# Geyer

##### Geyer's Saturation Point Process Model

Creates an instance of Geyer's ``saturation point process'' model which can then be fitted to point pattern data.

- Keywords
- spatial

##### Usage

`Geyer(r,sat)`

##### Arguments

- r
- Interaction radius. A positive real number.
- sat
- Saturation threshold. A positive real number.

##### Details

Geyer (1999) introduced the ``saturation process'', a modification of the
Strauss process (see `Strauss`

)
in which the total contribution
to the potential from each point (from its pairwise interaction with all
other points) is trimmed to a maximum value $s$.
This model is implemented in the function `Geyer()`

.

The saturation point process with interaction radius $r$, saturation threshold $s$, and parameters $\beta$ and $\gamma$, is the point process in which each point $x_i$ in the pattern $X$ contributes a factor $$\beta \gamma^{\min(s, t(x_i, X))}$$ to the probability density of the point pattern, where $t(x_i, X)$ denotes the number of ``close neighbours'' of $x_i$ in the pattern $X$. A close neighbour of $x_i$ is a point $x_j$ with $j \neq i$ such that the distance between $x_i$ and $x_j$ is less than or equal to $r$.

If the saturation threshold $s$ is set to infinity,
this model reduces to the Strauss process (see `Strauss`

)
with interaction parameter $\gamma^2$.
If $s = 0$, the model reduces to the Poisson point process.
If $s$ is a finite positive number, then the interaction parameter
$\gamma$ may take any positive value (unlike the case
of the Strauss process), with
values $\gamma < 1$
describing an ``ordered'' or ``inhibitive'' pattern,
and
values $\gamma > 1$
describing a ``clustered or ``attractive'' pattern.
The nonstationary saturation process is similar except that
the value $\beta$
is replaced by a function $\beta(x_i)$
of location.
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 saturation process interaction is
yielded by `Geyer(r, sat)`

where the
arguments `r`

and `sat`

specify
the Strauss interaction radius $r$ and the saturation threshold
$s$, respectively. See the examples below.
Note the only arguments are the interaction radius `r`

and the saturation threshold `sat`

.
When `r`

and `sat`

are fixed,
the model becomes an exponential family.
The canonical parameters $\log(\beta)$
and $\log(\gamma)$
are estimated by `ppm()`

, not fixed in
`Geyer()`

.

##### Value

- An object of class
`"interact"`

describing the interpoint interaction structure of Geyer's ``saturation point process'' with interaction radius $r$ and saturation threshold`sat`

.

##### References

Geyer, C.J. (1999)
Likelihood Inference for Spatial Point Processes.
Chapter 3 in
O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. Van Lieshout (eds)
*Stochastic Geometry: Likelihood and Computation*,
Chapman and Hall / CRC,
Monographs on Statistics and Applied Probability, number 80.
Pages 79--140.

##### See Also

##### Examples

```
data(cells)
ppm(cells, ~1, Geyer(r=0.07, sat=2), rbord=0.07)
# fit the stationary saturation process to `cells'
```

*Documentation reproduced from package spatstat, version 1.6-9, License: GPL version 2 or newer*