Here is a “solution” for practical 5. As with any data analysis, there is no correct answer, but this shows how I would approach this analysis. The analysis here is conditional on selecting a detection function and DSM in the previous exercises; I’ve shown a variety of models selected in the previous solutions to show the differences between models.

Much of the text below is as in the exercise itself, so it should be relatively easy to navigate.

Additional text and code is highlighted using boxes like this.

Now we’ve fitted some models and estimated abundance, we can estimate the variance associated with the abundance estimate (and plot it).

`library(dsm)`

```
## Loading required package: mgcv
## Loading required package: nlme
## This is mgcv 1.8-7. For overview type 'help("mgcv-package")'.
## Loading required package: mrds
## This is mrds 2.1.15
## Built: R 3.2.2; ; 2015-11-24 17:46:38 UTC; unix
## This is dsm 2.2.11
## Built: R 3.2.2; ; 2015-11-24 18:04:45 UTC; unix
```

`library(raster)`

```
## Loading required package: sp
##
## Attaching package: 'raster'
##
## The following object is masked from 'package:nlme':
##
## getData
```

```
library(ggplot2)
library(viridis)
library(plyr)
library(knitr)
library(rgdal)
```

```
## rgdal: version: 1.0-7, (SVN revision 559)
## Geospatial Data Abstraction Library extensions to R successfully loaded
## Loaded GDAL runtime: GDAL 1.11.3, released 2015/09/16
## Path to GDAL shared files: /usr/local/Cellar/gdal/1.11.3/share/gdal
## Loaded PROJ.4 runtime: Rel. 4.9.2, 08 September 2015, [PJ_VERSION: 491]
## Path to PROJ.4 shared files: (autodetected)
## Linking to sp version: 1.2-0
```

Load the models and prediction grid:

```
load("dsms.RData")
load("dsms-xy.RData")
load("predgrid.RData")
```

Depending on the model response (count or Horvitz-Thompson) we can use either `dsm.var.prop`

or `dsm.var.gam`

, respectively. `dsm_nb_xy_ms`

doesn’t include any covariates at the observer level in the detection function, so we can use the variance propagation method and estimate the uncertainty in detection function parameters in one step.

```
# need to remove the NAs as we did when plotting
predgrid_var <- predgrid[!is.na(predgrid$Depth),]
# now estimate variance
var_nb_xy_ms <- dsm.var.prop(dsm_nb_xy_ms, predgrid_var, off.set=predgrid_var$off.set)
```

To summarise the results of this variance estimate:

`summary(var_nb_xy_ms)`

```
## 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.
## -5.596e-06 -1.399e-07 -2.096e-08 1.690e-07 1.982e-07 7.159e-06
##
## Approximate asymptotic confidence interval:
## 2.5% Mean 97.5%
## 1005.079 1589.217 2512.848
## (Using log-Normal approximation)
##
## Point estimate : 1589.217
## Standard error : 328.2824
## Coefficient of variation : 0.2066
```

Try this out for some of the other models you’ve saved. Remember to use `dsm.var.gam`

when there are covariates in the detection function and `dsm.var.prop`

when there aren’t.

We’ll skip this and go straight to summarising all our models so far in the next section…

We can again summarise all the models, as we did with the DSMs and detection functions, now including the variance:

```
summarize_dsm_var <- function(model, predgrid){
summ <- summary(model)
vp <- summary(dsm.var.prop(model, predgrid, off.set=predgrid$off.set))
unconditional.cv.square <- vp$cv^2
asymp.ci.c.term <- exp(1.96*sqrt(log(1+unconditional.cv.square)))
asymp.tot <- c(vp$pred.est / asymp.ci.c.term,
vp$pred.est,
vp$pred.est * asymp.ci.c.term)
data.frame(response = model$family$family,
terms = paste(rownames(summ$s.table), collapse=", "),
AIC = AIC(model),
REML = model$gcv.ubre,
"Deviance_explained" = paste0(round(summ$dev.expl*100,2),"%"),
"lower_CI" = round(asymp.tot[1],2),
"Nhat" = round(asymp.tot[2],2),
"upper_CI" = round(asymp.tot[3],2)
)
}
```

```
# make a list of models
model_list <- list(dsm_nb_xy, dsm_nb_x_y, dsm_nb_xy_ms, dsm_nb_x_y_ms,
dsm_tw_xy, dsm_tw_x_y, dsm_tw_xy_ms, dsm_tw_x_y_ms)
# give the list names for the models, so we can identify them later
names(model_list) <- c("dsm_nb_xy", "dsm_nb_x_y", "dsm_nb_xy_ms", "dsm_nb_x_y_ms",
"dsm_tw_xy", "dsm_tw_x_y", "dsm_tw_xy_ms", "dsm_tw_x_y_ms")
per_model_var <- ldply(model_list, summarize_dsm_var, predgrid=predgrid_var)
```

