# F3est

##### Empty Space Function of a Three-Dimensional Point Pattern

Estimates the empty space function $F_3(r)$ from a three-dimensional point pattern.

- Keywords
- spatial, nonparametric

##### Usage

`F3est(X, ..., rmax = NULL, nrval = 128, vside = NULL, correction = c("rs", "km", "cs"))`

##### Arguments

- X
- Three-dimensional point pattern (object of class
`"pp3"`

). - ...
- Ignored.
- rmax
- Optional. Maximum value of argument $r$ for which $F_3(r)$ will be estimated.
- nrval
- Optional. Number of values of $r$ for which
$F_3(r)$ will be estimated. A large value of
`nrval`

is required to avoid discretisation effects. - vside
- Optional. Side length of the voxels in the discrete approximation.
- 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 empty space function is
$$F_3(r) = P(d(0,\Phi) \le r)$$
where $d(0,\Phi)$ denotes the distance from a fixed
origin $0$ to the nearest point of $\Phi$.
The three-dimensional point pattern `X`

is assumed to be a
partial realisation of a stationary point process $\Phi$.
The empty space function of $\Phi$ can then be estimated using
techniques described in the References.

The box containing the point
pattern is discretised into cubic voxels of side length `vside`

.
The distance function $d(u,\Phi)$ is computed for
every voxel centre point
$u$ using a three-dimensional version of the distance transform
algorithm (Borgefors, 1986). The empirical cumulative distribution
function of these values, with appropriate edge corrections, is the
estimate of $F_3(r)$.

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.
Analysis of a three-dimensional point pattern with replication.
*Applied Statistics* **42** (1993) 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.

Borgefors, G. (1986)
Distance transformations in digital images.
*Computer Vision, Graphics and Image Processing*
**34**, 344--371.

Chiu, S.N. and Stoyan, D. (1998)
Estimators of distance distributions for spatial patterns.
*Statistica Neerlandica* **52**, 239--246.

##### See Also

##### Examples

```
X <- rpoispp3(42)
Z <- F3est(X)
if(interactive()) plot(Z)
```

*Documentation reproduced from package spatstat, version 1.18-2, License: GPL (>= 2)*