# cauchy.estK

##### Fit the Neyman-Scott cluster process with Cauchy kernel

Fits the Neyman-Scott Cluster point process with Cauchy kernel to a point pattern dataset by the Method of Minimum Contrast.

##### Usage

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

##### 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.

- 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.

##### Details

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 \(K\) 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 Cauchy kernel to `X`

,
by finding the parameters of the Matern Cluster model
which give the closest match between the
theoretical \(K\) function of the Matern Cluster process
and the observed \(K\) 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 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.

##### Value

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.

##### References

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.

##### See Also

`kppm`

,
`cauchy.estpcf`

,
`lgcp.estK`

,
`thomas.estK`

,
`vargamma.estK`

,
`mincontrast`

,
`Kest`

,
`Kmodel`

.

`rCauchy`

to simulate the model.

##### Examples

```
# NOT RUN {
u <- cauchy.estK(redwood)
u
plot(u)
# }
```

*Documentation reproduced from package spatstat, version 1.57-1, License: GPL (>= 2)*