Inhomogeneous mark correlation functions for general marked point processes

Spatial phenomena in environmental and biological contexts often involve events that are unevenly distributed across space, carrying attributes whose associations/variations are space-dependent. In this paper, we introduce the class of inhomogeneous mark correlation functions, which capture mark associations/variations while explicitly accounting for spatial inhomogeneity. The proposed functions 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 patterns with clustering tendencies and sparse regions. Our simulations reveal the shortcomings of traditional methods under spatial inhomogeneity, underscoring the necessity of our approach. The results show that our estimators accurately identify both the positivity/negativity and the effective spatial range for detected mark associations/variations. Furthermore, we show that differences in how intensity is estimated generally have only a negligible influence on the empirical bias/variance of our proposed inhomogeneous mark correlation functions. 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 more detailed insights into tree growth patterns compared to traditional methods.