论文标题
测试在三维有限的凸形上的完全空间随机性
Testing for complete spatial randomness on three dimensional bounded convex shapes
论文作者
论文摘要
目前，理论上存在一个位于对象表面上的点模式的差距，研究人员专注于位于欧几里得空间（通常是平面和空间数据）中的模式。因此，平面数据和空间数据的方法依赖于欧几里得的几何形状，因此不适合分析在非欧几里得空间中观察到的点模式。最近，对球体上的点模式的分析已经扩展，但是，许多其他形状尚未探索。这部分是由于由于缺乏旋转和翻译异构体缺乏旋转和翻译异构体而定义在此类空间上存在的积分过程的平稳性概念的挑战。在这里，我们为在三个维度上定义的载体过程中构建功能汇总统计信息。使用映射定理，可以将泊松过程从任何凸形形状转换为单位球上的泊松过程，该过程具有旋转对称性，允许构建功能性摘要统计信息。我们介绍了此类摘要统计数据的第一阶和二阶属性，并证明了它们如何用于测试观察到的模式是在原始凸空间上表现出完全的空间随机性还是空间偏好。通过对不同维度的椭圆形的模拟来探讨对我们测试统计型的I型和II级错误的研究。
There is currently a gap in theory for point patterns that lie on the surface of objects, with researchers focusing on patterns that lie in a Euclidean space, typically planar and spatial data. Methodology for planar and spatial data thus relies on Euclidean geometry and is therefore inappropriate for analysis of point patterns observed in non-Euclidean spaces. Recently, there has been extensions to the analysis of point patterns on a sphere, however, many other shapes are left unexplored. This is in part due to the challenge of defining the notion of stationarity for a point process existing on such a space due to the lack of rotational and translational isometries. Here, we construct functional summary statistics for Poisson processes defined on convex shapes in three dimensions. Using the Mapping Theorem, a Poisson process can be transformed from any convex shape to a Poisson process on the unit sphere which has rotational symmetries that allow for functional summary statistics to be constructed. We present the first and second order properties of such summary statistics and demonstrate how they can be used to test whether an observed pattern exhibits complete spatial randomness or spatial preference on the original convex space. A study of the Type I and II errors of our test statistics are explored through simulations on ellipsoids of varying dimensions.
