), nrow=m); Y <- matrix(rnorm(n*k), ncol=n)
system.time(X %*% Y)

The result (‘elapsed’) should take a fraction of a second (e.g., 0.03 s), not multiple seconds.

Overview

Analyzing geostatistical data (coordinate-referenced observations from some underlying spatial process) is becoming increasingly common in ecology. sdmTMB implements geostatistical spatial and spatiotemporal GLMMs using TMB for model fitting and R-INLA to set up SPDE (stochastic partial differential equation) matrices. One common application is for species distribution models (SDMs), hence the package name. The goal of sdmTMB is to provide a fast, flexible, and user-friendly interface—similar to the popular R package glmmTMB—but with a focus on spatial and spatiotemporal models with an SPDE approach. We extend the generalized linear mixed models (GLMMs) familiar to ecologists to include the following optional features:

spatial random fields
spatiotemporal random fields that may be independent by year or modelled with random walks or autoregressive processes
smooth terms for covariates, using the familiar s() notation from mgcv
breakpoint (hockey-stick) or logistic covariates
time-varying covariates (coefficients modelled as random walks)
spatially varying coefficient models (SVCs)
interpolation or forecasting over missing or future time slices
a wide range of families: all standard R families plus tweedie(), nbinom1(), nbinom2(), lognormal(), and student(), plus some truncated and censored families
delta/hurdle models including delta_gamma(), delta_lognormal(), and delta_truncated_nbinom2()

Estimation is performed in sdmTMB via maximum marginal likelihood with the objective function calculated in TMB and minimized in R via stats::nlminb() with the random effects integrated over via the Laplace approximation. The sdmTMB package also allows for models to be passed to Stan via tmbstan, allowing for Bayesian model estimation.

See ?sdmTMB and ?predict.sdmTMB for the most complete examples. Also see the vignettes (‘Articles’) on the documentation site and the preprint and appendices linked to below.

Getting help

For questions about how to use sdmTMB or interpret the models, please post on the discussion board. If you email a question, we are likely to respond on the discussion board with an anonymized version of your question (and without data) if we think it could be helpful to others. Please let us know if you don’t want us to do that.

For bugs or feature requests, please post in the issue tracker.

Slides and recordings from a workshop on sdmTMB.

Citation

To cite sdmTMB in publications use:

citation("sdmTMB")

Anderson, S.C., E.J. Ward, P.A. English, L.A.K. Barnett., J.T. Thorson. 2024. sdmTMB: an R package for fast, flexible, and user-friendly generalized linear mixed effects models with spatial and spatiotemporal random fields. bioRxiv 2022.03.24.485545; doi: https://doi.org/10.1101/2022.03.24.485545

A list of (known) publications that use sdmTMB can be found here. Please use the above citation so we can track publications.

sdmTMB is heavily inspired by the VAST R package:

Thorson, J.T. 2019. Guidance for decisions using the Vector Autoregressive Spatio-Temporal (VAST) package in stock, ecosystem, habitat and climate assessments. Fisheries Research 210: 143–161. https://doi.org/10.1016/j.fishres.2018.10.013.

and the glmmTMB R package:

Brooks, M.E., Kristensen, K., van Benthem, K.J., Magnusson, A., Berg, C.W., Nielsen, A., Skaug, H.J., Maechler, M., and Bolker, B.M. 2017. glmmTMB balances speed and flexibility among packages for zero-inflated generalized linear mixed modeling. The R Journal 9(2): 378–400. https://doi.org/10.32614/rj-2017-066.

INLA and inlabru can fit many of the same models as sdmTMB (and many more) in an approximate Bayesian inference framework.

mgcv can fit similar SPDE-based Gaussian random field models with code included in Miller et al. (2019).

A table in the sdmTMB preprint describes functionality and timing comparisons between sdmTMB, VAST, INLA/inlabru, and mgcv and the discussion makes suggestions about when you might choose one package over another.

