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A J -function for Inhomogeneous Spatio-temporal Point Processes

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  • O. Cronie
  • M. N. M. Van Lieshout

Abstract

type="main" xml:id="sjos12123-abs-0001"> We propose a new summary statistic for inhomogeneous intensity-reweighted moment stationarity spatio-temporal point processes. The statistic is defined in terms of the n-point correlation functions of the point process, and it generalizes the J-function when stationarity is assumed. We show that our statistic can be represented in terms of the generating functional and that it is related to the spatio-temporal K-function. We further discuss its explicit form under some specific model assumptions and derive ratio-unbiased estimators. We finally illustrate the use of our statistic in practice. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics

Suggested Citation

  • O. Cronie & M. N. M. Van Lieshout, 2015. "A J -function for Inhomogeneous Spatio-temporal Point Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 562-579, June.
    • Handle: RePEc:bla:scjsta:v:42:y:2015:i:2:p:562-579
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    File URL: http://hdl.handle.net/10.1111/sjos.12123
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    References listed on IDEAS

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    1. A. J. Baddeley & J. Møller & R. Waagepetersen, 2000. "Non‐ and semi‐parametric estimation of interaction in inhomogeneous point patterns," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 54(3), pages 329-350, November.
    2. Cronie, Ottmar & Särkkä, Aila, 2011. "Some edge correction methods for marked spatio-temporal point process models," Computational Statistics & Data Analysis, Elsevier, vol. 55(7), pages 2209-2220, July.
    3. Grabarnik, Pavel & Myllymäki, Mari & Stoyan, Dietrich, 2011. "Correct testing of mark independence for marked point patterns," Ecological Modelling, Elsevier, vol. 222(23), pages 3888-3894.
    4. Marie-Colette N. M. Lieshout, 2012. "On Estimation of the Intensity Function of a Point Process," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 567-578, September.
    5. Edith Gabriel & Peter J. Diggle, 2009. "Second‐order analysis of inhomogeneous spatio‐temporal point process data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 63(1), pages 43-51, February.
    6. Gabriel, Edith & Rowlingson, Barry S. & Diggle, Peter J., 2013. "stpp: An R Package for Plotting, Simulating and Analyzing Spatio-Temporal Point Patterns," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 53(i02).
    7. Yosihiko Ogata, 1998. "Space-Time Point-Process Models for Earthquake Occurrences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(2), pages 379-402, June.
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    Cited by:

    1. Eckardt, Matthias & González, Jonatan A. & Mateu, Jorge, 2021. "Graphical modelling and partial characteristics for multitype and multivariate-marked spatio-temporal point processes," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    2. Mohammad Ghorbani & Ottmar Cronie & Jorge Mateu & Jun Yu, 2021. "Functional marked point processes: a natural structure to unify spatio-temporal frameworks and to analyse dependent functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 529-568, September.
    3. Ottmar Cronie & Julia Jansson & Konstantinos Konstantinou, 2024. "Discussion of the Paper “Marked Spatial Point Processes: Current State and Extensions to Point Processes on Linear Networks”," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 29(2), pages 379-388, June.
    4. Ghorbani, Mohammad & Vafaei, Nafiseh & Dvořák, Jiří & Myllymäki, Mari, 2021. "Testing the first-order separability hypothesis for spatio-temporal point patterns," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).

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