BadGey: 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

vector of interaction radii

sat

vector of saturation parameters, or a single common value of saturation parameter

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.

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.

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. http://www.jstatsoft.org/v55/i11/

Examples

Run this code

# NOT RUN {
   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
   
# }
# NOT RUN {
   ppm(cells ~polynom(x,y,3), BadGey(c(0.07, 0.1, 0.13), 2))
   
# }

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