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
nrvalis 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>