Multiscale Topological Inference for Marked Point Processes via Euler Characteristic Envelopes

2026-05-14

AI Summary

Traditional approaches struggle to effectively capture the high-order topological structures and nonlinear interactions between marks and spatial locations in point processes. This work proposes a multiscale topological inference framework that reconstructs the metric space via exponential mark-weighted distances and integrates Euler characteristic curves with nonparametric global envelope tests for statistical inference. By introducing mark-weighted filtrations to modulate connectivity within homogeneous neighborhoods and complementing this with local Z-score decomposition at critical scales, the method enables precise localization of topological signals. It represents the first organic integration of topological data analysis with classical point process statistics, demonstrating high sensitivity and specificity to attribute–spatial dependencies across diverse simulation scenarios while remaining robust to purely geometric effects.

Abstract

The statistical analysis of marked point processes requires disentangling complex spatial arrangements from attribute-dependent interactions. While classical summary statistics are effective for second-order dependencies, they frequently fail to capture higher-order topological structures and non-linear interactions between marks and space. In this work, we propose a novel multiscale topological inference framework for marked point processes by integrating mark-weighted filtrations with Euler Characteristic envelopes. We redefine the underlying metric space using an exponential mark-weighted distance, which modulates connectivity based on attribute similarity, effectively accelerating the merger of connected components among homophilic neighbors. To ensure rigorous statistical inference, we apply non-parametric global envelope tests to the resulting Euler Characteristic Curves, allowing for formal hypothesis testing against the null model of random labeling. Furthermore, we introduce a local decomposition of the topological signal via Z-scores at the critical filtration scale to identify and localize structural hubs and topological barriers. Systematic simulations across various scenarios demonstrate the framework's high specificity and sensitivity to attribute-space dependencies while remaining robust against purely geometric effects. This methodology provides a comprehensive and interpretable toolkit for identifying, quantifying, and localizing complex structural dependencies in marked spatial data, bridging the gap between topological data analysis and classical point process statistics.

Problem

• marked point processes
• topological inference
• Euler characteristic
• higher-order structures
• attribute-space dependencies

Innovation

• marked point processes
• topological data analysis
• Euler characteristic envelopes
• mark-weighted filtration
• global envelope test