[Paper Review] Topological Data Analysis (TDA) for Time Series

Налини Равишанкер, Renjie Chen|arXiv (Cornell University)|Sep 23, 2019

Topological and Geometric Data Analysis66 references18 citations

TL;DR

This paper presents a comprehensive review of Topological Data Analysis (TDA) for time series, focusing on using persistent homology via Takens’s embedding to extract topological features for classification, clustering, and structural break detection. It demonstrates that TDA-derived features outperform traditional methods like SVM in time series classification and effectively detect critical transitions in financial and biological time series.

ABSTRACT

The study of topology is strictly speaking, a topic in pure mathematics. However in only a few years, Topological Data Analysis (TDA), which refers to methods of utilizing topological features in data (such as connected components, tunnels, voids, etc.) has gained considerable momentum. More recently, TDA is being used to understand time series. This article provides a review of TDA for time series, with examples using R functions. Features derived from TDA are useful in classification and clustering of time series and in detecting breaks in patterns.

Motivation & Objective

• To provide a systematic review of TDA methods tailored for time series analysis, emphasizing practical implementation using R.
• To address the challenge of analyzing complex, non-linear, and non-stationary time series by leveraging topological features that capture shape and persistence.
• To demonstrate the utility of TDA in real-world applications such as financial market analysis, bioinformatics, and motion sensor data.
• To bridge theoretical concepts of computational topology with applied statistical learning in time series contexts.
• To highlight the advantages of TDA over conventional methods in detecting patterns, periodicity, and regime shifts in time series data.

Proposed method

• Transform univariate time series into a point cloud in phase space using Takens’s embedding with delay τ and embedding dimension d.
• Construct a Vietoris-Rips complex from the embedded point cloud to model the topological space at varying scales.
• Compute persistent homology to track the birth and death of topological features (e.g., connected components, loops) across scales, summarized in a persistence diagram.
• Derive summary statistics such as Betti sequences and persistence landscapes from the persistence diagram for downstream machine learning.
• Use L1 norm of persistence landscapes as a scalar feature for clustering and anomaly detection in sliding windows of time series.
• Integrate TDA features into supervised learning pipelines, such as one-dimensional CNNs, for time series classification.

Experimental results

Research questions

• RQ1 How can topological features derived from persistent homology improve time series classification compared to traditional methods?
• RQ2 To what extent can TDA detect structural breaks and critical transitions in non-stationary time series, such as cryptocurrency price data?
• RQ3 How do TDA features perform in capturing periodicity and hidden dynamics in time series with complex or noisy patterns?
• RQ4 What is the role of Takens’s embedding in preserving topological structure when transforming time series into point clouds?
• RQ5 Can TDA-based features be effectively used in unsupervised learning tasks like clustering to identify distinct dynamical regimes?

Key findings

• TDA features extracted via persistent homology outperformed support vector machines (SVM) in time series classification tasks, particularly when combined with one-dimensional convolutional neural networks.
• The L1 norm of the persistence landscape for 1st homology groups successfully identified topologically distinct regimes in cryptocurrency price data, enabling detection of critical transitions before market crashes.
• Structural break detection using sliding windows of embedded time series and persistence landscape features revealed regime shifts in highly erratic financial time series.
• TDA-based methods demonstrated robustness to data transformations and noise, as validated through randomness testing in EEG data analysis.
• Persistent homology on Morse functions derived from weighted Fourier series representations of EEG data preserved topological structure and enabled effective analysis of biological signals.
• The integration of TDA features into deep learning models improved classification accuracy across diverse domains, including motion sensor data and financial time series.