Subsampling Confidence Bound for Persistent Diagram via Time-delay Embedding

TL;DR

This work develops a topology-driven framework for detecting periodicity in time-series data via time-delay embeddings. It shows that the sliding-window embedding of a periodic signal is homotopy equivalent to $S^1$ while non-periodic signals are contractible, and it establishes a lower bound on the embedding's reach to ensure stable topological features. A subsampling-based confidence bound is derived for persistence diagrams, with asymptotic guarantees and a practical algorithm to compute a correction $c_\alpha$; a statistically valid periodicity test is also formulated. Through simulations and a BIDMC real-data study, the method demonstrates competitive periodicity detection against GLS and improved robustness to time-varying frequencies, offering a principled, uncertainty-aware alternative for topological time-series analysis.

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.

Figures (2)

• Figure 1: Example of $(\Xi,\epsilon)$-non-periodic function
• Figure 2: Diagram of Hypothesis Testing

Theorems & Definitions (41)

• Definition 2.1
• Theorem 2.2: Fasy_2014 Theorem 3
• Definition 3.1
• Definition 3.2
• Theorem 4.1
• Corollary 4.2
• Theorem 4.3
• Corollary 4.4
• Theorem 5.1
• Theorem 5.2