Inhomogeneous higher-order summary statistics for point processes on linear networks

Abstract

As a workaround for the lack of transitive transformations on linear network structures, which are required to consider different notions of distributional invariance, including stationarity, we introduce the notions of pseudostationarity and intensity reweighted moment pseudostationarity for point processes on linear networks. Moreover, using arbitrary so-called regular linear network distances, e.g. the Euclidean and the shortest-path distance, we further propose geometrically corrected versions of different higher-order summary statistics, including the inhomogeneous empty space function, the inhomogeneous nearest neighbour distance distribution function and the inhomogeneous J-function. Such summary statistics detect interactions of order higher than two. We also discuss their nonparametric estimators and through a simulation study, considering models with different types of spatial interaction and different networks, we study the performance of our proposed summary statistics by means of envelopes. Our summary statistic estimators manage to capture clustering, regularity as well as Poisson process independence. Finally, we make use of our new summary statistics to analyse two different datasets: motor vehicle traffic accidents and spiderwebs.

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Acknowledgements

The authors are grateful to the editor and two referees for constructive comments. J. Mateu is funded by Grant MTM2016-78917-R from the Spanish Ministry of Economy and Competitivity. All our calculations were performed in the R language (R Core Team 2020). Our code is provided as a part of the Electronic Supplementary Material (Online Resource 1). Further, the development version of our code can be found on https://github.com/Moradii/LinearJ.

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Correspondence to Mehdi Moradi.

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Cronie, O., Moradi, M. & Mateu, J. Inhomogeneous higher-order summary statistics for point processes on linear networks. Stat Comput 30, 1221–1239 (2020). https://doi.org/10.1007/s11222-020-09942-w

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Keywords

  • Inhomogeneous linear empty space function
  • Inhomogeneous linear J-function
  • Inhomogeneous linear nearest neighbour distance distribution function
  • Linear network
  • Pseudostationarity
  • Regular distance metric
  • Traffic accident data