A Persistent Homology Approach to Heart Rate Variability Analysis With an Application to Sleep-Wake Classification.

Yu-Min Chung,Hau-Tieng Wu,Yu-Lun Lo,Chuan-Shen Hu

doi:10.3389/fphys.2021.637684

Abstract

Persistent homology is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general pipeline to apply persistent homology to study time series, particularly the instantaneous heart rate time series for the heart rate variability (HRV) analysis. The first step is capturing the shapes of time series from two different aspects—the persistent homologies and hence persistence diagrams of its sub-level set and Taken's lag map. Second, we propose a systematic and computationally efficient approach to summarize persistence diagrams, which we coined persistence statistics. To demonstrate our proposed method, we apply these tools to the HRV analysis and the sleep-wake, REM-NREM (rapid eyeball movement and non rapid eyeball movement) and sleep-REM-NREM classification problems. The proposed algorithm is evaluated on three different datasets via the cross-database validation scheme. The performance of our approach is better than the state-of-the-art algorithms, and the result is consistent throughout different datasets.

Highlights

Heart rate variability (HRV) is the physiological phenomenon of variation in the lengths of consecutive cardiac cycles, or the rhythm of heart rate (Draghici and Taylor, 2016)
We mention that compared with existing topological data analysis (TDA) approach for time series analysis, our proposed persistence statistics features based on both sub-level set and Vietoris-Rips complexes filtrations are intuitive, straightforward to implement, and computationally efficient
We remark that while we focus on the heart rate variability (HRV) and sleep stage classification, the result indicates the potential of applying TDA-based approaches to study other complicated time series

Summary

INTRODUCTION

Heart rate variability (HRV) is the physiological phenomenon of variation in the lengths of consecutive cardiac cycles, or the rhythm of heart rate (Draghici and Taylor, 2016). The proposed TDA tool for HRV analysis has been applied to differentiating patients with the history of transient ischemic attack and hypertension In this article, motivated by the complicated interaction among different physiological systems over various scales and inter-individual variability, the need for a useful tool for the HRV analysis, and the numerical limitation of the recently developed TDA tools, we hypothesize that topological information could be useful to quantify the HRV, and propose a computationally efficient approach to analyze time series via TDA

Our Contribution

Application—Sleep Dynamics

Organization

MATHEMATICAL BACKGROUND

Simplicial Complexes

Homology and Betti Numbers

Persistent Homology

Persistence Diagram

Data Analysis With Persistence Diagram and Commonly Considered TDA Statistics

TDA FOR TIME SERIES ANALYSIS AND FEATURES EXTRACTION

First Useful Filtration—Sub-Level Set Filtration

Second Useful Filtration—Vietoris-Rips Complexes Filtration