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.
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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