Fits the Neyman-Scott cluster point process, with Variance Gamma kernel, to a point pattern dataset by the Method of Minimum Contrast.

```
vargamma.estK(X, startpar=c(kappa=1,scale=1), nu = -1/4, lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
```

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

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.

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 Variance Gamma kernel (Jalilian et al, 2013) to a point pattern dataset by the Method of Minimum Contrast, using the \(K\) function.

The argument `X`

can be either

- a point pattern:
An object of class

`"ppp"`

representing a point pattern dataset. The \(K\) function of the point pattern will be computed using`Kest`

, and the method of minimum contrast will be applied to this.- a summary statistic:
An object of class

`"fv"`

containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the \(K\) function, and this object should have been obtained by a call to`Kest`

or one of its relatives.

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 \(K\) function of the model
and the observed \(K\) 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 (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 have a common distribution described in Jalilian et al (2013).

The shape of the kernel is determined by the dimensionless
index `nu`

. This is the parameter
\(\nu^\prime = \alpha/2-1\) appearing in
equation (12) on page 126 of Jalilian et al (2013).
In previous versions of spatstat instead of specifying `nu`

(called `nu.ker`

at that time) the user could specify
`nu.pcf`

which is the parameter \(\nu=\alpha-1\)
appearing in equation (13), page 127 of Jalilian et al (2013).
These are related by `nu.pcf = 2 * nu.ker + 1`

and `nu.ker = (nu.pcf - 1)/2`

. This syntax is still supported but
not recommended for consistency across the package. In that case
exactly one of `nu.ker`

or `nu.pcf`

must be specified.

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.

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`

,
`vargamma.estpcf`

,
`lgcp.estK`

,
`thomas.estK`

,
`cauchy.estK`

,
`mincontrast`

,
`Kest`

,
`Kmodel`

.

`rVarGamma`

to simulate the model.

# NOT RUN { if(interactive()) { u <- vargamma.estK(redwood) print(u) plot(u) } # }