R: The Piecewise Constant Pairwise Interaction Point Process...

PairPiece {spatstat.core} | R Documentation

## The Piecewise Constant Pairwise Interaction Point Process Model

### Description

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.

### 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,…,x_n) = alpha . product { b(x[i]) } product { h(x_i, x_j) }

where x[1],…,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.

### 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.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

### References

Takacs, R. (1986) Estimator for the pair potential of a Gibbsian point process. Statistics 17, 429–433.

### See Also

`ppm`, `pairwise.family`, `ppm.object`, `Strauss` `rmh.ppm`

### 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)))
       # fit a stationary piecewise constant pairwise interaction process

       # ppm(cells ~polynom(x,y,3), PairPiece(c(0.05, 0.1)))
       # nonstationary process with log-cubic polynomial trend

[Package spatstat.core version 1.65-5 Index]