Beyond scalar-valued marks - Statistical analysis of object-valued marked spatial point processes
Final Report Abstract
The goal of the Walter Benjamin project Beyond scalar-valued marks - Statistical analysis of objectvalued marked spatial point processes was to extend the present literature of marked spatial point processes to non-scalar mark scenarios by introducing a new generic class of object-valued marks. In particular, we developed different mark summary characteristics and mark-weighted point characteristics for the analysis of planar point processes with non-scalar marks including compositions, functions, compositional functions and graphs as specific applications. All characteristics were defined through extended test functions and, as such, provide a direct link to the present methodological literature. Apart from auto-type characteristics we also considered cross-type mark summary characteristics, which could be used to analyse the mark variation or correlation between different type of points (e.g. tree species), and cross-type cross-mark characteristics, which control for both different types of points and at least two distinct object-valued marks (e.g. two distinct functions). In addition to the planar case, we also considered marked point processes on spatially embedded relational systems where the point locations are constrained to the edges (e.g. roads, trails, dendrites) of the network under study. After extending classic mark summary characteristics for real-valued marks to linear networks, we developed tools for network-constrained points with function-valued marks. All the developed tools allow to investigate the average pairwise variation or correlation of the marks as a function of the interpoint distance r for either planar or network-constrained data scenarios. To decide on deviations from the independent mark scenario, we further enriched all developed summary characteristics with global envelopes. The developed tools were implemented as open source solutions in R. We applied our methods to a wide range of data sets from scientific collaborations ranging from forestry and medicine through urban economics to automated system data.
