# densityVoronoi.lpp

##### Intensity Estimate of Point Pattern on Linear Network Using Voronoi-Dirichlet Tessellation

Computes an adaptive estimate of the intensity function of a point pattern on a linear network, using the Dirichlet-Voronoi tessellation on the network.

##### Usage

```
# S3 method for lpp
densityVoronoi(X, f = 1, …, nrep = 1, verbose = TRUE)
```

##### Arguments

- X
Point pattern on a linear network (object of class

`"lpp"`

).- f
Fraction (between 0 and 1 inclusive) of the data points that will be used to build a tessellation for the intensity estimate.

- …
Arguments passed to

`linim`

determining the pixel resolution of the result.- nrep
Number of independent repetitions of the randomised procedure.

- verbose
Logical value indicating whether to print progress reports.

##### Details

This function is an alternative to `density.lpp`

. It
computes an estimate of the intensity function of a point pattern
dataset on a linear network.
The result is a pixel image on the network, giving the estimated intensity.

This function is a method for the generic `densityVoronoi`

for the class `"lpp"`

of point patterns on a linear network.

If `f=1`

(the default), the Voronoi estimate (Barr and Schoenberg, 2010)
is computed: the point pattern `X`

is used to construct
a Voronoi/Dirichlet tessellation on the network
(see `lineardirichlet`

);
the lengths of the Dirichlet tiles are computed; the estimated intensity
in each tile is the reciprocal of the tile length.
The result is a pixel image
of intensity estimates which are constant on each tile of the tessellation.

If `f=0`

, the intensity estimate at every location is
equal to the average intensity (number of points divided by
network length). The result is a pixel image
of intensity estimates which are constant.

If `f`

is strictly between 0 and 1,
the smoothed Voronoi estimate (Moradi et al, 2019) is computed.
The dataset `X`

is randomly
thinned by deleting or retaining each point independently, with
probability `f`

of retaining a point.
The thinned pattern
is used to construct a Dirichlet tessellation and form the
Voronoi estimate, which is then
adjusted by a factor `1/f`

.
This procedure is repeated `nrep`

times and the results are
averaged to obtain the smoothed Voronoi estimate.

The value `f`

can be chosen automatically by bandwidth
selection using `bw.voronoi`

.

##### Value

Pixel image on a linear network (object of class `"linim"`

).

##### References

Moradi, M., Cronie, 0., Rubak, E., Lachieze-Rey, R.,
Mateu, J. and Baddeley, A. (2019)
Resample-smoothing of Voronoi intensity estimators.
*Statistics and Computing*, in press.

##### See Also

`densityVoronoi`

is the generic, with a method for
class `"ppp"`

.

`lineardirichlet`

computes the Dirichlet-Voronoi
tessellation on a network.

`bw.voronoi`

performs bandwidth selection of the fraction `f`

.

See also `density.lpp`

.

##### Examples

```
# NOT RUN {
nr <- if(interactive()) 100 else 3
plot(densityVoronoi(spiders, 0.1, nrep=nr))
# }
```

*Documentation reproduced from package spatstat, version 1.59-0, License: GPL (>= 2)*