Testing randomness of spatial point patterns with the Ripley statistic

Gabriel Lang; Eric Marcon

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 767-788
  • ISSN: 1292-8100

Abstract

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Aggregation patterns are often visually detected in sets of location data. These clusters may be the result of interesting dynamics or the effect of pure randomness. We build an asymptotically Gaussian test for the hypothesis of randomness corresponding to a homogeneous Poisson point process. We first compute the exact first and second moment of the Ripley K-statistic under the homogeneous Poisson point process model. Then we prove the asymptotic normality of a vector of such statistics for different scales and compute its covariance matrix. From these results, we derive a test statistic that is chi-square distributed. By a Monte-Carlo study, we check that the test is numerically tractable even for large data sets and also correct when only a hundred of points are observed.

How to cite

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Lang, Gabriel, and Marcon, Eric. "Testing randomness of spatial point patterns with the Ripley statistic." ESAIM: Probability and Statistics 17 (2013): 767-788. <http://eudml.org/doc/273635>.

@article{Lang2013,
abstract = {Aggregation patterns are often visually detected in sets of location data. These clusters may be the result of interesting dynamics or the effect of pure randomness. We build an asymptotically Gaussian test for the hypothesis of randomness corresponding to a homogeneous Poisson point process. We first compute the exact first and second moment of the Ripley K-statistic under the homogeneous Poisson point process model. Then we prove the asymptotic normality of a vector of such statistics for different scales and compute its covariance matrix. From these results, we derive a test statistic that is chi-square distributed. By a Monte-Carlo study, we check that the test is numerically tractable even for large data sets and also correct when only a hundred of points are observed.},
author = {Lang, Gabriel, Marcon, Eric},
journal = {ESAIM: Probability and Statistics},
keywords = {central limit theorem; goodness-of-fit test; Höffding decomposition; K-function; point pattern; Poisson process; U-statistic; Höffding decomposition; -function; -statistic},
language = {eng},
pages = {767-788},
publisher = {EDP-Sciences},
title = {Testing randomness of spatial point patterns with the Ripley statistic},
url = {http://eudml.org/doc/273635},
volume = {17},
year = {2013},
}

TY - JOUR
AU - Lang, Gabriel
AU - Marcon, Eric
TI - Testing randomness of spatial point patterns with the Ripley statistic
JO - ESAIM: Probability and Statistics
PY - 2013
PB - EDP-Sciences
VL - 17
SP - 767
EP - 788
AB - Aggregation patterns are often visually detected in sets of location data. These clusters may be the result of interesting dynamics or the effect of pure randomness. We build an asymptotically Gaussian test for the hypothesis of randomness corresponding to a homogeneous Poisson point process. We first compute the exact first and second moment of the Ripley K-statistic under the homogeneous Poisson point process model. Then we prove the asymptotic normality of a vector of such statistics for different scales and compute its covariance matrix. From these results, we derive a test statistic that is chi-square distributed. By a Monte-Carlo study, we check that the test is numerically tractable even for large data sets and also correct when only a hundred of points are observed.
LA - eng
KW - central limit theorem; goodness-of-fit test; Höffding decomposition; K-function; point pattern; Poisson process; U-statistic; Höffding decomposition; -function; -statistic
UR - http://eudml.org/doc/273635
ER -

