Subsampling Confidence Bound for Persistent Diagram via Time-delay Embedding

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.

💡 Research Summary

The paper addresses a fundamental gap in topological data analysis (TDA) for time‑series: while sliding‑window (time‑delay) embeddings and persistent homology can reveal cyclic structures, there has been no rigorous statistical framework to quantify uncertainty in the resulting persistence diagrams. The authors make three major contributions. First, they provide a precise topological characterization of sliding‑window embeddings. Under mild smoothness assumptions (f∈C²) and appropriate choices of embedding dimension m and delay τ, they prove that a truly periodic signal yields an embedding trajectory that is homotopy equivalent to a circle S¹, whereas a non‑periodic signal yields a contractible (interval‑like) trajectory. Moreover, they establish a lower bound on the reach of the embedded manifold, guaranteeing that the persistent homology of the distance‑filtration is stable against perturbations. Second, they adapt the subsampling confidence‑set methodology of Fasy et al. (2014) to persistence diagrams derived from time‑delay embeddings. By drawing subsamples of size b (with b→∞ and b/n→0) without replacement, computing the Hausdorff distance between each subsample point cloud and the full sample, and inverting the empirical tail function, they obtain a data‑driven confidence radius cα such that P(dB(ĤP,P) > cα) ≤ α + o(1). This provides an asymptotically valid (1‑α) confidence region for the true diagram of the underlying manifold. Third, they combine the topological insight and the confidence radius to construct a hypothesis test for periodicity. The test statistic is the maximal persistence (death‑birth) of any 1‑dimensional feature that exceeds the confidence radius. If such a feature exists, the null hypothesis of non‑periodicity is rejected. They prove that the test controls type‑I error at level α and, under the regularity conditions, achieves consistency (type‑II error → 0) as the sample size grows. The paper also details practical guidelines for choosing (m,τ) and the subsample size b, and discusses how the tubular neighborhoods of the embedding inherit the same homotopy type, ensuring that the persistence of loops is not an artifact of a particular scale. Experimental validation includes synthetic sine waves, noisy sine waves, and chirp signals with time‑varying frequency, as well as real electrocardiogram data from the BIDMC dataset. In simulations, the proposed method matches the detection power of the Generalized Lomb‑Scargle periodogram on pure periodic data, while dramatically reducing false positives on non‑periodic, frequency‑modulated signals. On the BIDMC data, the method successfully identifies normal cardiac cycles (persistent loops) and distinguishes them from arrhythmic recordings where the loops disappear within the confidence bound. Overall, the work bridges TDA and statistical inference, delivering a principled, robust, and theoretically grounded tool for periodicity detection in time‑series, and opens avenues for extensions to multivariate series, non‑stationary dynamics, and online implementations.