# adaptive.density: Intensity Estimate of Point Pattern Using Tessellation In spatstat: Spatial Point Pattern Analysis, Model-Fitting, Simulation, Tests

## Description

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

## Usage

 `1` ```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`.

\adrian

and \rolf

## 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

`density.ppp`, `dirichlet`, `im.object`.

## Examples

 ```1 2 3``` ``` plot(adaptive.density(nztrees, 1), main="Voronoi estimate") nr <- if(interactive()) 100 else 5 plot(adaptive.density(nztrees, nrep=nr), main="Adaptive estimate") ```

### Example output

```Loading required package: nlme
Loading required package: rpart

spatstat 1.51-0       (nickname: 'Poetic Licence')
For an introduction to spatstat, type 'beginner'

Note: spatstat version 1.51-0 is out of date by more than 14 weeks; a newer version should be available.

PLEASE NOTE:  The components "delsgs" and "summary" of the
object returned by deldir() are now DATA FRAMES rather than
matrices (as they were prior to release 0.0-18).
See help("deldir").

PLEASE NOTE: The process that deldir() uses for determining
duplicated points has changed from that used in version
0.0-9 of this package (and previously). See help("deldir").

Computing 5 intensity estimates...1, 2, 3, 4,  5.
Done.
```

spatstat documentation built on June 15, 2018, 9:04 a.m.