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Topological Periodicity Test (TopPT) via Confidence Bound of Time-Delay Embeddings

NeurIPS 2026 · Time series › Causal and generative time series · approach: Dynamical system reconstruction · 0 citations (OpenAlex)

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Proves time-delay embeddings of periodic signals are circle-like and non-periodic ones interval-like with positive reach, then builds a subsampling-based confidence radius for persistence diagrams and a hypothesis test (TopPT) for periodicity with asymptotic error control.

Problem

Periodicity detection from time series via persistent homology of sliding-window embeddings lacks rigorous topological characterization and statistical uncertainty quantification for persistence diagrams.

Approach

Shows sliding-window supports are S^1 for periodic and contractible/interval-like for non-periodic signals with a positive reach lower bound, then applies subsampling Hausdorff confidence radii to H1 persistence features to form a test whose type I/II error is asymptotically controlled, with an interpolation-error correction.

Key results

• On synthetic bounded-error signals, raw TopPT detects both periodic signals (rate 1.000) with zero false rejections on four non-periodic signals, while robust TopPT detects 0.699 and 0.457 of periodic cases.
• Scalar baselines (Lomb-Scargle, Fisher g, permutation) reject all four non-periodic synthetic signals (false rejection rate 1.000).
• TopPT rejects 35/40 Fantasia records (142/280 windows) and 51/53 BIDMC records (249/318 windows).

Surprising findings

• Scalar periodogram baselines reject all structured non-periodic synthetic signals, i.e. they flag false periodicity that TopPT avoids.
• Robust TopPT, which corrects for interpolation error, loses substantial power (0.457-0.699 vs 1.000) on periodic signals.

Limitations

• Theory restricted to globally periodic or globally non-periodic signals; partial recurrence and quasi-periodic dynamics excluded.
• Asymptotic guarantees with fixed m and tau; adaptive period selection on the same data not covered.
• Real-data experiments lack null labels so only detection/localization is assessed.

Experimental setup

Datasets: PhysioNet Fantasia BIDMC PPG and Respiration bounded-error synthetic signals

Benchmarks: bounded-error synthetic signals PhysioNet Fantasia BIDMC respiratory waveforms

Metrics: detection rate false rejection rate rejected records rejected windows

Baselines: Generalized Lomb-Scargle periodogram Fisher's g-test permutation max-periodogram test

Builds on: SW1Pers subsampling confidence bounds for persistence diagrams (Chazal et al.) Generalized Lomb-Scargle periodogram Vietoris-Rips / Cech stability theorem

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• Persistent Homology for High-dimensional Data Based on Spectral Methods (NeurIPS 2024 · similarity 0.87)
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