Interpretable Classification of Time Series Using Euler Characteristic Surfaces

TL;DR

This paper introduces Euler Characteristic Surfaces (ECS) as a fast, stable, and interpretable topological feature for time series classification, outperforming existing methods on biomedical datasets.

Contribution

The paper proposes ECS as an efficient, spatiotemporal topological signature for time series, with a stability guarantee and demonstrated superior classification performance.

Findings

ECS captures topological differences in dynamical systems like the Rössler system.

ECS-based classifier achieves 98% accuracy on ECG5000 dataset.

ECS outperforms PH-based methods and matches deep learning accuracy with interpretability.

Abstract

Persistent homology (PH) -- the conventional method in topological data analysis -- is computationally expensive, requires further vectorization of its signatures before machine learning (ML) can be applied, and captures information along only the spatial axis. For time series data, we propose Euler Characteristic Surfaces (ECS) as an alternative topological signature based on the Euler characteristic ($\chi$) -- a fundamental topological invariant. The ECS provides a computationally efficient, spatiotemporal, and inherently discretized feature representation that can serve as direct input to ML models. We prove a stability theorem guaranteeing that the ECS remains stable under small perturbations of the input time series. We first demonstrate that ECS effectively captures the nontrivial topological differences between the limit cycle and the strange attractor in the R\"{o}ssler system. We then develop an ECS-based classification framework and apply it to five benchmark biomedical datasets (four ECG, one EEG) from the UCR/UEA archive. On $\textit{ECG5000}$, our single-feature ECS classifier achieves $98\%$ accuracy with $O(n+R\cdot T)$ complexity, compared to $62\%$ reported by a recent PH-based method. An AdaBoost extension raises accuracy to $98.6\%$, matching the best deep learning results while retaining full interpretability. Strong results are also obtained on $\textit{TwoLeadECG}$ ($94.1\%$) and $\textit{Epilepsy2}$ ($92.6\%$).