Topological time-series analysis with delay-variant embedding

Identification of qualitative changes in time-series data offers insights into the dynamics behind these data. Such changes can be detected through topological approaches, which first embed these data into a high-dimensional space using a time-delay parameter and subsequently extract topological features describing the shape of the data from the embedded points. However, the essential topological features extracted at a single time-delay value are not sufficient for evaluating qualitative changes, even when well-chosen time-delay values are used. We therefore propose a delay-variant embedding scheme that constructs the extended topological features by regarding the time delay as a variable parameter, rather than as a single fixed value. This method reveals multiple-time-scale patterns in a time series by allowing observation of variations in topological features, with time delay serving as an extra dimension in topological-feature space. We theoretically prove the constructed topological features to be robust under noise perturbation of the time series. Furthermore, we combine these features with the kernel technique in the machine-learning algorithm to classify general time-series data. We demonstrate the effectiveness of this method in the classification of synthetic noisy biological and real electrocardiogram data. Our method outperforms that based on a single time-delay value and, surprisingly, achieves the highest classification accuracy on average among standard time-series analysis techniques.

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