spatstat (version 1.48-0)

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)

Arguments

r

circle radius

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 $gamma = exp(theta)$, 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.

See Also

ppm, ppm.object, Pairwise, AreaInter.

Examples

   fit <- ppm(cells ~ 1, Penttinen(0.07))
   fit
   reach(fit) # interaction range is circle DIAMETER