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

1 Citation (Scopus)

Abstract

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
Number of pages16
ISSN0960-3174
DOIs
Publication statusE-pub ahead of print - 19 Jan 2019

Fingerprint

Voronoi
Smoothing
Estimator
Linear networks
Highway accidents
Adaptive Estimator
Unbiasedness
Smoothing Techniques
Thinning
Nonparametric Estimator
Resampling
Point Process
Accidents
Data-driven
Cross-validation
Statistical property
Dirichlet
Choose
Traffic
Simulation Study

Keywords

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

Cite this

Moradi, M. Mehdi ; Cronie, Ottmar ; Rubak, Ege ; Lachieze-Rey, Raphael ; Mateu, Jorge ; Baddeley, Adrian. / Resample-smoothing of Voronoi intensity estimators. In: Statistics and Computing. 2019.
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abstract = "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).",
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Resample-smoothing of Voronoi intensity estimators. / Moradi, M. Mehdi; Cronie, Ottmar; Rubak, Ege; Lachieze-Rey, Raphael; Mateu, Jorge; Baddeley, Adrian.

In: Statistics and Computing, 19.01.2019.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Resample-smoothing of Voronoi intensity estimators

AU - Moradi, M. Mehdi

AU - Cronie, Ottmar

AU - Rubak, Ege

AU - Lachieze-Rey, Raphael

AU - Mateu, Jorge

AU - Baddeley, Adrian

PY - 2019/1/19

Y1 - 2019/1/19

N2 - 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).

AB - 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).

KW - Adaptive intensity estimation

KW - Independent thinning

KW - Machine learning

KW - Point process

KW - Resampling

KW - Voronoi intensity estimator

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