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A Third Order Point Process Characteristic

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  • K. Schladitz
  • A. J. Baddeley

Abstract

Second order characteristics, in particular Ripley's K‐function, are widely used for the statistical analysis of point patterns. We examine a third order analogue, namely the mean number T(r) of r‐close triples of points per unit area. Equivalently this is the expected number of r‐close point pairs in an r‐neighbourhood of the typical point. Various estimators for this function are proposed and compared, and we give an explicit formula for the isotropic edge correction. The theoretical value of T seems to be unobtainable for most point process models apart from the homogeneous Poisson process. However, simulation studies show that the function T discriminates well between different types of point processes. In particular it detects a clear difference between the Poisson process and the Baddeley–Silverman cell process whereas the K‐functions for these processes coincide.

Suggested Citation

  • K. Schladitz & A. J. Baddeley, 2000. "A Third Order Point Process Characteristic," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 657-671, December.
  • Handle: RePEc:bla:scjsta:v:27:y:2000:i:4:p:657-671
    DOI: 10.1111/1467-9469.00214
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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. Jalilian, Abdollah, 2016. "On the higher order product density functions of a Neyman–Scott cluster point process," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 144-150.
    3. Tscheschel, A. & Stoyan, D., 2006. "Statistical reconstruction of random point patterns," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 859-871, November.
    4. Carlos Comas & Jorge Mateu & Aila Särkkä, 2010. "A third‐order point process characteristic for multi‐type point processes," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(1), pages 19-44, February.

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