The Stationarity Test Of The Spatial Point Process

For testing stationarity of a given spatial point pattern,Guan(2008)proposed a model-free statistic,based on the deviations between observed and expected counts of points in expanding regions within the sampling window.On the strong mixing and weak dependence condition,the resulting test statistic converges in distribution to a functional of a two-dimensional Brownian motion.Chiu(2008)extends Guan's method to a general class of statistics by incorporating also such information when points are projected to the axes and by allowing different ways to construct regions in which the deviations are considered.The limiting distributions of the new statistics can be expressed in terms of integrals of a Brownian sheet and hence asymptotic critical values can be approximated.When applied to the longleaf pine data where Guan's test gave an inconclusive answer,Chiu's test indica.te a clear rejection of the stationarity hypothesis.We weaken the strong mixing condition used in Guan's test to ergodic condition,transform the general spatial point process into poisson point process through thinning and scale transformation,and test the stationarity.Then we use L-function to align the Poisson point process,and get the asymptotic critical value according to the nature of Poisson point process.Through the simulation of Poisson aggregation process and simple exclusion process,a more reasonable test effect is obtained.Applying it to the longleaf pine data,we also explicitly reject the stationarity hypothesis.