The fundamental functional summary statistics used for studying spatial point patterns are developed for marked homogeneous and inhomogeneous point processes on the surface of a sphere. These are extended to point processes on the surface of three dimensional convex shapes given the bijective mapping from the shape to the sphere is known. These functional summary statistics are used to test for independence between the marginals of multi-type spatial point processes with methods for sampling the null distribution proposed and discussed. This is illustrated on both simulated data and the RNGC galaxy point pattern, revealing attractive dependencies between different galaxy types.

翻译：用于研究空间点模式的基本功能性汇总统计量，是针对球面标记齐次与非齐次点过程而发展的。在已知三维凸形体到球面双射映射的条件下，这些统计量被推广至三维凸形体表面的点过程。这些功能性汇总统计量被用于检验多类型空间点过程边缘之间的独立性，文中提出并讨论了零分布抽样的方法。通过模拟数据和RNGC星系点模式的实例分析，揭示了不同星系类型之间存在吸引性依赖关系。