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Interpretable Classification of Time Series Using Euler Characteristic Surfaces

Salam Rabindrajit Luwang, Sushovan Majhi, Vishal Mandal, Atish J. Mitra, Md. Nurujjaman, Buddha Nath Sharma

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 $\textit{Epilepsy2}$ ($92.6\%$).

Interpretable Classification of Time Series Using Euler Characteristic Surfaces

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 () -- 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 , our single-feature ECS classifier achieves accuracy with complexity, compared to reported by a recent PH-based method. An AdaBoost extension raises accuracy to , matching the best deep learning results while retaining full interpretability. Strong results are also obtained on () and ().
Paper Structure (2 sections, 2 theorems, 24 equations, 14 figures, 3 tables)

This paper contains 2 sections, 2 theorems, 24 equations, 14 figures, 3 tables.

Table of Contents

  1. Standard AdaBoost procedure.
  2. ECS-specific design choices.

Key Result

Lemma 1

Let $\mathcal{D}=\{x_1,\ldots,x_M\}$ be a point cloud in $\mathbb{R}^3$. Then, for an arbitrary small $0< \epsilon < \frac{1}{2} \min\limits_{1\leq i<j\leq M} \lVert x_i-x_j\rVert_2$, there is $\delta > 0$ such that for any $\mathcal{D}'=\{x'_1,\ldots,x'_M\}$ with $\max_{1\leq i\leq M}\|x_i-x_i'\|_ where $\mathcal{K}(r)$ and $\mathcal{K}'(r)$ are the corresponding Alpha filtrations on $\mathcal{D

Figures (14)

  • Figure 1: Illustration of the (a) Voronoi diagram, (b) Nerve (Čech complex), and (c) Alpha complex for the hexagonal point set $S=\{(0, 2), (2, 1), (2, -1), (0, -2), (-2, -1), (-2, 1)\}$. In (a), green dashed lines delineate the Voronoi cells. In (b), the Čech complex at a fixed radius includes edges and triangles wherever the corresponding balls mutually intersect. In (c), the Alpha complex restricts the Čech complex to simplices consistent with the Voronoi decomposition, yielding a sparser complex that still captures the essential topology.
  • Figure 2: Schematic workflow for constructing Euler Characteristic Surface (ECS) from a time series that provides a "machine-learning-ready" topological feature vector.
  • Figure 3: Time series, Takens-embedded phase portraits, and Euler Characteristic Surfaces (ECSs) of the Rössler system for two values of the control parameter $c$. Column 1 (periodic, $c=2.3$): (a) $x$--component time series, (c) phase portrait in $\mathbb{R}^3$, (e) ECS. Column 2 (chaotic, $c=7.3$): (b) time series, (d) phase portrait, (f) ECS.
  • Figure 4: (a) Absolute difference $|\mathrm{ECS}_{\mathrm{chaotic}} - \mathrm{ECS}_{\mathrm{periodic}}|$ across the scale--time grid, with the annotated arrow indicating the coordinate of maximum divergence. (b) Heatmap of pairwise $L_{1}$ distances between ECSs constructed from perturbed time series of the periodic ($c=2.3$) and chaotic ($c=7.3$) regimes. Intra-regime distances (off- diagonal blocks) are markedly smaller than inter-regime distances (diagonal blocks).
  • Figure 5: The flowchart represents the ECS-based time series classification framework.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Lemma 1: Stability of Alpha Filtrations in $\mathbb{R}^3$
  • proof
  • Theorem 1: Stability of Takens-Embedded ECS
  • proof