lgcp.estK(X, startpar=c(sigma2=1,alpha=1),
covmodel=list(model="exponential"),
lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
optim
to control the optimisation algorithm. See Details. The argument X
can be either
[object Object],[object Object]
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
(
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. 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 $\alpha$. Then the remaining parameter $\mu$ is inferred from the estimated intensity $\lambda$.
The template covariance function $c(r)$ is specified
using the argument covmodel
. It may be any of the
covariance functions recognised by the command
Covariance
in the
covmodel
should be of the form
list(model="modelname", ...)
where
modelname
is the string name of one of the covariance models
recognised by the command
Covariance
in the
...
are arguments of the
form tag=value
giving the values of parameters controlling the
shape of these models. For example the exponential covariance is
specified by covmodel=list(model="exponential")
while the
Matern covariance with exponent $\nu=0.3$ is specified
by covmodel=list(model="matern", nu=0.3)
.
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.
}
"minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
"fv"
)
containing the observed values of the summary statistic
(observed
) and the theoretical values of the summary
statistic computed from the fitted model parameters.
}lgcp.estpcf
because of the computation time required for the integral
in the $K$-function.
Computation can be accelerated, at the cost of less accurate results,
by setting spatstat.options(fastK.lgcp=TRUE)
.
Waagepetersen, R. (2007)
An estimating function approach to inference for
inhomogeneous Neyman-Scott processes.
Biometrics 63, 252--258.
}
[object Object],[object Object]
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.
Covariance
in the
Kest
for the $K$ function.