Fits the Neyman-Scott Cluster point process with Cauchy kernel to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.
cauchy.estpcf(X, startpar=c(kappa=1,scale=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...,
pcfargs = list())Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details.
Vector of starting values for the parameters of the model.
Optional. An estimate of the intensity of the point process.
Optional. Exponents for the contrast criterion.
Optional. The interval of \(r\) values for the contrast criterion.
Optional arguments passed to optim
to control the optimisation algorithm. See Details.
Optional list containing arguments passed to pcf.ppp
to control the smoothing in the estimation of the
pair correlation function.
An object of class "minconfit". There are methods for printing
and plotting this object. It contains the following main components:
Vector of fitted parameter values.
Function value table (object of class "fv")
containing the observed values of the summary statistic
(observed) and the theoretical values of the summary
statistic computed from the fitted model parameters.
This algorithm fits the Neyman-Scott cluster point process model with Cauchy kernel to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.
The argument X can be either
An object of class "ppp"
representing a point pattern dataset.
The pair correlation function of the point pattern will be computed
using pcf, and the method of minimum contrast
will be applied to this.
An object of class "fv" containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the pair correlation function,
and this object should have been obtained by a call to
pcf or one of its relatives.
The algorithm fits the Neyman-Scott cluster point process
with Cauchy kernel to X,
by finding the parameters of the Matern Cluster model
which give the closest match between the
theoretical pair correlation function of the Matern Cluster process
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast.
The model is described in Jalilian et al (2013). It is a cluster process formed by taking a pattern of parent points, generated according to a Poisson process with intensity \(\kappa\), and around each parent point, generating a random number of offspring points, such that the number of offspring of each parent is a Poisson random variable with mean \(\mu\), and the locations of the offspring points of one parent follow a common distribution described in Jalilian et al (2013).
If the argument lambda is provided, then this is used
as the value of the point process intensity \(\lambda\).
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 corresponding model can be simulated using rCauchy.
For computational reasons, the optimisation procedure internally uses
the parameter eta2, which is equivalent to 4 * scale^2
where scale is the scale parameter for the model as used in
rCauchy.
Homogeneous or inhomogeneous Neyman-Scott/Cauchy models can also be
fitted using the function kppm and the fitted models
can be simulated using simulate.kppm.
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.
Ghorbani, M. (2012) Cauchy cluster process. Metrika, to appear.
Jalilian, A., Guan, Y. and Waagepetersen, R. (2013) Decomposition of variance for spatial Cox processes. Scandinavian Journal of Statistics 40, 119-137.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252--258.
kppm,
cauchy.estK,
lgcp.estpcf,
thomas.estpcf,
vargamma.estpcf,
mincontrast,
pcf,
pcfmodel.
rCauchy to simulate the model.
# NOT RUN {
u <- cauchy.estpcf(redwood)
u
plot(u, legendpos="topright")
# }
Run the code above in your browser using DataLab