This session is concerned with summary statistics for interpoint correlation (i.e.Â dependence between points) and simulation envelopes.

The lecturerâ€™s R script is available here (right click and save).

### Exercise 1

The `swedishpines`

dataset was recorded in a study plot in a large forest. We shall assume the pattern is stationary.

Calculate the estimate of the \(K\)-function using `Kest`

.

Plot \(\widehat K(r)\) against \(r\)

Plot \(\widehat K(r) - \pi r^2\) against \(r\).

Calculate the estimate of the \(L\)-function and plot it against \(r\).

Plot \(\widehat L(r) - r\) against \(r\).

Calculate and plot an estimate of the pair correlation function using `pcf`

.

Draw tentative conclusions from these plots about interpoint interaction in the data.

### Exercise 2

For the `swedishpines`

data:

Plot the \(K\) function along with pointwise envelopes from 39 simulations of CSR:

`plot(envelope(swedishpines, Kest, nsim=39))`

Plot the \(L\) function along with pointwise envelopes from 39 simulations of CSR.

Plot the \(L\) function along with **simultaneous** envelopes from **19** simulations of CSR, using `ginterval=c(0,0.5)`

.

Plot the \(L\) function for along with **simultaneous** envelopes from **99** simulations of CSR using `ginterval=c(0,0.5)`

. What is the significance level of the associated test?