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"))
Three-dimensional point pattern (object of class "pp3").
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 vector specifying the edge correction(s) to be applied. 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]


  • A function value table (object of class "fv") that can be plotted, printed or coerced to a data frame containing the function values.


A large value of nrval is required in order to avoid discretisation effects (due to the use of histograms in the calculation).


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.

  • F3est
X <- rpoispp3(42)
  Z <- F3est(X)
  if(interactive()) plot(Z)
Documentation reproduced from package spatstat, version 1.24-1, License: GPL (>= 2)

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