This session is concerned with Gibbs models for point patterns with interpoint interaction.
The lecturer’s R script is available here (right click and save).

Exercise 1

In this question we fit a Strauss point process model to the swedishpines data.

1. We need a guess at the interaction distance $$R$$. Compute and plot the $$L$$-function of the dataset and choose the value $$r$$ which maximises the discrepancy $$|L(r)-r|$$.

2. Fit the stationary Strauss model with the chosen interaction distance using

ppm(swedishpines ~ 1, Strauss(R))

where R is your chosen value.

1. Interpret the printout: how strong is the interaction?

2. Plot the fitted pairwise interaction function using plot(fitin(fit)).

Exercise 2

In Question 1 we guesstimated the Strauss interaction distance parameter. Alternatively this parameter could be estimated by profile pseudolikelihood.

1. Look again at the plot of the $$L$$-function of swedishpines and determine a plausible range of possible values for the interaction distance.

2. Generate a sequence of values equally spaced across this range, for example, if your range of plausible values was $$[5, 12]$$, then type

rvals <- seq(5, 12, by=0.5)
3. Construct a data frame, with one column named r (matching the argument name of Strauss), containing these values. For example

D <- data.frame(r = rvals)
4. Execute

fitp <- profilepl(D, Strauss, swedishpines ~ 1)

to find the maximum profile pseudolikelihood fit.

5. Print and plot the object fitp.

6. Compare the computed estimate of interaction distance $$r$$ with your guesstimate. Compare the corresponding estimates of the Strauss interaction parameter $$\gamma$$.

7. Extract the fitted Gibbs point process model from the object fitp as

bestfit <- as.ppm(fitp)

Exercise 3

For the Strauss model fitted in Question 1,

1. Generate and plot a simulated realisation of the fitted model using simulate.

2. Plot the $$L$$-function of the data along with the global simulation envelopes from 19 realisations of the fitted model.

Exercise 4

1. Read the help file for Geyer.

2. Fit a stationary Geyer saturation process to swedishpines, with the same interaction distance as for the Strauss model computed in Question 2, and trying different values of the saturation parameter sat = 1, 2, 3 say.

3. Fit the same model with the addition of a log-quadratic trend.

4. Plot the fitted trend and conditional intensity.

Exercise 5

Modify question 1 by using the Huang-Ogata approximate maximum likelihood algorithm (method="ho") instead of maximum pseudolikelihood (the default, method="mpl").

Exercise 6

Repeat Question 2 for the inhomogeneous Strauss process with log-quadratic trend. The corresponding call to profilepl is

fitp <- profilepl(D, Strauss, swedishpines ~ polynom(x,y,2))

Exercise 7

Repeat Question 3 for the inhomogeneous Strauss process with log-quadratic trend, using the inhomogeneous $$L$$-function Linhom in place of the usual $$L$$-function.