Strauss: The Strauss Point Process Model

Description

Creates an instance of the Strauss point process model which can then be fitted to point pattern data.

Usage

Strauss(r)

Arguments

The interaction radius of the Strauss process

Value

An object of class "interact" describing the interpoint interaction structure of the Strauss process with interaction radius $r$.

Details

The (stationary) Strauss process with interaction radius $r$ and parameters $beta$ and $gamma$ is the pairwise interaction point process in which each point contributes a factor $beta$ to the probability density of the point pattern, and each pair of points closer than $r$ units apart contributes a factor $gamma$ to the density.

Thus the probability density is $$ f(x_1,\ldots,x_n) = \alpha \beta^{n(x)} \gamma^{s(x)} $$ where $x[1],\ldots,x[n]$ represent the points of the pattern, $n(x)$ is the number of points in the pattern, $s(x)$ is the number of distinct unordered pairs of points that are closer than $r$ units apart, and $alpha$ is the normalising constant.

The interaction parameter $gamma$ must be less than or equal to $1$ so that this model describes an ``ordered'' or ``inhibitive'' pattern. The nonstationary Strauss 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 Strauss process pairwise interaction is yielded by the function Strauss(). See the examples below. Note the only argument is the interaction radius r. When r is fixed, the model becomes an exponential family. The canonical parameters $log(beta)$ and $log(gamma)$ are estimated by ppm(), not fixed in Strauss().

References

Kelly, F.P. and Ripley, B.D. (1976) On Strauss's model for clustering. Biometrika 63, 357--360.

Strauss, D.J. (1975) A model for clustering. Biometrika 62, 467--475.

Examples

Run this code

   Strauss(r=0.1)
   # prints a sensible description of itself
   data(cells)

   ## Not run: 
#    ppm(cells, ~1, Strauss(r=0.07))
#    # fit the stationary Strauss process to `cells'
#    ## End(Not run)


   ppm(cells, ~polynom(x,y,3), Strauss(r=0.07))
   # fit a nonstationary Strauss process with log-cubic polynomial trend

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