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

`Geyer(r,sat)`

r

Interaction radius. A positive real number.

sat

Saturation threshold. A non-negative real number.

An object of class `"interact"`

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

.

The value `sat=0`

is permitted by `Geyer`

,
but this is not very useful.
For technical reasons, when `ppm`

fits a
Geyer model with `sat=0`

, the default behaviour is to return
an “invalid” fitted model in which the estimate of
\(\gamma\) is `NA`

. In order to get a Poisson
process model returned when `sat=0`

,
you would need to set `emend=TRUE`

in
the call to `ppm`

.

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 \(s\).
The interaction structure of 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()`

.

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.

`ppm`

,
`pairwise.family`

,
`ppm.object`

,
`Strauss`

.

To make an interaction object like `Geyer`

but having
multiple interaction radii, see `BadGey`

or `Hybrid`

.

# NOT RUN { ppm(cells, ~1, Geyer(r=0.07, sat=2)) # fit the stationary saturation process to `cells' # }