Given a point process model fitted to a point pattern dataset,
this function computes the *compensator*
of the nearest neighbour distance distribution function \(G\)
based on the fitted model
(as well as the usual nonparametric estimates
of \(G\) based on the data alone).
Comparison between the nonparametric and model-compensated \(G\)
functions serves as a diagnostic for the model.

```
Gcom(object, r = NULL, breaks = NULL, ...,
correction = c("border", "Hanisch"),
conditional = !is.poisson(object),
restrict=FALSE,
model=NULL,
trend = ~1, interaction = Poisson(),
rbord = reach(interaction),
ppmcorrection="border",
truecoef = NULL, hi.res = NULL)
```

object

Object to be analysed.
Either a fitted point process model (object of class `"ppm"`

)
or a point pattern (object of class `"ppp"`

)
or quadrature scheme (object of class `"quad"`

).

r

Optional. Vector of values of the argument \(r\) at which the function \(G(r)\) should be computed. This argument is usually not specified. There is a sensible default.

breaks

This argument is for internal use only.

correction

Edge correction(s) to be employed in calculating the compensator.
Options are `"border"`

, `"Hanisch"`

and `"best"`

.
Alternatively `correction="all"`

selects all options.

conditional

Optional. Logical value indicating whether to compute the estimates for the conditional case. See Details.

restrict

Logical value indicating whether to compute
the restriction estimator (`restrict=TRUE`

) or
the reweighting estimator (`restrict=FALSE`

, the default).
Applies only if `conditional=TRUE`

. See Details.

model

Optional. A fitted point process model (object of
class `"ppm"`

) to be re-fitted to the data
using `update.ppm`

, if `object`

is a point pattern.
Overrides the arguments `trend,interaction,rbord,ppmcorrection`

.

trend,interaction,rbord

…

Extra arguments passed to `ppm`

.

ppmcorrection

The `correction`

argument to `ppm`

.

truecoef

Optional. Numeric vector. If present, this will be treated as
if it were the true coefficient vector of the point process model,
in calculating the diagnostic. Incompatible with `hi.res`

.

hi.res

Optional. List of parameters passed to `quadscheme`

.
If this argument is present, the model will be
re-fitted at high resolution as specified by these parameters.
The coefficients
of the resulting fitted model will be taken as the true coefficients.
Then the diagnostic will be computed for the default
quadrature scheme, but using the high resolution coefficients.

A function value table (object of class `"fv"`

),
essentially a data frame of function values.
There is a plot method for this class. See `fv.object`

.

This command provides a diagnostic for the goodness-of-fit of a point process model fitted to a point pattern dataset. It computes different estimates of the nearest neighbour distance distribution function \(G\) of the dataset, which should be approximately equal if the model is a good fit to the data.

The first argument, `object`

, is usually a fitted point process model
(object of class `"ppm"`

), obtained from the
model-fitting function `ppm`

.

For convenience, `object`

can also be a point pattern
(object of class `"ppp"`

).
In that case, a point process
model will be fitted to it,
by calling `ppm`

using the arguments
`trend`

(for the first order trend),
`interaction`

(for the interpoint interaction)
and `rbord`

(for the erosion distance in the border correction
for the pseudolikelihood). See `ppm`

for details
of these arguments.

The algorithm first extracts the original point pattern dataset
(to which the model was fitted) and computes the
standard nonparametric estimates of the \(G\) function.
It then also computes the *model-compensated*
\(G\) function. The different functions are returned
as columns in a data frame (of class `"fv"`

).
The interpretation of the columns is as follows
(ignoring edge corrections):

`bord`

:the nonparametric border-correction estimate of \(G(r)\), $$ \hat G(r) = \frac{\sum_i I\{ d_i \le r\} I\{ b_i > r \}}{\sum_i I\{ b_i > r\}} $$ where \(d_i\) is the distance from the \(i\)-th data point to its nearest neighbour, and \(b_i\) is the distance from the \(i\)-th data point to the boundary of the window \(W\).

`bcom`

:the model compensator of the border-correction estimate $$ {\bf C}\, \hat G(r) = \frac{\int \lambda(u,x) I\{ b(u) > r\} I\{ d(u,x) \le r\}}{ 1 + \sum_i I\{ b_i > r\} } $$ where \(\lambda(u,x)\) denotes the conditional intensity of the model at the location \(u\), and \(d(u,x)\) denotes the distance from \(u\) to the nearest point in \(x\), while \(b(u)\) denotes the distance from \(u\) to the boundary of the window\(W\).

`han`

:the nonparametric Hanisch estimate of \(G(r)\) $$ \hat G(r) = \frac{D(r)}{D(\infty)} $$ where $$ D(r) = \sum_i \frac{ I\{x_i \in W_{\ominus d_i}\} I\{d_i \le r\} }{ \mbox{area}(W_{\ominus d_i}) } $$ in which \(W_{\ominus r}\) denotes the erosion of the window \(W\) by a distance \(r\).

`hcom`

:the corresponding model-compensated function $$ {\bf C} \, G(r) = \int_W \frac{ \lambda(u,x) I(u \in W_{\ominus d(u)}) I(d(u) \le r) }{ \hat D(\infty) \mbox{area}(W_{\ominus d(u)}) + 1 } $$ where \(d(u) = d(u, x)\) is the (`empty space') distance from location \(u\) to the nearest point of \(x\).

If the fitted model is a Poisson point process, then the formulae above are exactly what is computed. If the fitted model is not Poisson, the formulae above are modified slightly to handle edge effects.

The modification is determined by the arguments
`conditional`

and `restrict`

.
The value of `conditional`

defaults to `FALSE`

for Poisson models
and `TRUE`

for non-Poisson models.
If `conditional=FALSE`

then the formulae above are not modified.
If `conditional=TRUE`

, then the algorithm calculates
the *restriction estimator* if `restrict=TRUE`

,
and calculates the *reweighting estimator* if `restrict=FALSE`

.
See Appendix E of Baddeley, Rubak and Moller (2011).
See also `spatstat.options('eroded.intensity')`

.
Thus, by default, the reweighting estimator is computed
for non-Poisson models.

The border-corrected and Hanisch-corrected estimates of \(G(r)\) are
approximately unbiased estimates of the \(G\)-function,
assuming the point process is
stationary. The model-compensated functions are unbiased estimates
*of the mean value of the corresponding nonparametric estimate*,
assuming the model is true. Thus, if the model is a good fit, the mean value
of the difference between the nonparametric and model-compensated
estimates is approximately zero.

To compute the difference between the nonparametric and model-compensated
functions, use `Gres`

.

Baddeley, A., Rubak, E. and Moller, J. (2011)
Score, pseudo-score and residual
diagnostics for spatial point process models.
*Statistical Science* **26**, 613--646.

Related functions:
`Gest`

,
`Gres`

.

Alternative functions:
`Kcom`

,
`psstA`

,
`psstG`

,
`psst`

.

Model fitting: `ppm`

.

# NOT RUN { data(cells) fit0 <- ppm(cells, ~1) # uniform Poisson G0 <- Gcom(fit0) G0 plot(G0) # uniform Poisson is clearly not correct # Hanisch estimates only plot(Gcom(fit0), cbind(han, hcom) ~ r) fit1 <- ppm(cells, ~1, Strauss(0.08)) plot(Gcom(fit1), cbind(han, hcom) ~ r) # Try adjusting interaction distance fit2 <- update(fit1, Strauss(0.10)) plot(Gcom(fit2), cbind(han, hcom) ~ r) G3 <- Gcom(cells, interaction=Strauss(0.12)) plot(G3, cbind(han, hcom) ~ r) # }