F3est

Empty Space Function of a Three-Dimensional Point Pattern

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

Usage

F3est(X, ..., rmax = NULL, nrval = 128, vside = NULL,
              correction = c("rs", "km", "cs"),
              sphere = c("fudge", "ideal", "digital"))

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.

• Ifsphere="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.
• Ifsphere="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.
• Ifsphere="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.

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

G3est, K3est, pcf3est.

Aliases

• F3est

Examples

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