Ktrans.sphwin: Transformations of Spherical K Function

In baddstats/spherstat: Spatial Point Pattern Analysis, Geometry and Trigonometry for the Sphere

Transformations of Spherical K Function

Description

Given a point pattern on the sphere, or on a region of the sphere, compute a transform of an estimate of the K-function.

Usage

  Kstab.sphwin(X, ...)
  Kinv.sphwin(X, ...)

Arguments

X	The observed point pattern, from which an estimate of K(r) will be computed. An object of class sp2 or sp3, or a 2 or 3 column matrix giving the locations of points in spherical coordinates.
...	Other arguments passed to Ksphere.

Details

This command computes an estimate of the variance-stabilising and inverse transformations of Ripley's (1977) K function for the spatial point pattern X. The variance-stabilising transformation of Ripley's K-function is

Kstab(r) = sqrt(K(r)/(2*pi*rho^2))

while the inverse transformation is

Kinv(r) = rho*arccos(K(r)/(2*pi*rho^2))

where K(r) is the K-function and rho is the radius of the sphere on which X is observed.

See Ksphere for information about Ripley's K-function. The variance-stabilising transformation was proposed by Besag (1977); the inverse transformation by Lawrence (2016).

The commands Kstab.sphwin and Kinv.sphwin first calls Ksphere to compute the estimate of the K-function, and then applies the appropriate transformation.

For a completely random (uniform Poisson) point pattern, the theoretical value of the inverse transformation function is Kinv(r) = r. The variance-stabilising transformation is more appropriate for use in simulation envelopes and hypothesis tests.

Other plots that can be useful include centring (include the argument .-theo~r) the function, or for either of K(r) or Kstab(r), plotting the estimated function against the theoretical value for a model (use e.g. the argument .~theo in the plot function).

See Ksphere for the list of arguments.

Value

An object of class "fv", see fv.object, which can be plotted directly using plot.fv.

Note

This function is the analogue for point processes on the sphere of the functions Kest and Kinhom in spatstat, which is the corresponding function for point processes in R^2. Hence elements of the code for Ksphere and help page have been taken from Kest and Kinhom with the permission of A. J. Baddeley. This enables the code to be highly efficient and give corresponding output to, and for the information on this help page to be consistent with that for the functions Kest and Kinhom. It is hoped that this will minimise or remove any confusion for users of both spatstat and spherstat.

Author(s)

Tom Lawrence.

References

Baddeley, A.J. Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and Hall, 1998. Chapter 2, pages 37–78.

Baddeley, A., Moller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.

Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.

Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.

Lawrence, T. (2017) Master's Thesis, University of Western Australia.

Ohser, J. (1983) On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 – 71.

Ohser, J. and Stoyan, D. (1981) On the second-order and orientation analysis of planar stationary point processes. Biometrical Journal 23, 523–533.

Raskin, R.G. (1994) Spatial analysis on the sphere: a review (94-7) NGCIA technical report (Available from http://eprints.cdlib.org/uc/item/5748n2xz). Accessed 4 November 2015.

Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society, Series B, 39, 172 – 212.

Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J. (1995) Stochastic geometry and its applications. 2nd edition. Springer Verlag.

Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

See Also

Fsphere, Gsphere, Jsphere, Kest, Kinhom, Fstab.sphwin, Finv.sphwin, Gstab.sphwin, Ginv.sphwin, Lest

Examples

   win <- sphwin()
   X <- runif.sphwin(100, win)
   rv <- seq(0, pi/2, length=512)
   K <- Ksphere(X, win, rv)
   Kstab <- Kstab.sphwin(X, win, rv)
   Kinv <- Kinv.sphwin(X, win, rv)
   plot(K)
   plot(K, .~theo)
   plot(K, .-theo~r)
   plot(Kstab)
   plot(Kstab, .~theo)
   plot(Kinv) 

baddstats/spherstat documentation built on Feb. 6, 2023, 1:45 a.m.