Ward Scott, Cohen Edward A K, Adams Niall
Department of Mathematics, Imperial College London, South Kensington, London, SW7 2AZ, United Kingdom.
Data Science Institute, Imperial College London, South Kensington, London, SW7 2AZ, United Kingdom.
Spat Stat. 2021 Mar;41:100489. doi: 10.1016/j.spasta.2020.100489.
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 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 construct a test statistics to determine whether an observed pattern exhibits complete spatial randomness or spatial preference on the original convex space. We compare this test statistic with one constructed from an analogue -function for inhomogeneous point processes on the sphere. A study of the Type I and II errors of our test statistics are explored through simulations on ellipsoids of varying dimensions.
目前，对于位于物体表面的点模式，理论上存在空白，研究人员主要关注位于欧几里得空间中的模式，通常是平面和空间数据。因此，平面和空间数据的方法依赖于欧几里得几何，所以不适用于分析在非欧几里得空间中观察到的点模式。最近，对球面上点模式的分析有了扩展，然而，许多其他形状仍未被探索。部分原因在于，由于缺乏旋转和平移等距性，为存在于此类空间中的点过程定义 概念具有挑战性。在这里，我们为三维凸形状上定义的泊松过程构建功能汇总统计量。利用映射定理，泊松过程可以从任何凸形状转换为单位球面上具有旋转对称性的泊松过程，这使得可以构建功能汇总统计量。我们展示了此类汇总统计量的一阶和二阶属性，并演示了如何使用它们来构建检验统计量，以确定观察到的模式在原始凸空间上是否表现出完全空间随机性或空间偏好。我们将这个检验统计量与从球面上非齐次点过程的类似 函数构建的检验统计量进行比较。通过对不同维度椭球体的模拟，探索了我们检验统计量的I型和II型错误。
