spatstat (version 1.12-6)

AreaInter: The Area Interaction Point Process Model

Description

Creates an instance of the Area Interaction point process model (marginal Widom-Rowlinson model) which can then be fitted to point pattern data.

Usage

AreaInter(r)

Arguments

r
The radius of the discs in the area interaction process

Value

  • An object of class "interact" describing the interpoint interaction structure of the area-interaction process with disc radius $r$.

Warnings

The interaction distance of this process is equal to 2 * r. Two discs of radius r overlap if their centres are closer than 2 * r units apart.

Details

The (stationary) area interaction process (Baddeley and Van Lieshout, 1995) with disc radius $r$, intensity parameter $\beta$ and interaction parameter $\gamma$ is a point process with probability density $$f(x_1,\ldots,x_n) = \alpha \beta^{n(x)} \gamma^{-A(x)}$$ where $x_1,\ldots,x_n$ represent the points of the pattern, $n(x)$ is the number of points in the pattern, and $A(x)$ is the area of the region formed by the union of discs of radius $r$ centred at the points $x_1,\ldots,x_n$. Here $\alpha$ is a normalising constant.

The interaction parameter $\gamma$ can be any positive number. If $\gamma = 1$ then the model reduces to a Poisson process. If $\gamma < 1$ then the process is regular, while if $\gamma > 1$ the process is clustered. The value $\gamma = 0$ corresponds to a hard core process with hard core radius $r$ (interaction distance $2r$).

In spatstat this model is reparametrised so that the interaction parameter is not affected by scale changes. In the new parametrisation, the density is $$f(x_1,\ldots,x_n) = \alpha \beta^{n(x)} \eta^{-B(x)}$$ where $\eta$ is the new interaction parameter, and $$B(x) = \frac{A(x)}{\pi r^2}$$ is the normalised area (so that the discs have unit area). The value $\eta = 1$ again corresponds to a Poisson process. The nonstationary area interaction process is similar except that the contribution of each individual point $x_i$ is a function $\beta(x_i)$ of location, rather than a constant beta. The function ppm(), which fits point process models to point pattern data, requires an argument of class "interact" describing the interpoint interaction structure of the model to be fitted. The appropriate description of the area interaction structure is yielded by the function AreaInter(). See the examples below. Note the only argument is the disc radius r. When r is fixed, the model becomes an exponential family. The canonical parameters $\log(\beta)$ and $\log(\eta)$ are estimated by ppm(), not fixed in AreaInter().

References

Baddeley, A.J. and Van Lieshout, M.N.M. (1995). Area-interaction point processes. Annals of the Institute of Statistical Mathematics 47 (1995) 601--619.

See Also

ppm, pairwise.family, ppm.object

Examples

Run this code
AreaInter(r=0.1)
   # prints a sensible description of itself
   data(swedishpines)
   # Fit the stationary area interaction process to Swedish Pines data   
   ppm(swedishpines, ~1, AreaInter(r=2), rbord=4)
   data(cells) 
   # Fit the stationary area interaction process to `cells'
   ppm(cells, ~1, AreaInter(r=0.03), rbord=0.06)
   # Zero value of eta indicates hard core process.

   ppm(swedishpines, ~polynom(x/10,y/10,3), AreaInter(r=2), rbord=4)
   # fit a nonstationary area interaction with log-cubic polynomial trend

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