Lecture 5: Making predictions, variance estimation, maps

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# Lecture 5: Predictions<br/>and<br/>variance

---

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# Let's talk about maps

---
# What does a map mean?

.pull-left[
![](dsm5-prediction-variance-advice_files/figure-html/predmap1-1.png)
]

.pull-right[
- Grids!
- Cells are abundance estimate
- "snapshot"
- Sum cells to get abundance
- Sum a subset?
]

---
# Going back to the formula

Count model ( `$j$` observations):

$$
n_j = A_j\hat{p}_j \exp\left[ \beta_0 + s(\text{y}_j) + s(\text{Depth}_j) \right] + \epsilon_j
$$

--
<br/>
Predictions (index `$r$`):

$$
\hat{n}_r = A_r \exp\left[ \hat{\beta}_0 + \hat{s}(\text{y}_r) + \hat{s}(\text{Depth}_r) \right]
$$

<br/>
Need to "fill-in" values for `$A_r$`, `$\text{y}_r$` and `$\text{Depth}_r$`.

---
# Predicting

- With these values can use `predict` in R
- `predict(model, newdata=data, off.set=off.set)`

- `off.set` gives the area of the grid cells
- more info in `?predict.dsm`

---
# Prediction data

```
##            x      y      Depth      SST      NPP DistToCAS
## 126 547984.6 788254  153.59825 12.04609 1462.521 11788.974
## 127 557984.6 788254  552.31067 12.81379 1465.410  5697.248
## 258 527984.6 778254   96.81992 12.90251 1429.432 13722.626
## 259 537984.6 778254  138.23763 13.21393 1424.862  9720.671
## 260 547984.6 778254  505.14386 13.75655 1379.351  8018.690
## 261 557984.6 778254 1317.59521 14.42525 1348.544  3775.462
##              EKE off.set      long      lat
## 126 0.0008329031   1e+08 -66.52252 40.94697
## 127 0.0009806611   1e+08 -66.40464 40.94121
## 258 0.0011575423   1e+08 -66.76551 40.86781
## 259 0.0013417297   1e+08 -66.64772 40.86227
## 260 0.0026881567   1e+08 -66.52996 40.85662
## 261 0.0045683752   1e+08 -66.41221 40.85087
```

---
# Predictors

---
# Making a prediction

- Add another column to the prediction data
- Plotting then easier (in R)

```r
predgrid$Nhat_tw <- predict(dsm_all_tw_rm,
                            predgrid,
                            off.set=predgrid$off.set)
```

---
# Maps of predictions

.pull-left[

![](dsm5-prediction-variance-advice_files/figure-html/predmap-1.png)
]
.pull-right[

```r
p <- ggplot(predgrid) +
      geom_tile(aes(x=x, y=y,
                    fill=Nhat_tw)) +
      scale_fill_viridis() +
      coord_equal()
print(p)
```
]

---
# Total abundance

Each cell has an abundance, sum to get total

```r
sum(predgrid$Nhat_tw)
```

```
## [1] 2491.863
```

---
# Subsetting

R subsetting lets you calculate "interesting" estimates:

```r
# how many sperm whales at depths shallower than 2500m?
sum(predgrid$Nhat_tw[predgrid$Depth < 2500])
```

```
## [1] 1006.27
```

```r
# how many sperm whales East of 0?
sum(predgrid$Nhat_tw[predgrid$x>0])
```

```
## [1] 1383.744
```

---
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# Extrapolation

---
# What do we mean by extrapolation?

.pull-left[

- Predicting at values outside those observed
- What does "outside" mean?
  - between transects?
  - outside "survey area"?
]
.pull-right[

![](dsm5-prediction-variance-advice_files/figure-html/plottracks-1.png)
]

---
# Extrapolation

- In general, try not to do it!
- Variance issues?
- Space-time interchangability?
- `dsmextra` package by Phil Bouchet
  - [https://densitymodelling.github.io/dsmextra/index.html](https://densitymodelling.github.io/dsmextra/index.html)

![dsm extra logo](images/dsmextra-hex.png)

---
# Prediction recap

* Using `predict`
* Getting "overall" abundance
* Subsetting
* Plotting in R
* Extrapolation (and its dangers)

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# Estimating variance

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# Now we can make predictions
## Now we are dangerous.

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# Predictions are useless without uncertainty

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# Where does uncertainty come from?

---
# Sources of uncertainty

- Detection function parameters
- GAM parameters
- (And more! But only looking at these 2 here!)

![](dsm5-prediction-variance-advice_files/figure-html/unc-sources-1.png)
---
# Uncertianty of what?

- Uncertainty from detection function + GAM
- Want to talk about `$\hat{N}$`, so need to do some maths
- `dsm` does this for you!
- Details in Miller et al (2013) appendix

---
# GAM + detection function uncertainty

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

<div style="font-size:150%">
$$
\text{CV}^2\left( \hat{N} \right) \approx \text{CV}^2\left( \text{GAM} \right) +
$$

$$
\text{CV}^2\left( \text{detection function}\right)
$$
</div>
<br/>
<br/>
<p align="center">the "delta method"</p>
---
# When can we use the delta method?

