A new family of random graphs for testing spatial segregation

The authors discuss a graph-based approach for testing spatial point patterns. This approach falls under the category of data-random graphs, which have been introduced and used for statistical pattern recognition in recent years. The authors address specifically the prob lem of testing complete spatial ran- domness against spatial patterns of segregation or association between two or more classes of points on the plane. To this end, they use a particular type of parameterized random digraph called a proximity catch digraph (PCD) which is based on relative positions of the data points from various classes. The statistic employed is the relative density of the PCD, which is a U -statistic when scaled properly. The authors de- rive the limiting distribution of the relative density, using the standard asympto tic theory of U -statistics. They evaluate the finite-sample performance of their test statistic by Monte C arlo simulations and assess its asymptotic performance via Pitman's asymptotic efficiency, thereby yie lding the optimal parameters for testing. They further stress that their methodology remains valid for data in higher dimensions. In this article, a graph-based approach for testing spatial point patterns is discussed. In the statis- tical literature, the analysis of spatial point patterns in natural populations has been extensively studied and has important implications in epidemiology, population biology, and ecology. The pattern of points from one class with respect to points from other classes, rather than the pattern of points from one class with respect to the ground, is investigated. The spatial relationships among two or more classes have important implications especially for plant species. See, for example, Pielou (1961) and Dixon (1994, 2002).