# distcdf

##### Distribution Function of Interpoint Distance

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)`

##### 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!

- regularise
Logical value indicating whether to smooth the results for very small distances, to avoid discretisation artefacts.

##### 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 7 pixels.

##### Value

An object of class `"fv"`

, see `fv.object`

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

.

##### See Also

##### Examples

```
# NOT RUN {
# The unit disc
B <- disc()
plot(distcdf(B))
# }
```

*Documentation reproduced from package spatstat, version 1.63-0, License: GPL (>= 2)*