Testing Uniformity of a Spatial Point Pattern

For a spatial point pattern observed in a bounded window, we propose using discrepancies, which are measures of uniformity in the quasi-Monte Carlo method, to test the complete spatial randomness hypothesis. Tests using these discrepancies are in fact goodness-of-fit tests for uniform distribution. The discrepancies are free from edge effects and, unlike the popular maximum absolute pointwise difference statistic of a summary function over a suitably chosen range, do not have an arbitrary parameter. Simulation studies show that they are often more powerful when a given pattern is a realization of a process with long-range interaction or a nonstationary process.

[1]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[2]  Tony Warnock,et al.  Computational investigations of low-discrepancy point-sets. , 1972 .

[3]  Debashis Kushary,et al.  Bootstrap Methods and Their Application , 2000, Technometrics.

[4]  B. Ripley.  Tests of 'Randomness' for Spatial Point Patterns , 1979 .

[5]  Fred J. Hickernell,et al.  A generalized discrepancy and quadrature error bound , 1998, Math. Comput..

[6]  E. Thönnes,et al.  A comparative study on the power of van Lieshout and Baddeley's J-function , 1998 .

[7]  J. Symanzik.  Statistical Analysis of Spatial Point Patterns (2nd ed.) , 2005 .

[8]  F. Kelly,et al.  A note on Strauss's model for clustering , 1976 .

[9]  D. Stoyan,et al.  Stochastic Geometry and Its Applications , 1989 .

[10]  Ikuho Yamada,et al.  An Empirical Comparison of Edge Effect Correction Methods Applied to K -function Analysis , 2003 .

[11]  J. Møller,et al.  Statistical Inference and Simulation for Spatial Point Processes , 2003 .

[12]  Dale L. Zimmerman,et al.  A Bivariate Cramer-von Mises Type of Test for Spatial Randomness , 1993 .

[13]  F. J. Hickernell.  What Affects the Accuracy of Quasi-Monte Carlo Quadrature? , 2000 .

[14]  H. Keng,et al.  Applications of number theory to numerical analysis , 1981 .

[15]  C. Duby,et al.  Comparing the Performances of Diggle's Tests of Spatial Randomness for Small Samples with and without Edge‐Effect Correction: Application to Ecological Data , 1999, Biometrics.

[16]  F. J. Hickernell.  Lattice rules: how well do they measure up? in random and quasi-random point sets , 1998 .

[17]  L. Heinrich.  On a test of randomness of spatial point patterns , 1984 .

[18]  P. Haase.  Spatial pattern analysis in ecology based on Ripley's K-function: Introduction and methods of edge correction , 1995 .

[19]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[20]  Fred J. Hickernell,et al.  Goodness-of-fit statistics, discrepancies and robust designs , 1999 .

[21]  D. J. Strauss.  A model for clustering , 1975 .

[22]  van Marie-Colette Lieshout,et al.  Markov Point Processes and Their Applications , 2000 .

[23]  H. Weyl.  Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

[24]  Jürgen Symanzik,et al.  Statistical Analysis of Spatial Point Patterns , 2005, Technometrics.

[25]  Peter J. Diggle,et al.  On parameter estimation and goodness-of-fit testing for spatial point patterns , 1979 .