# 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,eta2=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
[object Object],[object Object]

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 (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
follow 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 `rCauchy`

.

For computational reasons, the optimisation procedure uses the parameter
`eta2`

, which is equivalent to `4 * omega^2`

where `omega`

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: 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

Ghorbani, M. (2012) Cauchy cluster process.
*Metrika*, to appear.

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`

,
`cauchy.estpcf`

,
`lgcp.estK`

,
`thomas.estK`

,
`vargamma.estK`

,
`mincontrast`

,
`Kest`

,
`Kmodel`

.

`rCauchy`

to simulate the model.

##### Examples

```
u <- cauchy.estK(redwood)
u
plot(u)
```

*Documentation reproduced from package spatstat, version 1.31-2, License: GPL (>= 2)*