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

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

Generate Fry Plots for the Swedish Pines data using the two commands

```
fryplot(swedishpines)
fryplot(swedishpines, width=50)
```

What can you interpret from these plots?

### Exercise 3

The `japanesepines`

dataset is believed to exhibit spatial inhomogeneity. The question is whether, after allowing for inhomogeneity, there is still evidence of interpoint interaction. We will use the inhomogeneous \(K\)-function.

Compute the inhomogeneous \(K\) function using the default estimate of intensity (a leave-one-out kernel smoother) with heavy smoothing:

```
KiS <- Kinhom(japanesepines, sigma=0.1)
plot(KiS)
```

Fit a parametric trend to the data, and use this to compute the inhomogeneous \(K\) function:

```
fit <- ppm(japanesepines ~ polynom(x,y,2))
lambda <- predict(fit, type="trend")
KiP <- Kinhom(japanesepines, lambda)
plot(KiP)
```

Plot corresponding estimates of the inhomogeneous \(L\) function, using either `Linhom`

or

`plot(KiS, sqrt(./pi) ~ r)`

and similarly for `KiP`

.

Draw tentative conclusions about interpoint interaction.

### Exercise 4

To understand the difficulties with the \(K\)-function when the point pattern is not spatially homogeneous, try the following experiment.

Generate a simulated realisation of an inhomogeneous Poisson process, e.g.

`X <- rpoispp(function(x,y){ 200 * exp(-3 * x) })`

Plot the \(K\)-function or \(L\)-function. It will most likely appear to show evidence of clustering.

### Exercise 5

The cell process (`rcell`

) has the same theoretical \(K\)-function as the uniform Poisson process.

Read the help file

Generate a simulated realisation of the cell process with a 10x10 grid of cells and plot it.

Plot the \(K\) or \(L\)-function for this pattern, and determine whether it is distinguishable from a Poisson process.