Basic use

An sdmTMB model requires a data frame that contains a response column, columns for any predictors, and columns for spatial coordinates. It usually makes sense to convert the spatial coordinates to an equidistant projection such as UTMs such that distance remains constant throughout the study region [e.g., using sf::st_transform()]. Here, we illustrate a spatial model fit to Pacific cod (Gadus macrocephalus) trawl survey data from Queen Charlotte Sound, BC, Canada. Our model contains a main effect of depth as a penalized smoother, a spatial random field, and Tweedie observation error. Our data frame pcod (built into the package) has a column year for the year of the survey, density for density of Pacific cod in a given survey tow, present for whether density > 0, depth for depth in meters of that tow, and spatial coordinates X and Y, which are UTM coordinates in kilometres.

library(dplyr)
library(ggplot2)
library(sdmTMB)
head(pcod)

#> # A tibble: 3 × 6
#>    year density present depth     X     Y
#>   <int>   <dbl>   <dbl> <dbl> <dbl> <dbl>
#> 1  2003   113.        1   201  446. 5793.
#> 2  2003    41.7       1   212  446. 5800.
#> 3  2003     0         0   220  449. 5802.

We start by creating a mesh object that contains matrices to apply the SPDE approach.

mesh <- make_mesh(pcod, xy_cols = c("X", "Y"), cutoff = 10)

Here, cutoff defines the minimum allowed distance between points in the units of X and Y (km). Alternatively, we could have created any mesh via the fmesher or INLA packages and supplied it to make_mesh(). We can inspect our mesh object with the associated plotting method plot(mesh).

Fit a spatial model with a smoother for depth:

fit <- sdmTMB(
  density ~ s(depth),
  data = pcod,
  mesh = mesh,
  family = tweedie(link = "log"),
  spatial = "on"
)

Print the model fit:

fit
#> Spatial model fit by ML ['sdmTMB']
#> Formula: density ~ s(depth)
#> Mesh: mesh (isotropic covariance)
#> Data: pcod
#> Family: tweedie(link = 'log')
#>  
#>             coef.est coef.se
#> (Intercept)     2.37    0.21
#> sdepth          0.62    2.53
#> 
#> Smooth terms:
#>            Std. Dev.
#> sds(depth)     13.93
#> 
#> Dispersion parameter: 12.69
#> Tweedie p: 1.58
#> Matérn range: 16.39
#> Spatial SD: 1.86
#> ML criterion at convergence: 6402.136
#> 
#> See ?tidy.sdmTMB to extract these values as a data frame.

The output indicates our model was fit by maximum (marginal) likelihood (ML). We also see the formula, mesh, fitted data, and family. Next we see any estimated main effects including the linear component of the smoother (sdepth), the standard deviation on the smoother weights (sds(depth)), the Tweedie dispersion and power parameters, the Matérn range distance (distance at which points are effectively independent), the marginal spatial field standard deviation, and the negative log likelihood at convergence.

We can extract parameters as a data frame:

tidy(fit, conf.int = TRUE)
#> # A tibble: 1 × 5
#>   term        estimate std.error conf.low conf.high
#>   <chr>          <dbl>     <dbl>    <dbl>     <dbl>
#> 1 (Intercept)     2.37     0.215     1.95      2.79

tidy(fit, effects = "ran_pars", conf.int = TRUE)
#> # A tibble: 4 × 5
#>   term      estimate std.error conf.low conf.high
#>   <chr>        <dbl>     <dbl>    <dbl>     <dbl>
#> 1 range        16.4    4.47        9.60     28.0 
#> 2 phi          12.7    0.406      11.9      13.5 
#> 3 sigma_O       1.86   0.218       1.48      2.34
#> 4 tweedie_p     1.58   0.00998     1.56      1.60

Run some basic sanity checks on our model:

sanity(fit)
#> ✔ Non-linear minimizer suggests successful convergence
#> ✔ Hessian matrix is positive definite
#> ✔ No extreme or very small eigenvalues detected
#> ✔ No gradients with respect to fixed effects are >= 0.001
#> ✔ No fixed-effect standard errors are NA
#> ✔ No standard errors look unreasonably large
#> ✔ No sigma parameters are < 0.01
#> ✔ No sigma parameters are > 100
#> ✔ Range parameter doesn't look unreasonably large

Use the ggeffects package to plot the smoother effect:

ggeffects::ggpredict(fit, "depth [50:400, by=2]") |> plot()

If the depth effect was parametric and not a penalized smoother, we could have alternatively used ggeffects::ggeffect() for a fast marginal effect plot.

