Lab 10: Gibbs processes

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 γ.

  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.