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

1. Calculate the estimate of the $$K$$-function using Kest.

2. Plot $$\widehat K(r)$$ against $$r$$

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

4. Calculate the estimate of the $$L$$-function and plot it against $$r$$.

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

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

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

### Exercise 2

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

fryplot(swedishpines)
fryplot(swedishpines, width=50)
2. 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.

1. 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)
2. 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)
3. Plot corresponding estimates of the inhomogeneous $$L$$ function, using either Linhom or

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

and similarly for KiP.

4. 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.

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

X <- rpoispp(function(x,y){ 200 * exp(-3 * x) })
2. 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.

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