随机$r$ -因子邻近捕捉有向图检验空间分布格局的相对密度
Relative Density of the Random $r$-Factor Proximity Catch Digraph for Testing Spatial Patterns of Segregation and Association
摘要：
基于数据随机图的统计模式分类方法是近年来出现的一种新方法。在这种方法中，使用来自各种类别的数据点的相对位置从数据构造随机有向图。不同的随机图由与每个数据点相关联的邻近区域的不同定义产生，并且不同的图统计可以用于数据简化。在这篇文章中使用的方法是基于一个参数化的家庭的邻近映射确定一个相关的家庭的数据随机有向图。有向图的相对弧密度被用作汇总统计量，提供了一种替代以前采用的控制数。相对弧密度的一个重要优点是，适当地重新缩放，它是一个$U$ -统计，便于分析研究其渐近分布使用标准的$U$ -统计中心极限理论。该方法说明了应用程序的隔离和关联的空间模式的测试。渐近分布的知识允许Pitman和Hodges-Lehmann渐近功效的评估，以及邻近图参数的选择以优化效率。此外，这里提出的方法还具有对任何维度的数据有效的优点。
英文摘要：
Statistical pattern classification methods based on data-random graphs were introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the data points from various classes. Different random graphs result from different definitions of the proximity region associated with each data point and different graph statistics can be employed for data reduction. The approach used in this article is based on a parameterized family of proximity maps determining an associated family of data-random digraphs. The relative arc density of the digraph is used as the summary statistic, providing an alternative to the domination number employed previously. An important advantage of the relative arc density is that, properly re-scaled, it is a $U$-statistic, facilitating analytic study of its asymptotic distribution using standard $U$-statistic central limit theory. The approach is illustrated with an application to the testing of spatial patterns of segregation and association. Knowledge of the asymptotic distribution allows evaluation of the Pitman and Hodges-Lehmann asymptotic efficacies, and selection of the proximity map parameter to optimize efficiency. Furthermore the approach presented here also has the advantage of validity for data in any dimension.
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