References

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  1. [1] A.J. Baddeley, M. Kerscher, K. Schladitz and B.T. Scott, Estimating the J function without edge correction. Research report of the department of mathematics, University of Western Australia (1997). Zbl1018.62085
  2. [2] J-M. Bardet, P. Doukhan, G. Lang and N. Ragache, Dependent Lindeberg central limit theorem and some applications. ESAIM: PS 12 (2008) 154–172. Zbl1187.60013MR2374636
  3. [3] S. Bernstein, Quelques remarques sur le théorème limite Liapounoff. C.R. (Dokl.) Acad. Sci. URSS 24 (1939) 3–8. Zbl0022.06104JFM65.0556.01
  4. [4] J.E. Besag, Comments on Ripley’s paper. J. Roy. Statist. Soc. Ser. B39 (1977) 193–195. 
  5. [5] S.N. Chiu, Correction to Koen’s critical values in testing spatial randomness. J. Stat. Comput. Simul.77 (2007) 1001–1004. Zbl1131.62085MR2416479
  6. [6] S.N. Chiu and K.I. Liu, Generalized Cramér-von Mises goodness-of-fit tests for multivariate distributions. Comput. Stat. Data Anal.53 (2009) 3817–3834. Zbl05689138MR2749926
  7. [7] N.A. Cressie, Statistics for spatial data. John Wiley and Sons, New York (1993). Zbl0799.62002MR1239641
  8. [8] P.J. Diggle, Statistical analysis of spatial point patterns. Academic Press, London (1983). Zbl1021.62076MR743593
  9. [9] M. Fromont, B. Laurent and P. Reynaud-Bouret, Adaptive tests of homogeneity for a Poisson process. Ann. I.H.P. (B) 47 (2011) 176–213. Zbl1207.62161MR2779402
  10. [10] P. Grabarnik and S.N. Chiu, Goodness-of-fit test for complete spatial randomness against mixtures of regular and clustured spatial point processes. Biometrika89 (2002) 411–421. Zbl1019.62091MR1913968
  11. [11] J. Gignoux, C. Duby and S. Barot, Comparing the performances of Diggle’s tests of spatial randomness for small samples with and without edge effect correction: application to ecological data. Biometrics55 (1999) 156–164. Zbl1059.62651
  12. [12] Y. Guan, On nonparametric variance estimation for second-order statistics of inhomogeneous spatial point Processes with a known parametric intensity form. J. Am. Stat. Ass.104 (2009) 1482–1491. Zbl1205.62140MR2750573
  13. [13] L.P. Ho and S.N. Chiu, Testing Uniformity of a Spatial Point Pattern. J. Comput. Graph. Stat. 16 2 (2007) 378–398. MR2325346
  14. [14] L. Heinrich, Goodness-of-fit tests for the second moment function of a stationary multidimensional Poisson process. Statistics22 (1991) 245–268. Zbl0809.62075MR1097377
  15. [15] J. Illian, A. Penttinen, H. Stoyan and D. Stoyan, Statistical analysis and modelling of spatial point patterns. Wiley-Interscience, Chichester (2008). Zbl1197.62135MR2384630
  16. [16] C. Koen, Approximate confidence bounds for Ripley’s statistic for random points in a square. Biom. J.33 (1991) 173–177. 
  17. [17] E. Marcon and F. Puech, Evaluating the geographic concentration of industries using distance-based methods. J. Econom. Geogr.3 (2003) 409–428. 
  18. [18] J. Møller and R.P. Waagepetersen, Statistical inference and simulation for spatial point processes, vol. 100 of Monographs on statistics and applied probability. Chapman and Hall/CRC, Boca Raton (2004). Zbl1044.62101MR2004226
  19. [19] R Development Core Team (2012). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. http://www.R-project.org. 
  20. [20] B.D. Ripley, The second-order analysis of stationary point processes. J. Appl. Probab.13 (1976) 255–266. Zbl0364.60087MR402918
  21. [21] B.D. Ripley, Modelling spatial patterns. J. Roy. Statist. Soc. Ser. B 39 2 (1977) 172–212. Zbl0369.60061MR488279
  22. [22] B.D. Ripley, Tests of randomness for spatial point patterns. J. Roy. Statist. Soc. Ser. B 41 3 (1979) 368–374. Zbl0427.62065
  23. [23] B.D. Ripley, Spatial statistics. John Wiley and Sons, New York (1981). Zbl0583.62087MR624436
  24. [24] R. Saunders and G.M. Funk, Poisson limits for a clustering model of Strauss. J. Appl. Probab.14 (1977) 776–784. Zbl0379.60032MR458649
  25. [25] D. Stoyan, W.S. Kendall and J. Mecke, Stochastic geometry and its applications. Akademie-Verlag, Berlin (1987). Zbl0622.60019MR879119
  26. [26] D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields. Methods of Geometrical Statistics. John Wiley and Sons, New York (1994). Zbl0828.62085MR1297125
  27. [27] C.C. Taylor, I.L. Dryden and R. Farnoosh, The K function for nearly regular point processes. Biometrics57 (2000) 224–231. Zbl1209.62210MR1821339
  28. [28] M. Thomas, A generalization of Poisson’s binomial limit for use in ecology. Biometrika36 (1949) 18–25. MR33999
  29. [29] E. Thönnes and M.-C. van Lieshout, A comparative study on the power of van Lieshout and Baddeley’s J function. Biom. J.41 (1999) 721–734. Zbl1055.62507
  30. [30] J.S. Ward and F.J. Ferrandino, New derivation reduces bias and increases power of Ripley’s L index. Ecological Modelling116 (1999) 225–236. 

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