Modeling Dynamic Mixtures of Time-Delay Systems from Streaming Time Series

A new online framework called DelayMix models streaming time-series data as dynamic mixtures of time-delay systems, maintaining robust model tracking while reducing memory usage, according to research submitted May 25, 2026 [1][2]. The framework addresses a core challenge in adaptive modeling: rapid system changes, or regime shifts, caused by environmental factors or input delay changes that degrade model performance [2]. DelayMix constructs a summary system tensor using the system's Markov parameter series, capturing both dynamic behavior and delay characteristics [2]. A tensor decomposition algorithm can then extract relevant past models from the tensor to select the system that best fits the current regime, enabling rapid adaptation to environmental changes with computational efficiency [2]. Tests on real datasets showed DelayMix consistently outperformed other methods, achieving superior forecast accuracy and faster adaptation to delays, especially for highly non-stationary data [2]. The approach is relevant to domains where streaming data exhibits sudden shifts. In traffic flow, for instance, the complex interactions of vehicles can display behaviors such as cluster formation and shock wave propagation, and bottlenecks—which the Federal Highway Authority attributes 40% of congestion to—significantly disrupt flow [3]. Classical traffic flow theories include the Lighthill-Whitham-Richards model and various car-following models, but a universally satisfactory theory applicable to real-world conditions remains elusive [3]. Similarly, chaotic behavior exists in many natural systems, including fluid flow and road traffic, where small differences in initial conditions can yield widely diverging outcomes, rendering long-term prediction impossible in general [4]. Other modeling frameworks tackle different dynamic systems. The EPA's Storm Water Management Model (SWMM) simulates rainfall-runoff and subsurface runoff quantity and quality from urban areas, tracking flow rate, depth, and water quality through pipes and channels during multi-time-step simulations [6]. In combustion engineering, homogeneous charge compression ignition (HCCI) requires microprocessor control and physical understanding of the ignition process to manage auto-ignition timing [7]. DelayMix's online framework, which summarizes past regimes using a fixed-length representation, offers a contrasting data-driven method for systems where input-output delays shift over time [2]. The research was submitted on May 25, 2026, and is available through arXivLabs, a framework allowing collaborators to develop and share new arXiv features [1].