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), fmesher, 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
- Overview
- Getting help
- Citation
- Basic use
- Advanced functionality
- Time-varying coefficients
- Spatially varying coefficients (SVC)
- Random intercepts
- Breakpoint and threshold effects
- Simulating data
- Sampling from the joint precision matrix
- Calculating uncertainty on spatial predictions
- Cross validation
- Priors
- Bayesian MCMC sampling with Stan
- Turning off random fields
- Using a custom fmesher mesh
- Barrier meshes
- Related software
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::pak("sdmTMB/sdmTMB", dependencies = TRUE)There are some extra utilities in the sdmTMBextra package.
For large models, 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 (other than R 4.5.0) or with OpenBLAS on a Mac, 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 > 1 second.
Analyzing geostatistical data (coordinate-referenced observations from some underlying spatial process) is becoming increasingly common in many fields. sdmTMB implements geostatistical spatial and spatiotemporal GLMMs using TMB for model fitting and fmesher to set up SPDE matrices (for the stochastic partial differential equation approach; a computationally efficient method for modeling spatial correlation). 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 common generalized linear mixed models (GLMMs) 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(),student(),gengamma(), plus some truncated and censored families - delta/hurdle models including
delta_gamma(),delta_lognormal(), anddelta_truncated_nbinom2()
Estimation is via maximum marginal likelihood (with random effects
integrated out) 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 listed below.
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.
There have been several past sdmTMB workshops. Slides and exercises from the latest workshop are available here. Recordings from an older workshop are also available.
To cite sdmTMB in publications, please use:
citation("sdmTMB")Anderson, S.C., E.J. Ward, P.A. English, L.A.K. Barnett., J.T. Thorson. 2025. sdmTMB: an R package for fast, flexible, and user-friendly generalized linear mixed effects models with spatial and spatiotemporal random fields. In press at Journal of Statistical Software. bioRxiv preprint: 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.
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 1 km remains the same distance
throughout the study region (unlike latitude/longitude) [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 a triangular network used to approximate the spatial field (a “finite element mesh”).
mesh <- make_mesh(pcod, xy_cols = c("X", "Y"), cutoff = 10)Here, cutoff defines the minimum allowed distance between mesh
vertices in the units of X and Y (km). Smaller values create finer
meshes but increase computation time. Alternatively, we could have
created a 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')
#>
#> Conditional model:
#> coef.est coef.se
#> (Intercept) 2.37 0.21
#> sdepth 0.62 2.53
#>
#> Smooth terms:
#> Std. Dev.
#> sd__s(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 shows our model was fit by maximum marginal likelihood
(ML), followed by the formula, mesh, data, and family. The main
effects section includes the linear component of the depth smoother
(sdepth) and the standard deviation on the smoother weights
(sds(depth)). The Tweedie dispersion (phi) and power parameters
control the distribution’s mean-variance relationship. The Matérn range
is the distance at which spatial correlation becomes negligible (~0.13
correlation). The marginal spatial field standard deviation (sigma_O)
represents unexplained spatial variation. The log likelihood represents
the objective function value at convergence.
We can extract parameters as a data frame:
tidy(fit, conf.int = TRUE)
#> # A tibble: 2 × 5
#> term estimate std.error conf.low conf.high
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 (Intercept) 2.37 0.215 1.95 2.79
#> 2 sdepth 0.62 2.53 -4.34 5.58
tidy(fit, effects = "ran_pars", conf.int = TRUE)
#> # A tibble: 5 × 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
#> 5 sd__s(depth) 13.9 NA 7.54 25.7Run 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 largeUse 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
We exponentiate the predictions with exp() to transform from log-link
space back to the density scale:
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 (a
time series of abundance accounting for spatial variation in sampling),
we could predict on a grid covering the entire survey (qcs_grid) with
grid cell area 4 km² (2 x 2 km) and pass the predictions to
get_index():
grid_yrs <- replicate_df(qcs_grid, "year", unique(pcod$year))
p_st <- predict(fit_spatiotemporal, newdata = grid_yrs,
return_tmb_object = TRUE)
index <- get_index(p_st, area = rep(4, nrow(grid_yrs)))
ggplot(index, aes(year, est)) +
geom_ribbon(aes(ymin = lwr, ymax = upr), fill = "grey90") +
geom_line(lwd = 1, colour = "grey30") +
labs(x = "Year", y = "Biomass (kg)")
Or the center of gravity (mean location of the population, useful for detecting distributional shifts):
cog <- get_cog(p_st, format = "wide")
ggplot(cog, aes(est_x, est_y, colour = year)) +
geom_pointrange(aes(xmin = lwr_x, xmax = upr_x)) +
geom_pointrange(aes(ymin = lwr_y, ymax = upr_y)) +
scale_colour_viridis_c()
For more on these basic features, see the vignettes Intro to modelling with sdmTMB and Index standardization with sdmTMB.
