Interpretable Classification of Time Series Using Euler Characteristic Surfaces

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

cs.LGMar 16, 2026

Interpretable Classification of Time Series Using Euler Characteristic Surfaces

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

Organizations: National Institute of Technology, Department of Physics, Ravangla, 737139, India · George Washington University, Data science program, Washington, DC, 20052, USA · Montana Technological University, Department of Mathematical Sciences, Butte, 59701, USA

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

Figures & tables

Panel of Figure 1 — 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.

Panel of Figure 2 — 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).

Panel of Figure 5 — Figure 5: The flowchart represents the ECS-based time series classification framework.

Panel of Figure 6 — Figure 6: Flowchart of the ECS-based AdaBoost classification framework. The number of weak learners $T$ is set equal to $K\times R$ , matching the dimensionality of the ECS feature vector.

Figure 7: Mean cross-validated AUC of the best ECS feature as a function of the grid size (

K

) used to construct the ECS for the ECG5000 dataset.

Figure 8: Panels (a) and (b) show representative ECG time series for classes 1 and 2, respectively. Panels (c) and (d) show the corresponding ECSs.

Figure 9: Topological comparison of ECS representations constructed from the ECG5000 training set. Panel (a) shows the pointwise absolute difference $|\Delta\chi|$ between the mean ECS of each class, plotted over the scale–window grid; the annotated maximum identifies the $(k,r)$ coordinate at which the two classes are most topologically distinguishable. Panel (b) shows the pairwise $L_{1}$ distances between the ECSs of all training samples, visualized as a heatmap.

Figure 10: (a) Training-set density of feature 72 for both classes after polarity reversal (

p=-1

), with the Youden-optimal threshold

\theta=3

. (b) Test-set ROC curve obtained with feature 72.

Figure 11: Heatmap of the total $\alpha$ contribution of each ECS feature in the AdaBoost ensemble trained on the ECG5000 dataset ( $6\times 6$ grid, $T=36$ stumps).

Metric	Single Feature	AdaBoost
ROC-AUC	0.9937	0.9988
Accuracy	0.9803	0.9855
Precision	0.9852	0.9848
Recall (Sensitivity)	0.9623	0.9767
Specificity	0.9912	0.9909
F1 Score	0.9736	0.9807

Table 1: Classification performance on the ECG5000 test set (4,217 samples). The single-feature model uses a

10\times 10

ECS grid; the AdaBoost model uses a

6\times 6

grid with

T=36

stumps.

Figure 12: Panels (a) and (b) show representative EEG time series for the seizure and non-seizure classes, respectively. Panels (c) and (d) show the corresponding ECSs.

Figure 13: Topological comparison of ECS representations constructed from the Epilepsy2 training set. Panel (a) shows the point-wise absolute difference $|\Delta\chi|$ between the mean ECS of each class, plotted over the scale–window grid; the annotated maximum identifies the $(k,r)$ coordinate at which the two classes are most topologically distinguishable. Panel (b) shows the pairwise $L_{1}$ distance between the ECSs of all training samples, displayed as a heatmap.

Metric	Single Feature	AdaBoost
ROC-AUC	0.8347	0.9628
Accuracy	0.8761	0.9256
Precision	0.9398	0.9631
Recall (Sensitivity)	0.9034	0.9433
F1 Score	0.9212	0.9531
True Positives	8,275	8,641

Table 2: Classification performance on the Epilepsy2 test set (11,420 samples). The ECS was constructed with a

9\times 9

grid (

m=4

\tau=1

) and the AdaBoost ensemble comprised

T=81

stumps.

Figure 14: Heatmap of the total $\alpha$ contribution of each ECS feature in the AdaBoost ensemble trained on the Epilepsy2 dataset ( $9\times 9$ grid, $T=81$ stumps).

Dataset	Method	AUC	Accuracy	Precision	Recall	Specificity	F1
TwoLeadECG	Single Feature	0.9742	0.9263	0.8894	0.9737	0.8787	0.9296
TwoLeadECG	AdaBoost	0.9780	0.9412	0.9736	0.9070	0.9754	0.9391
ECG200	Single Feature	0.8516	0.7800	0.9565	0.6875	0.9444	0.8000
ECG200	AdaBoost	0.8811	0.8000	0.8143	0.8906	0.6389	0.8507
ECGFiveDays	Single Feature	0.7949	0.7851	0.7431	0.8753	0.6939	0.8038
ECGFiveDays	AdaBoost	0.8494	0.7933	0.7904	0.8014	0.7850	0.7959

Table 3: Classification performance across three additional ECG datasets from the UCR archive. All metrics are reported on the held-out test set. The optimal ECS grid size was determined independently for each dataset by 5-fold cross-validation.

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TopTimeNet: Topologically-assisted time-series classification model

Sharareh Sayyad, Sophia Bazzi

Department of Mathematics and Statistics, Washington State University, Pullman, Washington 99164-3113, USA · European Molecular Biology Laboratory, EMBL Hamburg, c/o DESY, Building 25A, Notkestraße 85, 22603 Hamburg, Germany

Time Series Classification Classification

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Distinguishing periodic from chaotic dynamics in a time series is a fundamental challenge in both physics and engineering. Yet, end-to-end learned architectures must discover both a representation and a decision boundary from data, at substantial cost. We introduce TopTimeNet, which decouples these tasks: a fixed, non-learned stage extracts a

42

-dimensional geometric and topological descriptor from Takens delay embeddings and persistent homology, and a lightweight learnable stage performs classification. On a benchmark of

49

nonlinear dynamical systems, a

1{,}638

-parameter configuration matches the mean accuracy of one with

33\times

more trainable parameters. Additionally, this approach delivers mean accuracy comparable to convolutional neural networks and surpasses the average performance of converged Transformer models, while requiring three to four orders of magnitude fewer trainable parameters. Robustness also depends sharply on where noise is introduced: TopTimeNet degrades gracefully under perturbations to its precomputed features, but degrades sharply when noise is introduced into the raw signal and the full feature-extraction pipeline is recomputed, showing that robustness to perturbations of the precomputed features does not imply robustness of the complete raw-signal-to-prediction pipeline. These results show that decoupling fixed geometric and topological feature construction from a lightweight discriminative stage can achieve comparable classification accuracy with substantially fewer trainable parameters.

Abstract

Interpretable Time-Series Classification

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