Subsampling Confidence Bound for Persistent Diagram via Time-delay Embedding

Abstract

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.