Inhomogeneous mark correlation functions for general marked point processes

Abstract: Spatial phenomena in environmental and biological contexts often involve events that are unevenly distributed across space and carry attributes, whose associations/variations are space-dependent. In this paper, we introduce the class of inhomogeneous mark correlation functions, capturing mark associations/variations, while explicitly accounting for the spatial inhomogeneity of events. The proposed functions are designed to quantify how, on average, marks vary or associate with one another as a function of pairwise spatial distances. We develop nonparametric estimators and evaluate their performance through simulation studies covering a range of scenarios with mark association or variation, spanning from nonstationary point patterns without spatial interaction to those characterised by clustering tendencies. Our simulations reveal the shortcomings of traditional methods in the presence of spatial inhomogeneity, underscoring the necessity of our approach. Furthermore, the results show that our estimators accurately identify both the positivity/negativity and effective spatial range for detected mark associations/variations. The proposed inhomogeneous mark correlation functions are then applied to two distinct forest ecosystems: Longleaf pine trees in southern Georgia, USA, marked by their diameter at breast height, and Scots pine trees in Pfynwald, Switzerland, marked by their height. Our findings reveal that the inhomogeneous mark correlation functions provide deeper and more detailed insights into tree growth patterns compared to traditional methods