arXiv Machine Learning
5d ago

Persistent Homology of Time Series through Complex Networks

The paper introduces a unified pipeline that classifies univariate time series by first converting them into graphs using one of five constructions from three families (visibility, transition, proximity). The resulting graph is turned into a dissimilarity matrix, from which a Vietoris–Rips filtration produces persistence diagrams that are vectorized via persistence landscapes and topological summary statistics. Experiments on twelve UCR benchmarks reveal that no single graph construction dominates, diffusion distance consistently outperforms shortest-path metrics, and persistence-based features remain robust to noise.

By \.Ismail G\"uzel
arXiv Statistics ML
Aug 24

Topological Detection of Hopf Bifurcations via Persistent Homology: A Functional Criterion from Time Series

The paper introduces a topological method for detecting Hopf bifurcations directly from scalar time series. It combines delay-coordinate reconstruction with persistent homology, using the maximum persistence of one‑dimensional homology classes as a scalar descriptor of cyclic structure. Finite‑resolution persistence bounds are derived for the supercritical Hopf case, and a derivative‑based estimator localizes the critical parameter; the method is tested on the Hopf normal form, Lorenz system, and a reduced Belousov–Zhabotinsky model.

By Jhonathan Barrios, Y\'asser Ech\'avez, Carlos F. \'Alvarez