# G3est

##### Nearest Neighbour Distance Distribution Function of a Three-Dimensional Point Pattern

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

- Keywords
- spatial, nonparametric

##### Usage

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

##### Arguments

- 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.

##### Details

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: [object Object],[object Object],[object Object]

##### Value

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

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

##### Warnings

A large value of `nrval`

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

##### References

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.

##### See Also

##### Examples

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

*Documentation reproduced from package spatstat, version 1.31-2, License: GPL (>= 2)*