Softcore
The Soft Core Point Process Model
Creates an instance of the Soft Core point process model which can then be fitted to point pattern data.
Usage
Softcore(kappa)
Arguments
- kappa
- The exponent $\kappa$ of the Soft Core interaction
Details
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.
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 interpoint interaction decreasing smoothly with distance.
The strength of interaction is controlled by the
parameter $\sigma$, a positive real number,
with larger values corresponding
to stronger interaction; and by the exponent $\kappa$
in the range $(0,1)$, with larger values corresponding to
weaker interaction.
If $\sigma = 0$
the model reduces to the Poisson point process.
If $\sigma > 0$,
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 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.
Note the only 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()
.
Value
- An object of class
"interact"
describing the interpoint interaction structure of the Soft Core process with exponent $\kappa$.
References
Ogata, Y, and Tanemura, M. (1981). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Annals of the Institute of Statistical Mathematics, B 33, 315--338.
Ogata, Y, and Tanemura, M. (1984). Likelihood analysis of spatial point patterns. Journal of the Royal Statistical Society, series B 46, 496--518.
See Also
Examples
data(cells)
ppm(cells, ~1, Softcore(kappa=0.5), correction="isotropic")
# fit the stationary Soft Core process to `cells'