# 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, sigma0=NA)`

##### Arguments

- kappa
- The exponent $\kappa$ of the Soft Core interaction
- sigma0
- Optional. Initial estimate of the parameter $\sigma$. A positive number.

##### 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.
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 $\sigma$.

##### 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'
```

*Documentation reproduced from package spatstat, version 1.31-3, License: GPL (>= 2)*