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())
```

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.

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.

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

- a point pattern:
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.- 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 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")
# }
```

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