Beyond scalar-valued marks - Statistical analysis of object-valued marked spatial point processes

虽然近年来在标记空间点过程的分析方面取得了一些进展，但许多现代现实世界空间点过程场景的统计调查和表征仍然面临着巨大的挑战。许多现代的，通常是高维的，应用程序越来越需要调查的标记空间点的过程设置（i）多个重合点的位置报告和（ii）的属性远远超出标量值的情况下。特别是，由于巨大的技术进步和存储容量的增加，研究人员现在面临着越来越多的更复杂的空间点处理场景，其中点和/或标记位于更复杂的域上。 虽然扩展的空间点过程的时空点过程设置，网络结构和领域已被提出，在最近几年，目前的文献集中在很大程度上只扩展的点，但没有标记。因此，空间点处理场景的扩展仍然非常有限，其中标记本身是生活在更复杂空间（即对象值标记）而不是标量值数量上的数据对象。特别是，仍然缺乏相应的和计算效率的扩展二阶标记的特点，这样的非标量标记scenaries.The项目的目标是推进领域的标记空间点过程与对象值标记无论是在理论和现代真实的数据分析问题的适用方法。不同的技术，结构分析和量化的潜在平面相互依存关系的对象值标记的空间点过程在不同的域将开发和实施，远远超出了目前的理论和方法治疗标记的空间点过程。除了引入对象值标记空间点过程的一般理论外，主要关注的是二阶标记特征的扩展，包括例如标记相关函数或标记变差函数。从逐点标记特征的发展开始，将引入基于与不同对象值标记之间的接近度相关的度量和半度量距离的特征。所开发的工具将允许调查和量化的点属性之间的成对关系，同时占在平面区域或网络结构的点的分布研究。所有开发都将作为开源实现提供，并应用于不同的空间数据来源。

While there has been some progress in the analysis of marked spatial point processes within recent years, the statistical investigation and characterisation of many modern real-world spatial point process scenarios still poses substantial challenges. Many modern, commonly high dimensional, applications increasingly require the investigation of marked spatial point process settings where (i) multiple coincident point locations are reported and (ii) the attributes go far beyond the scalar-valued case. In particular, due to the immense technological progress and increase in storage capacities, researchers are now facing an ever-increasing amount of even more complex spatial point process scenarios, where the points and/ or the marks live on more complex domains. While extensions of spatial point processes to spatio-temporal point process settings, network structures and the sphere have been proposed in recent years, the present literature has so far focussed to a large extend only on extensions for the points but not the marks. In consequence, extensions of spatial point process scenarios where the marks themselves are data objects living on more complicated spaces (i.e. object-valued marks) rather than scalar-valued quantities, remain extremely limited. In particular, there is still a lack of corresponding and computationally efficient extensions of second-order mark characteristics to such non-scalar mark scenarios.The goal of this project is to advance the field of marked spatial point processes with object-valued marks both in terms of theory and in terms of applicable methods for modern real data analysis problems. Different techniques for the structural analysis and quantification of potential planar interdependencies for object-valued marked spatial point processes on different domains will be developed and implemented that go well beyond the present theoretical and methodology treatment of marked spatial point processes. Apart from the introduction of a general theory of object-valued marked spatial point processes, the main focus is put on extensions of second-order mark characteristics including e.g. the mark correlation function or the mark variogram to the present context. Starting with the development of point-wise mark characteristics, characteristics based on metric and semimetric distances related to the proximity between distinct object-valued marks will be introduced. The developed tools will allow to investigate and quantify the pairwise relationships between the point attributes while accounting for the distribution of the points over the planar region or the network structure under study. All developments will be provided as open source implementations and be applied to different sources of spatial data.