Interpretable time series analysis with Gumbel dynamics

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 (e.g., non-instantaneous state transitions), 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. Breaking from established literature, this new class uses a stochastic relaxed state to directly combine reusable dynamical primitives and govern observations, without a separate stochastic continuous trajectory. 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.