[Paper Review] Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis

José A. Perea, John Harer|arXiv (Cornell University)|Jul 23, 2013

Topological and Geometric Data Analysis19 references4 citations

TL;DR

This paper introduces SW1PerS, a novel method that uses sliding window embeddings and 1-dimensional persistent homology to detect and quantify periodicity in time series. By showing that maximum persistence corresponds to the signal's natural frequency, it provides a theoretically grounded, topology-driven approach that outperforms traditional methods in identifying periodic and quasi-periodic patterns in synthetic and biological data.

ABSTRACT

We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window (or time-delay) embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets, and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS which stands for Sliding Windows and 1-dimensional Persistence Scoring.

Motivation & Objective

• To develop a topological framework for analyzing time series using sliding window embeddings and persistent homology.
• To establish a theoretical link between maximum persistence in persistence diagrams and the natural frequency of periodic signals.
• To quantify how persistence depends on window size and embedding dimension in sliding window point clouds.
• To provide a new, topology-based method—SW1PerS—for detecting periodicity that is agnostic to signal shape and robust to noise.
• To demonstrate the method's effectiveness on synthetic and biological time series, particularly in gene expression analysis.

Proposed method

• Construct sliding window embeddings of a time series by sampling the signal at regular intervals τ over M+1 consecutive time points, forming a point cloud in ℝᴹ⁺¹.
• Apply 1-dimensional persistent homology to the resulting point cloud to detect circular structures indicative of periodicity.
• Use maximum persistence (mp) as a topological measure of 'roundness' in the point cloud, which correlates with signal periodicity.
• Prove that maximum persistence is maximized when the window size τ matches the signal's period, establishing a theoretical foundation for window selection.
• Derive explicit convergence and approximation theorems for persistence diagrams under perturbations of the signal and window parameters.
• Use Fourier approximation to model the signal and compute the persistence diagram of the resulting parametrized point cloud, enabling analytical treatment of topological features.

Experimental results

Research questions

• RQ1 How does the maximum persistence in a 1D persistence diagram of a sliding window embedding relate to the periodicity of the underlying signal?
• RQ2 What is the dependence of persistence on window size τ and embedding dimension M in sliding window point clouds?
• RQ3 Can topological features derived from sliding window embeddings reliably detect periodicity in noisy or quasi-periodic signals?
• RQ4 How does the theoretical structure of the persistence diagram of a sliding window embedding behave for signals derived from truncated Fourier series?
• RQ5 Can the method be generalized to detect non-sinusoidal periodic patterns, and what are the implications for signal sampling and experimental design?

Key findings

• Maximum persistence (mp) in the 1D persistence diagram of a sliding window embedding is maximized when the window size τ matches the period of the signal, providing a theoretical basis for window selection.
• The maximum persistence satisfies mp(dgm) = 2d_B(dgm, dgm_Δ), where dgm_Δ is the diagonal diagram, linking topological persistence to the bottleneck distance.
• The q-Wasserstein distance W_q(dgm, dgm_Δ) provides a smooth, parameterized family of features for periodicity, with larger q emphasizing larger-scale topological features.
• The method is robust to noise and effective on signals with complex periodic structures, outperforming traditional cosine-based methods in detecting non-sinusoidal periodicity.
• Theoretical analysis shows that the persistence diagram of a sliding window embedding of a truncated Fourier series can be explicitly computed, enabling precise study of topological features.
• The approach reveals that persistent homology with coefficients in ℤ (not just 𝔽₂) is necessary for full topological insight, and the paper identifies which primes to avoid in such computations.