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Residual analysis for spatial point processes (with discussion)

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  • A. Baddeley
  • R. Turner
  • J. Møller
  • M. Hazelton

Abstract

Summary. We define residuals for point process models fitted to spatial point pattern data, and we propose diagnostic plots based on them. The residuals apply to any point process model that has a conditional intensity; the model may exhibit spatial heterogeneity, interpoint interaction and dependence on spatial covariates. Some existing ad hoc methods for model checking (quadrat counts, scan statistic, kernel smoothed intensity and Berman's diagnostic) are recovered as special cases. Diagnostic tools are developed systematically, by using an analogy between our spatial residuals and the usual residuals for (non‐spatial) generalized linear models. The conditional intensity λ plays the role of the mean response. This makes it possible to adapt existing knowledge about model validation for generalized linear models to the spatial point process context, giving recommendations for diagnostic plots. A plot of smoothed residuals against spatial location, or against a spatial covariate, is effective in diagnosing spatial trend or co‐variate effects. Q–Q‐plots of the residuals are effective in diagnosing interpoint interaction.

Suggested Citation

  • A. Baddeley & R. Turner & J. Møller & M. Hazelton, 2005. "Residual analysis for spatial point processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 617-666, November.
    • Handle: RePEc:bla:jorssb:v:67:y:2005:i:5:p:617-666
    DOI: 10.1111/j.1467-9868.2005.00519.x
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    Citations

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    Cited by:

    1. Jean-François Coeurjolly & Ege Rubak, 2013. "Fast Covariance Estimation for Innovations Computed from a Spatial Gibbs Point Process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 669-684, December.
    2. Alex Reinhart & Joel Greenhouse, 2018. "Self‐exciting point processes with spatial covariates: modelling the dynamics of crime," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 67(5), pages 1305-1329, November.
    3. A. Baddeley & J. Møller & A. Pakes, 2008. "Properties of residuals for spatial point processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(3), pages 627-649, September.
    4. Nicoletta D’Angelo & Marianna Siino & Antonino D’Alessandro & Giada Adelfio, 2022. "Local spatial log-Gaussian Cox processes for seismic data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(4), pages 633-671, December.
    5. Andrew J Edelman, 2012. "Positive Interactions between Desert Granivores: Localized Facilitation of Harvester Ants by Kangaroo Rats," PLOS ONE, Public Library of Science, vol. 7(2), pages 1-9, February.
    6. Amanda S. Hering & Sean Bair, 2014. "Characterizing spatial and chronological target selection of serial offenders," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 63(1), pages 123-140, January.
    7. Jean-François Coeurjolly, 2017. "Median-based estimation of the intensity of a spatial point process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 303-331, April.
    8. Tonglin Zhang & Ge Lin, 2009. "Cluster Detection Based on Spatial Associations and Iterated Residuals in Generalized Linear Mixed Models," Biometrics, The International Biometric Society, vol. 65(2), pages 353-360, June.
    9. Shaochuan Lu, 2012. "Markov modulated Poisson process associated with state-dependent marks and its applications to the deep earthquakes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(1), pages 87-106, February.
    10. Coeurjolly, Jean-François, 2015. "Almost sure behavior of functionals of stationary Gibbs point processes," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 241-246.
    11. Coeurjolly, Jean-François & Reynaud-Bouret, Patricia, 2019. "A concentration inequality for inhomogeneous Neyman–Scott point processes," Statistics & Probability Letters, Elsevier, vol. 148(C), pages 30-34.
    12. Giada Adelfio & Marcello Chiodi, 2021. "Including covariates in a space-time point process with application to seismicity," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(3), pages 947-971, September.
    13. Daniel, Jeffrey & Horrocks, Julie & Umphrey, Gary J., 2018. "Penalized composite likelihoods for inhomogeneous Gibbs point process models," Computational Statistics & Data Analysis, Elsevier, vol. 124(C), pages 104-116.
    14. Lothar Heinrich & Stella Klein, 2014. "Central limit theorems for empirical product densities of stationary point processes," Statistical Inference for Stochastic Processes, Springer, vol. 17(2), pages 121-138, July.
    15. Kenneth A. Flagg & Andrew Hoegh & John J. Borkowski, 2020. "Modeling Partially Surveyed Point Process Data: Inferring Spatial Point Intensity of Geomagnetic Anomalies," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 25(2), pages 186-205, June.
    16. D'Angelo, Nicoletta & Adelfio, Giada & Mateu, Jorge, 2023. "Locally weighted minimum contrast estimation for spatio-temporal log-Gaussian Cox processes," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).
    17. Jesper Møller & Carlos Díaz‐Avalos, 2010. "Structured Spatio‐Temporal Shot‐Noise Cox Point Process Models, with a View to Modelling Forest Fires," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(1), pages 2-25, March.
    18. Yongtao Guan, 2008. "Variance estimation for statistics computed from inhomogeneous spatial point processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 175-190, February.
    19. Dereudre, D. & Lavancier, F., 2011. "Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 498-519, January.
    20. Janine B. Illian & David F. R. P. Burslem, 2017. "Improving the usability of spatial point process methodology: an interdisciplinary dialogue between statistics and ecology," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 101(4), pages 495-520, October.
    21. Yehua Li & Yongtao Guan, 2014. "Functional Principal Component Analysis of Spatiotemporal Point Processes With Applications in Disease Surveillance," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1205-1215, September.
    22. Heinrich Lothar & Klein Stella, 2011. "Central limit theorem for the integrated squared error of the empirical second-order product density and goodness-of-fit tests for stationary point processes," Statistics & Risk Modeling, De Gruyter, vol. 28(4), pages 359-387, December.
    23. Tonglin Zhang & Ge Lin, 2008. "Identification of local clusters for count data: a model-based Moran's I test," Journal of Applied Statistics, Taylor & Francis Journals, vol. 35(3), pages 293-306.
    24. Miguel Gómez-Antonio & Stuart Sweeney, 2021. "Testing the role of intra-metropolitan local factors on knowledge-intensive industries’ location choices," The Annals of Regional Science, Springer;Western Regional Science Association, vol. 66(3), pages 699-728, June.
    25. Ian W. Renner & David I. Warton, 2013. "Equivalence of MAXENT and Poisson Point Process Models for Species Distribution Modeling in Ecology," Biometrics, The International Biometric Society, vol. 69(1), pages 274-281, March.

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