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.
BadGey(r, sat)vector of interaction radii
vector of saturation parameters, or a single common value of saturation parameter
An object of class "interact"
describing the interpoint interaction
structure of a point process.
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.
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.
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. 10.18637/jss.v055.i11
# 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
# ppm(cells ~polynom(x,y,3), BadGey(c(0.07, 0.1, 0.13), 2))
# }
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