Spatially varying coefficients with sdmTMB

class: center, middle, inverse, title-slide

# Spatially varying coefficients with sdmTMB
## NOAA PSAW Seminar Series
### 
### March 9, 2022

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# What is a spatially varying coefficient?

* Describes how the effect of a variable varies over space

* e.g., how does the temporal trend in fish density vary among locations?
  
  * e.g., how does fish distribution look when the PDO is high vs. low?
  
---

# Mathematically

Minimal model:

`$$g(u_{s}) = \omega_{s} + X_{\zeta}\zeta_{s}$$`
* `$g()$` is an inverse link function and `$u$` a linear predictor 
* `$\omega_{s}$` represents spatial field (intercept)  
* `$X_{\zeta}$` is design matrix of covariates (usually varying by time, but constant in space)
* `$\zeta_{s}$` is estimated spatially varying coefficient

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# When might we want a spatially varying coefficient?

* When we suspect non-local effects of a regional forcing, that varies spatially

* e.g., the influence of ENSO depends on latitude; influence of Bering Sea cold pool on pollock
  
--

* When the question requires evaluating change at fine spatial scales
  * e.g., there is no directional trend in species distribution shifts at broad scales, due to nuanced or opposing patterns over space being obscured 
---

# Spatially varying coefficients in sdmTMB

Spatially varying effect of time on cod density:

```r
pcod$year_scaled <- as.numeric(scale(pcod$year))
mesh <- make_mesh(pcod, c("X", "Y"), cutoff = 10)
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"
)
```

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# Local trends in cod population density

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# Scale-dependent interpretation of species distribution shifts

.tiny[
[Barnett, L.A.K., Ward, E.J. & Anderson, S.C. Improving estimates of species distribution change by incorporating local trends. Ecography, 44: 427-439. (2021)](https://doi.org/10.1111/ecog.05176)
]

<!-- To see how the interpretation of change in distribution depends on spatial scale, lets compare our fine-scale interpretation to a coarse-scale interpretation using a time series of the coastwide COG. These are results for arrowtooth flounder on the US west coast, where I am showing maps of the trend in population density over time, clusters of locations with similar trends, and the prediction of mean density over all years, which shows us that arrowtooth are most prevalent in the northern half of the region.  However, the trend map indicates that densities are mostly decreasing in the north and increasing in the central area toward the southern end of its core range.  Thus, it seems that arrowtooth are expanding southward as a traveling wave at the leading edge of the range.

On the far right panel, you can see that the COG also indicates a southward shift, yet it is slight and only detectable in this case because of a narrow 95% CI.  From this alone it would be hard to say whether the change is really due to increased southward movement or productivity, or a decrease in productivity in the north. -->

* Potential applied uses
  * Determining spatial structure of assessment model inputs
  * Determining quota allocation over space

---

# Spatially varying effect of NAO on Snowy Owls

.small[

```r
mesh <- make_mesh(snow, c("X", "Y"), cutoff = 1.5)
fit_owls <- sdmTMB(
  count ~ 1 + nao + (1 | year_factor),
* spatial_varying = ~ 0 + nao,
  family = nbinom2(link = "log"), 
  data = snow, 
  mesh = mesh,
  time = "year", 
  spatial = "on",
  spatiotemporal = "iid"
)
```
]

<!-- Description: spatially varying coefficient for effect of mean annual NAO (North Atlantic Oscillation) on counts of Snowy Owls observed on annual citizen science Christmas Bird Counts 1979--2020 in Canada and the US.

Our only fixed effect will be the NAO climate index (`nao`), but because of the wide spatial extent of these surveys, we also expect that the effect of NAO might not be consistent throughout North America. Therefore, the effect of NAO will also be allowed to vary spatially using this formula: `spatial_varying = ~ 0 + nao`.

Because there does not appear to be much correlation between years, year will be included as a random factor: `(1 | year_factor)`, allowing the mean estimate for each of these spatiotemporal fields to vary. 
-->

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# Influence of NAO on Snowy Owls is greatest in southeast

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# Tips and tricks

* Unlike `time_varying`, likely *do* want the same coefficient in the main `formula` effects
  * `spatial_varying` represents a random field with mean zero
--

* Predictor should be roughly mean zero and SD of 1 to help with estimation
--

* Current sdmTMB limited to one effect; an unlimited version is in development in a branch
--

See vignette: [Fitting spatial trend models with sdmTMB](https://pbs-assess.github.io/sdmTMB/articles/spatial-trend-models.html)

See sdmTMB paper appendix for Snowy Owl model