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

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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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References

  • Anderes E, Møller J, Rasmussen JG (2020) Isotropic covariance functions on graphs and their edges. Ann. Stat. (Forthcoming). arXiv:1710.01295)

  • Ang, Q.W., Baddeley, A., Nair, G.: Geometrically corrected second order analysis of events on a linear network, with applications to ecology and criminology. Scand. J. Stat. 39(4), 591–617 (2012)

    MathSciNet  MATH  Google Scholar 

  • Baddeley, A., Turner, R.: spatstat: an R package for analyzing spatial point patterns. J. Stat. Softw. 12(6), 1–42 (2005)

    Google Scholar 

  • Baddeley, A., Møller, J., Waagepetersen, R.: Non-and semi-parametric estimation of interaction in inhomogeneous point patterns. Stat. Neerl. 54(3), 329–350 (2000)

    MathSciNet  MATH  Google Scholar 

  • Baddeley, A., Turner, R., Møller, J., Hazelton, M.: Residual analysis for spatial point processes. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 67(5), 617–666 (2005). (with discussion)

    MathSciNet  MATH  Google Scholar 

  • Baddeley, A., Jammalamadaka, A., Nair, G.: Multitype point process analysis of spines on the dendrite network of a neuron. J. R. Stat. Soc. Ser. C (Appl. Stat.) 63(5), 673–694 (2014)

    MathSciNet  Google Scholar 

  • Baddeley, A., Rubak, E., Turner, R.: Spatial Point Patterns: Methodology and Applications with R. CRC Press, Boca Raton (2015)

    MATH  Google Scholar 

  • Baddeley, A., Nair, G., Rakshit, S., McSwiggan, G.: “Stationary” point processes are uncommon on linear networks. Stat 6(1), 68–78 (2017)

    MathSciNet  Google Scholar 

  • Bartlett, M.: The spectral analysis of two-dimensional point processes. Biometrika 51(3/4), 299–311 (1964)

    MathSciNet  MATH  Google Scholar 

  • Biscio, C.A.N., Poinas, A., Waagepetersen, R.: A note on gaps in proofs of central limit theorems. Stat. Probab. Lett. 135, 7–10 (2018)

    MathSciNet  MATH  Google Scholar 

  • Borruso, G.: Network density estimation: analysis of point patterns over a network. In: Computational Science and Its Applications—ICCSA 2005, pp. 126–132. Springer, Berlin (2005)

  • Borruso, G.: Network density estimation: a GIS approach for analysing point patterns in a network space. Trans. GIS 12(3), 377–402 (2008)

    Google Scholar 

  • Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications. Wiley, New York (2013)

    MATH  Google Scholar 

  • Christensen, H.S., Møller, J.: Modelling Spine Locations on Dendrite Trees Using Inhomogeneous Cox Point Processes. arXiv preprint arXiv:1907.12283 (2019)

  • Cronie, O., van Lieshout, M.N.M.: A J-function for inhomogeneous spatio-temporal point processes. Scand. J. Stat. 42, 562–579 (2015)

    MathSciNet  MATH  Google Scholar 

  • Cronie, O., van Lieshout, M.N.M.: Summary statistics for inhomogeneous marked point processes. Ann. Inst. Stat. Math. 68, 905–928 (2016)

    MathSciNet  MATH  Google Scholar 

  • Cronie, O., van Lieshout, M.N.M.: A non-model-based approach to bandwidth selection for kernel estimators of spatial intensity functions. Biometrika 105(2), 455–462 (2018)

    MathSciNet  MATH  Google Scholar 

  • Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure, 2nd edn. Springer, New York (2008)

    MATH  Google Scholar 

  • Diggle, P.: On parameter estimation and goodness-of-fit testing for spatial point patterns. Biometrics 35(1), 87–101 (1979)

    MATH  Google Scholar 

  • Eckardt, M., Mateu, J.: Point patterns occurring on complex structures in space and space-time: an alternative network approach. J. Comput. Graph. Stat. 27(2), 312–322 (2018)

    MathSciNet  Google Scholar 

  • Iftimi, A., Cronie, O., Montes, F.: Second-order analysis of marked inhomogeneous spatiotemporal point processes: applications to earthquake data. Scandinavian J. Stat. 46(3), 661–685 (2019)

    MathSciNet  MATH  Google Scholar 

  • Illian, J., Penttinen, A., Stoyan, H., Stoyan, D.: Statistical Analysis and Modelling of Spatial Point Patterns. Wiley, New York (2008)

    MATH  Google Scholar 

  • Kallenberg, O.: Random Measures, Theory and Applications. Springer, Berlin (2017)

    MATH  Google Scholar 

  • Levine, N.: Houston, Texas, Metropolitan Traffic Safety Planning Program. Transp. Res. Rec. J. Transp. Res. Board 1969, 92–100 (2006)

    Google Scholar 

  • Levine, N.: A motor vehicle safety planning support system: the Houston experience. In: Geertman, S., Stillwell, J. (eds.) Planning Support Systems Best Practice and New Methods, pp. 93–111. Springer, Dordrecht (2009)

