The Strauss / Hard Core Point Process Model
Creates an instance of the “Strauss/ hard core” point process model which can then be fitted to point pattern data.
StraussHard(r, hc=NA)
r |
The interaction radius of the Strauss interaction |
hc |
The hard core distance. Optional. |
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,…,x_n) = alpha . beta^n(x) gamma^s(x)
where x[1],…,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.
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.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
and Rolf Turner r.turner@auckland.ac.nz
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.
StraussHard(r=1,hc=0.02) # prints a sensible description of itself data(cells) # ppm(cells, ~1, StraussHard(r=0.1, hc=0.05)) # fit the stationary Strauss/hard core process to `cells' 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
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