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Persistent homology (PH) is a recently developed theory in the field ofalgebraic topology to study shapes of datasets. It is an effective dataanalysis tool that is robust to noise and has been widely applied. Wedemonstrate a general pipeline to apply PH to study time series; particularlythe instantaneous heart rate time series for the heart rate variability (HRV)analysis. The first step is capturing the shapes of time series from twodifferent aspects -- {the PH's and hence persistence diagrams of its} sub-levelset and Taken's lag map. Second, we propose a systematic {and computationallyefficient} approach to summarize persistence diagrams, which we coined {\empersistence statistics}. To demonstrate our proposed method, we apply thesetools to the HRV analysis and the sleep-wake, REM-NREM (rapid eyeball movementand non rapid eyeball movement) and sleep-REM-NREM classification problems. Theproposed algorithm is evaluated on three different datasets via thecross-database validation scheme. The performance of our approach is betterthan the state-of-the-art algorithms, and the result is consistent throughoutdifferent datasets.

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