densityVoronoi {spatstat.explore} | R Documentation |
Intensity Estimate of Point Pattern Using Voronoi-Dirichlet Tessellation
Description
Computes an adaptive estimate of the intensity function of a point pattern using the Dirichlet-Voronoi tessellation.
Usage
densityVoronoi(X, ...)
## S3 method for class 'ppp'
densityVoronoi(X, f = 1, ...,
counting=FALSE,
fixed=FALSE,
nrep = 1, verbose=TRUE)
Arguments
X |
Point pattern dataset (object of class |
f |
Fraction (between 0 and 1 inclusive) of the data points that will be used to build a tessellation for the intensity estimate. |
... |
Arguments passed to |
counting |
Logical value specifying the choice of estimation method. See Details. |
fixed |
Logical. If |
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 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 datasetX
is randomly thinned by deleting or retaining each point independently, with probabilityf
of retaining a point. -
if
fixed=TRUE
, a random sample of fixed sizem
is taken from the datasetX
, wherem
is the largest integer less than or equal tof*n
andn
is the number of points inX
.
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 factor1/f
orn/m
as appropriate. to obtain an estimate of the intensity ofX
in the tile.if
counting = TRUE
, the randomly selected subsetA
is used to construct a Dirichlet tessellation, while the complementary subsetB
(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 byA
, we count the number of points ofB
falling in the tile, and divide by the area of the same tile, to obtain an estimate of the intensity of the patternB
in the tile. This estimate is adjusted by1/(1-f)
orn/(n-m)
as appropriate to obtain an estimate of the intensity ofX
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.
Value
A pixel image (object of class "im"
) whose values are
estimates of the intensity of X
.
Author(s)
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner rolfturner@posteo.net and Ege Rubak rubak@math.aau.dk and Mehdi Moradi m2.moradi@yahoo.com.
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.
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.
See Also
adaptive.density
,
density.ppp
,
dirichlet
,
im.object
.
Examples
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")