    Google Scholar 

  • Mateu, J., Moradi, M., Cronie, O.:Spatio-temporal point patterns on linear networks: Pseudo-separable intensity estimation. Spatial Stat. (2019). https://doi.org/10.1016/j.spasta.2019.100400

  • McSwiggan, G., Baddeley, A., Nair, G.: Kernel density estimation on a linear network. Scand. J. Stat. 44(2), 324–345 (2017)

    MathSciNet  MATH  Google Scholar 

  • Møller, J., Waagepetersen, R.: Statistical Inference and Simulation for Spatial Point Processes. CRC Press, Boca Raton (2004)

    MATH  Google Scholar 

  • Møller, J., Syversveen, A.R., Waagepetersen, R.P.: Log Gaussian Cox processes. Scand. J. Stat. 25(3), 451–482 (1998)

    MathSciNet  MATH  Google Scholar 

  • Moradi, M.M.: Spatial and Spatio-Temporal Point Patterns on Linear Networks. PhD dissertation, University Jaume I (2018)

  • Moradi, M.M., Mateu, J.: First and second-order characteristics of spatio-temporal point processes on linear networks. J. Comput. Graph. Stat. (2019). https://doi.org/10.1080/10618600.2019.1694524

    Article  Google Scholar 

  • Moradi, M.M., Rodriguez-Cortes, F., Mateu, J.: On kernel-based intensity estimation of spatial point patterns on linear networks. J. Comput. Graph. Stat. 27(2), 302–311 (2018)

    MathSciNet  Google Scholar 

  • Moradi, M.M., Cronie, O., Rubak, E., Lachieze-Rey, R., Mateu, J., Baddeley, A.: Resample-smoothing of Voronoi intensity estimators. Stat. Comput. 29(5), 995–1010 (2019)

    MathSciNet  MATH  Google Scholar 

  • Okabe, A., Sugihara, K.: Spatial Analysis along Networks: Statistical and Computational Methods. Wiley, New York (2012)

    MATH  Google Scholar 

  • Okabe, A., Yamada, I.: The K-function method on a network and its computational implementation. Geogr. Anal. 33(3), 271–290 (2001)

    Google Scholar 

  • Okabe, A., Satoh, T., Sugihara, K.: A kernel density estimation method for networks, its computational method and a GIS-based tool. Int. J. Geogr. Inf. Sci. 23(1), 7–32 (2009)

    Google Scholar 

  • Paloheimo, J.: On a theory of search. Biometrika 58(1), 61–75 (1971)

    MathSciNet  MATH  Google Scholar 

  • Peebles, P.J.E.: The Large-Scale Structure of The Universe. Princeton University Press, Princeton (1980)

    MATH  Google Scholar 

  • R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2020)

  • Rakshit, S., Nair, G., Baddeley, A.: Second-order analysis of point patterns on a network using any distance metric. Spat. Stat. 22, 129–154 (2017)

    MathSciNet  Google Scholar 

  • Rakshit, S., Baddeley, A., Nair, G.: Efficient code for second order analysis of events on a linear network. J. Stat. Softw. 90(1), 1–37 (2019a)

    Google Scholar 

  • Rakshit, S., Davies, T.M., Moradi, M.M., McSwiggan, G., Nair, G., Mateu, J., Baddeley, A.: Fast kernel smoothing of point patterns on a large network using two-dimensional convolution. Int. Stat. Rev. 87(3), 531–556 (2019b)

    MathSciNet  Google Scholar 

  • Scott, D.W.: Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, New York (2015)

    MATH  Google Scholar 

  • van Lieshout, M.N.M.: A J-function for marked point patterns. Ann. Inst. Stat. Math. 58(2), 235 (2006)

    MathSciNet  MATH  Google Scholar 

  • van Lieshout, M.N.M.: A J-function for inhomogeneous point processes. Stat. Neerl. 65(2), 183–201 (2011)

    MathSciNet  Google Scholar 

  • van Lieshout, M.N.M., Baddeley, A.: A nonparametric measure of spatial interaction in point patterns. Stat. Neerl. 50(3), 344–361 (1996)

    MathSciNet  MATH  Google Scholar 

  • Voss, S.: Habitat Preferences and Spatial Dynamics of the Urban Wall Spider: Oecobius Annulipes Lucas. Ph.D. thesis, Honours thesis, Department of Zoology, University of Western Australia (1999)

  • Xie, Z., Yan, J.: Kernel density estimation of traffic accidents in a network space. Comput. Environ. Urban Syst. 32(5), 396–406 (2008)

    Google Scholar 

  • Zessin, H.: The method of moments for random measures. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 62(3), 395–409 (1983)

    MathSciNet  MATH  Google Scholar 

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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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Authors and Affiliations

  1. Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden

    Ottmar Cronie

  2. Biostatistics School of Public Health and Community Medicine Institute of Medicine University of Gothenburg, Gothenburg, Sweden

    Ottmar Cronie

  3. Department of Statistics, Computer Science, and Mathematics, Public University of Navarre, 31006, Pamplona, Spain

    Mehdi Moradi

  4. Department of Mathematics, University Jaume I, Castellón, Spain

    Jorge Mateu

Authors
  1. Ottmar Cronie
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  2. Mehdi Moradi
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  3. Jorge Mateu
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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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