# adaptive.density

##### Intensity Estimate of Point Pattern Using Tessellation

Computes an adaptive estimate of the intensity function of a point pattern.

##### Usage

`adaptive.density(X, f = 0.1, ..., nrep = 1, verbose=TRUE)`

##### Arguments

- X
Point pattern dataset (object of class

`"ppp"`

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

- …
Arguments passed to

`as.im`

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.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 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.

If `f=0`

, the intensity estimate at every location is
equal to the average intensity (number of points divided by window area).

If `f`

is strictly between 0 and 1,
the dataset `X`

is randomly split into two patterns `A`

and
`B`

containing a fraction `f`

and `1-f`

, respectively,
of the original data. The subpattern `A`

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

is retained for counting. For each tile of the Dirichlet
tessellation, 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 divided by `1-f`

to obtain an estimate
of the intensity of `X`

in the tile. The result is a pixel image
of intensity estimates which are constant on each tile of the tessellation.

If `nrep`

is greater than 1, this randomised procedure is
repeated `nrep`

times, and the results are averaged.

This technique has been used by Ogata et al. (2003), Ogata (2004) and Baddeley (2007).

##### Value

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

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

.

##### References

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.

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.

##### See Also

##### Examples

```
# NOT RUN {
plot(adaptive.density(nztrees, 1), main="Voronoi estimate")
nr <- if(interactive()) 100 else 5
plot(adaptive.density(nztrees, nrep=nr), main="Adaptive estimate")
# }
```

*Documentation reproduced from package spatstat, version 1.55-1, License: GPL (>= 2)*