The point pattern in Figure 1.1 (following page)
Figure 1.1: Simulated point pattern.
is made-up data. It is a simulation
of the saturation model with r = .05, c = 4.5, 
, and
. The simulation was actually done in a toroidal region
with side 1.2, and the points outside the unit square were thrown away.
We pretend Figure 1.1 is a data set of unknown origin to which we
will fit several models.
The first model we will fit is the saturation model with r = .05 and
c = 4.5 (the same as the simulation) with 
an unknown parameter.
In this section we consider the process to be defined in the unit
square with free boundary conditions, which is not the same as the simulation.
The observed value of the canonical statistic t(x) for the pattern in
Figure 1.1 is (372.0, 1321.5).
We start with a sample of size 
with spacing between samples 100.
At the MLE, the observed and expected values of the canonical statistic
are equal, so the the mean number of points is 372. A spacing of 100
only attempts to update about one quarter of the points between sampled
point patterns. This spacing is not optimal. We shall see how to choose
better spacing presently.
Figure 1.2 (following page)
Figure 1.2: Simulated values of the canonical statistic.
is a scatter plot of the the values of the canonical statistic
t(x) = (n(x), u(x)) for this simulation. The large dot is the the
observed value of the canonical statistic.
If the simulation parameter 
were the MLE, the scatter
plot would be centered at the observed value. If the observed value were
outside the convex hull of the simulated values, the Monte Carlo likelihood
would have no maximum, and if the observed value were inside the convex
hull but very close to the boundary, the MCL approximation would be very bad.
In either case, we would have to use stochastic approximation
(Section 1.11) or a trust region (Geyer and Thompson, 1992)
to get closer to the MLE. For the simulation in Figure 1.2 
is close enough to the true MLE 
so that the MCL approximation
gives a good estimate 
of the MLE.
Using the code described earlier in this section, we obtain
as the MCMLE, the maximizer of the Monte Carlo
likelihood . For comparison the MCNR update gives (4.21, 0.364).