# thomas.estpcf

##### Fit the Thomas Point Process by Minimum Contrast

Fits the Thomas point process to a point pattern dataset by the Method of Minimum Contrast using the pair correlation function.

##### Usage

```
thomas.estpcf(X, startpar=c(kappa=1,sigma2=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
```

##### Arguments

- X
- Data to which the Thomas model will be fitted. Either a point pattern or a summary statistic. See Details.
- startpar
- Vector of starting values for the parameters of the Thomas 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. - pcfargs
- Optional list containing arguments passed to
`pcf.ppp`

to control the smoothing in the estimation of the pair correlation function.

##### Details

This algorithm fits the Thomas point process model to a point pattern dataset
by the Method of Minimum Contrast, using the pair correlation function
`pcf`

.

The argument `X`

can be either
[object Object],[object Object]

The algorithm fits the Thomas point process to `X`

,
by finding the parameters of the Thomas model
which give the closest match between the
theoretical pair correlation function of the Thomas process
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see `mincontrast`

.
The Thomas point process is described in Moller and Waagepetersen
(2003, pp. 61--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 isotropically Normally distributed around the parent
point with standard deviation $\sigma$.

The theoretical pair correlation function of the Thomas process is $$g(r) = 1 + \frac 1 {4\pi \kappa \sigma^2} \exp(-\frac{r^2}{4\sigma^2})).$$ The theoretical intensity of the Thomas 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 $\sigma^2$. 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 Thomas process can be simulated, using `rThomas`

.

Homogeneous or inhomogeneous Thomas process 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. (2007)
An estimating function approach to inference for
inhomogeneous Neyman-Scott processes.
*Biometrics* **63**, 252--258.

##### See Also

`thomas.estK`

`mincontrast`

,
`pcf`

,
`rThomas`

to simulate the fitted model.

##### Examples

```
data(redwood)
u <- thomas.estpcf(redwood, c(kappa=10, sigma2=0.1))
u
plot(u, legendpos="topright")
u2 <- thomas.estpcf(redwood, c(kappa=10, sigma2=0.1),
pcfargs=list(stoyan=0.12))
```

*Documentation reproduced from package spatstat, version 1.25-5, License: GPL (>= 2)*