R: Geyer's Saturation Point Process Model

Geyer {spatstat.core} | R Documentation

## Geyer's Saturation Point Process Model

### Description

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

### Usage

      Geyer(r,sat)

### Arguments

`r` | Interaction radius. A positive real number.
`sat` | Saturation threshold. A non-negative 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. 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 != 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`.

### Zero saturation

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`.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz

### 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

`ppm`, `pairwise.family`, `ppm.object`, `Strauss`.

To make an interaction object like `Geyer` but having multiple interaction radii, see `BadGey` or `Hybrid`.

### Examples

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

[Package spatstat.core version 2.0-0 Index]