Intensity Estimate of Point Pattern Using Voronoi-Dirichlet Tessellation

Computes an adaptive estimate of the intensity function of a point pattern using the Dirichlet-Voronoi tessellation.

methods, smooth, spatial
densityVoronoi(X, …)

# S3 method for ppp densityVoronoi(X, f = 1, …, counting=FALSE, fixed=FALSE, nrep = 1, verbose=TRUE)


Point pattern dataset (object of class "ppp").


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 determining the pixel resolution of the result.


Logical value specifying the choice of estimation method. See Details.


Logical. If FALSE (the default), the data points are independently randomly thinned, so the number of data points that are retained is random. If TRUE, the number of data points retained is fixed. See Details.


Number of independent repetitions of the randomised procedure.


Logical value indicating whether to print progress reports.


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.


A pixel image (object of class "im") whose values are estimates of the intensity of X.


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, in press.

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

densityVoronoi.lpp, adaptive.density, density.ppp, dirichlet, im.object.

  • densityVoronoi
  • densityVoronoi.ppp
  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")
# }
Documentation reproduced from package spatstat, version 1.59-0, License: GPL (>= 2)

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