This session is concerned with summary statistics for interpoint spacing and distances.

The lecturer’s R script is available here (right click and save).

### Exercise 1

The command `rThomas`

generates simulated realisations of the Thomas model (‘modified Thomas cluster process’).

Read the help file.

Type `plot(rThomas(10, 0.05, 8))`

a few times, and interpret the results.

- Experiment with the arguments of
`rThomas`

to obtain point patterns that
- consist of a few, well-separated, very tight clusters of points;
- look similar to realisations of a uniform Poisson process.

### Exercise 2

Read the help file for `kppm`

.

Fit the Thomas model to the `redwood`

data by the method of minimum contrast:

```
fit <- kppm(redwood ~ 1, clusters="Thomas")
fit
plot(fit)
```

Read off the parameters of the fitted model, and generate a simulated realisation of the fitted model using `rThomas`

.

Type `plot(simulate(fit))`

to generate a simulated realisation of the fitted model automatically.

Try the command

`fit2 <- kppm(redwood ~ 1, clusters="Thomas", startpar=c(kappa=10, scale=0.1))`

and briefly explore the fitting algorithm’s sensitivity to the initial guesses at the parameter values `kappa`

and `scale`

.

Generate and plot several simulated realisations of the fitted model, to assess whether it is plausible.

Extract and plot the fitted pair correlation function by

```
pcffit <- pcfmodel(fit)
plot(pcffit, xlim = c(0, 0.3))
```

Type `plot(envelope(fit, Lest, nsim=39))`

to generate simulation envelopes of the \(L\) function from this fitted model. Do they suggest the model is plausible?

### Exercise 3

Fit a Matern cluster process to the `redwood`

data.

Use `vcov`

to estimate the covariance matrix of the parameter estimates.

Compare with the covariance matrix obtained when fitting a homogeneous Poisson model.