Testing randomness of spatial point patterns with the Ripley statistic
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 767-788.

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.

DOI : 10.1051/ps/2012027
Classification : 60G55, 60F05, 62F03
Mots-clés : central limit theorem, goodness-of-fit test, Höffding decomposition, K-function, point pattern, Poisson process, U-statistic
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     title = {Testing randomness of spatial point patterns with the {Ripley} statistic},
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     url = {http://www.numdam.org/articles/10.1051/ps/2012027/}
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Lang, Gabriel; Marcon, Eric. Testing randomness of spatial point patterns with the Ripley statistic. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 767-788. doi : 10.1051/ps/2012027. http://www.numdam.org/articles/10.1051/ps/2012027/

[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). | Zbl

[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. | Numdam | MR | Zbl

[3] S. Bernstein, Quelques remarques sur le théorème limite Liapounoff. C.R. (Dokl.) Acad. Sci. URSS 24 (1939) 3-8. | JFM

[4] J.E. Besag, Comments on Ripley's paper. J. Roy. Statist. Soc. Ser. B 39 (1977) 193-195.

[5] S.N. Chiu, Correction to Koen's critical values in testing spatial randomness. J. Stat. Comput. Simul. 77 (2007) 1001-1004. | MR | Zbl

[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. | MR

[7] N.A. Cressie, Statistics for spatial data. John Wiley and Sons, New York (1993). | MR | Zbl

[8] P.J. Diggle, Statistical analysis of spatial point patterns. Academic Press, London (1983). | MR | Zbl

[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. | Numdam | MR | Zbl

[10] P. Grabarnik and S.N. Chiu, Goodness-of-fit test for complete spatial randomness against mixtures of regular and clustured spatial point processes. Biometrika 89 (2002) 411-421. | MR | Zbl

[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. Biometrics 55 (1999) 156-164. | Zbl

[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. | MR | Zbl

[13] L.P. Ho and S.N. Chiu, Testing Uniformity of a Spatial Point Pattern. J. Comput. Graph. Stat. 16 2 (2007) 378-398. | MR

[14] L. Heinrich, Goodness-of-fit tests for the second moment function of a stationary multidimensional Poisson process. Statistics 22 (1991) 245-268. | MR | Zbl

[15] J. Illian, A. Penttinen, H. Stoyan and D. Stoyan, Statistical analysis and modelling of spatial point patterns. Wiley-Interscience, Chichester (2008). | MR | Zbl

[16] C. Koen, Approximate confidence bounds for Ripley's statistic for random points in a square. Biom. J. 33 (1991) 173-177.

[17] E. Marcon and F. Puech, Evaluating the geographic concentration of industries using distance-based methods. J. Econom. Geogr. 3 (2003) 409-428.

[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). | MR | Zbl

[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] B.D. Ripley, The second-order analysis of stationary point processes. J. Appl. Probab. 13 (1976) 255-266. | MR | Zbl

[21] B.D. Ripley, Modelling spatial patterns. J. Roy. Statist. Soc. Ser. B 39 2 (1977) 172-212. | MR | Zbl

[22] B.D. Ripley, Tests of randomness for spatial point patterns. J. Roy. Statist. Soc. Ser. B 41 3 (1979) 368-374. | Zbl

[23] B.D. Ripley, Spatial statistics. John Wiley and Sons, New York (1981). | MR | Zbl

[24] R. Saunders and G.M. Funk, Poisson limits for a clustering model of Strauss. J. Appl. Probab. 14 (1977) 776-784. | MR | Zbl

[25] D. Stoyan, W.S. Kendall and J. Mecke, Stochastic geometry and its applications. Akademie-Verlag, Berlin (1987). | MR | Zbl

[26] D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields. Methods of Geometrical Statistics. John Wiley and Sons, New York (1994). | MR | Zbl

[27] C.C. Taylor, I.L. Dryden and R. Farnoosh, The K function for nearly regular point processes. Biometrics 57 (2000) 224-231. | MR | Zbl

[28] M. Thomas, A generalization of Poisson's binomial limit for use in ecology. Biometrika 36 (1949) 18-25. | MR

[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. | Zbl

[30] J.S. Ward and F.J. Ferrandino, New derivation reduces bias and increases power of Ripley's L index. Ecological Modelling 116 (1999) 225-236.

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