复曲面上点过程的K函数
$K$-functions for point processes on complex surfaces
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中文总结 AI 辅助
本文针对复曲面上的点过程，提出并对比多种扩展经典K函数的方法，其中表面积K函数表现最优，将欧氏球计数替换为曲面测地球计数，自变量改为面积，通过模拟与真实数据验证性能。
中文摘要 AI 辅助
K函数是评估二维或三维欧氏空间中点过程聚类或规则性的基础汇总统计量。但实际应用中，许多平面点模式源于将三维空间曲面上的物体位置投影到二维空间，例如景观中的事件或物体常被忽略高程信息，这可能导致对生成点模式的点过程性质得出错误结论。将经典K函数扩展到复曲面上的点模式并无唯一方法，本文从理论和计算性质出发，提出、探究并讨论了多种方法。表现最佳的方法被命名为表面积K函数，可视为经典K函数的类似物，将欧氏球内的点计数替换为曲面测地球内的点计数，但重要区别在于，表面积K函数的自变量是面积而非测地球的半径。本文将各类曲面K函数应用于模拟数据和真实数据，对比了它们的性能。
英文摘要
The $K$-function is a fundamental summary statistic for assessing clustering or regularity of point processes in two or three dimensional Euclidean space. In practice, however, many planar point patterns arise from projecting locations of objects on a surface in three dimensional space to two dimensional space. For example, when events or objects occur in a landscape their elevation is often ignored. This can lead to erroneous conclusions regarding properties of the point process generating the point pattern. There is not a unique way to extend the classical $K$-function to point patterns on a complex surface. In this paper we propose, explore, and discuss several approaches in terms of their theoretical and computational properties. The best performing approach, coined the surface area $K$-function, can be viewed is an analogue of the classical $K$-function replacing counts of points in Euclidean balls with counts of points in surface geodesic balls. However, an important distinction is that the argument of our surface area $K$-function is area instead of radius of geodesic balls. The performances of the various surface $K$-functions are compared in applications to simulated and real data.
