F3est {spatstat}R Documentation

Empty Space Function of a Three-Dimensional Point Pattern

Description

Estimates the empty space function F3(r) from a three-dimensional point pattern.

Usage

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

Arguments

X

Three-dimensional point pattern (object of class "pp3").

...

Ignored.

rmax

Optional. Maximum value of argument r for which F3(r) will be estimated.

nrval

Optional. Number of values of r for which F3(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.

Details

For a stationary point process Phi in three-dimensional space, the empty space function is

F3(r) = P(d(0,Phi) <= 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 F3(r).

The available edge corrections are:

"rs":

the reduced sample (aka minus sampling, border correction) estimator (Baddeley et al, 1993)

"km":

the three-dimensional version of the Kaplan-Meier estimator (Baddeley and Gill, 1997)

"cs":

the three-dimensional generalisation of the Chiu-Stoyan or Hanisch estimator (Chiu and Stoyan, 1998).

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

Author(s)

Adrian Baddeley Adrian.Baddeley@csiro.au http://www.maths.uwa.edu.au/~adrian/ and Rana Moyeed.

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.

Examples

  X <- rpoispp3(42)
  Z <- F3est(X)
  if(interactive()) plot(Z)

[Package spatstat version 1.25-3 Index]