# matclust.estK

##### Fit the Matern Cluster Point Process by Minimum Contrast

Fits the Matern Cluster point process to a point pattern dataset by the Method of Minimum Contrast.

##### Usage

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

##### Arguments

- X
- Data to which the Matern Cluster model will be fitted. Either a point pattern or a summary statistic. See Details.
- startpar
- Vector of starting values for the parameters of the Matern Cluster process.
- 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 Matern Cluster point process model 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 Matern Cluster point process 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 Matern Cluster point process is described in Moller and Waagepetersen
(2003, p. 62). 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
are independent and uniformly distributed inside a circle of radius
$R$ centred on the parent point.

The theoretical $K$-function of the Matern Cluster process is $$K(r) = \pi r^2 + \frac 1 \kappa h(\frac{r}{2R})$$ where $$h(z) = 2 + \frac 1 \pi [ ( 8 z^2 - 4 ) \mbox{arccos}(z) - 2 \mbox{arcsin}(z) + 4 z \sqrt{(1 - z^2)^3} - 6 z \sqrt{1 - z^2} ]$$ for $z <= 1$,="" and="" $h(z)="1$" for="" $z=""> 1$. The theoretical intensity of the Matern Cluster process is $\lambda = \kappa \mu$.

In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters $\kappa$ and $R$. Then the remaining parameter $\mu$ is inferred from the estimated intensity $\lambda$.

If the argument `lambda`

is provided, then this is used
as the value of $\lambda$. 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 Matern Cluster process can be simulated, using
`rMatClust`

.

Homogeneous or inhomogeneous Matern Cluster models can also be
fitted using the function `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

Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.

Waagepetersen, R. (2006). An estimation function approach to inference for inhomogeneous Neyman-Scott processes. Submitted.

##### See Also

`kppm`

,
`lgcp.estK`

,
`thomas.estK`

,
`mincontrast`

,
`Kest`

,
`rMatClust`

to simulate the fitted model.

##### Examples

```
data(redwood)
u <- matclust.estK(redwood, c(kappa=10, R=0.1))
u
plot(u)
```

*Documentation reproduced from package spatstat, version 1.18-1, License: GPL (>= 2)*