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PairPiece(r)
"interact"
describing the interpoint interaction
structure of a point process. The process is a pairwise interaction process,
whose interaction potential is piecewise constant, with jumps
at the distances given in the vector $r$.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 function mpl()
, 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 $s > r_{max}$
where $r_{max}$ is the maximum value in r
. Thus the
model has as many regular parameters (see mpl
)
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 reduces to the
Strauss process, see Strauss
.
The analogue of this model for multitype point processes has not yet been implemented.
mpl
,
pairwise.family
,
ppm.object
library(spatstat)
PairPiece(c(0.1,0.2))
# prints a sensible description of itself
data(cells)
mpl(cells, ~1, PairPiece(r = c(0.05, 0.1, 0.2)), rbord=0.2)
# fit a stationary piecewise constant pairwise interaction process
mpl(cells, ~polynom(x,y,3), PairPiece(c(0.05, 0.1)), rbord=0.1)
# nonstationary process with log-cubic polynomial trend
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