Kest: K-function

Description

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

Usage

Kest(X)
  Kest(X, r)
  Kest(X, breaks)

Arguments

The observed point pattern, from which an estimate of $K(r)$ will be computed. An object of class "ppp", or data in any format acceptable to as.ppp().

vector of values for the argument $r$ at which $K(r)$ should be evaluated. There is a sensible default.

breaks

Value

A data frame containing
rthe vector of values of the argument $r$ at which the function $K$ has been estimated
borderthe ``reduced sample'' or ``border corrected'' estimator of $K(r)$
theothe theoretical value $K(r) = \pi r^2$ for a stationary Poisson process

synopsis

Kest(X, r=NULL, breaks=NULL, slow, both, ...)

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.

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 correction implemented here is the border method or ``reduced sample'' estimator (Ripley, 1988). Although this is less efficient than some alternative corrections, it has the advantage that it can be computed for a window of arbitrary shape.

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

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'' $$g(r) = \frac{K'(r)}{2\pi r}$$ where $K'(r)$ is the derivative of $K(r)$. See pcf on how to estimate this function.

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.

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.

Examples

Run this code

library(spatstat)
 pp <- runifpoint(50)
 Kpp <- Kest(pp)
 plot(Kpp$r, Kpp$border, type="l", xlab="r", ylab="K(r)", ylim=c(0,1),
       main = "K-function")
 r <- Kpp$r
 lines(r, Kpp$theo, lty=2)
 legend(0.5, 2, c("empirical (border correction)", "Poisson process"), lty=c(1,2))
 data(cells)
 K <- Kest(cells)
 plot(K$r, K$border, type="l")
 plot(border ~ r, type="l", data=K)
 # restrict the plot to values of r less than 0.1
 plot(border ~ r, type="l", data=K[K$r <= 0.1, ])
 plot(border ~ r, type="l", data=K, subset=(r <= 0.1))

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