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).
In this question we fit a Strauss point process model to the swedishpines
data.
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 | . |
Fit the stationary Strauss model with the chosen interaction distance using
ppm(swedishpines ~ 1, Strauss(R))
where R
is your chosen value.
Interpret the printout: how strong is the interaction?
Plot the fitted pairwise interaction function using plot(fitin(fit))
.
In Question 1 we guesstimated the Strauss interaction distance parameter. Alternatively this parameter could be estimated by profile pseudolikelihood.
Look again at the plot of the L-function of swedishpines
and determine a plausible range of possible values for the interaction distance.
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)
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)
Execute
fitp <- profilepl(D, Strauss, swedishpines ~ 1)
to find the maximum profile pseudolikelihood fit.
Print and plot the object fitp
.
Compare the computed estimate of interaction distance r with your guesstimate. Compare the corresponding estimates of the Strauss interaction parameter γ.
Extract the fitted Gibbs point process model from the object fitp
as
bestfit <- as.ppm(fitp)
For the Strauss model fitted in Question 1,
Generate and plot a simulated realisation of the fitted model using simulate
.
Plot the L-function of the data along with the global simulation envelopes from 19 realisations of the fitted model.
Read the help file for Geyer
.
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.
Fit the same model with the addition of a log-quadratic trend.
Plot the fitted trend and conditional intensity.
Modify question 1 by using the Huang-Ogata approximate maximum likelihood algorithm (method="ho"
) instead of maximum pseudolikelihood (the default, method="mpl"
).
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))
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.