The (stationary)
Soft Core point process with parameters \(\beta\) and
\(\sigma\) and exponent \(\kappa\)
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
contributes a factor
$$
\exp \left\{ - \left( \frac{\sigma}{d} \right)^{2/\kappa} \right\}
$$
to the density, where \(d\) is the distance between the two points.
See the Examples for a plot of this interaction curve.
Thus the process has probability density
$$
f(x_1,\ldots,x_n) =
\alpha \beta^{n(x)}
\exp \left\{ - \sum_{i < j} \left(
\frac{\sigma}{||x_i-x_j||}
\right)^{2/\kappa} \right\}
$$
where \(x_1,\ldots,x_n\) represent the
points of the pattern, \(n(x)\) is the number of points in the
pattern, \(\alpha\) is the normalising constant,
and the sum on the right hand side is
over all unordered pairs of points of the pattern.
This model describes an ``ordered'' or ``inhibitive'' process,
with the strength of inhibition decreasing smoothly with distance.
The interaction is controlled by the parameters \(\sigma\)
and \(\kappa\).
The spatial scale of interaction is controlled by the
parameter \(\sigma\), which is a positive real number
interpreted as a distance, expressed in the same units of distance as
the spatial data. The parameter \(\sigma\) is the distance at which the
pair potential reaches the threshold value 0.37.
The shape of the interaction function
is controlled by the exponent
\(\kappa\) which is a dimensionless number
in the range \((0,1)\), with larger values corresponding to
a flatter shape (or a more gradual decay rate).
The process is well-defined only for \(\kappa\) in
\((0,1)\).
The limit of the model as \(\kappa \to 0\) is the
hard core process with hard core distance \(h=\sigma\).
The “strength” of the interaction is determined by both of the
parameters \(\sigma\) and \(\kappa\).
The larger the value of \(\kappa\), the wider the range of
distances over which the interaction has an effect.
If \(\sigma\) is very small, the interaction is very weak
for all practical purposes (theoretically if \(\sigma = 0\)
the model reduces to the Poisson point process).
The nonstationary Soft Core 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 Soft Core process pairwise interaction is
yielded by the function Softcore(). See the examples below.
The main argument is the exponent kappa.
When kappa is fixed, the model becomes an exponential family
with canonical parameters \(\log \beta\)
and $$
\log \gamma = \frac{2}{\kappa} \log\sigma
$$
The canonical parameters are estimated by ppm(), not fixed in
Softcore().
The optional argument sigma0 can be used to improve
numerical stability. If sigma0 is given, it should be a positive
number, and it should be a rough estimate of the
parameter \(\sigma\).