Multiscale Topological Inference for Marked Point Processes via Euler Characteristic Envelopes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918501581586432 |
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| author | Eckardt, Matthias Moradi, Mehdi |
| author_facet | Eckardt, Matthias Moradi, Mehdi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14647 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiscale Topological Inference for Marked Point Processes via Euler Characteristic Envelopes Eckardt, Matthias Moradi, Mehdi Methodology Applications 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. |
| title | Multiscale Topological Inference for Marked Point Processes via Euler Characteristic Envelopes |
| topic | Methodology Applications |
| url | https://arxiv.org/abs/2605.14647 |