`kable(per_model_var)`

.id | response | terms | AIC | REML | Deviance_explained | lower_CI | Nhat | upper_CI |
---|---|---|---|---|---|---|---|---|

dsm_nb_xy | Negative Binomial(0.105) | s(x,y) | 775.2593 | 392.6460 | 40.49% | 1081.45 | 1687.29 | 2632.52 |

dsm_nb_x_y | Negative Binomial(0.085) | s(x), s(y) | 789.7554 | 395.8609 | 31.11% | 1025.49 | 1586.08 | 2453.12 |

dsm_nb_xy_ms | Negative Binomial(0.108) | s(x,y), s(Depth) | 758.1052 | 384.7781 | 37.5% | 1064.59 | 1589.22 | 2372.38 |

dsm_nb_x_y_ms | Negative Binomial(0.098) | s(y), s(Depth) | 762.5970 | 386.1100 | 35.73% | 1070.52 | 1607.30 | 2413.24 |

dsm_tw_xy | Tweedie(p=1.29) | s(x,y) | 1249.3796 | 394.8611 | 34.55% | 1209.91 | 1742.96 | 2510.86 |

dsm_tw_x_y | Tweedie(p=1.306) | s(x), s(y) | 1252.2601 | 399.8421 | 27.27% | 1154.66 | 1670.76 | 2417.53 |

dsm_tw_xy_ms | Tweedie(p=1.268) | s(x,y), s(Depth) | 1229.8875 | 389.8638 | 37.82% | 1168.36 | 1645.05 | 2316.22 |

dsm_tw_x_y_ms | Tweedie(p=1.282) | s(Depth), s(NPP) | 1227.3139 | 387.8614 | 34.54% | 715.43 | 1201.75 | 2018.64 |

We can plot a map of the coefficient of variation, but we first need to estimate the variance per prediction cell, rather than over the whole area. This calculation takes a while!

```
# use the split function to make each row of the predictiond data.frame into
# an element of a list
predgrid_var_split <- split(predgrid_var, 1:nrow(predgrid_var))
var_split_nb_xy_ms <- dsm.var.prop(dsm_nb_xy_ms, predgrid_var_split, off.set=predgrid_var$off.set)
```

Now we have the per-cell coefficients of variation, we assign that to a column of the prediction grid data and plot it as usual:

```
predgrid_var_map <- predgrid_var
cv <- sqrt(var_split_nb_xy_ms$pred.var)/unlist(var_split_nb_xy_ms$pred)
predgrid_var_map$CV <- cv
p <- ggplot(predgrid_var_map) +
geom_tile(aes(x=x, y=y, fill=CV, width=10*1000, height=10*1000)) +
scale_fill_viridis() +
coord_equal() +
geom_point(aes(x=x,y=y, size=count), data=dsm_nb_xy_ms$data[dsm_nb_xy_ms$data$count>0,])
print(p)
```

Note that here we overplot the segments where sperm whales were observed (and scale the size of the point according to the number observed), using `geom_point()`

.

We can also overplot the effort, which can be a useful way to see what the cause of uncertainty is. Though it may not only be caused by lack of effort but also covariate coverage, this can be useful to see.

First we need to load the segment data from the `gdb`

`tracks <- readOGR("Analysis.gdb", "Segments")`

```
## OGR data source with driver: OpenFileGDB
## Source: "Analysis.gdb", layer: "Segments"
## with 949 features
## It has 8 fields
```

`tracks <- fortify(tracks)`

We can then just add this to the plot object we have built so far (with `+`

), but this looks a bit messy with the observations, so let’s start from scratch:

```
p <- ggplot(predgrid_var_map) +
geom_tile(aes(x=x, y=y, fill=CV, width=10*1000, height=10*1000)) +
scale_fill_viridis() +
coord_equal() +
geom_path(aes(x=long,y=lat, group=group), data=tracks)
print(p)
```

Try this with the other models you fitted and see what the differences are between the maps of coefficient of variation.

We can use a similar technique as we did in the prediction exercises to get coefficient of variation maps for all of the models…

**Note this can take a long time!**

```
# make a function that calculates the CV, adds them to a column named CV
# and adds a column called "model" that stores the model name, then returns the
# data.frame.
make_cv_dat <- function(model_name, predgrid_split, predgrid){
# we use get() here to grab the object with the name of its argument
var_obj <- dsm.var.prop(get(model_name), predgrid_split, off.set=predgrid$off.set)
predgrid[["CV"]] <- sqrt(var_obj$pred.var)/unlist(var_obj$pred)
predgrid[["model"]] <- model_name
return(predgrid)
}
# load plyr and apply to a list of the names of the models, make_cv_dat returns
# a data.frame (hence this is an "ld" function: list->data.frame) that it then binds
# together
library(plyr)
big_cv <- ldply(list("dsm_nb_xy", "dsm_nb_x_y", "dsm_nb_xy_ms", "dsm_nb_x_y_ms",
"dsm_tw_xy", "dsm_tw_x_y", "dsm_tw_xy_ms", "dsm_tw_x_y_ms"),
make_cv_dat, predgrid_split=predgrid_var_split, predgrid=predgrid_var_map)
```

```
p <- ggplot(big_cv) +
geom_tile(aes(x=x, y=y, fill=CV, width=10*1000, height=10*1000)) +
scale_fill_viridis() +
coord_equal() +
facet_wrap(~model, nrow=2) +
geom_path(aes(x=long,y=lat, group=group), data=tracks)
print(p)
```