Online Time Series Forecasting

Completed forecasts provide residual feedback, but retaining full residual blocks increases auxiliary state. We propose Endpoint-Preserving Online Correction (EPOC) with a compressed residual state. It stores low-order discrete cosine transform (DCT) coefficients and the final value of the preceding residual block. Each channel shares the endpoint across component-wise online ridge regressions using current-forecast coefficients. We evaluate eight multivariate series, DLinear and PatchTST, three seeds, and two training variants: 96 matched fixed-base conditions at a 24-step horizon. EPOC achieves mean condition-wise reductions in mean squared error (MSE) and mean absolute error (MAE) of 15.40% and 9.35% from the uncorrected base, respectively, with a median of 6,352 B in retained auxiliary arrays. EPOC also outperforms the $δ$-Adapter, COSA, FAC, and OMPB in paired MSE on most conditions while retaining less state. Full ELF achieves the largest mean MSE reduction, 19.29%, but its median retained state is 474,048 B ($\times$75 relative to EPOC). Equal-size summary controls favor the endpoint by 1.65--2.20% in paired MSE; a coefficient-reconstructed endpoint yields similar accuracy to the observed endpoint, highlighting its role as a shared input. Increasing the retained DCT component count from 4 to 8 adds 1.00 percentage point of MSE reduction for 5,728 B. On jointly trained bases, EPOC lowers MSE by 16.69--20.15% relative to globally blended TEFL-style adapters applied to the same base. Code and numerical records are available at https://github.com/keiotakmin/endpoint-preserving-residual-correction.

Agentic time series forecasting concerns systems whose underlying mechanisms evolve, making the relative effectiveness of numerical models, reasoning strategies, and intervention rules inherently time-varying. Consequently, a time series agent must adapt the forecasts it produces and the orchestration policy that determines which components to trust and how to coordinate them. The deployment process naturally provides supervision for this adaptation as forecast horizons elapse and realized targets reveal the effectiveness of earlier decisions. Committing all numerical expert forecasts and candidate agent paths before target observation allows each realized outcome to evaluate the entire alternative set, providing delayed feedback without additional annotation. However, existing time series agents primarily incorporate prior experience through forecast refinement, reflection, or retrieval, without systematically converting realized outcomes into persistent updates to the joint orchestration policy governing later origins. To exploit this delayed feedback systematically, we introduce TimEvolve, a frozen-backbone time series agent that converts each realized outcome into persistent joint updates of expert trust, agent path selection, and intervention strength. A temporally ordered predict, reveal, and update protocol applies this feedback to subsequent forecasts. Experiments across eight Time-MMD domains show that TimEvolve achieves the best average MSE and MAE ranks among fifteen methods and the lowest errors on both metrics in seven domains. These results demonstrate the value of learning forecasting policies from the futures encountered during deployment.

Reservoir computing (RC) trains only a linear readout over a fixed recurrent layer, making it fast and data-efficient for online prediction. However, a static reservoir degrades under system drift, readout-only adaptation is then insufficient, and unconstrained reservoir adaptation can destroy the echo-state and incremental stability properties that make RC reliable. This paper proposes LoRA-RC, which adapts the recurrent matrix through a low-rank correction driven by streaming prediction errors. The base reservoir and adaptation bases are fixed offline; a small core matrix is adapted online, projected onto a spectral-norm ball, and low-pass filtered at each step. The projection guarantees that every applied recurrent matrix remains within a certified contraction set, and an incremental input-to-state stability bound is established for the reservoir along each online adaptation path, with path-independent rate and gain. On a Lorenz system with an abrupt parameter drift, LoRA-RC cuts post-drift prediction error by 56% versus a fixed RC and 51% versus readout-only adaptation; ablations over 20 seeds show that removing the projection inflates this error by more than a factor of 40.

To address the sequential and evolving nature of time series, the Online Time Series Forecasting (OTSF) task has been extensively studied in multiple domains. Existing research focuses on adapting to non-stationary environments by employing memory buffer-based retrieval strategies. However, we observe that such frameworks struggle with long-term adaptation and fail to generalize to unseen patterns. To this end, we introduce CoSPOT, an LLM-based online time series forecasting framework that leverages a pre-trained LLM as the backbone online forecaster, motivated by its strong few-shot capabilities. For efficient online adaptation, CoSPOT keeps the LLM frozen and employs compositional spectral prompts grounded in frequency-domain bases to guide the model with the overall distribution of the input, thereby substantially reducing the number of parameters updated during the online phase. Specifically, CoSPOT decomposes time series into frequency bases and composes the corresponding spectral basis prompts according to their amplitudes, allowing unseen patterns to be represented as new combinations of learned basis prompts. Our extensive experiments on real-world datasets demonstrate the superiority and practicality of CoSPOT across challenging online scenarios, including extended online phases and cross-dataset settings with substantial distribution shifts. Our code is available at https://github.com/seungyoon-Choi/CoSPOT.

