This function is an experimental extension to the
point process model fitting command ppm
.
The extension allows the trend of the model to include irregular parameters,
which will be maximised by a Newton-type iterative
method, using nlm
.
For the sake of explanation,
consider a Poisson point process with intensity function
\(\lambda(u)\) at location \(u\). Assume that
$$
\lambda(u) = \exp(\alpha + \beta Z(u)) \, f(u, \gamma)
$$
where \(\alpha,\beta,\gamma\) are
parameters to be estimated, \(Z(u)\) is a spatial covariate
function, and \(f\) is some known function.
Then the parameters
\(\alpha,\beta\) are called regular because they
appear in a loglinear form; the parameter
\(\gamma\) is called irregular.
To fit this model using ippm
, we specify the
intensity using the trend
formula
in the same way as usual for ppm
.
The trend formula is a representation of the log intensity.
In the above example the log intensity is
$$
\log\lambda(u) = \alpha + \beta Z(u) + \log f(u, \gamma)
$$
So the model above would be encoded with the trend formula
~Z + offset(log(f))
. Note that the irregular part of the model
is an offset term, which means that it is included in the log trend
as it is, without being multiplied by another regular parameter.
The optimisation runs faster if we specify the derivative
of \(\log f(u,\gamma)\) with
respect to \(\gamma\). We call this the
irregular score. To specify this, the user must write an R function
that computes the irregular score for any value of
\(\gamma\) at any location (x,y)
.
Thus, to code such a problem,
The argument trend
should define the
log intensity, with the irregular part as an offset;
The argument start
should be a list
containing initial values of each of the irregular parameters;
The argument iScore
, if provided,
must be a list (with one entry
for each entry of start
) of functions
with arguments x,y,…
, that evaluate the partial derivatives
of \(\log f(u,\gamma)\) with
respect to each irregular parameter.
The coded example below illustrates the model with two irregular
parameters \(\gamma,\delta\) and irregular term
$$
f((x,y), (\gamma, \delta)) = 1 + \exp(\gamma - \delta x^3)
$$
Arguments …
passed to ppm
may
also include interaction
. In this case the model is not
a Poisson point process but a more general Gibbs point process;
the trend formula trend
determines the first-order trend
of the model (the first order component of the conditional intensity),
not the intensity.