Guided by Stars: Interpretable Concept Learning Over Time Series via Temporal Logic Semantics

TL;DR

STELLE introduces a neuro-symbolic framework that embeds time series into a shared semantic space of Signal Temporal Logic (STL) concepts, enabling end-to-end interpretable classification. By extending an STL-based kernel to embed both formulas and raw trajectories, STELLE learns concept relevance and discriminability, producing local explanations as concise STL conditions and global explanations as class-level temporal patterns. The approach achieves competitive accuracy on diverse multivariate time series benchmarks while providing intrinsic symbolic explanations, a capability not offered by standard black-box models. This integration of temporal-logic semantics with neural learning holds potential for regulatory compliance and safer deployment in safety-critical domains, balancing predictive performance with explanatory fidelity.

Abstract

Time series classification is a task of paramount importance, as this kind of data often arises in safety-critical applications. However, it is typically tackled with black-box deep learning methods, making it hard for humans to understand the rationale behind their output. To take on this challenge, we propose a novel approach, STELLE (Signal Temporal logic Embedding for Logically-grounded Learning and Explanation), a neuro-symbolic framework that unifies classification and explanation through direct embedding of trajectories into a space of temporal logic concepts. By introducing a novel STL-inspired kernel that maps raw time series to their alignment with predefined STL formulae, our model jointly optimises accuracy and interpretability, as each prediction is accompanied by the most relevant logical concepts that characterise it. This yields (i) local explanations as human-readable STL conditions justifying individual predictions, and (ii) global explanations as class-characterising formulae. Experiments demonstrate that STELLE achieves competitive accuracy while providing logically faithful explanations, validated on diverse real-world benchmarks.

Figures (4)

• Figure 1: STELLE architecture. Input trajectories $\mathbf{x}$ are embedded via STL concepts formulae $\Phi_C$, creating $\mathcal{H}(\mathbf{x})$. This gets scaled, creating $\gamma(\mathbf{x})$, and integrated with class-specific scores $\mathcal{G}(\mathbf{x})$ for classification $\hat{y}$ and generation of local explanations, which can be aggregated into global explanations.
• Figure 2: Local explanations for two trajectories in the ERing dataset, showing the target trajectory (in red) and the trajectories from other classes (in gray) across its four channels. As title, the true and predicted class, and the proposed postprocessed explanation.
• Figure 3: Global explanation for class 2 in the ERing dataset, showing the target trajectories (in red) taken from the test set wrt their predicted class, and the training trajectories from other classes (in gray) across its four channels. As title, the class and the proposed postprocessed explanation.
• Figure : Postprocess formulae based on robustness separation