PairPiece
The Piecewise Constant Pairwise Interaction Point Process Model
Creates an instance of a pairwise interaction point process model with piecewise constant potential function. The model can then be fitted to point pattern data.
- Keywords
- spatial
Usage
PairPiece(r)Arguments
- r
- vector of jump points for the potential function
Details
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 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 $s > r_{max}$
where $r_{max}$ 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.
Value
- An object of class
"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 $r$.
See Also
Examples
PairPiece(c(0.1,0.2))
# prints a sensible description of itself
data(cells)
ppm(cells, ~1, PairPiece(r = c(0.05, 0.1, 0.2)), rbord=0.2)
# fit a stationary piecewise constant pairwise interaction process
ppm(cells, ~polynom(x,y,3), PairPiece(c(0.05, 0.1)), rbord=0.1)
# nonstationary process with log-cubic polynomial trend