# 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"),
sphere = c("fudge", "ideal", "digital"))
```

##### 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.
- sphere
- Optional. Character string specifying how to calculate the theoretical value of $F_3(r)$ for a Poisson process. 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]

The result includes a column `theo`

giving the
theoretical value of $F_3(r)$ for
a uniform Poisson process (Complete Spatial Randomness).
This value depends on the volume of the sphere of radius `r`

measured in the discretised distance metric.
The argument `sphere`

determines how this will be calculated.

- If
`sphere="ideal"`

the calculation will use the volume of an ideal sphere of radius$r$namely$(4/3) \pi r^3$. This is not recommended because the theoretical values of$F_3(r)$are inaccurate. - If
`sphere="fudge"`

then the volume of the ideal sphere will be multiplied by 0.78, which gives the approximate volume of the sphere in the discretised distance metric. - If
`sphere="digital"`

then the volume of the sphere in the discretised distance metric is computed exactly using another distance transform. This takes longer to compute, but is exact.

##### 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 small value of `vside`

and a large value of `nrval`

are required for reasonable accuracy.

The default value of `vside`

ensures that the total number of
voxels is `2^22`

or about 4 million.
To change the default number of voxels, see
`spatstat.options("nvoxel")`

.

##### 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

```
<testonly>op <- spatstat.options(nvoxel=2^18)</testonly>
X <- rpoispp3(42)
Z <- F3est(X)
if(interactive()) plot(Z)
<testonly>spatstat.options(op)</testonly>
```

*Documentation reproduced from package spatstat, version 1.31-3, License: GPL (>= 2)*