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

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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.

DOI : 10.1051/ps/2012027
Classification : 60G55, 60F05, 62F03
Keywords: central limit theorem, goodness-of-fit test, Höffding decomposition, K-function, point pattern, Poisson process, U-statistic 
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     author = {Lang, Gabriel and Marcon, Eric},
     title = {Testing randomness of spatial point patterns with the {Ripley} statistic},
     journal = {ESAIM: Probability and Statistics},
     pages = {767--788},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2012027},
     mrnumber = {3126161},
     language = {en},
     url = {http://geodesic.mathdoc.fr/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

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