Index standardization with sdmTMB

If the code in this vignette has not been evaluated, a rendered version is available on the documentation site under ‘Articles’.

library(ggplot2)
library(dplyr)
library(sdmTMB)

Let’s perform index standardization with the built-in data for Pacific cod.

The density units should be kg/km².
Here, X and Y are coordinates in UTM zone 9.

glimpse(pcod)
#> Rows: 2,143
#> Columns: 12
#> $ year          <int> 2003, 2003, 2003, 2003, 2003, 2003, 2003, 2003, 2003, 20…
#> $ X             <dbl> 446.4752, 446.4594, 448.5987, 436.9157, 420.6101, 417.71…
#> $ Y             <dbl> 5793.426, 5800.136, 5801.687, 5802.305, 5771.055, 5772.2…
#> $ depth         <dbl> 201, 212, 220, 197, 256, 293, 410, 387, 285, 270, 381, 1…
#> $ density       <dbl> 113.138476, 41.704922, 0.000000, 15.706138, 0.000000, 0.…
#> $ present       <dbl> 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
#> $ lat           <dbl> 52.28858, 52.34890, 52.36305, 52.36738, 52.08437, 52.094…
#> $ lon           <dbl> -129.7847, -129.7860, -129.7549, -129.9265, -130.1586, -…
#> $ depth_mean    <dbl> 5.155194, 5.155194, 5.155194, 5.155194, 5.155194, 5.1551…
#> $ depth_sd      <dbl> 0.4448783, 0.4448783, 0.4448783, 0.4448783, 0.4448783, 0…
#> $ depth_scaled  <dbl> 0.3329252, 0.4526914, 0.5359529, 0.2877417, 0.8766077, 1…
#> $ depth_scaled2 <dbl> 0.11083919, 0.20492947, 0.28724555, 0.08279527, 0.768440…

First we will create our SPDE mesh. We will use a relatively course mesh for a balance between speed and accuracy in this vignette (cutoff = 10 where cutoff is in the units of X and Y (km here) and represents the minimum distance between points before a new mesh vertex is added). You will likely want to use a higher resolution mesh for applied scenarios. You will want to make sure that increasing the number of knots does not change the conclusions.

pcod_spde <- make_mesh(pcod, c("X", "Y"), cutoff = 10)
#> as(<dgCMatrix>, "dgTMatrix") is deprecated since Matrix 1.5-0; do as(., "TsparseMatrix") instead
plot(pcod_spde)

Let’s fit a GLMM. Note that if you want to use this model for index standardization then you will likely want to include 0 + as.factor(year) or -1 + as.factor(year) so that there is a factor predictor that represents the mean estimate for each time slice.

m <- sdmTMB(
  data = pcod, 
  formula = density ~ 0 + as.factor(year),
  time = "year", mesh = pcod_spde, family = tweedie(link = "log"))
#> Warning in checkMatrixPackageVersion(): Package version inconsistency detected.
#> TMB was built with Matrix version 1.5.3
#> Current Matrix version is 1.5.1
#> Please re-install 'TMB' from source using install.packages('TMB', type = 'source') or ask CRAN for a binary version of 'TMB' matching CRAN's 'Matrix' package

We can inspect randomized quantile residuals:

pcod$resids <- residuals(m) # randomized quantile residuals
# pcod$resids <- residuals(m, type = "mle-mcmc") # better but slower
hist(pcod$resids)

qqnorm(pcod$resids)
abline(a = 0, b = 1)

ggplot(pcod, aes(X, Y, col = resids)) + scale_colour_gradient2() +
  geom_point() + facet_wrap(~year) + coord_fixed()

Now we want to predict on a fine-scale grid on the entire survey domain. There is a grid built into the package for Queen Charlotte Sound named qcs_grid. Our prediction grid also needs to have all the covariates that we used in the model above.

glimpse(qcs_grid)
#> Rows: 65,826
#> Columns: 6
#> $ X             <dbl> 456, 458, 460, 462, 464, 466, 468, 470, 472, 474, 476, 4…
#> $ Y             <dbl> 5636, 5636, 5636, 5636, 5636, 5636, 5636, 5636, 5636, 56…
#> $ depth         <dbl> 347.08345, 223.33479, 203.74085, 183.29868, 182.99983, 1…
#> $ depth_scaled  <dbl> 1.56081222, 0.56976987, 0.36336929, 0.12570465, 0.122036…
#> $ depth_scaled2 <dbl> 2.436134789, 0.324637708, 0.132037240, 0.015801658, 0.01…
#> $ year          <int> 2003, 2003, 2003, 2003, 2003, 2003, 2003, 2003, 2003, 20…

