distcdf: Distribution Function of Interpoint Distance

In spatstat.explore: Exploratory Data Analysis for the 'spatstat' Family

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,
          regularise=TRUE, savedenom=FALSE, delta=NULL)

Arguments

W	A window (object of class "owin") containing the first random point.
V	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) or a pixel image or a numeric value. The default corresponds to a uniform distribution over the window.
nr	Integer. The number of values of interpoint distance r for which the CDF will be computed. Should be a large value. Alternatively if nr=NULL, a good default value will be chosen, depending on the pixel resolution.
regularise	Logical value indicating whether to smooth the results for very small distances, to avoid discretisation artefacts.
savedenom	Logical value indicating whether to save the denominator of the double integral as an attribute of the result.
delta	Optional. A positive number. The maximum permitted spacing between values of the function argument.

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 values of dW and dV must be finite and nonnegative.

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.

The arguments W or V may also be point patterns (objects of class "ppp"). The result is the cumulative distribution function of the distance from a randomly selected point in the point pattern, to a randomly selected point in the other point pattern or window.

If regularise=TRUE (the default), values of the cumulative distribution function for very short distances are smoothed to avoid discretisation artefacts. Smoothing is applied to all distances shorter than the width of 10 pixels.

Numerical accuracy of some calculations requires very fine spacing of the values of the function argument r. If the argument delta is given, then after the cumulative distribution function has been calculated, it will be interpolated onto a finer grid of r values with spacing less than or equal to delta.

Value

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

Author(s)

\spatstatAuthors

.

See Also

setcov, as.mask.

Examples

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

spatstat.explore documentation built on Oct. 22, 2024, 9:07 a.m.