Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T13:24:44.991Z Has data issue: false hasContentIssue false
  • English
  • Français

Testing randomness of spatial point patterns with the Ripleystatistic

Published online by Cambridge University Press:  04 November 2013

Gabriel Lang  and
Eric Marcon
Gabriel Lang
Affiliation:
AgroParisTech, UMR 518 Mathématique et Informatique Appliquées, 19 avenue du Maine, 75732 Paris Cedex 15, France. [email protected]
Eric Marcon
Affiliation:
AgroParisTech, UMR 745 Ecologie des Forêts de Guyane, Campus agronomique BP 316, 97379 Kourou Cedex, France; [email protected]
Get access

Abstract

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

Keywords

Central limit theoremgoodness-of-fit testHöffding decompositionK-functionpoint patternPoisson processU-statistic
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 767 - 788
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
Bardet, J-M., Doukhan, P., Lang, G. and Ragache, N., Dependent Lindeberg central limit theorem and some applications. ESAIM: PS 12 (2008) 154–172. Google Scholar
Bernstein, S., Quelques remarques sur le théorème limite Liapounoff. C.R. (Dokl.) Acad. Sci. URSS 24 (1939) 3–8. Google Scholar
Besag, J.E., Comments on Ripley’s paper. J. Roy. Statist. Soc. Ser. B 39 (1977) 193–195. Google Scholar
Chiu, S.N., Correction to Koen’s critical values in testing spatial randomness. J. Stat. Comput. Simul. 77 (2007) 1001–1004. Google Scholar
Chiu, S.N. and Liu, K.I., Generalized Cramér-von Mises goodness-of-fit tests for multivariate distributions. Comput. Stat. Data Anal. 53 (2009) 3817–3834. Google Scholar
N.A. Cressie, Statistics for spatial data. John Wiley and Sons, New York (1993).
P.J. Diggle, Statistical analysis of spatial point patterns. Academic Press, London (1983).
Fromont, M., Laurent, B. and Reynaud-Bouret, P., Adaptive tests of homogeneity for a Poisson process. Ann. I.H.P. (B) 47 (2011) 176–213. Google Scholar
Grabarnik, P. and Chiu, S.N., Goodness-of-fit test for complete spatial randomness against mixtures of regular and clustured spatial point processes. Biometrika 89 (2002) 411–421. Google Scholar
Gignoux, J., Duby, C. and Barot, S., 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. Google ScholarPubMed
Guan, Y., 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. Google ScholarPubMed
Ho, L.P. and Chiu, S.N., Testing Uniformity of a Spatial Point Pattern. J. Comput. Graph. Stat. 16 2 (2007) 378–398. Google Scholar
Heinrich, L., Goodness-of-fit tests for the second moment function of a stationary multidimensional Poisson process. Statistics 22 (1991) 245–268. Google Scholar
J. Illian, A. Penttinen, H. Stoyan and D. Stoyan, Statistical analysis and modelling of spatial point patterns. Wiley-Interscience, Chichester (2008).
Koen, C., Approximate confidence bounds for Ripley’s statistic for random points in a square. Biom. J. 33 (1991) 173–177. Google Scholar
Marcon, E. and Puech, F., Evaluating the geographic concentration of industries using distance-based methods. J. Econom. Geogr. 3 (2003) 409–428. Google Scholar
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).
R Development Core Team (2012). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. http://www.R-project.org.
Ripley, B.D., The second-order analysis of stationary point processes. J. Appl. Probab. 13 (1976) 255–266. Google Scholar
Ripley, B.D., Modelling spatial patterns. J. Roy. Statist. Soc. Ser. B 39 2 (1977) 172–212. Google Scholar
Ripley, B.D., Tests of randomness for spatial point patterns. J. Roy. Statist. Soc. Ser. B 41 3 (1979) 368–374. Google Scholar
B.D. Ripley, Spatial statistics. John Wiley and Sons, New York (1981).
Saunders, R. and Funk, G.M., Poisson limits for a clustering model of Strauss. J. Appl. Probab. 14 (1977) 776–784. Google Scholar
D. Stoyan, W.S. Kendall and J. Mecke, Stochastic geometry and its applications. Akademie-Verlag, Berlin (1987).
D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields. Methods of Geometrical Statistics. John Wiley and Sons, New York (1994).
Taylor, C.C., Dryden, I.L. and Farnoosh, R., The K function for nearly regular point processes. Biometrics 57 (2000) 224–231. Google Scholar
Thomas, M., A generalization of Poisson’s binomial limit for use in ecology. Biometrika 36 (1949) 18–25. Google Scholar
Thönnes, E. and van Lieshout, M.-C., A comparative study on the power of van Lieshout and Baddeley’s J function. Biom. J. 41 (1999) 721–734. Google Scholar
Ward, J.S. and Ferrandino, F.J., New derivation reduces bias and increases power of Ripley’s L index. Ecological Modelling 116 (1999) 225–236. Google Scholar