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.
- Installation
- Overview
- Basic use
- Advanced functionality
- Time-varying coefficients
- Spatially varying coefficients (SVC)
- Random intercepts
- Breakpoint and theshold 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 INLA mesh
- Barrier meshes
Assuming you have a C++ compiler installed, you can install sdmTMB:
# install.packages("remotes")
remotes::install_github("pbs-assess/sdmTMB", dependencies = TRUE)To cite sdmTMB in publications use:
citation("sdmTMB")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
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(), andstudent(), 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.
See
?sdmTMB
and
?predict.sdmTMB
for the most complete examples. Also see the vignettes (‘Articles’) on
the documentation
site.
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 INLA package 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, k = 5),
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, k = 5)
#> Mesh: mesh
#> Data: pcod
#> Family: tweedie(link = 'log')
#> coef.est coef.se
#> (Intercept) 2.34 0.23
#> sdepth -26.96 21.52
#>
#> Smooth terms:
#> Std. Dev.
#> sds(depth) 34.03
#>
#> Dispersion parameter: 12.70
#> Tweedie p: 1.58
#> Matern range: 17.47
#> Spatial SD: 1.86
#> ML criterion at convergence: 6403.637
#>
#> 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) 2.34 0.227 1.89 2.78
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 17.5 NA 10.5 29.1
#> 2 phi 12.7 NA 11.9 13.5
#> 3 sigma_O 1.86 NA 1.51 2.29
#> 4 tweedie_p 1.58 NA 1.57 1.60Plot the smoother effect:
plot_smooth(fit, ggplot = TRUE)
Predict on new data:
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. -2.32 -2.34 0.0155 0.0155
#> 2 458 5636 223. 2.06 2.02 0.0475 0.0475
#> 3 460 5636 204. 3.10 3.02 0.0796 0.0796
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, k = 5),
data = pcod,
mesh = mesh,
family = binomial(link = "logit")
)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():
p_st <- predict(fit_spatiotemporal, newdata = qcs_grid,
return_tmb_object = TRUE, area = 4)
index <- get_index(p_st)
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.
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 effect of time:
pcod$year_scaled <- as.numeric(scale(pcod$year))
fit <- sdmTMB(
density ~ s(depth, k = 5) + year_scaled,
spatial_varying = ~ 0 + 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).
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)) + 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 0.00000Using those simulations to check DHARMa residuals:
# dharma_residuals(s, fit)
# or with the pipe, %>%:
simulate(fit, nsim = 500) %>%
dharma_residuals(fit)
See 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 with `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] -6750.892
# Expected log pointwise predictive density from left-out data:
# (average likelihood density)
m_cv$elpd
#> [1] -1.008653See
?sdmTMB_cv
for more details.
Priors/penalties can be placed on most parameters. For example, here we
place a PC (penalized complexity) prior on the Matérn random field
parameters, a standard normal prior on the effect of depth, a Normal(0,
10^2) prior on the intercept, and a half-normal prior on the Tweedie
dispersion parameter (phi):
mesh <- make_mesh(pcod, c("X", "Y"), cutoff = 10)
fit <- sdmTMB(
density ~ depth_scaled,
data = pcod, mesh = mesh,
family = tweedie(),
priors = sdmTMBpriors(
matern_s = pc_matern(range_gt = 10, sigma_lt = 5),
b = normal(c(0, 0), c(1, 10)),
phi = halfnormal(0, 15)
)
)We can visualize the PC Matérn prior:
plot_pc_matern(range_gt = 10, sigma_lt = 5)
See
?sdmTMBpriors
for more details.
The fitted model can be passed to the tmbstan package to sample from the
posterior with Stan. Note this can be slow for large or poorly
identified models. See examples of fixing parameters in
?extract_mcmc.
# only 1 chain and 400 iterations for speed:
fit_mcmc <- tmbstan::tmbstan(fit$tmb_obj, chains = 1, iter = 400)Internal parameter posteriors:
print(fit_mcmc, pars = c("b_j", "omega_s[1]"))
#> 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[1] -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:
pred_mcmc <- predict(fit, newdata = qcs_grid, tmbstan_model = fit_mcmc)
# Each row has 200 posterior samples for a row of the `newdata` data frame:
dim(pred_mcmc)
#> [1] 65826 200See
?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-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 the example in
?add_barrier_mesh.