Multiscale Topological Inference for Marked Point Processes via Euler Characteristic Envelopes

TL;DR

This paper introduces a multiscale topological inference framework for marked point processes that combines Euler Characteristic envelopes with mark-weighted filtrations to detect complex spatial and attribute-dependent structures.

Contribution

It presents a novel integration of topological data analysis with statistical hypothesis testing for marked point processes, including local decomposition for identifying structural features.

Findings

High sensitivity and specificity in detecting attribute-space dependencies

Robustness against purely geometric effects demonstrated through simulations

Effective localization of structural hubs and barriers using Z-scores

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.