Next, we can predict on new data. We will use a data frame qcs_grid from the package, which contains all the locations (and covariates) at which we wish to predict. Here, these newdata are a grid, or raster, covering our survey.

p <- predict(fit, newdata = qcs_grid)

head(p)

#> # A tibble: 3 × 7
#>       X     Y depth   est est_non_rf est_rf omega_s
#>   <dbl> <dbl> <dbl> <dbl>      <dbl>  <dbl>   <dbl>
#> 1   456  5636  347. -3.06      -3.08 0.0172  0.0172
#> 2   458  5636  223.  2.03       1.99 0.0460  0.0460
#> 3   460  5636  204.  2.89       2.82 0.0747  0.0747

ggplot(p, aes(X, Y, fill = exp(est))) + geom_raster() +
  scale_fill_viridis_c(trans = "sqrt")

We could switch to a presence-absence model by changing the response column and family:

fit <- sdmTMB(
  present ~ s(depth),
  data = pcod, 
  mesh = mesh,
  family = binomial(link = "logit")
)

Or a hurdle/delta model by changing the family:

fit <- sdmTMB(
  density ~ s(depth),
  data = pcod,
  mesh = mesh,
  family = delta_gamma(link1 = "logit", link2 = "log"),
)

We could instead fit a spatiotemporal model by specifying the time column and a spatiotemporal structure:

fit_spatiotemporal <- sdmTMB(
  density ~ s(depth, k = 5), 
  data = pcod, 
  mesh = mesh,
  time = "year",
  family = tweedie(link = "log"), 
  spatial = "off", 
  spatiotemporal = "ar1"
)

If we wanted to create an area-weighted standardized population index, we could predict on a grid covering the entire survey (qcs_grid) with grid cell area 4 (2 x 2 km) and pass the predictions to get_index():

grid_yrs <- replicate_df(qcs_grid, "year", unique(pcod$year))
p_st <- predict(fit_spatiotemporal, newdata = grid_yrs, 
  return_tmb_object = TRUE)
index <- get_index(p_st, area = rep(4, nrow(grid_yrs)))
ggplot(index, aes(year, est)) +
  geom_ribbon(aes(ymin = lwr, ymax = upr), fill = "grey90") +
  geom_line(lwd = 1, colour = "grey30") +
  labs(x = "Year", y = "Biomass (kg)")

Or the center of gravity:

cog <- get_cog(p_st, format = "wide")
ggplot(cog, aes(est_x, est_y, colour = year)) +
  geom_pointrange(aes(xmin = lwr_x, xmax = upr_x)) +
  geom_pointrange(aes(ymin = lwr_y, ymax = upr_y)) +
  scale_colour_viridis_c()

For more on these basic features, see the vignettes Intro to modelling with sdmTMB and Index standardization with sdmTMB.

Advanced functionality

Time-varying coefficients

Time-varying intercept:

fit <- sdmTMB(
  density ~ 0 + s(depth, k = 5), 
  time_varying = ~ 1, 
  data = pcod, mesh = mesh,
  time = "year",  
  family = tweedie(link = "log"),
  silent = FALSE # see progress
)

Time-varying (random walk) effect of depth:

fit <- sdmTMB(
  density ~ 1, 
  time_varying = ~ 0 + depth_scaled + depth_scaled2,
  data = pcod, mesh = mesh,
  time = "year",
  family = tweedie(link = "log"),
  spatial = "off",
  spatiotemporal = "ar1",
  silent = FALSE
)

See the vignette Intro to modelling with sdmTMB for more details.

Spatially varying coefficients (SVC)

Spatially varying effect of time:

pcod$year_scaled <- as.numeric(scale(pcod$year))
fit <- sdmTMB(
  density ~ s(depth, k = 5) + year_scaled,
  spatial_varying = ~ year_scaled, 
  data = pcod, mesh = mesh, 
  time = "year",
  family = tweedie(link = "log"),
  spatiotemporal = "off"
)

See zeta_s in the output, which represents the coefficient varying in space. You’ll want to ensure you set up your model such that it ballpark has a mean of 0 (e.g., by including it in formula too).

grid_yrs <- replicate_df(qcs_grid, "year", unique(pcod$year))
grid_yrs$year_scaled <- (grid_yrs$year - mean(pcod$year)) / sd(pcod$year)
p <- predict(fit, newdata = grid_yrs) %>% 
  subset(year == 2011) # any year
ggplot(p, aes(X, Y, fill = zeta_s_year_scaled)) + geom_raster() +
  scale_fill_gradient2()

See the vignette on Fitting spatial trend models with sdmTMB for more details.

