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

```
distcdf(W, V=W, ..., dW=1, dV=dW, nr=1024,
regularise=TRUE, savedenom=FALSE, delta=NULL)
```

An object of class `"fv"`

, see `fv.object`

,
which can be plotted directly using `plot.fv`

.

- 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.

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

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`

.

`setcov`

,
`as.mask`

.

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

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