# A persistent homology approach to heart rate variability analysis with an application to sleep-wake classification

arXiv: https://arxiv.org/abs/1908.06856 · HTML: https://mlsanity.com/paper/1908.06856/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Aug 2019.

## Built on by (most cited first, 12 of 12)

- [Stability and Machine Learning Applications of Persistent Homology Using the Delaunay-Rips Complex](https://mlsanity.com/paper/2303.01501/) (2023)
- [Synthetic Data Generation and Deep Learning for the Topological Analysis of 3D Data](https://mlsanity.com/paper/2309.16968/) (2023)
- [Time-Series Forecasting via Topological Information Supervised Framework with Efficient Topological Feature Learning](https://mlsanity.com/paper/2503.23757/) (2025)
- [Time Series Classification via Topological Data Analysis](https://mlsanity.com/paper/2102.01956/) (2021)
- [Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measures](https://mlsanity.com/paper/2104.01672/) (2021)
- [Persistent Homology of Coarse Grained State Space Networks](https://mlsanity.com/paper/2206.02530/) (2022)
- [Emotion recognition in talking-face videos using persistent entropy and neural networks](https://mlsanity.com/paper/2110.13571/) (2021)
- [A Pipeline for Data-Driven Learning of Topological Features with Applications to Protein Stability Prediction](https://mlsanity.com/paper/2408.04847/) (2024)
- [Two-Stage Hierarchical and Explainable Feature Selection Framework for Dimensionality Reduction in Sleep Staging](https://mlsanity.com/paper/2409.00565/) (2024)
- [Notes on Applicability of Explainable AI Methods to Machine Learning Models Using Features Extracted by Persistent Homology](https://mlsanity.com/paper/2310.09780/) (2023)
- [Topology-enhanced machine learning for speech signal processing](https://mlsanity.com/paper/2311.15210/) (2023)
- [Challenges in 3D Data Synthesis for Training Neural Networks on Topological Features](https://mlsanity.com/paper/2511.04972/) (2025)

## Abstract (arXiv)

Persistent homology (PH) 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 PH 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 PH's 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 em 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.