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.

  1. Read the help file

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

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