spatstat (version 1.56-1)

StraussHard: The Strauss / Hard Core Point Process Model

Description

Creates an instance of the ``Strauss/ hard core'' point process model which can then be fitted to point pattern data.

Usage

StraussHard(r, hc=NA)

Arguments

r

The interaction radius of the Strauss interaction

hc

The hard core distance. Optional.

Value

An object of class "interact" describing the interpoint interaction structure of the ``Strauss/hard core'' process with Strauss interaction radius \(r\) and hard core distance hc.

Details

A Strauss/hard core process with interaction radius \(r\), hard core distance \(h < r\), and parameters \(\beta\) and \(\gamma\), is a pairwise interaction point process in which

• distinct points are not allowed to come closer than a distance \(h\) apart

• each pair of points closer than \(r\) units apart contributes a factor \(\gamma\) to the probability density.

This is a hybrid of the Strauss process and the hard core process.

The probability density is zero if any pair of points is closer than \(h\) units apart, and otherwise equals $$ f(x_1,\ldots,x_n) = \alpha \beta^{n(x)} \gamma^{s(x)} $$ 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 \(\gamma\) may take any positive value (unlike the case for the Strauss process). If \(\gamma < 1\), the model describes an ``ordered'' or ``inhibitive'' pattern. If \(\gamma > 1\), the model is ``ordered'' or ``inhibitive'' up to the distance \(h\), but has an ``attraction'' between points lying at distances in the range between \(h\) and \(r\).

If \(\gamma = 1\), the process reduces to a classical hard core process with hard core distance \(h\). If \(\gamma = 0\), the process reduces to a classical hard core process with hard core distance \(r\).

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/hard core process pairwise interaction is yielded by the function StraussHard(). See the examples below.

The canonical parameter \(\log(\gamma)\) is estimated by ppm(), not fixed in StraussHard().

If the hard core distance argument hc is missing or NA, it will be estimated from the data when ppm is called. The estimated value of hc is the minimum nearest neighbour distance multiplied by \(n/(n+1)\), where \(n\) is the number of data points.

References

Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283--322.

Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.

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

See Also

ppm, pairwise.family, ppm.object

Examples

# NOT RUN {
   StraussHard(r=1,hc=0.02)
   # prints a sensible description of itself

   data(cells)

   
# }
# NOT RUN {
   ppm(cells, ~1, StraussHard(r=0.1, hc=0.05))
   # fit the stationary Strauss/hard core  process to `cells'
   
# }
# NOT RUN {
   ppm(cells, ~ polynom(x,y,3), StraussHard(r=0.1, hc=0.05))
   # fit a nonstationary Strauss/hard core process
   # with log-cubic polynomial trend
# }