K-function

Description

Estimates the reduced second moment function K(r) from a point pattern in a window of arbitrary shape.

Usage

  Kest(X, ..., r=NULL, breaks=NULL, 
     correction=c("border", "isotropic", "Ripley", "translate"),
    nlarge=3000)

Arguments

Details

The K function (variously called ``Ripley's K-function'' and the ``reduced second moment function'') of a stationary point process X is defined so that lambda K(r) equals the expected number of additional random points within a distance r of a typical random point of X. Here lambda is the intensity of the process, i.e. the expected number of points of X per unit area. The K function is determined by the second order moment properties of X.

An estimate of K derived from a spatial point pattern dataset can be used in exploratory data analysis and formal inference about the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1988). In exploratory analyses, the estimate of K is a useful statistic summarising aspects of inter-point ``dependence'' and ``clustering''. For inferential purposes, the estimate of K is usually compared to the true value of K for a completely random (Poisson) point process, which is K(r) = pi * r^2. Deviations between the empirical and theoretical K curves may suggest spatial clustering or spatial regularity.

This routine Kest estimates the K function of a stationary point process, given observation of the process inside a known, bounded window. The argument X is interpreted as a point pattern object (of class "ppp", see ppp.object) and can be supplied in any of the formats recognised by as.ppp().

The estimation of K is hampered by edge effects arising from the unobservability of points of the random pattern outside the window. An edge correction is needed to reduce bias (Baddeley, 1998; Ripley, 1988). The corrections implemented here are

border

the border method or ``reduced sample'' estimator (see Ripley, 1988). This is the least efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary shape.

isotropic/Ripley

Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented for rectangular and polygonal windows (not for binary masks).

translate

Translation correction (Ohser, 1983). Implemented for all window geometries, but slow for complex windows.

Note that the estimator assumes the process is stationary (spatially homogeneous). For inhomogeneous point patterns, see Kinhom.

If the point pattern X contains more than about 3000 points, the isotropic and translation edge corrections can be computationally prohibitive. The computations for the border method are much faster, and are statistically efficient when there are large numbers of points. Accordingly, if the number of points in X exceeds the threshold nlarge, then only the border correction will be computed. Setting nlarge=Inf will prevent this from happening. Setting nlarge=0 is equivalent to selecting only the border correction with correction="border".

For instructional purposes, you can also set correction="none" to compute an estimate of the K function without edge correction. This estimate is biased and should not be used for data analysis.

The estimator Kest ignores marks. Its counterparts for multitype point patterns are Kcross, Kdot, and for general marked point patterns see Kmulti.

Some writers, particularly Stoyan (1994, 1995) advocate the use of the ``pair correlation function''

g(r) = K'(r)/ ( 2 * pi * r)

where K'(r) is the derivative of K(r). See pcf on how to estimate this function.

Value

An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing columns

together with columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function K(r) obtained by the edge corrections named.

Warnings

The estimator of K(r) is approximately unbiased for each fixed r. Bias increases with r and depends on the window geometry. For a rectangular window it is prudent to restrict the r values to a maximum of 1/4 of the smaller side length of the rectangle. Bias may become appreciable for point patterns consisting of fewer than 15 points.

While K(r) is always a non-decreasing function, the estimator of K is not guaranteed to be non-decreasing. This is rarely a problem in practice.

Author(s)

Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf

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.

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.

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

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

Fest, Gest, Jest, pcf, reduced.sample, Kcross, Kdot, Kinhom, Kmulti

Examples

 pp <- runifpoint(50)
 K <- Kest(pp)
 data(cells)
 K <- Kest(cells, correction="isotropic")
 plot(K)
 plot(K, main="K function for cells")
 # plot the L function
 plot(K, sqrt(iso/pi) ~ r)
 plot(K, sqrt(./pi) ~ r, ylab="L(r)", main="L function for cells")