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

`densityVoronoi(X, ...)`# S3 method for ppp
densityVoronoi(X, f = 1, ...,
counting=FALSE,
fixed=FALSE,
nrep = 1, verbose=TRUE)

A pixel image (object of class `"im"`

) whose values are
estimates of the intensity of `X`

.

- X
Point pattern dataset (object of class

`"ppp"`

).- 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

`as.im`

determining the pixel resolution of the result.- counting
Logical value specifying the choice of estimation method. See Details.

- fixed
Logical. If

`FALSE`

(the default), the data points are independently randomly thinned, so the number of data points that are retained is random. If`TRUE`

, the number of data points retained is fixed. See Details.- nrep
Number of independent repetitions of the randomised procedure.

- verbose
Logical value indicating whether to print progress reports.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk and Mehdi Moradi m2.moradi@yahoo.com.

This function is an alternative to `density.ppp`

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

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 (see `dirichlet`

);
the areas of the Dirichlet tiles are computed; the estimated intensity
in each tile is the reciprocal of the tile area.
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 window area).
The result is a pixel image
of intensity estimates which are constant.

If `f`

is strictly between 0 and 1,
the estimation method is applied to a random subset of `X`

.
This randomised procedure is repeated `nrep`

times,
and the results are averaged.
The subset is selected as follows:

if

`fixed=FALSE`

, the dataset`X`

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

of retaining a point.if

`fixed=TRUE`

, a random sample of fixed size`m`

is taken from the dataset`X`

, where`m`

is the largest integer less than or equal to`f*n`

and`n`

is the number of points in`X`

.

Then the intensity estimate is calculated as follows:

if

`counting = FALSE`

(the default), the thinned pattern is used to construct a Dirichlet tessellation and form the Voronoi estimate (Barr and Schoenberg, 2010) which is then adjusted by a factor`1/f`

or`n/m`

as appropriate. to obtain an estimate of the intensity of`X`

in the tile.if

`counting = TRUE`

, the randomly selected subset`A`

is used to construct a Dirichlet tessellation, while the complementary subset`B`

(consisting of points that were not selected in the sample) is used for counting to calculate a quadrat count estimate of intensity. For each tile of the Dirichlet tessellation formed by`A`

, we count the number of points of`B`

falling in the tile, and divide by the area of the same tile, to obtain an estimate of the intensity of the pattern`B`

in the tile. This estimate is adjusted by`1/(1-f)`

or`n/(n-m)`

as appropriate to obtain an estimate of the intensity of`X`

in the tile.

Ogata et al. (2003) and Ogata (2004) estimated intensity using the
Dirichlet-Voronoi tessellation in a modelling context.
Baddeley (2007) proposed intensity estimation by subsampling
with `0 < f < 1`

, and used the technique described above
with `fixed=TRUE`

and `counting=TRUE`

.
Barr and Schoenberg (2010) described and analysed the
Voronoi estimator (corresponding to `f=1`

).
Moradi et al (2019) developed the subsampling technique with
`fixed=FALSE`

and `counting=FALSE`

and called it the
*smoothed Voronoi estimator*.

Baddeley, A. (2007)
Validation of statistical models for spatial point patterns.
In J.G. Babu and E.D. Feigelson (eds.)
*SCMA IV: Statistical Challenges in Modern Astronomy IV*,
volume 317 of Astronomical Society of the Pacific Conference Series,
San Francisco, California USA, 2007. Pages 22--38.

Barr, C., and Schoenberg, F.P. (2010).
On the Voronoi estimator for the intensity of an inhomogeneous
planar Poisson process. *Biometrika* **97** (4), 977--984.

Moradi, M., Cronie, 0., Rubak, E., Lachieze-Rey, R.,
Mateu, J. and Baddeley, A. (2019)
Resample-smoothing of Voronoi intensity estimators.
*Statistics and Computing* **29** (5) 995--1010.

Ogata, Y. (2004)
Space-time model for regional seismicity and detection of crustal
stress changes.
*Journal of Geophysical Research*, **109**, 2004.

Ogata, Y., Katsura, K. and Tanemura, M. (2003).
Modelling heterogeneous space-time occurrences of earthquakes and its
residual analysis.
*Applied Statistics* **52** 499--509.

`adaptive.density`

,
`density.ppp`

,
`dirichlet`

,
`im.object`

.

```
plot(densityVoronoi(nztrees, 1, f=1), main="Voronoi estimate")
nr <- if(interactive()) 100 else 5
plot(densityVoronoi(nztrees, f=0.5, nrep=nr), main="smoothed Voronoi estimate")
```

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