Computes an adaptive estimate of the intensity function of a point pattern.
Point pattern dataset (object of class
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
Number of independent repetitions of the randomised procedure.
Logical value indicating whether to print progress reports.
This function is an alternative to
computes an estimate of the intensity function of a point pattern
dataset. The result is a pixel image giving the estimated intensity,
f=1, the Voronoi estimate (Barr and Schoenberg, 2010)
is computed: the point pattern
X is used to construct
a Voronoi/Dirichlet tessellation (see
the areas of the Dirichlet tiles are computed; the estimated intensity
in each tile is the reciprocal of the tile area.
f=0, the intensity estimate at every location is
equal to the average intensity (number of points divided by window area).
f is strictly between 0 and 1,
X is randomly split into two patterns
B containing a fraction
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.
nrep is greater than 1, this randomised procedure is
nrep times, and the results are averaged.
This technique has been used by Ogata et al. (2003), Ogata (2004) and Baddeley (2007).
A pixel image (object of class
"im") whose values are
estimates of the intensity of
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.
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