Distribution shift metrics with sdmTMB

class: center, middle, inverse, title-slide

.title[
# Distribution shift metrics with sdmTMB
]
.subtitle[
## DFO DSAF workshop
]
.author[
### 
]
.date[
### January 12–16 2026
]

---

# Metrics we'll cover

.large[
Center of gravity

Density weighted habitat variables

Range edges

Effective area occupied

Spatially varying trends
]

---

# General idea throughout

.large[
1. Fit a reasonable spatiotemporal model (😮‍💨)

2. Predict to a constant grid over the spatial domain
(standardizes for spatial variation in sampling, combining data sources, etc.)

3. Calculate metrics from those predictions with uncertainty
]

---

# Center of gravity

.small[
Density-weighted mean latitude and longitude

Tells us how the center of the distribution is changing over time

Can obscure smaller-scale variation

Several spatial patterns can create the same center of gravity
]

.tiny[
See for review of 1980s/90s citations: Woillez, M., Poulard, J.-C., Rivoirard, J., Petitgas, P., and Bez, N. 2007. Indices for capturing spatial patterns and their evolution in time, with application to European hake (Merluccius merluccius) in the Bay of Biscay. ICES J Mar Sci 64(3): 537–550. <https://doi.org/10.1093/icesjms/fsm025>.

The GMRF model-based version we're doing here: Thorson, J.T., Pinsky, M.L., and Ward, E.J. 2016. Model-based inference for estimating shifts in species distribution, area occupied and centre of gravity. Methods Ecol Evol 7(8): 990–1002. <https://doi.org/10.1111/2041-210X.12567>.
]

---

# Density weighted habitat variables

Exactly what the center of gravity is, but applied to other habitat variables

E.g., density-weighted mean depth

E.g., density-weighted mean temperature

For dynamic habitat variables (e.g., temperature) consider comparing to pattern with static distribution.

---

### Density-weighted habitat variables example

.tiny[
Davidson, L.N.K., English, P.A., King, J., Grant, P.B.C., Taylor, I.G., Barnett, L.A.K., Gertseva, V., Tribuzio, C.A., and Anderson, S.C. 2026. Mystery of the disappearing dogfish: transboundary analyses reveal steep population declines across the northeast pacific with little evidence for regional redistribution. Fish and Fisheries 27(1): 1–12. <https://doi.org/10.1111/faf.70028>.
]

---

### Density-weighted habitat variables example

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# Range edges

.small[
Accumulate grid cells in one direction (e.g., south to north) and calculate the point at which you accumulate quantiles of biomass or abundance density

Median is a robust version similar in spirit to center of gravity

Tail quantiles (e.g. 2.5% and 97.5%) represent range edges

This isn't actually a range edge if the survey(s) doesn't capture the edges of the population!
]

.tiny[
Fredston‐Hermann, A., Selden, R., Pinsky, M., Gaines, S.D., and Halpern, B.S. 2020. Cold range edges of marine fishes track climate change better than warm edges. Glob Change Biol 26(5): 2908–2922. <https://doi.org/10.1111/gcb.15035>.
]

---

# Effective area occupied

What area would be required to contain the population if it were spread uniformly at the expected density experienced by a randomly chosen individual

Measures how concentrated the species density is

Effective area occupied is maximized when individuals are spread uniformly and decreases as individuals become more spatially concentrated

.tiny[
Thorson, J.T., Rindorf, A., Gao, J., Hanselman, D.H., and Winker, H. 2016. Density-dependent changes in effective area occupied for sea-bottom-associated marine fishes. Proceedings of the Royal Society B: Biological Sciences 283(1840): 20161853. Royal Society. doi:10.1098/rspb.2016.1853
]

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# Effective area occupied

Assume space is divided into grid cells `$i = 1, \ldots, n$`, each with area `$a$` and density `$D_i$`

$$
\text{EAO} = \frac{(\sum_i D_i a)^2}{\sum_i D_i^2 a} \quad 😲
$$
--

.small[
$$
\text{EAO} = \frac{\text{total biomass}}{\text{expected density experienced by an individual}}
$$
]

See `exercises/understanding-eao.R`

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# Effective area occupied (concentrated)

.small[
Area occupied if density is uniformly distributed at the mean density that a randomly chosen individual experiences
]

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# Effective area occupied

.small[

``` r
D <- c(10, 2, 0) # densities in 3 grid cells
a <- 1  # area of each cell

# Total biomass (or abundance)
(b <- sum(D * a))
#> [1] 12

# Probability a random individual is in a cell
(p <- D * a / b)
#> [1] 0.8333333 0.1666667 0.0000000

# Expected density experienced by an individual:
(m <- sum(p * D))
#> [1] 8.666667

# Effective area occupied
b / m
#> [1] 1.384615

sum(D * a)^2 / sum(D^2 * a)
#> [1] 1.384615
```
]

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# Effective area occupied (uniform)

.small[
If distribution is uniform, EAO maximized (total survey area)
]

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# Spatially varying trends

Center of gravity can mask complex and interesting local patterns

Many different different local patterns can give rise to the same center of gravity

We often care about fine-scale spatial management

Solution: fit spatially varying coefficient (SVC) models with time as the predictor

.xsmall[
Barnett, L.A.K., Ward, E.J., and Anderson, S.C. 2021. Improving estimates of species distribution change by incorporating local trends. Ecography 44(3): 427–439. <https://doi.org/10.1111/ecog.05176>.
]

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# Spatially varying trends example

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# Subregional indices of abundance

Split up the surveyed area into areas of interest

E.g., north/south, or clusters from spatially varying trends ([Barnett et al. (2021)](https://doi.org/10.1111/ecog.05176))

Run `predict()` and `get_index()` (or `get_cog()` etc.) on each subsetted grid

Integrates index, or center of gravity, etc. over smaller areas to tell a nuanced story