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Detecting dependence between marks and locations of marked point processes

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  • Martin Schlather
  • Paulo J. Ribeiro
  • Peter J. Diggle

Abstract

Summary. We introduce two characteristics for stationary and isotropic marked point proces‐ ses, E(h) and V(h), and describe their use in investigating mark–point interactions. These quantities are functions of the interpoint distance h and denote the conditional expectation and the conditional variance of a mark respectively, given that there is a further point of the process a distance h away. We present tests based on E and V for the hypothesis that the values of the marks can be modelled by a random field which is independent of the unmarked point process. We apply the methods to two data sets in forestry.

Suggested Citation

  • Martin Schlather & Paulo J. Ribeiro & Peter J. Diggle, 2004. "Detecting dependence between marks and locations of marked point processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 79-93, February.
    • Handle: RePEc:bla:jorssb:v:66:y:2004:i:1:p:79-93
    DOI: 10.1046/j.1369-7412.2003.05343.x
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    Cited by:

    1. Renato Assunção & Alexandra Maia, 2007. "A Note on Testing Separability in Spatial-Temporal Marked Point Processes," Biometrics, The International Biometric Society, vol. 63(1), pages 290-294, March.
    2. Lothar Heinrich & Stella Klein & Martin Moser, 2014. "Empirical Mark Covariance and Product Density Function of Stationary Marked Point Processes—A Survey on Asymptotic Results," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 283-293, June.
    3. Jiří Dvořák & Tomáš Mrkvička & Jorge Mateu & Jonatan A. González, 2022. "Nonparametric Testing of the Dependence Structure Among Points–Marks–Covariates in Spatial Point Patterns," International Statistical Review, International Statistical Institute, vol. 90(3), pages 592-621, December.
    4. Pawlas, Zbynek, 2009. "Empirical distributions in marked point processes," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4194-4209, December.
    5. Peng Shi & Glenn M. Fung & Daniel Dickinson, 2022. "Assessing hail risk for property insurers with a dependent marked point process," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(1), pages 302-328, January.
    6. Renshaw, Eric & Mateu, Jorge & Saura, Fuensanta, 2007. "Disentangling mark/point interaction in marked-point processes," Computational Statistics & Data Analysis, Elsevier, vol. 51(6), pages 3123-3144, March.
    7. Ho, Lai Ping & Stoyan, D., 2008. "Modelling marked point patterns by intensity-marked Cox processes," Statistics & Probability Letters, Elsevier, vol. 78(10), pages 1194-1199, August.
    8. Duncan Lee & Claire Ferguson & E. Marian Scott, 2011. "Constructing representative air quality indicators with measures of uncertainty," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 174(1), pages 109-126, January.
    9. Peter J. Diggle & Raquel Menezes & Ting‐li Su, 2010. "Geostatistical inference under preferential sampling," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 59(2), pages 191-232, March.
    10. Elvan Ceyhan, 2009. "Class‐specific tests of spatial segregation based on nearest neighbor contingency tables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 63(2), pages 149-182, May.
    11. Yongtao Guan, 2006. "Tests for Independence between Marks and Points of a Marked Point Process," Biometrics, The International Biometric Society, vol. 62(1), pages 126-134, March.
    12. Alexander Malinowski & Martin Schlather & Zhengjun Zhang, 2016. "Intrinsically weighted means and non-ergodic marked point processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 1-24, February.
    13. Sarkka, Aila & Renshaw, Eric, 2006. "The analysis of marked point patterns evolving through space and time," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1698-1718, December.
    14. Jakub Staněk & Ondřej Šedivý & Viktor Beneš, 2014. "On Random Marked Sets with a Smaller Integer Dimension," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 397-410, June.
    15. C. Comas & P. Delicado & J. Mateu, 2011. "A second order approach to analyse spatial point patterns with functional marks," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 503-523, November.

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