Random intercepts

We can use the same syntax (1 | group) as lme4 or glmmTMB to fit random intercepts:

pcod$year_factor <- as.factor(pcod$year)
fit <- sdmTMB(
  density ~ s(depth, k = 5) + (1 | year_factor),
  data = pcod, mesh = mesh,
  time = "year",
  family = tweedie(link = "log")
)

Breakpoint and threshold effects

fit <- sdmTMB(
  present ~ 1 + breakpt(depth_scaled), 
  data = pcod, mesh = mesh,
  family = binomial(link = "logit")
)

fit <- sdmTMB(
  present ~ 1 + logistic(depth_scaled), 
  data = pcod, mesh = mesh,
  family = binomial(link = "logit")
)

See the vignette on Threshold modeling with sdmTMB for more details.

Simulating data

Simulating data from scratch

predictor_dat <- expand.grid(
  X = seq(0, 1, length.out = 100), Y = seq(0, 1, length.out = 100)
)
mesh <- make_mesh(predictor_dat, xy_cols = c("X", "Y"), cutoff = 0.05)
sim_dat <- sdmTMB_simulate(
  formula = ~ 1,
  data = predictor_dat,
  mesh = mesh,
  family = poisson(link = "log"),
  range = 0.3,
  sigma_O = 0.4,
  seed = 1,
  B = 1 # B0 = intercept
)
head(sim_dat)
#> # A tibble: 6 × 7
#>        X     Y omega_s    mu   eta observed `(Intercept)`
#>    <dbl> <dbl>   <dbl> <dbl> <dbl>    <dbl>         <dbl>
#> 1 0          0  -0.154  2.33 0.846        1             1
#> 2 0.0101     0  -0.197  2.23 0.803        0             1
#> 3 0.0202     0  -0.240  2.14 0.760        2             1
#> 4 0.0303     0  -0.282  2.05 0.718        2             1
#> 5 0.0404     0  -0.325  1.96 0.675        3             1
#> 6 0.0505     0  -0.367  1.88 0.633        2             1


# sample 200 points for fitting:
set.seed(1)
sim_dat_obs <- sim_dat[sample(seq_len(nrow(sim_dat)), 200), ]

ggplot(sim_dat, aes(X, Y)) +
  geom_raster(aes(fill = exp(eta))) + # mean without observation error
  geom_point(aes(size = observed), data = sim_dat_obs, pch = 21) +
  scale_fill_viridis_c() +
  scale_size_area() +
  coord_cartesian(expand = FALSE)

Fit to the simulated data:

mesh <- make_mesh(sim_dat_obs, xy_cols = c("X", "Y"), cutoff = 0.05)
fit <- sdmTMB(
  observed ~ 1,
  data = sim_dat_obs,
  mesh = mesh,
  family = poisson()
)

See ?sdmTMB_simulate for more details.

Simulating from an existing fit

s <- simulate(fit, nsim = 500)
dim(s)
#> [1] 969 500
s[1:3,1:4]
#>      [,1]     [,2]     [,3]     [,4]
#> [1,]    0 59.40310 83.20888  0.00000
#> [2,]    0 34.56408  0.00000 19.99839
#> [3,]    0  0.00000  0.00000  0.00000

See the vignette on Residual checking with sdmTMB, ?simulate.sdmTMB, and ?dharma_residuals for more details.

Sampling from the joint precision matrix

We can take samples from the implied parameter distribution assuming an MVN covariance matrix on the internal parameterization:

samps <- gather_sims(fit, nsim = 1000)
ggplot(samps, aes(.value)) + geom_histogram() +
  facet_wrap(~.variable, scales = "free_x")
#> `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

See ?gather_sims and ?get_index_sims for more details.

