Marked spatial point processes: current state and extensions to point processes on linear networks

TL;DR

This paper reviews current methods for analyzing marked spatial point processes in Euclidean space and introduces new tools for processes on linear networks, highlighting the importance of network structure in understanding mark interactions.

Contribution

It extends existing marked point process analysis to linear networks by proposing novel summary characteristics and demonstrates their importance through simulations and real-world applications.

Findings

Ignoring network structure leads to incorrect conclusions about mark interactions.

New summary characteristics improve analysis of marked point processes on networks.

Applications show the relevance of network-aware analysis in ecological and urban studies.

Abstract

Within the applications of spatial point processes, it is increasingly becoming common that events are labeled 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. Second, we 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 types of butterflies in Melbourne, Australia, and the locations of public trees along the street network of Vancouver, Canada, where trees are labeled by their diameters at breast height.