vargamma.estpcf
Fit the Neyman-Scott Cluster Point Process with Variance Gamma kernel
Fits the Neyman-Scott cluster point process, with Variance Gamma kernel, to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.
Usage
vargamma.estpcf(X, startpar=c(kappa=1,eta=1), nu.ker = -1/4, lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs = list())
Arguments
- X
- Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details.
- startpar
- Vector of starting values for the parameters of the model.
- nu.ker
- Numerical value controlling the shape of the tail of the clusters.
A number greater than
-1/2
. - lambda
- Optional. An estimate of the intensity of the point process.
- q,p
- Optional. Exponents for the contrast criterion.
- rmin, rmax
- Optional. The interval of $r$ values for the contrast criterion.
- ...
- Optional arguments passed to
optim
to control the optimisation algorithm. See Details. - pcfargs
- Optional list containing arguments passed to
pcf.ppp
to control the smoothing in the estimation of the pair correlation function.
Details
This algorithm fits the Neyman-Scott Cluster point process model with Variance Gamma kernel (Jalilian et al, 2011) to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.
The argument X
can be either
[object Object],[object Object]
The algorithm fits the Neyman-Scott Cluster point process
with Variance Gamma kernel to X
,
by finding the parameters of the model
which give the closest match between the
theoretical pair correlation function of the model
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast
.
The Neyman-Scott cluster point process with Variance Gamma
kernel is described in Jalilian et al (2011).
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
have a common distribution described in Jalilian et al (2011).
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 rVarGamma
.
The parameter eta
appearing in startpar
is equivalent to the
scale parameter omega
used in rVarGamma
.
Homogeneous or inhomogeneous Neyman-Scott/VarGamma 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.
Value
- An object of class
"minconfit"
. There are methods for printing and plotting this object. It contains the following main components: par Vector of fitted parameter values. fit 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.
References
Jalilian, A., Guan, Y. and Waagepetersen, R. (2011) 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.
See Also
kppm
,
vargamma.estK
,
lgcp.estpcf
,
thomas.estpcf
,
cauchy.estpcf
,
mincontrast
,
pcf
,
pcfmodel
.
rVarGamma
to simulate the model.
Examples
u <- vargamma.estpcf(redwood)
u
plot(u, legendpos="topright")