# SatPiece: Piecewise Constant Saturated Pairwise Interaction Point... In spatstat: Spatial Point Pattern Analysis, Model-Fitting, Simulation, Tests

## Description

Creates an instance of a saturated pairwise interaction point process model with piecewise constant potential function. The model can then be fitted to point pattern data.

## Usage

 `1` ``` SatPiece(r, sat) ```

## Arguments

 `r` vector of jump points for the potential function `sat` vector of saturation values, or a single saturation value

## Details

This is a generalisation of the Geyer saturation point process model, described in `Geyer`, to the case of multiple interaction distances. It can also be described as the saturated analogue of a pairwise interaction process with piecewise-constant pair potential, described in `PairPiece`.

The saturated point process with interaction radii r[1], ..., r[k], saturation thresholds s[1],...,s[k], intensity parameter beta and interaction parameters gamma[1], ..., gamma[k], is the point process in which each point x[i] in the pattern X contributes a factor

beta gamma[1]^v(1, x_i, X) ... gamma[k]^v(k, x_i, X)

to the probability density of the point pattern, where

v(j, x_i, X) = min(s[j], t(j, x_i, X))

where t(j,x[i],X) denotes the number of points in the pattern X which lie at a distance between r[j-1] and r[j] from the point x[i]. We take r[0] = 0 so that t(1, x[i], X) is the number of points of X that lie within a distance r[1] of the point x[i].

`SatPiece` is used to fit this model to data. 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 Saturated pairwise interaction is yielded by the function `SatPiece()`. See the examples below.

Simulation of this point process model is not yet implemented. This model is not locally stable (the conditional intensity is unbounded).

The argument `r` specifies the vector of interaction distances. The entries of `r` must be strictly increasing, positive numbers.

The argument `sat` specifies the vector of saturation parameters. It should be a vector of the same length as `r`, and its entries should be nonnegative numbers. Thus `sat[1]` corresponds to the distance range from `0` to `r[1]`, and `sat[2]` to the distance range from `r[1]` to `r[2]`, etc. Alternatively `sat` may be a single number, and this saturation value will be applied to every distance range.

Infinite values of the saturation parameters are also permitted; in this case v(j, x_i, X) = t(j, x_i, X) and there is effectively no ‘saturation’ for the distance range in question. If all the saturation parameters are set to `Inf` then the model is effectively a pairwise interaction process, equivalent to `PairPiece` (however the interaction parameters gamma obtained from `SatPiece` are the square roots of the parameters gamma obtained from `PairPiece`).

If `r` is a single number, this model is virtually equivalent to the Geyer process, see `Geyer`.

## Value

An object of class `"interact"` describing the interpoint interaction structure of a point process.

## Author(s)

and \rolf

in collaboration with Hao Wang and Jeff Picka

`ppm`, `pairsat.family`, `Geyer`, `PairPiece`, `BadGey`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ``` SatPiece(c(0.1,0.2), c(1,1)) # prints a sensible description of itself SatPiece(c(0.1,0.2), 1) data(cells) ppm(cells, ~1, SatPiece(c(0.07, 0.1, 0.13), 2)) # fit a stationary piecewise constant Saturated pairwise interaction process ## Not run: ppm(cells, ~polynom(x,y,3), SatPiece(c(0.07, 0.1, 0.13), 2)) # nonstationary process with log-cubic polynomial trend ## End(Not run) ```