# rThomas

##### Simulate Thomas Process

Generate a random point pattern, a realisation of the Thomas cluster process.

##### Usage

`rThomas(kappa, sigma, mu, win = owin(c(0,1),c(0,1)))`

##### Arguments

- kappa
- Intensity of the Poisson process of cluster centres. A single positive number.
- sigma
- Standard deviation of displacement of a point from its cluster centre.
- mu
- Expected number of points per cluster.
- win
- Window in which to simulate the pattern.
An object of class
`"owin"`

or something acceptable to`as.owin`

.

##### Details

This algorithm generates a realisation of the
Thomas process, a special case of the Neyman-Scott process.
The algorithm
generates a uniform Poisson point process of ``parent'' points
with intensity `kappa`

. Then each parent point is
replaced by a random cluster of points, the number of points
per cluster being Poisson (`mu`

) distributed, and their
positions being isotropic Gaussian displacements from the
cluster parent location.

This classical model can be fitted to data by the method of minimum contrast,
using `thomas.estK`

or `kppm`

.
The algorithm can also generate spatially inhomogeneous versions of
the Thomas process:

- The parent points can be spatially inhomogeneous.
If the argument
`kappa`

is a`function(x,y)`

or a pixel image (object of class`"im"`

), then it is taken as specifying the intensity function of an inhomogeneous Poisson process that generates the parent points. - The offspring points can be inhomogeneous. If the
argument
`mu`

is a`function(x,y)`

or a pixel image (object of class`"im"`

), then it is interpreted as the reference density for offspring points, in the sense of Waagepetersen (2006). For a given parent point, the offspring constitute a Poisson process with intensity function equal to`mu(x,y) * f(x,y)`

where`f`

is the Gaussian density centred at the parent point.

`kappa`

is a single number)
and the offspring are inhomogeneous (`mu`

is a
function or pixel image), the model can be fitted to data
using `kppm`

, or
using `thomas.estK`

applied to the inhomogeneous
$K$ function.
##### Value

- The simulated point pattern (an object of class
`"ppp"`

).Additionally, some intermediate results of the simulation are returned as attributes of this point pattern. See

`rNeymanScott`

.

##### References

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

##### See Also

`rpoispp`

,
`rMatClust`

,
`rGaussPoisson`

,
`rNeymanScott`

,
`thomas.estK`

,
`kppm`

##### Examples

```
#homogeneous
X <- rThomas(10, 0.2, 5)
#inhomogeneous
Z <- as.im(function(x,y){ 5 * exp(2 * x - 1) }, owin())
Y <- rThomas(10, 0.2, Z)
```

*Documentation reproduced from package spatstat, version 1.20-2, License: GPL (>= 2)*