Intensity Estimate of Point Pattern Using Tessellation

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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.

Author(s)

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

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