Now make the predictions on new data.

predictions <- predict(m, newdata = qcs_grid, return_tmb_object = TRUE)

Let’s make a small function to make maps.

plot_map <- function(dat, column) {
  ggplot(dat, aes(X, Y, fill = {{ column }})) +
    geom_raster() +
    facet_wrap(~year) +
    coord_fixed()
}

There are four kinds of predictions that we get out of the model. First we will show the predictions that incorporate all fixed effects and random effects:

plot_map(predictions$data, exp(est)) +
  scale_fill_viridis_c(trans = "sqrt") +
  ggtitle("Prediction (fixed effects + all random effects)")

We can also look at just the fixed effects, here year:

plot_map(predictions$data, exp(est_non_rf)) +
  ggtitle("Prediction (fixed effects only)") +
  scale_fill_viridis_c(trans = "sqrt")

We can look at the spatial random effects that represent consistent deviations in space through time that are not accounted for by our fixed effects. In other words, these deviations represent consistent biotic and abiotic factors that are affecting biomass density but are not accounted for in the model.

plot_map(predictions$data, omega_s) +
  ggtitle("Spatial random effects only") +
  scale_fill_gradient2()

And finally we can look at the spatiotemporal random effects that represent deviation from the fixed effect predictions and the spatial random effect deviations. These represent biotic and abiotic factors that are changing through time and are not accounted for in the model.

plot_map(predictions$data, epsilon_st) +
  ggtitle("Spatiotemporal random effects only") +
  scale_fill_gradient2()

When we ran our predict.sdmTBM() function, it also returned a report from TMB in the output because we included return_tmb_object = TRUE. We can then run our get_index() function to extract the total biomass calculations and standard errors.

We will need to set the area argument to 4 km² since our grid cells are 2 km x 2 km. If some grid cells were not fully in the survey domain (or were on land), we could feed a vector of grid areas to the area argument that matched the number of grid cells.

index <- get_index(predictions, area = 4, bias_correct = TRUE)

ggplot(index, aes(year, est)) + geom_line() +
  geom_ribbon(aes(ymin = lwr, ymax = upr), alpha = 0.4) +
  xlab('Year') + ylab('Biomass estimate (kg)')

These are our biomass estimates:

mutate(index, cv = sqrt(exp(se^2) - 1)) %>% 
  select(-log_est, -se) %>%
  knitr::kable(format = "pandoc", digits = c(0, 0, 0, 0, 2))

year	est	lwr	upr	cv
2003	936192	653708	1340745	0.18
2004	1832132	1359017	2469952	0.15
2005	1757226	1224183	2522371	0.19
2007	452108	328782	621693	0.16
2009	722994	518720	1007712	0.17
2011	1357912	1028869	1792187	0.14
2013	1422651	1037720	1950367	0.16
2015	1487470	1116296	1982060	0.15
2017	750066	543631	1034891	0.17

We can also calculate an index for part of the survey domain. We’ll make an index for everything south of UTM 5700 by subsetting our prediction grid. For more complicated spatial polygons you could intersect the polygon on the prediction grid using something like sf::st_intersects().

qcs_grid_south <- qcs_grid[qcs_grid$Y < 5700, ]
predictions_south <- predict(m, newdata = qcs_grid_south, 
  return_tmb_object = TRUE)
index_south <- get_index(predictions_south, area = 4, bias_correct = TRUE)
head(index_south)
#>   year      est      lwr      upr  log_est        se
#> 1 2003 264684.1 156370.3 448024.2 12.48629 0.2685305
#> 2 2004 602467.2 403036.3 900580.8 13.30879 0.2051092
#> 3 2005 411737.5 263137.1 644256.2 12.92814 0.2284279
#> 4 2007 184944.2 117597.2 290860.3 12.12781 0.2310190
#> 5 2009 316488.6 203603.5 491961.5 12.66504 0.2250618
#> 6 2011 432250.2 292913.5 637868.3 12.97676 0.1985380

We can visually compare the two indexes:

mutate(index, region = "all") %>% 
  bind_rows(mutate(index_south, region = "south")) %>% 
  ggplot(aes(year, est)) + 
  geom_line(aes(colour = region)) +
  geom_ribbon(aes(ymin = lwr, ymax = upr, fill = region), alpha = 0.4)

2023-01-03