# 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, a function, or a pixel image.
- sigma
- Standard deviation of random displacement (along each coordinate axis) of a point from its cluster centre.
- mu
- Mean number of points per cluster (a single positive number) or reference intensity for the cluster points (a function or a pixel image).
- 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 (`modified')
Thomas process, a special case of the Neyman-Scott process,
inside the window `win`

.

In the simplest case, where `kappa`

and `mu`

are single numbers, the algorithm
generates a uniform Poisson point process of `kappa`

. Then each parent point is
replaced by a random cluster of `mu`

)
distributed, and their
positions being isotropic Gaussian displacements from the
cluster parent location. The resulting point pattern
is a realisation of the classical
`win`

.
This point process has intensity `kappa * mu`

.

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 (2007). For a given parent point, the offspring constitute a Poisson process with intensity function equal to`mu * f`

, where`f`

is the Gaussian probability density centred at the parent point. Equivalently we first generate, for each parent point, a Poisson (`mumax`

) random number of offspring (where$M$is the maximum value of`mu`

) with independent Gaussian displacements from the parent location, and then randomly thin the offspring points, with retention probability`mu/M`

. - Both the parent points and the offspring points can be spatially inhomogeneous, as described above.

Note that if `kappa`

is a pixel image, its domain must be larger
than the window `win`

. This is because an offspring point inside
`win`

could have its parent point lying outside `win`

.
In order to allow this, the simulation algorithm
first expands the original window `win`

by a distance `4 * sigma`

and generates the Poisson process of
parent points on this larger window. If `kappa`

is a pixel image,
its domain must contain this larger window.

The intensity of the Thomas process is `kappa * mu`

if either `kappa`

or `mu`

is a single number. In the general
case the intensity is an integral involving `kappa`

, `mu`

and `f`

.

The Thomas process with homogeneous parents
(i.e. where `kappa`

is a single number)
can be fitted to data using `kppm`

or related functions.
Currently it is not possible to fit the Thomas model
with inhomogeneous parents.

##### 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

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.

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

`rpoispp`

,
`rMatClust`

,
`rGaussPoisson`

,
`rNeymanScott`

,
`thomas.estK`

,
`thomas.estpcf`

,
`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 1.34-1, License: GPL (>= 2)*