Using the built-in British Columbia Queen Charlotte Sound Pacific Cod dataset, we might be interested in fitting a model that describes spatially varying trends through time. The data are as follows:

  • There are columns for depth and depth squared.
  • Depth was centred and scaled by its standard deviation and we’ve included those in the data frame so that they could be used to similarly scale the prediction grid.
  • The density units should be kg/km2.
  • Here, X and Y are coordinates in UTM zone 9.

We will set up our SPDE mesh with a relatively coarse resolution so that this vignette builds quickly:

pcod_spde <- make_mesh(pcod, c("X", "Y"), cutoff = 12)
plot(pcod_spde)

We will fit a model that includes a slope for ‘year’, an intercept spatial random field, and another random field for spatially varying slopes the represent trends over time in space (spatial_varying argument). Our model just estimates an intercept and accounts for all other variation through the random effects.

First, we will set up a column for time that is Normal(0, 1) to help with estimation:

d <- pcod
d$scaled_year <- (pcod$year - mean(pcod$year)) / sd(pcod$year) + 3

Now fit a model using spatial_varying ~ 0 + scaled_year:

(The 0 + drops the intercept, although sdmTMB would take care of that anyways here.)

m1 <- sdmTMB(density ~ scaled_year, data = d,
  mesh = pcod_spde, family = tweedie(link = "log"),
  spatial_varying = ~ 0 + scaled_year, time = "year",
  spatiotemporal = "off")

We have turned off spatiotemporal random fields for this example for simplicity, but they also could be IID or AR1.

Let’s extract some parameter estimates. Look for sigma_Z:

tidy(m1, conf.int = TRUE)
#>          term   estimate std.error   conf.low  conf.high
#> 1 (Intercept)  3.4251463 0.3897312  2.6612872 4.18900536
#> 2 scaled_year -0.2140606 0.1164730 -0.4423436 0.01422232
tidy(m1, "ran_pars", conf.int = TRUE)
#>        term   estimate std.error  conf.low  conf.high
#> 1     range 29.7381725        NA 20.775802 42.5667765
#> 3       phi 14.0667396        NA 13.226193 14.9607048
#> 4   sigma_O  1.9167968        NA  1.489051  2.4674171
#> 5   sigma_Z  0.5442651        NA  0.398302  0.7437182
#> 6 tweedie_p  1.5952763        NA  1.575281  1.6149607

Let’s look at the predictions and estimates of the spatially varying coefficients on a grid:

plot_map_raster <- function(dat, column = "est") {
  ggplot(dat, aes_string("X", "Y", fill = column)) +
    geom_raster() +
    facet_wrap(~year) +
    coord_fixed() +
    scale_fill_viridis_c()
}

First, we need to predict on a grid. We also need to add a column for scaled_year to match the fitting:

nd <- qcs_grid
nd$scaled_year <- (nd$year - mean(pcod$year)) / sd(pcod$year)
p1 <- predict(m1, newdata = nd)

First let’s look at the spatial trends.

We will just pick out a single year to plot since they should all be the same for the slopes. Note that these are in log space. zeta_s are the spatially varying coefficients.

plot_map_raster(filter(p1, year == 2003), "zeta_s_scaled_year")

This is the spatially varying intercept:

plot_map_raster(filter(p1, year == 2003), "omega_s") + scale_fill_gradient2()

These are the predictions including all fixed and random effects plotted in log space.

plot_map_raster(filter(p1, year == 2003), "est")

And we can look at just the spatiotemporal random effects for models 2 and 3 (intercept + slope combined):

plot_map_raster(filter(p1, year == 2003), "est_rf")