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.