Resample-smoothing of Voronoi intensity estimators

M. Mehdi Moradi, Ottmar Cronie, Ege Rubak, Raphael Lachieze-Rey, Jorge Mateu, Adrian Baddeley

Research output: Contribution to journalJournal articleResearchpeer-review

21 Citations (Scopus)
8 Downloads (Pure)


Voronoi estimators are non-parametric and adaptive estimators of the intensity of a point process. The intensity estimate at a given location is equal to the reciprocal of the size of the Voronoi/Dirichlet cell containing that location. Their major drawback is that they tend to paradoxically under-smooth the data in regions where the point density of the observed point pattern is high, and over-smooth where the point density is low. To remedy this behaviour, we propose to apply an additional smoothing operation to the Voronoi estimator, based on resampling the point pattern by independent random thinning. Through a simulation study we show that our resample-smoothing technique improves the estimation substantially. In addition, we study statistical properties such as unbiasedness and variance, and propose a rule-of-thumb and a data-driven cross-validation approach to choose the amount of smoothing to apply. Finally we apply our proposed intensity estimation scheme to two datasets: locations of pine saplings (planar point pattern) and motor vehicle traffic accidents (linear network point pattern).
Original languageEnglish
JournalStatistics and Computing
Issue number5
Pages (from-to)995-1010
Number of pages16
Publication statusPublished - 11 Sep 2019


  • Adaptive intensity estimation
  • Independent thinning
  • Machine learning
  • Point process
  • Resampling
  • Voronoi intensity estimator


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