基于比例边邻近捕捉有向图的底层图的相对边密度，用于测试二元空间模式（技术报告）
Relative Edge Density of the Underlying Graphs Based on Proportional-Edge Proximity Catch Digraphs for Testing Bivariate Spatial Patterns (Technical Report)
摘要：
数据随机图在空间模式统计检验中的应用是近年来出现的。在这种方法中，使用来自各种类别的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模式测试。本文中使用的方法是基于一个家庭的数据随机有向图，这是由一个家庭的参数化邻近映射确定的基础图。的AND-和OR-基础图形的相对边缘密度被用作汇总统计量，提供了一种替代的相对弧密度和控制数的有向图先前采用。适当缩放后，底层图的相对边密度是U统计量，便于使用标准U统计量中心极限理论对其渐进分布进行分析研究。该方法说明了应用程序的二元空间聚类模式的隔离和关联的测试。渐近分布的知识允许皮特曼渐近效率的评价，因此选择的邻近映射参数，以优化效率。渐近效率和Monte Carlo模拟分析表明，与-底层版本是更好的（在功率和效率方面）的隔离替代，而或-底层版本是更好的关联替代。这里提出的方法也适用于更高维度的数据。
英文摘要：
The use of data-random graphs in statistical testing of spatial patterns is introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the 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 pattern testing. The approach used in this article is based on underlying graphs of a family of data-random digraphs which is determined by a family of parameterized proximity maps. The relative edge density of the AND- and OR-underlying graphs is used as the summary statistic, providing an alternative to the relative arc density and domination number of the digraph employed previously. Properly scaled, relative edge density of the underlying graphs 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 bivariate spatial clustering patterns of segregation and association. Knowledge of the asymptotic distribution allows evaluation of the Pitman asymptotic efficiency, hence selection of the proximity map parameter to optimize efficiency. Asymptotic efficiency and Monte Carlo simulation analysis indicate that the AND-underlying version is better (in terms of power and efficiency) for the segregation alternative, while the OR-underlying version is better for the association alternative. The approach presented here is also valid for data in higher dimensions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
