# Penttinen

0th

Percentile

##### Penttinen Interaction

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

Keywords
models, spatial
##### Usage
Penttinen(r)
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

ppm, ppm.object, Pairwise, AreaInter.

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

Documentation reproduced from package spatstat, version 1.64-1, License: GPL (>= 2)

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