A pairwise interaction point process in a bounded region
is a stochastic point process with probability density of the form
$$
f(x_1,\ldots,x_n) =
\alpha \prod_i b(x_i) \prod_{i < j} h(x_i, x_j)
$$
where $x[1],\ldots,x[n]$ represent the
points of the pattern. The first product on the right hand side is
over all points of the pattern; the second product is over all
unordered pairs of points of the pattern. Thus each point $x[i]$ of the pattern contributes a factor
$b(x[i])$ to the probability density, and each pair of
points $x[i], x[j]$ contributes a factor
$h(x[i], x[j])$ to the density.
The pairwise interaction term $h(u, v)$ is called
piecewise constant
if it depends only on the distance between $u$ and $v$,
say $h(u,v) = H(||u-v||)$, and $H$ is a piecewise constant
function (a function which is constant except for jumps at a finite
number of places). The use of piecewise constant interaction terms
was first suggested by Takacs (1986).
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 pairwise
interaction is yielded by the function PairPiece().
See the examples below.
The entries of r must be strictly increasing, positive numbers.
They are interpreted as the points of discontinuity of $H$.
It is assumed that $H(s) =1$ for all $s > rmax$
where $rmax$ is the maximum value in r. Thus the
model has as many regular parameters (see ppm)
as there are entries in r. The $i$-th regular parameter
$theta[i]$ is the logarithm of the value of the
interaction function $H$ on the interval
$[r[i-1],r[i])$.
If r is a single number, this model is similar to the
Strauss process, see Strauss. The difference is that
in PairPiece the interaction function is continuous on the
right, while in Strauss it is continuous on the left.
The analogue of this model for multitype point processes
has not yet been implemented.