Spatial and spatiotemporal GLMMs with TMB
sdmTMB is an R package that fits spatial and spatiotemporal predictive-process 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 also the documentation site.
Assuming you have a C++ compiler installed, you can install sdmTMB:
# install.packages("remotes") remotes::install_github("pbs-assess/sdmTMB", dependencies = TRUE)
If you have problems installing INLA, try installing it directly first.
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:
s()notation from mgcv
student(), plus some truncated and censored families
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.
To cite sdmTMB in publications use:
Anderson, S.C., E.J. Ward, P.A. English, L.A.K. Barnett. 2022. 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
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.
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.
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
Y, which are UTM coordinates in kilometres.
We start by creating a mesh object that contains matrices to apply the SPDE approach.
cutoff defines the minimum allowed distance between points in the units of
Y (km). Alternatively, we could have created any mesh via the INLA package and supplied it to
make_mesh(). We can inspect our mesh object with the associated plotting method
Fit a spatial model with a smoother for depth:
Print the model fit:
fit #> Spatial model fit by ML ['sdmTMB'] #> Formula: density ~ s(depth) #> Mesh: mesh #> Data: pcod #> Family: tweedie(link = 'log') #> #> coef.est coef.se #> (Intercept) 2.37 0.21 #> sdepth 6.17 25.17 #> #> Smooth terms: #> Std. Dev. #> sds(depth) 13.93 #> #> Dispersion parameter: 12.69 #> Tweedie p: 1.58 #> Matern 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> <lgl> <dbl> <dbl> #> 1 range 16.4 NA 9.60 28.0 #> 2 phi 12.7 NA 11.9 13.5 #> 3 sigma_O 1.86 NA 1.48 2.34 #> 4 tweedie_p 1.58 NA 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 eigen values detected #> ✓ No gradients with respect to fixed effects are >= 0.001 #> ✓ No fixed-effect standard errors are NA #> ✓ No fixed-effect standard errors look unreasonably large #> ✓ No sigma parameters are < 0.001 #> ✓ Range parameter doesn't look unreasonably large
Use the visreg package to plot the smoother effect in link space with randomized quantile partial residuals:
Or on the response scale:
Predict on new data:
p <- predict(fit, newdata = qcs_grid)
We could switch to a presence-absence model by changing the response column and family:
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:
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
p_st <- predict(fit_spatiotemporal, newdata = qcs_grid, return_tmb_object = TRUE) index <- get_index(p_st, area = rep(4, nrow(qcs_grid))) 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()
Time-varying (random walk) effect of depth:
See the vignette Intro to modelling with sdmTMB for more details.
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" )
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
qcs_grid$year_scaled <- (qcs_grid$year - mean(pcod$year)) / sd(pcod$year) p <- predict(fit, newdata = qcs_grid) %>% 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:
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:
?sdmTMB_simulate for more details.
Using those simulations to check DHARMa residuals:
# dharma_residuals(s, fit) # or with the pipe, %>%: simulate(fit, nsim = 500) %>% dharma_residuals(fit)
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`.
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 #>  -7122.779 # Expected log pointwise predictive density from left-out data: # (average likelihood density) m_cv$elpd #>  -1.005114
?sdmTMB_cv for more details.
Priors/penalties can be placed on most parameters. For example, here we place a PC (penalized complexity) prior on the Matérn random field parameters, a standard normal prior on the effect of depth, a Normal(0, 10^2) prior on the intercept, and a half-normal prior on the Tweedie dispersion parameter (
mesh <- make_mesh(pcod, c("X", "Y"), cutoff = 10) fit <- sdmTMB( density ~ depth_scaled, data = pcod, mesh = mesh, family = tweedie(), priors = sdmTMBpriors( matern_s = pc_matern(range_gt = 10, sigma_lt = 5), b = normal(c(0, 0), c(1, 10)), phi = halfnormal(0, 15) ) )
We can visualize the PC Matérn prior:
plot_pc_matern(range_gt = 10, sigma_lt = 5)
?sdmTMBpriors for more details.
The fitted model can be passed to the tmbstan package to sample from the posterior with Stan. Note this can be slow for large or poorly identified models. See examples of fixing parameters in
# only 1 chain and 400 iterations for speed: fit_mcmc <- tmbstan::tmbstan(fit$tmb_obj, chains = 1, iter = 400)
Internal parameter posteriors:
print(fit_mcmc, pars = c("b_j", "omega_s")) #> Inference for Stan model: sdmTMB. #> 1 chains, each with iter=400; warmup=200; thin=1; #> post-warmup draws per chain=200, total post-warmup draws=200. #> #> mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat #> b_j 0.99 0.03 0.15 0.62 0.93 1.00 1.06 1.27 35 1.00 #> omega_s -0.07 0.03 0.23 -0.50 -0.23 -0.06 0.10 0.33 63 1.01 #> #> Samples were drawn using NUTS(diag_e). #> For each parameter, n_eff is a crude measure of effective sample size, #> and Rhat is the potential scale reduction factor on split chains (at #> convergence, Rhat=1).
Predicting with the Stan/tmbstan model:
?extract_mcmc for more details.
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
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)
A barrier mesh limits correlation across barriers (e.g., land or water). See the example in