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.

Keywords: central limit theorem, goodness-of-fit test, Höffding decomposition, K-function, point pattern, Poisson process, U-statistic

@article{PS_2013__17__767_0, 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://www.numdam.org/articles/10.1051/ps/2012027/} }

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 SP - 767 EP - 788 VL - 17 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2012027/ DO - 10.1051/ps/2012027 LA - en ID - PS_2013__17__767_0 ER -

%0 Journal Article %A Lang, Gabriel %A Marcon, Eric %T Testing randomness of spatial point patterns with the Ripley statistic %J ESAIM: Probability and Statistics %D 2013 %P 767-788 %V 17 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2012027/ %R 10.1051/ps/2012027 %G en %F PS_2013__17__767_0

Lang, Gabriel; Marcon, Eric. Testing randomness of spatial point patterns with the Ripley statistic. ESAIM: Probability and Statistics, Volume 17 (2013), pp. 767-788. doi : 10.1051/ps/2012027. http://www.numdam.org/articles/10.1051/ps/2012027/

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