Inhomogeneous mark correlation functions for general marked point processes

2025-05-30

AI Summary

Spatial events in ecological and environmental data often exhibit non-uniform distributions and carry attributes (e.g., diameter-at-breast-height, tree height), rendering quantification of mark dependence or variation challenging under conventional stationary assumptions. Method: We propose the first **non-stationary mark correlation function** for general marked point processes, explicitly modeling spatial non-stationarity while simultaneously identifying the direction (positive/negative) and effective range of mark dependence. The method employs a non-parametric estimation framework. Contribution/Results: Through simulation studies, we demonstrate its superior accuracy over existing approaches. Applied to two real-world forest datasets—longleaf pine in the U.S. and Scots pine in Switzerland—the method uncovers finer-scale spatial dependence structures in tree growth. Compared with traditional stationary methods, it significantly improves estimation precision and ecological interpretability, offering a novel statistical tool for analyzing forest spatial patterns and underlying ecological processes.

Technology Category

• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Kernel Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

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

Problem

• Quantify space-dependent mark associations in unevenly distributed events
• Develop estimators for mark variations in nonstationary spatial patterns
• Analyze tree growth patterns using inhomogeneous mark correlation functions

Innovation

• Introduces inhomogeneous mark correlation functions
• Develops nonparametric estimators for mark associations
• Applies functions to analyze tree growth patterns