Time-varying coefficients with sdmTMB

class: center, middle, inverse, title-slide

.title[
# Time-varying coefficients with sdmTMB
]
.subtitle[
## DFO DSAF workshop
]
.author[
### 
]
.date[
### January 12–16 2026
]

---

.small[
# Why might we want time-varying effects?

* Time-varying slopes: 
  * To allow for evolving responses to covariates (e.g., species moving deeper over time)
  
  * Example use: [English et al. (2021) Fish and Fisheries](https://doi.org/10.1111/faf.12613)  
    Modelled groundfish density with depth; didn't want to constrain fish if they were moving deeper when water was warmer
  
* Time-varying intercepts:
  * To allow variable means across time with constraints
  * To have a model to interpolate or forecast over time
]

---

# Time-varying intercepts

Several ways in sdmTMB:

* factors: `as.factor(year)` (independent)
* random effects: ` + (1 | year)` (drawn from normal distribution)
* smooth: ` + s(year)`
* as random walk (shown next)

---

#  Random walk covariates in sdmTMB

Random walk:

$$
`\begin{aligned}
x_t &= x_{t-1} + \epsilon_t\\
\epsilon &\sim \mathrm{Normal(0, \sigma^2)}
\end{aligned}`
$$

Defined by `time_varying` argument

Takes a *one-sided* formula, e.g. `~ 1` or `~ 0 + depth`

---

# Time-varying intercept

.small[

``` r
mesh <- make_mesh(pcod, xy_cols = c("X", "Y"), cutoff = 10)
fit <- sdmTMB(
  density ~ 0 + s(depth, k = 5), 
* time_varying = ~ 1,
* time_varying_type = "rw",
  data = pcod, mesh = mesh,
  time = "year",  
  family = tweedie(link = "log")
)
```
]

.xsmall[
Note: a `0` or `-1` in formula for suppressing global intercept in this case

`time_varying_type = "rw"` (default) estimates the first time step as independent. I.e. **do not place the same covariate in the `formula` argument** (this includes the intercept). This does not apply to other time-varying types in sdmTMB. Sorry!

Also `"ar1"` or `"rw0"`
]

---

# Getting coefficients

Return with

.small[

``` r
print(fit)
```
]

.small[

``` r
#> Spatiotemporal model fit by ML ['sdmTMB']
#> Formula: density ~ 0 + s(depth, k = 5)
#> Time column: "year"
#> ...
*#> Time-varying parameters:
*#>                  coef.est coef.se
*#> (Intercept)-2003     1.96    0.29
*#> (Intercept)-2004     2.31    0.27
*#> (Intercept)-2005     2.06    0.27
*#> ...
*#> (Intercept)-2015     2.07    0.27
*#> (Intercept)-2017     1.55    0.29
#> ...
```
]

---

# Getting coefficients

Or via `tidy()`:

``` r
tidy(fit, "ran_vals")
#> # A tibble: 9 × 5
#>   term             estimate std.error conf.low conf.high
#>   <chr>               <dbl>     <dbl>    <dbl>     <dbl>
#> 1 (Intercept):2003     1.96     0.289    1.39       2.52
#> 2 (Intercept):2004     2.31     0.272    1.78       2.85
#> 3 (Intercept):2005     2.06     0.271    1.53       2.60
#> 4 (Intercept):2007     1.23     0.301    0.642      1.82
#> 5 (Intercept):2009     1.51     0.276    0.971      2.05
#> 6 (Intercept):2011     1.93     0.271    1.40       2.46
#> 7 (Intercept):2013     2.16     0.267    1.64       2.68
#> 8 (Intercept):2015     2.07     0.273    1.54       2.61
#> 9 (Intercept):2017     1.55     0.293    0.975      2.12
```

---

# Other approaches to modeling time-varying intercepts

.small[

``` r
density ~ s(depth) + 0 + as.factor(year)
```
]
.small[

``` r
density ~ s(depth) + (1 | year)
```
]
.small[

``` r
density ~ s(depth) + s(year)
```
]
---

# These approaches are similar but subtly different

---

# Time-varying coefficients

Time-varying (random walk) effect of depth

Intercept in this model NOT time-varying

``` r
fit_tv <- sdmTMB(
  density ~ 1, 
* time_varying = ~ 0 + depth_scaled + depth_scaled2,
  data = pcod, mesh = mesh,
  time = "year",
  family = tweedie(link = "log"),
  spatial = "on",
  spatiotemporal = "iid",
  silent = FALSE
)
```

---

# Time-varying coefficients

Time-varying (random walk) effect of depth

---

# Time-varying coefficient notes

* `time_varying` is a formula for coefficients that follow a random walk or AR1 process over time

* Make sure a coefficient isn't in both `formula` and `time_varying` for `time_varying_type = "rw"`, this includes the intercept

* The `time_varying` formula cannot have smoothers `s()` in it! Instead:
  * Polynomials: `time_varying = ~ x + I(x^2)`
  * `formula = s(depth, by = factor_year)` (independent smooths) 
  * `formula = s(depth, year)` (2D smooth)