- Assumes detection function and GAM are **independent**
- This is okay if:
  - no detection function covariates *or*
  - detection covariates won't be correlated with spatial model

![](images/delta_when.png)
---
# Variance propagation

- When detection function is not independent
- Uncertainty "propagated" through the model
- Refit both models together
- [Bravington, Miller and Hedley (2021)](https://link.springer.com/article/10.1007%2Fs13253-021-00438-2)

---
# In R...

- 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
  - covariates can only vary at segment level

---
# Variance of abundance

Using `dsm.var.gam`

```r
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:
##     2.5%     Mean    97.5% 
## 1539.017 2491.863 4034.641 
## (Using log-Normal approximation)
## 
## Point estimate                 : 2491.863 
## CV of detection function       : 0.2113123 
## CV from GAM                    : 0.1329 
## Total standard error           : 622.0386 
## Total coefficient of variation : 0.2496
```

---
# Plotting - data processing

- Calculate uncertainty per-cell
- `dsm.var.*` thinks `predgrid` is one "region"
- Need to split data into cells (using `split()`)
- Need `width` and `height` of cells for plotting

---
# Plotting (code)

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

```
## $`1`
##            x      y    Depth      SST      NPP DistToCAS
## 126 547984.6 788254 153.5983 12.04609 1462.521  11788.97
##              EKE off.set      long      lat    Nhat_tw
## 126 0.0008329031   1e+08 -66.52252 40.94697 0.01417646
##     height width
## 126  10000 10000
## 
## $`2`
##            x      y    Depth      SST     NPP DistToCAS
## 127 557984.6 788254 552.3107 12.81379 1465.41  5697.248
##              EKE off.set      long      lat    Nhat_tw
## 127 0.0009806611   1e+08 -66.40464 40.94121 0.05123446
##     height width
## 127  10000 10000
## 
## $`3`
##            x      y    Depth      SST      NPP DistToCAS
## 258 527984.6 778254 96.81992 12.90251 1429.432  13722.63
##             EKE off.set      long      lat    Nhat_tw
## 258 0.001157542   1e+08 -66.76551 40.86781 0.01118852
##     height width
## 258  10000 10000
```

---
# CV plot

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

.pull-left[
![](dsm5-prediction-variance-advice_files/figure-html/plotit-1.png)
]
.pull-right[
```
p <- plot(dsm_tw_var_map,
          observations=FALSE,
          plot=FALSE) + 
      coord_equal() +
      scale_fill_viridis()
print(p)
```
]

---
# Interpreting CV plots

- 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

---
# Effort overplotted

![](dsm5-prediction-variance-advice_files/figure-html/plottracksCV-1.png)

---
# Big CVs

- Here CVs are "well behaved"
- Not always the case (huge CVs possible)
- These can be a pain to plot
- Use `cut()` in R to make categorical variable
  - e.g. `c(seq(0,1, len=10), 2:4, Inf)` or somesuch
- (Example in practical)

---
# Uncertainty recap

- How does uncertainty arise in a DSM?
- Estimate variance of abundance estimate
- Map coefficient of variation

---
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# Practical advice

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# Pilot studies and "you get what you pay for"

- Designing surveys is hard
- Designing surveys is essential

- Better to fail one season than fail for 5, 10 years
- Get information early, get it cheap
  - Inform design from a pilot study

---

# Sometimes things are complicated

- Weather has a big effect on detectability
- Need to record during survey
- Disambiguate between distribution/detectability
- Potential confounding can be BAD

![weather or density?](images/weather_or_density.png)

---
# Visibility during POWER 2014

Thanks to Hiroto Murase and co. for this data!

---
# Covariates can make a big difference!

- Same data, same spatial model
- With weather covariates and without
<img src="images/covar-inout.png" width="100%">

---
# Disappointment
- Sometimes you don't have enough data
- Or, enough coverage
- Or, the right covariates
<br/>
<br/>
<br/>
<div class="medq", align="center"><b>Sometimes, you can't build a spatial model</b></div>

---
# Segmenting

- Example on [course site](http://workshops.distancesampling.org/online-dsm-2020/data-format/dsm-data-formatting.html)
- Length of `$\approx 2 w$` is reasonable
- Too big: no detail
- Too small: all 0/1
- See also [Redfern et al., (2008)](http://www.int-res.com/abstracts/meps/v363/p1-14/)

---
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# Getting help

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# Resources

- [Course reading list](http://workshops.distancesampling.org/online-dsm-2020/reading_list.html) has pointers to these topics
- [DenMod wiki](https://osf.io/5eza8/wiki/home/) with FAQ and more
- Distance sampling Google Group
  - Friendly, helpful, low traffic
  - see [distancesampling.org/distancelist.html](http://distancesampling.org/distancelist.html)