Empty Space Function of a Three-Dimensional Point Pattern

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

spatial, nonparametric
F3est(X, ..., rmax = NULL, nrval = 128, vside = NULL,
              correction = c("rs", "km", "cs"),
              sphere = c("fudge", "ideal", "digital"))
Optional. Maximum value of argument $r$ for which $F_3(r)$ will be estimated.
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.
Optional. Side length of the voxels in the discrete approximation.
Optional. Character string specifying how to calculate the theoretical value of $F_3(r)$ for a Poisson process. See 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.




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").


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.

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

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