Interpretable Classification of Time Series Using Euler Characteristic Surfaces

2026-03-16

AI Summary

This work addresses the limitations of traditional persistent homology for time series analysis—namely its high computational cost, exclusive focus on spatial information, and reliance on post-hoc vectorization—by introducing the Euler Characteristic Surface (ECS), a novel discrete topological feature that extends the Euler characteristic to jointly capture spatiotemporal dynamics. ECS is directly compatible with machine learning models and offers computational efficiency, sensitivity to topological structure, theoretical stability, and intrinsic interpretability. Evaluated using a single-feature classifier combined with AdaBoost, the method achieves classification accuracies of 98.6%, 94.1%, and 92.6% on the ECG5000, TwoLeadECG, and Epilepsy2 datasets, respectively, matching the performance of state-of-the-art deep learning approaches.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Machine learning and data science for the Web

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 ($χ$) -- 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 Rössler 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 $\textit{Epilepsy2}$ ($92.6\%$).

Problem

• time series classification
• topological data analysis
• interpretability
• Euler characteristic
• biomedical signals

Innovation

• Euler Characteristic Surfaces
• topological data analysis
• time series classification
• interpretability
• computational efficiency