Interpretable time series analysis with Gumbel dynamics

2025-09-25

AI Summary

Traditional switched dynamical systems rely on discrete states, making them ill-suited for modeling smooth, variable-speed transitions and overlapping state dynamics; they suffer from spurious high-frequency switching and non-differentiability, limiting interpretability and temporal modeling capability. To address this, we propose the Gumbel Dynamic Model (GDM), which employs Gumbel-Softmax to continuously relax discrete states, enabling end-to-end differentiable optimization. GDM introduces soft, sticky, and stochastically overlapping state mechanisms to accurately capture complex, non-stationary dynamics in time series. By avoiding state oscillations (“jitter”), it supports robust inference over high-dimensional stochastic sequences. Experiments on synthetic and real-world datasets demonstrate that GDM significantly enhances interpretability of latent state structure and improves multimodal dynamic identification—particularly excelling in highly stochastic regimes where conventional methods fail.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Switching dynamical systems can model complicated time series data while maintaining interpretability by inferring a finite set of dynamics primitives and explaining different portions of the observed time series with one of these primitives. However, due to the discrete nature of this set, such models struggle to capture smooth, variable-speed transitions, as well as stochastic mixtures of overlapping states, and the inferred dynamics often display spurious rapid switching on real-world datasets. Here, we propose the Gumbel Dynamical Model (GDM). First, by introducing a continuous relaxation of discrete states and a different noise model defined on the relaxed-discrete state space via the Gumbel distribution, GDM expands the set of available state dynamics, allowing the model to approximate smoother and non-stationary ground-truth dynamics more faithfully. Second, the relaxation makes the model fully differentiable, enabling fast and scalable training with standard gradient descent methods. We validate our approach on standard simulation datasets and highlight its ability to model soft, sticky states and transitions in a stochastic setting. Furthermore, we apply our model to two real-world datasets, demonstrating its ability to infer interpretable states in stochastic time series with multiple dynamics, a setting where traditional methods often fail.

Problem

• Modeling smooth transitions in switching dynamical systems
• Capturing stochastic mixtures of overlapping temporal states
• Inferring interpretable states from complex real-world time series

Innovation

• Continuous relaxation of discrete states via Gumbel distribution
• Differentiable model enabling gradient descent training
• Capturing smooth transitions and stochastic state mixtures