# 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=NA)`

##### Arguments

- r
The interaction radius of the Strauss interaction

- hc
The hard core distance. Optional.

##### 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.

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 $$ f(x_1,\ldots,x_n) = \alpha \beta^{n(x)} \gamma^{s(x)} $$ 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 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.

##### 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* **62**, 467--475.

##### See Also

##### Examples

```
# NOT RUN {
StraussHard(r=1,hc=0.02)
# prints a sensible description of itself
data(cells)
# }
# NOT RUN {
ppm(cells, ~1, StraussHard(r=0.1, hc=0.05))
# fit the stationary Strauss/hard core process to `cells'
# }
# NOT RUN {
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 1.63-3, License: GPL (>= 2)*