spatstat.core (version 2.1-2)

# Penttinen: Penttinen Interaction

## Description

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

## Usage

Penttinen(r)

r

## Value

An object of class "interact" describing the interpoint interaction structure of a point process.

## 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.

## 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.

ppm, ppm.object, Pairwise, AreaInter.
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