Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.06324 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908977216880640 |
|---|---|
| author | Park, Donghyun An, Junhyun Kim, Taehyoung Kim, Jisu |
| author_facet | Park, Donghyun An, Junhyun Kim, Taehyoung Kim, Jisu |
| contents | Time-delay embedding is a fundamental technique in Topological Data Analysis (TDA) for reconstructing the phase space dynamics of time-series data. Persistent homology effectively identifies global topological features, such as loops associated with periodicity. Nevertheless, a statistically rigorous way to quantify uncertainty in the resulting topological features has remained underdeveloped -- a problem that we aim to challenge. First, we analyze the topological characterization of time-delay embeddings under both periodic and non-periodic conditions. Precisely, the embedded trajectory is homotopy equivalent to a circle ($S^1$) for periodic signals and is contractible for non-periodic ones. We also prove that the reach of the sliding window embedding is lower-bounded, ensuring stable persistence features. Next, we propose a subsampling-based method to construct confidence bounds for persistence diagrams derived from time-delay embeddings. Specifically, we derive confidence bounds with asymptotic guarantees, under the assumption that the support satisfies standard manifold regularity. Integrating the results, we propose a statistical testing framework to determine the periodicity of the underlying sampling function. This framework provides a principled statistical test for periodicity with asymptotically controlled type I and type II error rates. Simulation studies demonstrate that our method achieves detection performance comparable to the Generalized Lomb-Scargle Periodogram on periodic data while exhibiting superior robustness in distinguishing non-periodic signals with time-varying frequencies, such as chirp signals. Finally, it successfully captured the periodicity when applied to the BIDMC dataset. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06324 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subsampling Confidence Bound for Persistent Diagram via Time-delay Embedding Park, Donghyun An, Junhyun Kim, Taehyoung Kim, Jisu Statistics Theory Time-delay embedding is a fundamental technique in Topological Data Analysis (TDA) for reconstructing the phase space dynamics of time-series data. Persistent homology effectively identifies global topological features, such as loops associated with periodicity. Nevertheless, a statistically rigorous way to quantify uncertainty in the resulting topological features has remained underdeveloped -- a problem that we aim to challenge. First, we analyze the topological characterization of time-delay embeddings under both periodic and non-periodic conditions. Precisely, the embedded trajectory is homotopy equivalent to a circle ($S^1$) for periodic signals and is contractible for non-periodic ones. We also prove that the reach of the sliding window embedding is lower-bounded, ensuring stable persistence features. Next, we propose a subsampling-based method to construct confidence bounds for persistence diagrams derived from time-delay embeddings. Specifically, we derive confidence bounds with asymptotic guarantees, under the assumption that the support satisfies standard manifold regularity. Integrating the results, we propose a statistical testing framework to determine the periodicity of the underlying sampling function. This framework provides a principled statistical test for periodicity with asymptotically controlled type I and type II error rates. Simulation studies demonstrate that our method achieves detection performance comparable to the Generalized Lomb-Scargle Periodogram on periodic data while exhibiting superior robustness in distinguishing non-periodic signals with time-varying frequencies, such as chirp signals. Finally, it successfully captured the periodicity when applied to the BIDMC dataset. |
| title | Subsampling Confidence Bound for Persistent Diagram via Time-delay Embedding |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2512.06324 |