distcdf: Distribution Function of Interpoint Distance

Description

Computes the cumulative distribution function of the distance between two independent random points in a given window or windows.

Usage

distcdf(W, V=W, ..., dW=1, dV=dW, nr=1024)

Arguments

A window (object of class "owin") containing the first random point.

Optional. Another window containing the second random point. Defaults to W.

...

Arguments passed to as.mask to determine the pixel resolution for the calculation.

dV, dW

Optional. Probability densities (not necessarily normalised) for the first and second random points respectively. Data in any format acceptable to as.im, for example, a function(x,y)

Integer. The number of values of interpoint distance $r$ for which the CDF will be computed. Should be a large value!

Value

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

Details

This command computes the Cumulative Distribution Function $CDF(r) = Prob(T \le r)$ of the Euclidean distance $T = \|X_1 - X_2\|$ between two independent random points $X_1$ and $X_2$.

In the simplest case, the command distcdf(W), the random points are assumed to be uniformly distributed in the same window W.

Alternatively the two random points may be uniformly distributed in two different windows W and V.

In the most general case the first point $X_1$ is random in the window W with a probability density proportional to dW, and the second point $X_2$ is random in a different window V with probability density proportional to dV.

The calculation is performed by numerical integration of the set covariance function setcov for uniformly distributed points, and by computing the covariance function imcov in the general case. The accuracy of the result depends on the pixel resolution used to represent the windows: this is controlled by the arguments ... which are passed to as.mask. For example use eps=0.1 to specify pixels of size 0.1 units.

Examples

Run this code

# The unit disc
 B <- disc()
 plot(distcdf(B))