$K$-functions for point processes on complex surfaces

2026-08-21

AI Summary

本文针对复杂表面上点过程的聚类或规则性评估问题，提出并探讨了几种扩展经典K-函数的方法，其中表现最佳的是基于表面面积的K-函数。

Abstract

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.

Problem

• point processes
• complex surfaces
• K-function
• projection
• geodesic balls

Innovation

• surface area K-function
• geodesic balls
• point processes on complex surfaces