# 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.

##### 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 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 \(s > r_{max}\)
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 \(r\).

##### References

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

##### See Also

##### Examples

```
# NOT RUN {
PairPiece(c(0.1,0.2))
# prints a sensible description of itself
data(cells)
# }
# NOT RUN {
ppm(cells, ~1, PairPiece(r = c(0.05, 0.1, 0.2)))
# fit a stationary piecewise constant pairwise interaction process
# }
# NOT RUN {
ppm(cells, ~polynom(x,y,3), PairPiece(c(0.05, 0.1)))
# nonstationary process with log-cubic polynomial trend
# }
```

*Documentation reproduced from package spatstat, version 1.59-0, License: GPL (>= 2)*