lgcp.estK(X, startpar=c(var=1,scale=1), covmodel=list(model="exponential"), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)optim
to control the optimisation algorithm. See Details.
"minconfit". There are methods for printing
and plotting this object. It contains the following main components:The shape of the covariance of the LGCP must be specified: the default is the exponential covariance function, but other covariance models can be selected.
The argument X can be either
The algorithm fits a log-Gaussian Cox point process (LGCP)
model to X, by finding the parameters of the LGCP model
which give the closest match between the
theoretical $K$ function of the LGCP model
and the observed $K$ function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast.
The model fitted is a stationary, isotropic log-Gaussian Cox process (Moller and Waagepetersen, 2003, pp. 72-76). To define this process we start with a stationary Gaussian random field $Z$ in the two-dimensional plane, with constant mean $mu$ and covariance function $C(r)$. Given $Z$, we generate a Poisson point process $Y$ with intensity function $lambda(u) = exp(Z(u))$ at location $u$. Then $Y$ is a log-Gaussian Cox process.
The $K$-function of the LGCP is
$$
K(r) = \int_0^r 2\pi s \exp(C(s)) \, {\rm d}s.
$$
The intensity of the LGCP is
$$
\lambda = \exp(\mu + \frac{C(0)}{2}).
$$
The covariance function $C(r)$ is parametrised in the form
$$
C(r) = \sigma^2 c(r/\alpha)
$$
where $sigma^2$ and $alpha$ are parameters
controlling the strength and the scale of autocorrelation,
respectively, and $c(r)$ is a known covariance function
determining the shape of the covariance.
The strength and scale parameters
$sigma^2$ and $alpha$
will be estimated by the algorithm as the values
var and scale respectively.
The template covariance function $c(r)$ must be specified
as explained below.
In this algorithm, the Method of Minimum Contrast is first used to find
optimal values of the parameters $sigma^2$
and $alpha^2$. Then the remaining parameter
$mu$ is inferred from the estimated intensity
$lambda$.
The template covariance function $c(r)$ is specified
using the argument covmodel. This should be of the form
list(model="modelname", ...) where
modelname is a string identifying the template model
as explained below, and ... are optional arguments of the
form tag=value giving the values of parameters controlling the
shape of the template model.
The default is the exponential covariance
$c(r) = e^(-r)$
so that the scaled covariance is
$$
C(r) = \sigma^2 e^{-r/\alpha}.
$$
To determine the template model, the string "modelname" will be
prefixed by "RM" and the code will search for
a function of this name in the RandomFields package.
For a list of available models see
RMmodel in the
RandomFields package. For example the
Matern covariance with exponent $nu = 0.3$ is specified
by covmodel=list(model="matern", nu=0.3) corresponding
to the function RMmatern in the RandomFields package.
If the argument lambda is provided, then this is used
as the value of $lambda$. Otherwise, if X is a
point pattern, then $lambda$
will be estimated from X.
If X is a summary statistic and lambda is missing,
then the intensity $lambda$ cannot be estimated, and
the parameter $mu$ will be returned as NA.
The remaining arguments rmin,rmax,q,p control the
method of minimum contrast; see mincontrast.
The optimisation algorithm can be controlled through the
additional arguments "..." which are passed to the
optimisation function optim. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B" to select an optimisation
algorithm that respects box constraints, and use the arguments
lower and upper to specify (vectors of) minimum and
maximum values for each parameter.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252--258.
lgcp.estpcf for alternative method of fitting LGCP.
matclust.estK,
thomas.estK for other models.
mincontrast for the generic minimum contrast
fitting algorithm, including important parameters that affect
the accuracy of the fit.
RMmodel in the
RandomFields package, for covariance function models.
Kest for the $K$ function.
if(interactive()) {
u <- lgcp.estK(redwood)
} else {
# slightly faster - better starting point
u <- lgcp.estK(redwood, c(var=1, scale=0.1))
}
u
plot(u)
if(FALSE) {
## takes several minutes!
lgcp.estK(redwood, covmodel=list(model="matern", nu=0.3))
}
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