David L Miller

Now we are dangerous.

- We are doing statistics
- We want to know about
**uncertainty** - This is the most useful part of the analysis

- Variance of total abundance
- Map of uncertainty (coefficient of variation)

- Detection function
- GAM parameters

- Dashed lines are +/- 2 standard errors
- How do we translate to \( \hat{N} \)?

- Before we expressed smooths as:
- \( s(x) = \sum_{k=1}^K \beta_k b_k(x) \)

- Theory tells us that:
- \( \boldsymbol{\beta} \sim N(\boldsymbol{\hat{\beta}}, \mathbf{V}_\boldsymbol{\beta}) \)
- where \( \mathbf{V}_\boldsymbol{\beta} \) is a bit complicated

- Apply parameter variance to \( \hat{N} \)

- “map” data onto fitted values \( \mathbf{X}\boldsymbol{\beta} \)
- “map” prediction matrix to predictions \( \mathbf{X}_p \boldsymbol{\beta} \)
- Here \( \mathbf{X}_p \) need to take smooths into account
- pre-/post-multiply by \( \mathbf{X}_p \) to “transform variance”
- \( \Rightarrow \mathbf{X}_p^\text{T}\mathbf{V}_\boldsymbol{\beta} \mathbf{X}_p \)
- link scale, need to do another transform for response

(Getting a little fast-and-loose with the mathematics)

From previous lectures we know:

\[
\text{CV}^2\left( \hat{N} \right) \approx \text{CV}^2\left( \text{GAM} \right) +\\
\text{CV}^2\left( \text{detection function}\right)
\]

- Assumes detection function and GAM are
**independent** - Maybe this is okay?

- Include the detectability as a “fixed” term in GAM
- Mean effect is zero
- Variance effect included
- Uncertainty “propagated” through the model
- Details in bibliography (too much to detail here)

- Functions in
`dsm`

to do this `dsm.var.gam`

- assumes spatial model and detection function are independent

`dsm.var.prop`

- propagates uncertainty from detection function to spatial model
- only works for
`count`

models (more or less)

Using `dsm.var.gam`

```
dsm_tw_var_ind <- dsm.var.gam(dsm_all_tw_rm, predgrid, off.set=predgrid$off.set)
summary(dsm_tw_var_ind)
```

```
Summary of uncertainty in a density surface model calculated
analytically for GAM, with delta method
Approximate asymptotic confidence interval:
5% Mean 95%
1538.968 2491.864 4034.773
(Using delta method)
Point estimate : 2491.864
Standard error : 331.1575
Coefficient of variation : 0.2496
```

Using `dsm.var.prop`

```
dsm_tw_var <- dsm.var.prop(dsm_all_tw_rm, predgrid, off.set=predgrid$off.set)
summary(dsm_tw_var)
```

```
Summary of uncertainty in a density surface model calculated
by variance propagation.
Quantiles of differences between fitted model and variance model
Min. 1st Qu. Median Mean 3rd Qu. Max.
-4.665e-04 -3.535e-05 -4.358e-06 -3.991e-06 2.095e-06 1.232e-03
Approximate asymptotic confidence interval:
5% Mean 95%
1460.721 2491.914 4251.075
(Using delta method)
Point estimate : 2491.914
Standard error : 691.8776
Coefficient of variation : 0.2776
```

- Calculate uncertainty per-cell
`dsm.var.*`

thinks`predgrid`

is one “region”- Need to split data into cells (using
`split()`

) - (Could be arbitrary sets of cells, see exercises)
- Need
`width`

and`height`

of cells for plotting

```
predgrid$width <- predgrid$height <- 10*1000
predgrid_split <- split(predgrid, 1:nrow(predgrid))
head(predgrid_split,3)
```

```
$`1`
x y Depth SST NPP off.set height width
126 547984.6 788254 153.5983 9.04917 1462.521 1e+08 10000 10000
$`2`
x y Depth SST NPP off.set height width
127 557984.6 788254 552.3107 9.413981 1465.41 1e+08 10000 10000
$`3`
x y Depth SST NPP off.set height width
258 527984.6 778254 96.81992 9.699239 1429.432 1e+08 10000 10000
```

```
dsm_tw_var_map <- dsm.var.prop(dsm_all_tw_rm, predgrid_split,
off.set=predgrid$off.set)
```

```
p <- plot(dsm_tw_var_map, observations=FALSE, plot=FALSE) +
coord_equal() +
scale_fill_viridis()
print(p)
```

- Plotting coefficient of variation
- Standardise standard deviation by mean
- \( \text{CV} = \text{se}(\hat{N})/\hat{N} \) (per cell)
- Can be useful to overplot survey effort