Topological time-series analysis with delay-variant embedding.

Identifying the qualitative changes in time-series data provides insights into the dynamics associated with such data. Such qualitative changes can be detected through topological approaches, which first embed the 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 that are extracted using a single time delay are considered to be insufficient for evaluating the aforementioned qualitative changes, even when a well-selected time delay is used. We therefore propose a delay-variant embedding method that constructs the extended topological features by considering the time delay as a variable parameter instead of considering it as a single fixed value. This delay-variant embedding method reveals multiple-timescale patterns in a time series by allowing the observation of the variations in topological features, with the time delay serving as an additional dimension in the topological feature space. We theoretically prove that the constructed topological features are robust when the time series is perturbed by noise. Furthermore, we combine these features with the kernel technique in machine learning algorithms to classify the general time-series data. We demonstrate the effectiveness of our method for classifying the synthetic noisy biological and real time-series data. Our method outperforms a method that is based on a single time delay and, surprisingly, achieves the highest classification accuracy on an average among the standard time-series analysis techniques.