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

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.
https://www.jstatsoft.org/v55/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)) # }