R: Distribution Function of Interpoint Distance

distcdf {spatstat.core} | R Documentation

## 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 ≤ r) of the Euclidean distance T = |X1-X2| between two independent random points X1 and X2.

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 X1 is random in the window `W` with a probability density proportional to `dW`, and the second point X2 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)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### See Also

`setcov`, `as.mask`.

### Examples

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

[Package spatstat.core version 2.4-4 Index]