BadGey: Hybrid Geyer Point Process Model

In spatstat.model: Parametric Statistical Modelling and Inference for the 'spatstat' Family

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,\ldots,r_k, saturation thresholds s_1,\ldots,s_k, intensity parameter \beta and interaction parameters \gamma_1,\ldots,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)} \ldots 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.

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

and \rolf 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))
   

spatstat.model documentation built on Sept. 30, 2024, 9:26 a.m.