# StraussHard

##### 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.

##### Usage

`StraussHard(r, hc)`

##### Arguments

- r
- The interaction radius of the Strauss interaction
- hc
- The hard core distance

##### 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.

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 process reduces to a classical
hard core 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 $r$.
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()`

.

##### 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`

.

##### 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* **63**, 467--475.

##### See Also

##### Examples

```
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
```

*Documentation reproduced from package spatstat, version 1.23-3, License: GPL (>= 2)*