# LGCPs - An example in two dimensions

#### David Borchers and Finn Lindgren

#### Generated on 2023-12-03

Source:`vignettes/articles/2d_lgcp.Rmd`

`2d_lgcp.Rmd`

## Introduction

For this vignette we are going to be working with a dataset obtained
from the `R`

package `spatstat`

. We will set up a
two-dimensional LGCP to estimate Gorilla abundance.

## Get the data

For the next few practicals we are going to be working with a dataset
obtained from the `R`

package `spatstat`

, which
contains the locations of 647 gorilla nests. We load the dataset like
this:

`data(gorillas, package = "inlabru")`

This dataset is a list containing a number of `R`

objects,
including the locations of the nests, the boundary of the survey area
and an `INLA`

mesh - see `help(gorillas)`

for
details. Extract the the objects we need from the list, into other
objects, so that we don’t have to keep typing
‘`gorillas$`

’:

```
nests <- gorillas$nests
mesh <- gorillas$mesh
boundary <- gorillas$boundary
```

Plot the points (the nests).

## Fiting the model

Fit an LGCP model to the locations of the gorilla nests, predict on the survey region, and produce a plot of the estimated density - which should look like the plot shown below.

Recall that the steps to specifying, fitting and predicting are:

Specify a model, comprising (for 2D models)

`coordinates`

on the left of`~`

and an SPDE`+ Intercept(1)`

on the right. Please use the SPDE prior specification stated below.Call

`lgcp( )`

, passing it (with 2D models) the model components, the`SpatialPointsDataFrame`

containing the observed points and the`SpatialPolygonsDataFrame`

defining the survey boundary using the`samplers`

argument.Call

`predict( )`

, passing it the fitted model from 2., locations at which to predict and an appropriate predictor specification. The locations at which to predict can be a`SpatialPixelsDataFrame`

covering the mesh, obtained by calling`fm_pixels(mesh, format = "sp")`

.

```
matern <- inla.spde2.pcmatern(
mesh,
prior.sigma = c(0.1, 0.01),
prior.range = c(5, 0.01)
)
cmp <- coordinates ~
mySmooth(coordinates, model = matern) +
Intercept(1)
fit <- lgcp(cmp, nests, samplers = boundary, domain = list(coordinates = mesh))
```

## Predicting intensity

You should get a plot like that below (the command below assumes that
the prediction is in an object called `lambda`

):

```
pred <- predict(
fit,
fm_pixels(mesh, mask = boundary, format = "sp"),
~ data.frame(
lambda = exp(mySmooth + Intercept),
loglambda = mySmooth + Intercept
)
)
pl1 <- ggplot() +
gg(pred$lambda) +
gg(boundary) +
ggtitle("LGCP fit to Points", subtitle = "(Response Scale)")
pl2 <- ggplot() +
gg(pred$loglambda) +
gg(boundary, alpha = 0) +
ggtitle("LGCP fit to Points", subtitle = "(Linear Predictor Scale)")
multiplot(pl1, pl2, cols = 2)
```

You can plot the median, lower 95% and upper 95% density surfaces as
follows (assuming that the predicted intensity is in object
`lambda`

).

```
ggplot() +
gg(cbind(pred$lambda, data.frame(property = "q0.500")), aes(fill = median)) +
gg(cbind(pred$lambda, data.frame(property = "q0.025")), aes(fill = q0.025)) +
gg(cbind(pred$lambda, data.frame(property = "q0.975")), aes(fill = q0.975)) +
coord_equal() +
facet_wrap(~property)
```

## SPDE parameters

Plot the SPDE parameter and fixed effect parameter posteriors.

```
int.plot <- plot(fit, "Intercept")
spde.range <- spde.posterior(fit, "mySmooth", what = "range")
spde.logvar <- spde.posterior(fit, "mySmooth", what = "log.variance")
range.plot <- plot(spde.range)
var.plot <- plot(spde.logvar)
multiplot(range.plot, var.plot, int.plot)
```

Look at the correlation function if you want to:

```
corplot <- plot(spde.posterior(fit, "mySmooth", what = "matern.correlation"))
covplot <- plot(spde.posterior(fit, "mySmooth", what = "matern.covariance"))
multiplot(covplot, corplot)
```

## Estimating Abundance

Finally, estimate abundance using the `predict`

function.
As a first step we need an estimate for the integrated lambda. The
integration `weight`

values are contained in the
`fm_int()`

output.

```
Lambda <- predict(
fit,
fm_int(mesh, boundary),
~ sum(weight * exp(mySmooth + Intercept))
)
Lambda
#> mean sd q0.025 q0.5 q0.975 median mean.mc_std_err
#> 1 670.0177 30.3882 614.7392 671.14 729.3213 671.14 3.03882
#> sd.mc_std_err
#> 1 2.050789
```

Given some generous interval boundaries (500, 800) for lambda we can estimate the posterior abundance distribution via

```
Nest <- predict(
fit, fm_int(mesh, boundary),
~ data.frame(
N = 500:800,
dpois(500:800,
lambda = sum(weight * exp(mySmooth + Intercept))
)
)
)
```

Get its quantiles via

```
inla.qmarginal(c(0.025, 0.5, 0.975), marginal = list(x = Nest$N, y = Nest$mean))
#> [1] 599.8821 673.8480 753.1823
```

… the mean via

```
inla.emarginal(identity, marginal = list(x = Nest$N, y = Nest$mean))
#> [1] 674.4301
```

and plot posteriors:

```
Nest$plugin_estimate <- dpois(Nest$N, lambda = Lambda$mean)
ggplot(data = Nest) +
geom_line(aes(x = N, y = mean, colour = "Posterior")) +
geom_line(aes(x = N, y = plugin_estimate, colour = "Plugin"))
```

The true number of nests in 647; the mean and median of the posterior distribution of abundance should be close to this if you have not done anything wrong!