Calculating uncertainty on spatial predictions

The fastest way to get point-wise prediction uncertainty is to use the MVN samples:

p <- predict(fit, newdata = predictor_dat, nsim = 500)
predictor_dat$se <- apply(p, 1, sd)
ggplot(predictor_dat, aes(X, Y, fill = se)) +
  geom_raster() +
  scale_fill_viridis_c(option = "A") +
  coord_cartesian(expand = FALSE)

Cross validation

sdmTMB has built-in functionality for cross-validation. If we were to set a future::plan(), the folds would be fit in parallel:

mesh <- make_mesh(pcod, c("X", "Y"), cutoff = 10)
## Set parallel processing if desired:
# library(future)
# plan(multisession)
m_cv <- sdmTMB_cv(
  density ~ s(depth, k = 5),
  data = pcod, mesh = mesh,
  family = tweedie(link = "log"), k_folds = 2
)
#> Running fits with `future.apply()`.
#> Set a parallel `future::plan()` to use parallel processing.

# Sum of log likelihoods of left-out data:
m_cv$sum_loglik
#> [1] -6756.28

See ?sdmTMB_cv for more details.

Priors

Priors/penalties can be placed on most parameters. For example, here we place a PC (penalized complexity) prior on the Matérn random field parameters, a standard normal prior on the effect of depth, a Normal(0, 10^2) prior on the intercept, and a half-normal prior on the Tweedie dispersion parameter (phi):

mesh <- make_mesh(pcod, c("X", "Y"), cutoff = 10)
fit <- sdmTMB(
  density ~ depth_scaled,
  data = pcod, mesh = mesh,
  family = tweedie(),
  priors = sdmTMBpriors(
    matern_s = pc_matern(range_gt = 10, sigma_lt = 5),
    b = normal(c(0, 0), c(1, 10)),
    phi = halfnormal(0, 15)
  )
)

We can visualize the PC Matérn prior:

plot_pc_matern(range_gt = 10, sigma_lt = 5)

See ?sdmTMBpriors for more details.

Bayesian MCMC sampling with Stan

The fitted model can be passed to the tmbstan package to sample from the posterior with Stan. See the Bayesian vignette.

Turning off random fields

We can turn off the random fields for model comparison:

fit_sdmTMB <- sdmTMB(
  present ~ poly(depth_scaled, 2),
  data = pcod, mesh = mesh,
  spatial = "off",
  family = binomial()
)
fit_glm <- glm(
  present ~ poly(depth_scaled, 2),
  data = pcod,
  family = binomial()
)

tidy(fit_sdmTMB)
#> # A tibble: 3 × 3
#>   term                   estimate std.error
#>   <chr>                     <dbl>     <dbl>
#> 1 (Intercept)              -0.426    0.0573
#> 2 poly(depth_scaled, 2)1  -31.7      3.03  
#> 3 poly(depth_scaled, 2)2  -66.9      4.09

broom::tidy(fit_glm)
#> # A tibble: 3 × 5
#>   term                   estimate std.error statistic  p.value
#>   <chr>                     <dbl>     <dbl>     <dbl>    <dbl>
#> 1 (Intercept)              -0.426    0.0573     -7.44 1.03e-13
#> 2 poly(depth_scaled, 2)1  -31.7      3.03      -10.5  1.20e-25
#> 3 poly(depth_scaled, 2)2  -66.9      4.09      -16.4  3.50e-60

Using a custom fmesher mesh

Defining a mesh directly with INLA:

bnd <- INLA::inla.nonconvex.hull(cbind(pcod$X, pcod$Y), convex = -0.1)
mesh_inla <- INLA::inla.mesh.2d(
  boundary = bnd,
  max.edge = c(25, 50)
)
mesh <- make_mesh(pcod, c("X", "Y"), mesh = mesh_inla)
plot(mesh)

fit <- sdmTMB(
  density ~ s(depth, k = 5),
  data = pcod, mesh = mesh,
  family = tweedie(link = "log")
)

Barrier meshes

A barrier mesh limits correlation across barriers (e.g., land or water). See add_barrier_mesh() in sdmTMBextra.

sdmTMB

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