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A nonparametric measure of spatial interaction in point patterns

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  • M. N. M. van Lieshout
  • A. J. Baddeley

Abstract

The strength and range of interpoint interactions in a spatial point process can be quantified by the function J = (1 ‐ G)/(1 ‐ F), where G is the nearest‐neighbour distance distribution function and F the empty space function of the process. J(r) is identically equal to 1 for a Poisson process; values of J(r) smaller or larger than 1 indicate clustering or regularity, respectively. We show that, for a large class of point processes, J(r) is constant for distances r greater than the range of spatial interaction. Hence both the range and type of interaction can be inferred from J without parametric model assumptions. It is also possible to evaluate J(r) explicitly for many point process models, so that J is also useful for parameter estimation. Various properties are derived, including the fact that the J function of the superposition of independent point processes is a weighted mean of the J functions of the individual processes. Estimators of J can be constructed from standard estimators of F and G. We compute estimates of J for several standard point pattern datasets and implement a Monte Carlo test for complete spatial randomness.

Suggested Citation

  • M. N. M. van Lieshout & A. J. Baddeley, 1996. "A nonparametric measure of spatial interaction in point patterns," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 50(3), pages 344-361, November.
  • Handle: RePEc:bla:stanee:v:50:y:1996:i:3:p:344-361
    DOI: 10.1111/j.1467-9574.1996.tb01501.x
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    Cited by:

    1. Wu, Liu-Cang & Li, Hui-Qiong, 2009. "Summary statistics for measuring the relationship among three types of points in multivariate point patterns," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 2809-2816, June.
    2. Sze Him Leung & Ji Meng Loh & Chun Yip Yau & Zhengyuan Zhu, 2021. "Spatial Sampling Design Using Generalized Neyman–Scott Process," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 26(1), pages 105-127, March.
    3. repec:jss:jstsof:12:i06 is not listed on IDEAS
    4. Ushio Tanaka & Yosihiko Ogata, 2014. "Identification and estimation of superposed Neyman–Scott spatial cluster processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 687-702, August.
    5. Ninna Vihrs & Jesper Møller & Alan E. Gelfand, 2022. "Approximate Bayesian inference for a spatial point process model exhibiting regularity and random aggregation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 185-210, March.
    6. M. Lieshout, 2006. "A J-Function for Marked Point Patterns," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(2), pages 235-259, June.
    7. Jonatan A. González & Bernardo M. Lagos-Álvarez & Jorge Mateu, 2021. "Two-way layout factorial experiments of spatial point pattern responses in mineral flotation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(4), pages 1046-1075, December.

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