R: Hybrid Geyer Point Process Model

BadGey {spatstat.core} | R Documentation

## Hybrid Geyer Point Process Model

### Description

Creates an instance of the Baddeley-Geyer point process model, defined as a hybrid of several Geyer interactions. The model can then be fitted to point pattern data.

### Usage

      BadGey(r, sat)

### Arguments

`r` | vector of interaction radii
`sat` | vector of saturation parameters, or a single common value of saturation parameter

### Details

This is Baddeley's generalisation of the Geyer saturation point process model, described in `Geyer`, to a process with multiple interaction distances.

The BadGey point process with interaction radii r[1], …, r[k], saturation thresholds s[1],…,s[k], intensity parameter β and interaction parameters γ[1], …, γ[k], is the point process in which each point x[i] in the pattern X contributes a factor

β γ[1]^v(1, x_i, X) … γ[k]^v(k, x_i, X)

to the probability density of the point pattern, where

v(j, x[i], X) = min(s[j], t(j, x[i], X))

where t(j,x[i],X) denotes the number of points in the pattern X which lie within a distance r[j] from the point x[i].

`BadGey` is used to fit this model to data. 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 piecewise constant Saturated pairwise interaction is yielded by the function `BadGey()`. See the examples below.

The argument `r` specifies the vector of interaction distances. The entries of `r` must be strictly increasing, positive numbers.

The argument `sat` specifies the vector of saturation parameters that are applied to the point counts t(j,x[i],X). It should be a vector of the same length as `r`, and its entries should be nonnegative numbers. Thus `sat[1]` is applied to the count of points within a distance `r[1]`, and `sat[2]` to the count of points within a distance `r[2]`, etc. Alternatively `sat` may be a single number, and this saturation value will be applied to every count.

Infinite values of the saturation parameters are also permitted; in this case v(j, x[i], X) = t(j, x[i], X) and there is effectively no ‘saturation’ for the distance range in question. If all the saturation parameters are set to `Inf` then the model is effectively a pairwise interaction process, equivalent to `PairPiece` (however the interaction parameters γ obtained from `BadGey` have a complicated relationship to the interaction parameters γ obtained from `PairPiece`).

If `r` is a single number, this model is virtually equivalent to the Geyer process, see `Geyer`.

### Value

An object of class `"interact"` describing the interpoint interaction structure of a point process.

### Hybrids

A ‘hybrid’ interaction is one which is built by combining several different interactions (Baddeley et al, 2013). The `BadGey` interaction can be described as a hybrid of several `Geyer` interactions.

The `Hybrid` command can be used to build hybrids of any interactions. If the `Hybrid` operator is applied to several `Geyer` models, the result is equivalent to a `BadGey` model. This can be useful for incremental model selection.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz in collaboration with Hao Wang and Jeff Picka

### References

Baddeley, A., Turner, R., Mateu, J. and Bevan, A. (2013) Hybrids of Gibbs point process models and their implementation. Journal of Statistical Software 55:11, 1–43. doi: 10.18637/jss.v055.i11

### See Also

`ppm`, `pairsat.family`, `Geyer`, `PairPiece`, `SatPiece`, `Hybrid`

### Examples

       BadGey(c(0.1,0.2), c(1,1))
       # prints a sensible description of itself
       BadGey(c(0.1,0.2), 1)

       # fit a stationary Baddeley-Geyer model
       ppm(cells ~1, BadGey(c(0.07, 0.1, 0.13), 2))

       # nonstationary process with log-cubic polynomial trend
       # ppm(cells ~polynom(x,y,3), BadGey(c(0.07, 0.1, 0.13), 2))

[Package spatstat.core version 2.3-1 Index]