Marked Spatial Point Processes: Current State and Extensions to Point Processes on Linear Networks

Within the applications of spatial point processes, it is increasingly becoming common that events are labelled by marks, prompting an exploration beyond the spatial distribution of events by incorporating the marks in the undertaken analysis. In this paper, we first consider marked spatial point processes in ℝ^2 , where marks are either integer-valued, real-valued, or object-valued, and review the state-of-the-art to analyze the spatial structure and type of interaction/correlation between marks. More specifically, we review cross/dot-type summary characteristics, mark-weighted summary characteristics, various mark correlation functions, and frequency domain approaches. We also propose novel cross/dot-type higher-order summary characteristics, mark-weighted summary characteristics, and mark correlation functions for marked point processes on linear networks. Through a simulation study, we show that ignoring the underlying network gives rise to erroneous conclusions about the interaction/correlation between marks. Finally, we consider two applications: the locations of two different proteins on the membranes of cells infected with the influenza virus and the locations of public trees along the street network of Vancouver, Canada, where trees are labelled by their diameters at breast height.