BadGey {spatstat}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 beta and interaction parameters gamma[1], ..., gamma[k], is the point process in which each point x[i] in the pattern X contributes a factor

beta gamma[1]^v(1, x_i, X) ... gamma[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 gamma obtained from BadGey have a complicated relationship to the interaction parameters gamma 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.

Author(s)

Adrian Baddeley Adrian.Baddeley@csiro.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner r.turner@auckland.ac.nz in collaboration with Hao Wang and Jeff Picka

See Also

ppm, pairsat.family, Geyer, PairPiece, SatPiece

Examples

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

   # 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
   ## Not run: 
   ppm(cells, ~polynom(x,y,3), BadGey(c(0.07, 0.1, 0.13), 2))
   
## End(Not run)

[Package spatstat version 1.25-3 Index]