Estimates the nearest-neighbour distance distribution function \(G_3(r)\) from a three-dimensional point pattern.

`G3est(X, ..., rmax = NULL, nrval = 128, correction = c("rs", "km", "Hanisch"))`

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

) that can be
plotted, printed or coerced to a data frame containing the function values.

- X
Three-dimensional point pattern (object of class

`"pp3"`

).- ...
Ignored.

- rmax
Optional. Maximum value of argument \(r\) for which \(G_3(r)\) will be estimated.

- nrval
Optional. Number of values of \(r\) for which \(G_3(r)\) will be estimated. A large value of

`nrval`

is required to avoid discretisation effects.- correction
Optional. Character vector specifying the edge correction(s) to be applied. See Details.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rana Moyeed.

A large value of `nrval`

is required in order to avoid
discretisation effects (due to the use of histograms in the
calculation).

For a stationary point process \(\Phi\) in three-dimensional
space, the nearest-neighbour function
is
$$
G_3(r) = P(d^\ast(x,\Phi) \le r \mid x \in \Phi)
$$
the cumulative distribution function of the distance
\(d^\ast(x,\Phi)\) from a typical point \(x\)
in \(\Phi\) to its nearest neighbour, i.e.
to the nearest *other* point of \(\Phi\).

The three-dimensional point pattern `X`

is assumed to be a
partial realisation of a stationary point process \(\Phi\).
The nearest neighbour function of \(\Phi\) can then be estimated using
techniques described in the References. For each data point, the
distance to the nearest neighbour is computed.
The empirical cumulative distribution
function of these values, with appropriate edge corrections, is the
estimate of \(G_3(r)\).

The available edge corrections are:

`"rs"`

:the reduced sample (aka minus sampling, border correction) estimator (Baddeley et al, 1993)

`"km"`

:the three-dimensional version of the Kaplan-Meier estimator (Baddeley and Gill, 1997)

`"Hanisch"`

:the three-dimensional generalisation of the Hanisch estimator (Hanisch, 1984).

Alternatively `correction="all"`

selects all options.

Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A. (1993)
Analysis of a three-dimensional point pattern with replication.
*Applied Statistics* **42**, 641--668.

Baddeley, A.J. and Gill, R.D. (1997)
Kaplan-Meier estimators of interpoint distance
distributions for spatial point processes.
*Annals of Statistics* **25**, 263--292.

Hanisch, K.-H. (1984)
Some remarks on estimators of the distribution function
of nearest neighbour distance in stationary spatial point patterns.
*Mathematische Operationsforschung und Statistik, series Statistics*
**15**, 409--412.

`pp3`

to create a three-dimensional point
pattern (object of class `"pp3"`

).

`F3est`

,
`K3est`

,
`pcf3est`

for other summary functions of
a three-dimensional point pattern.

`Gest`

to estimate the empty space function of
point patterns in two dimensions.

```
X <- rpoispp3(42)
Z <- G3est(X)
if(interactive()) plot(Z)
```

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