# 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

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

`ppm(cells ~1, DiggleGratton(0.05, 0.1))`

*Documentation reproduced from package spatstat, version 1.42-2, License: GPL (>= 2)*