Hybrid Geyer Point Process Model
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)
r |
vector of interaction radii |
sat |
vector of saturation parameters, or a single common value of saturation parameter |
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.
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.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz in collaboration with Hao Wang and Jeff Picka
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. https://www.jstatsoft.org/v55/i11/
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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