# DiggleGratton

0th

Percentile

##### Diggle-Gratton model

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

Keywords
spatial
##### Usage
DiggleGratton(delta, 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.

##### 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.

mpl, ppm.object, Pairwise
data(cells)
mpl(cells, ~1, DiggleGratton(0.05, 0.1))