Epidemic LLM forecasters are usually trained and evaluated as static supervised models, whereas operational pandemic forecasting is a streaming process in which labels arrive after predictions and disease regimes shift over time. We study this mismatch in weekly COVID-19 hospitalization trend forecasting across five variant regimes. We introduce EpiEvolve, a self-evolving agent that wraps an LLM forecaster trained on the warm-start period and keeps its weights fixed during streaming. EpiEvolve adapts by storing forecast outcomes in a hierarchical episodic memory, reflecting on delayed labels, retrieving cases relevant to the current regime, and distilling recurring errors into strategic rules. The resulting context lets the forecaster reuse its own past predictions and outcomes in later weeks while following a chronological protocol that prevents future leakage. On the streaming dataset, EpiEvolve reaches $0.629$ average accuracy, compared with $0.561$ for the static backbone and $0.325$ for the external CDC ensemble, and reduces recovery lag after regime shifts from $5$ to $2$ weeks. Ablations show that reflection, strategic memory, and regime-aware retrieval each contribute to the gains.

Irregular multivariate time series forecasting is critical in many real-world applications, where time series are irregularly sampled and exhibit dynamically evolving missingness patterns. Although existing methods perform well in offline settings, they often suffer from significant performance degradation when deployed online due to dynamic shifts in data distribution. Maintaining forecasting capability in such dynamic scenarios typically necessitates online adaptation techniques. Since irregular sampling fundamentally undermines temporal continuity and periodicity, we cannot leverage these widely studied characteristics from regular MTS for online learning. To this end, we study the problem of online IMTS forecasting and propose Under-Cali, an uncertainty-driven dual-expert calibration framework consisting of three core components: an uncertainty estimator, a dual-expert calibration module, and an adaptive routing module. We design an uncertainty estimator that serves as the core control signal to jointly manage inference and adaptation processes. In our framework, the uncertainty estimator first assesses uncertainty for each incoming batch. The adaptive routing module then directs samples with high uncertainty to the unreliable expert for calibration, while low uncertainty samples remain with the reliable expert. Subsequently, the system updates the reliable expert and the uncertainty estimator using well-calibrated reliable samples, and updates the unreliable expert with challenging samples, enabling stable and efficient online learning. Under-Cali keeps the source forecasting model frozen and performs adaptation only through a lightweight, model-agnostic calibration module, enabling efficient adaptation. Extensive experiments on IMTS benchmarks demonstrate consistent improvements with low computational cost. Our code is available at https://github.com/HaonanWen/Under-Cali.

We propose a novel adaptive Mixture-of-Experts (MoE) framework for time series forecasting that addresses the optimization problem arising from small gating weights by incorporating expert-specific losses, which provide each expert with a direct learning signal independent of the gate-assigned weight. Specifically, the overall objective comprises the base forecasting loss and expert-specific losses, allowing individual expert prediction errors to directly influence parameter updates alongside the aggregate forecasting error. The framework also encourages different experts to learn from different temporal segments of the data. The proposed framework is further combined with a partial online learning strategy that enables efficient incremental updates of model parameters. By integrating expert-level loss information with partial online optimization, the proposed method improves forecasting performance while retaining computational efficiency. Empirical results across economic, tourism, and energy datasets with different sampling frequencies show that the proposed approach generally outperforms state-of-the-art supervised neural forecasting models, including Transformer-based architectures such as PatchTST, as well as zero-shot time-series foundation models such as TimeMoE. Furthermore, ablation studies confirm the effectiveness of the expert-specific loss integration strategy, highlighting its contribution to enhancing predictive performance.

Online Time Series Forecasting (OTSF) requires models to continuously adapt to concept drift. However, existing methods often treat concept drift as a monolithic phenomenon. To address this limitation, we first redefine concept drift by categorizing it into two distinct types: Recurring Drift, where previously seen patterns reappear, and Emergent Drift, where entirely new patterns emerge. We then propose DynaME (Dynamic Multi-period Experts), a novel hybrid framework designed to effectively address this dual nature of drift. For Recurring Drift, DynaME employs a committee of specialized experts that are dynamically fitted to the most relevant historical periodic patterns at each time step. For Emergent Drift, the framework detects high-uncertainty scenarios and shifts reliance to a stable, general expert. Extensive experiments on several benchmark datasets and backbones demonstrate that DynaME effectively adapts to both concept drifts and significantly outperforms existing baselines.