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Fast Kernel Smoothing of Point Patterns on a Large Network using Two‐dimensional Convolution

Author

Listed:
  • Suman Rakshit
  • Tilman Davies
  • M. Mehdi Moradi
  • Greg McSwiggan
  • Gopalan Nair
  • Jorge Mateu
  • Adrian Baddeley

Abstract

We propose a computationally efficient and statistically principled method for kernel smoothing of point pattern data on a linear network. The point locations, and the network itself, are convolved with a two‐dimensional kernel and then combined into an intensity function on the network. This can be computed rapidly using the fast Fourier transform, even on large networks and for large bandwidths, and is robust against errors in network geometry. The estimator is consistent, and its statistical efficiency is only slightly suboptimal. We discuss bias, variance, asymptotics, bandwidth selection, variance estimation, relative risk estimation and adaptive smoothing. The methods are used to analyse spatially varying frequency of traffic accidents in Western Australia and the relative risk of different types of traffic accidents in Medellín, Colombia.

Suggested Citation

  • Suman Rakshit & Tilman Davies & M. Mehdi Moradi & Greg McSwiggan & Gopalan Nair & Jorge Mateu & Adrian Baddeley, 2019. "Fast Kernel Smoothing of Point Patterns on a Large Network using Two‐dimensional Convolution," International Statistical Review, International Statistical Institute, vol. 87(3), pages 531-556, December.
    • Handle: RePEc:bla:istatr:v:87:y:2019:i:3:p:531-556
    DOI: 10.1111/insr.12327
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    Cited by:

    1. Federico Ferraccioli & Eleonora Arnone & Livio Finos & James O. Ramsay & Laura M. Sangalli, 2021. "Nonparametric density estimation over complicated domains," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(2), pages 346-368, April.
    2. Andrea Gilardi & Jorge Mateu & Riccardo Borgoni & Robin Lovelace, 2022. "Multivariate hierarchical analysis of car crashes data considering a spatial network lattice," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(3), pages 1150-1177, July.
    3. S Ward & H S Battey & E A K Cohen, 2023. "Nonparametric estimation of the intensity function of a spatial point process on a Riemannian manifold," Biometrika, Biometrika Trust, vol. 110(4), pages 1009-1021.
    4. Arnone, Eleonora & Ferraccioli, Federico & Pigolotti, Clara & Sangalli, Laura M., 2022. "A roughness penalty approach to estimate densities over two-dimensional manifolds," Computational Statistics & Data Analysis, Elsevier, vol. 174(C).
    5. Ottmar Cronie & Mehdi Moradi & Christophe A N Biscio, 2024. "A cross-validation-based statistical theory for point processes," Biometrika, Biometrika Trust, vol. 111(2), pages 625-641.

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