Fast Topological Data Analysis Features for Nonstationary Time Series

Topological Data Analysis (TDA) combines methods from algebraic topology and modern mathematics to provide features enabling the characterization of point clouds. Often, these features are used with a machine learning algorithm to resolve classification problems. While results in the literature show that TDA features can be powerful in analyzing complex systems, their extraction requires the construction of persistence diagrams or charts that are computationally demanding. Our primary focus is the real-time identification of nonstationary time series in the sub-millisecond or high-rate realm, where TDA is difficult to apply. Here, we propose a novel approach to TDA feature extraction termed “Fast TDA.” Fast TDA consists of extracting features from time series inspired by traditional TDA methods. Our method discovers 0-dimensional holes (H0) through the Euclidean distances between consecutive neighbors in the point cloud, and 1-dimensional holes (H1) from the minor axis of ellipses fitted to the point cloud. This reduces computational time compared to conventional TDA by 90%. Our method’s accuracy and computing cost are analyzed using laboratory datasets extracted from the Dynamic Reproduction of Projectiles in Ballistic Environments for Advanced Research (DROPBEAR) testbed, a dynamic system compromising a single dominating frequency. Results show a high correlation between (H1) and Fast TDA features in detecting the location of a moving boundary condition. We also provide an initial framework for two dominating frequencies implementing synthetic data. We found that Fast TDA outperforms traditional TDA in handling noisy signals, such as those from DROPBEAR, by reducing the mean response error by 40% and performing adequately over a dual harmonic signal by identifying 6 out of 7 key features. Lastly, the computing time for this algorithm is approximately 1900 times faster than for traditional TDA.