# 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,scale=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

- 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 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
\(\sigma\) which is equal to the parameter `scale`

. The
named vector of stating values can use either `sigma2`

(\(\sigma^2\)) or `scale`

as the name of the second
component, but the latter is recommended for consistency with other
cluster models.

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:

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

Diggle, P. J., Besag, J. and Gleaves, J. T. (1976)
Statistical analysis of spatial point patterns by
means of distance methods. *Biometrics* **32** 659--667.

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

Thomas, M. (1949) A generalisation of Poisson's binomial limit for use
in ecology. *Biometrika* **36**, 18--25.

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

```
# NOT RUN {
data(redwood)
u <- thomas.estpcf(redwood, c(kappa=10, scale=0.1))
u
plot(u, legendpos="topright")
u2 <- thomas.estpcf(redwood, c(kappa=10, scale=0.1),
pcfargs=list(stoyan=0.12))
# }
```

*Documentation reproduced from package spatstat, version 1.56-1, License: GPL (>= 2)*