# DiggleGratton

##### Diggle-Gratton model

Creates an instance of the Diggle-Gratton pairwise interaction point process model, which can then be fitted to point pattern data.

##### Usage

`DiggleGratton(delta=NA, rho)`

##### Arguments

- delta
lower threshold \(\delta\)

- rho
upper threshold \(\rho\)

##### Details

Diggle and Gratton (1984, pages 208-210) introduced the pairwise interaction point process with pair potential \(h(t)\) of the form $$ h(t) = \left( \frac{t-\delta}{\rho-\delta} \right)^\kappa \quad\quad \mbox{ if } \delta \le t \le \rho $$ with \(h(t) = 0\) for \(t < \delta\) and \(h(t) = 1\) for \(t > \rho\). Here \(\delta\), \(\rho\) and \(\kappa\) are parameters.

Note that we use the symbol \(\kappa\) where Diggle and Gratton (1984) and Diggle, Gates and Stibbard (1987) use \(\beta\), since in spatstat we reserve the symbol \(\beta\) for an intensity parameter.

The parameters must all be nonnegative, and must satisfy \(\delta \le \rho\).

The potential is inhibitory, i.e.\ this model is only appropriate for regular point patterns. The strength of inhibition increases with \(\kappa\). For \(\kappa=0\) the model is a hard core process with hard core radius \(\delta\). For \(\kappa=\infty\) the model is a hard core process with hard core radius \(\rho\).

The irregular parameters
\(\delta, \rho\) must be given in the call to
`DiggleGratton`

, while the
regular parameter \(\kappa\) will be estimated.

If the lower threshold `delta`

is missing or `NA`

,
it will be estimated from the data when `ppm`

is called.
The estimated value of `delta`

is the minimum nearest neighbour distance
multiplied by \(n/(n+1)\), where \(n\) is the
number of data points.

##### Value

An object of class `"interact"`

describing the interpoint interaction
structure of a point process.

##### References

Diggle, P.J., Gates, D.J. and Stibbard, A. (1987)
A nonparametric estimator for pairwise-interaction point processes.
*Biometrika* **74**, 763 -- 770.

Diggle, P.J. and Gratton, R.J. (1984)
Monte Carlo methods of inference for implicit statistical models.
*Journal of the Royal Statistical Society, series B*
**46**, 193 -- 212.

##### See Also

##### Examples

```
# NOT RUN {
ppm(cells ~1, DiggleGratton(0.05, 0.1))
# }
```

*Documentation reproduced from package spatstat, version 1.64-1, License: GPL (>= 2)*