Time-varying coefficients allow parameters to change over time as random walks. This is useful when relationships may shift gradually (e.g., due to environmental change).
Time-varying intercept:
fit <- sdmTMB(
density ~ 0 + s(depth, k = 5),
time_varying = ~ 1,
data = pcod, mesh = mesh,
time = "year",
family = tweedie(link = "log"),
silent = FALSE # see progress
)Time-varying (random walk) effect of depth:
fit <- sdmTMB(
density ~ 1,
time_varying = ~ 0 + depth_scaled + depth_scaled2,
data = pcod, mesh = mesh,
time = "year",
family = tweedie(link = "log"),
spatial = "off",
spatiotemporal = "ar1",
silent = FALSE
)See the vignette Intro to modelling with sdmTMB for more details.
Spatially varying coefficients allow the effect of a predictor to differ across space, revealing spatial heterogeneity in relationships (e.g., trends that are positive in some areas and negative in others).
Spatially varying effect of time:
pcod$year_scaled <- as.numeric(scale(pcod$year))
fit <- sdmTMB(
density ~ s(depth, k = 5) + year_scaled,
spatial_varying = ~ year_scaled,
data = pcod, mesh = mesh,
time = "year",
family = tweedie(link = "log"),
spatiotemporal = "off"
)See zeta_s in the output, which represents spatial variation in the
coefficient. Ensure the SVC covariate is centered (mean ≈ 0) and include
it in the main formula too so that zeta_s represents deviations from
the average effect.
grid_yrs <- replicate_df(qcs_grid, "year", unique(pcod$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.
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")
)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.
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.
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 0.00000See the vignette on Residual checking with
sdmTMB,
?simulate.sdmTMB,
and
?dharma_residuals
for more details.
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 `binwidth`.
See
?gather_sims
and
?get_index_sims
for more details.
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$se <- apply(p, 1, sd)
ggplot(predictor_dat, aes(X, Y, fill = se)) +
geom_raster() +
scale_fill_viridis_c(option = "A") +
coord_cartesian(expand = FALSE)
sdmTMB has built-in functionality for cross-validation. If we were to
set a future::plan(), the folds would be fit in parallel:
mesh <- make_mesh(pcod, c("X", "Y"), cutoff = 10)
## Set parallel processing if desired:
# library(future)
# plan(multisession)
m_cv <- sdmTMB_cv(
density ~ s(depth, k = 5),
data = pcod, mesh = mesh,
family = tweedie(link = "log"), k_folds = 2
)
#> Running fits with `future.apply()`.
#> Set a parallel `future::plan()` to use parallel processing.
# Sum of log likelihoods of left-out data:
m_cv$sum_loglik
#> [1] -7219.976See
?sdmTMB_cv
for more details.
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.
The fitted model can be passed to the tmbstan package to sample from the posterior with Stan. See the Bayesian vignette.
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 × 5
#> term estimate std.error conf.low conf.high
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 (Intercept) -0.426 0.0573 -0.538 -0.314
#> 2 poly(depth_scaled, 2)1 -31.7 3.03 -37.6 -25.8
#> 3 poly(depth_scaled, 2)2 -66.9 4.09 -74.9 -58.9
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-60Defining 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")
)A barrier mesh limits correlation across barriers (e.g., land or water).
See add_barrier_mesh() in
sdmTMBextra.
sdmTMB is heavily inspired by the VAST and the glmmTMB R packages.
The newer tinyVAST R package can fit many of the models that VAST and sdmTMB can with an interface similar to sdmTMB. Generally, we recommend tinyVAST for multivariate applications or for (dynamic) structural equation modelling with optional spatial and/or spatiotemporal components.
INLA and inlabru can fit many of the same models as sdmTMB (and more) in an approximate Bayesian inference framework.
mgcv can fit similar SPDE-based Gaussian Markov random field models with code included in Miller et al. (2019), but this will be slower for large spatial datasets.
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.