Prompted by modern technologies in data acquisition, the statistical analysis of spatially distributed function-valued quantities has attracted a lot of attention in recent years. In particular, combinations of functional variables and spatial point processes yield a highly challenging instance of such modern spatial data applications. Indeed, the analysis of spatial random point configurations, where the point attributes themselves are functions rather than scalar-valued quantities, is just in its infancy, and extensions to function-valued quantities still remain limited. In this view, we extend current existing first- and second-order summary characteristics for real-valued point attributes to the case where in addition to every spatial point location a set of distinct function-valued quantities are available. Providing a flexible treatment of more complex point process scenarios, we build a framework to consider points with multivariate function-valued marks, and develop sets of different cross-function (cross-type and also multi-function cross-type) versions of summary characteristics that allow for the analysis of highly demanding modern spatial point process scenarios. We consider estimators of the theoretical tools and analyse their behaviour through a simulation study and two real data applications.

翻译：受现代数据采集技术的推动，近年来对空间分布函数值量的统计分析引起了广泛关注。特别是，函数变量与空间点过程的组合构成了此类现代空间数据应用中极具挑战性的实例。事实上，对点属性本身为函数而非标量值的空间随机点构型进行分析仍处于起步阶段，且向函数值量的扩展仍然有限。基于此，我们将现有用于实值点属性的一阶和二阶摘要特征扩展至每个空间点位置除自身外还可获取一组不同函数值量的情形。通过提供对更复杂点过程情景的灵活处理，我们构建了一个考虑具有多元函数值标记的点的框架，并开发了不同的交叉函数（交叉类型及多函数交叉类型）版本的摘要特征集，从而能够分析要求严苛的现代空间点过程情景。我们考虑了这些理论工具的估计量，并通过模拟研究及两项实际数据应用分析了其性能。