# Penttinen

##### Penttinen Interaction

Creates an instance of the Penttinen pairwise interaction point process model, which can then be fitted to point pattern data.

##### Usage

`Penttinen(r)`

##### Arguments

- r
circle radius

##### Details

Penttinen (1984, Example 2.1, page 18), citing Cormack (1979), described the pairwise interaction point process with interaction factor $$ h(d) = e^{\theta A(d)} = \gamma^{A(d)} $$ between each pair of points separated by a distance $d$. Here \(A(d)\) is the area of intersection between two discs of radius \(r\) separated by a distance \(d\), normalised so that \(A(0) = 1\).

The scale of interaction is controlled by the disc radius \(r\): two points interact if they are closer than \(2 r\) apart. The strength of interaction is controlled by the canonical parameter \(\theta\), which must be less than or equal to zero, or equivalently by the parameter \(\gamma = e^\theta\), which must lie between 0 and 1.

The potential is inhibitory, i.e.\ this model is only appropriate for regular point patterns. For \(\gamma=0\) the model is a hard core process with hard core diameter \(2 r\). For \(\gamma=1\) the model is a Poisson process.

The irregular parameter
\(r\) must be given in the call to
`Penttinen`

, while the
regular parameter \(\theta\) will be estimated.

This model can be considered as a pairwise approximation
to the area-interaction model `AreaInter`

.

##### Value

An object of class `"interact"`

describing the interpoint interaction
structure of a point process.

##### References

Cormack, R.M. (1979)
Spatial aspects of competition between individuals.
Pages 151--212 in *Spatial and Temporal Analysis in Ecology*,
eds. R.M. Cormack and J.K. Ord, International Co-operative
Publishing House, Fairland, MD, USA.

Penttinen, A. (1984)
*Modelling Interaction in Spatial Point Patterns:
Parameter Estimation by the Maximum Likelihood Method.*
Jyvaskyla
Studies in Computer Science, Economics and Statistics **7**,
University of Jyvaskyla, Finland.

##### See Also

##### Examples

```
# NOT RUN {
fit <- ppm(cells ~ 1, Penttinen(0.07))
fit
reach(fit) # interaction range is circle DIAMETER
# }
```

*Documentation reproduced from package spatstat, version 1.57-1, License: GPL (>= 2)*