This paper is concerned with online time series forecasting, where unknown distribution shifts occur over time, i.e., latent variables influence the mapping from historical to future observations. To develop an automated way of online time series forecasting, we propose a Theoretical framework for Online Time-series forecasting (TOT in short) with theoretical guarantees. Specifically, we prove that supplying a forecaster with latent variables tightens the Bayes risk--the benefit endures under estimation uncertainty of latent variables and grows as the latent variables achieve a more precise identifiability. To better introduce latent variables into online forecasting algorithms, we further propose to identify latent variables with minimal adjacent observations. Based on these results, we devise a modelagnostic blueprint by employing a temporal decoder to match the distribution of observed variables and two independent noise estimators to model the causal inference of latent variables and mixing procedures of observed variables, respectively. Experiment results on synthetic data support our theoretical claims. Moreover, plugin implementations built on several baselines yield general improvement across multiple benchmarks, highlighting the effectiveness in real-world applications.

Online continual learning (OCL) methods adapt to changing environments without forgetting past knowledge. Similarly, online time series forecasting (OTSF) is a real-world problem where data evolve in time and success depends on both rapid adaptation and long-term memory. Indeed, time-varying and regime-switching forecasting models have been extensively studied, offering a strong justification for the use of OCL in these settings. Building on recent work that applies OCL to OTSF, this paper aims to strengthen the theoretical and practical connections between time series methods and OCL. First, we reframe neural network optimization as a parameter filtering problem, showing that natural gradient descent is a score-driven method and proving its information-theoretic optimality. Then, we show that using a Student's t likelihood in addition to natural gradient induces a bounded update, which improves robustness to outliers. Finally, we introduce Natural Score-driven Replay (NatSR), which combines our robust optimizer with a replay buffer and a dynamic scale heuristic that improves fast adaptation at regime drifts. Empirical results demonstrate that NatSR achieves stronger forecasting performance than more complex state-of-the-art methods.

This paper is concerned with online time series forecasting, where unknown distribution shifts occur over time, i.e., latent variables influence the mapping from historical to future observations. To develop an automated way of online time series forecasting, we propose a Theoretical framework for Online Time-series forecasting (TOT in short) with theoretical guarantees. Specifically, we prove that supplying a forecaster with latent variables tightens the Bayes risk, the benefit endures under estimation uncertainty of latent variables and grows as the latent variables achieve a more precise identifiability. To better introduce latent variables into online forecasting algorithms, we further propose to identify latent variables with minimal adjacent observations. Based on these results, we devise a model-agnostic blueprint by employing a temporal decoder to match the distribution of observed variables and two independent noise estimators to model the causal inference of latent variables and mixing procedures of observed variables, respectively. Experiment results on synthetic data support our theoretical claims. Moreover, plug-in implementations built on several baselines yield general improvement across multiple benchmarks, highlighting the effectiveness in real-world applications.

Online updating of time series forecasting models aims to tackle the challenge of concept drifting by adjusting forecasting models based on streaming data. While numerous algorithms have been developed, most of them focus on model design and updating. In practice, many of these methods struggle with continuous performance regression in the face of accumulated concept drifts over time. To address this limitation, we present a novel approach, Concept \textbf{D}rift \textbf{D}etection an\textbf{D} \textbf{A}daptation (D3A), that first detects drifting conception and then aggressively adapts the current model to the drifted concepts after the detection for rapid adaption. To best harness the utility of historical data for model adaptation, we propose a data augmentation strategy introducing Gaussian noise into existing training instances. It helps mitigate the data distribution gap, a critical factor contributing to train-test performance inconsistency. The significance of our data augmentation process is verified by our theoretical analysis. Our empirical studies across six datasets demonstrate the effectiveness of D3A in improving model adaptation capability. Notably, compared to a simple Temporal Convolutional Network (TCN) baseline, D3A reduces the average Mean Squared Error (MSE) by $43.9\%$. For the state-of-the-art (SOTA) model, the MSE is reduced by $33.3\%$.

In recent years, both online and offline deep learning models have been developed for time series forecasting. However, offline deep forecasting models fail to adapt effectively to changes in time-series data, while online deep forecasting models are often expensive and have complex training procedures. In this paper, we reframe the online nonlinear time-series forecasting problem as one of linear hyperdimensional time-series forecasting. Nonlinear low-dimensional time-series data is mapped to high-dimensional (hyperdimensional) spaces for linear hyperdimensional prediction, allowing fast, efficient and lightweight online time-series forecasting. Our framework, TSF-HD, adapts to time-series distribution shifts using a novel co-training framework for its hyperdimensional mapping and its linear hyperdimensional predictor. TSF-HD is shown to outperform the state of the art, while having reduced inference latency, for both short-term and long-term time series forecasting. Our code is publicly available at http://github.com/tsfhd2024/tsf-hd.git

The fast adaptation capability of deep neural networks in non-stationary environments is critical for online time series forecasting. Successful solutions require handling changes to new and recurring patterns. However, training deep neural forecaster on the fly is notoriously challenging because of their limited ability to adapt to non-stationary environments and the catastrophic forgetting of old knowledge. In this work, inspired by the Complementary Learning Systems (CLS) theory, we propose Fast and Slow learning Networks (FSNet), a holistic framework for online time-series forecasting to simultaneously deal with abrupt changing and repeating patterns. Particularly, FSNet improves the slowly-learned backbone by dynamically balancing fast adaptation to recent changes and retrieving similar old knowledge. FSNet achieves this mechanism via an interaction between two complementary components of an adapter to monitor each layer's contribution to the lost, and an associative memory to support remembering, updating, and recalling repeating events. Extensive experiments on real and synthetic datasets validate FSNet's efficacy and robustness to both new and recurring patterns. Our code is available at \url{https://github.com/salesforce/fsnet}.

Over the past years Robust PCA has been established as a standard tool for reliable low-rank approximation of matrices in the presence of outliers. Recently, the Robust PCA approach via nuclear norm minimization has been extended to matrices with linear structures which appear in applications such as system identification and data series analysis. At the same time it has been shown how to control the rank of a structured approximation via matrix factorization approaches. The drawbacks of these methods either lie in the lack of robustness against outliers or in their static nature of repeated batch-processing. We present a Robust Structured Low-Rank Approximation method on the Grassmannian that on the one hand allows for fast re-initialization in an online setting due to subspace identification with manifolds, and that is robust against outliers due to a smooth approximation of the $\ell_p$-norm cost function on the other hand. The method is evaluated in online time series forecasting tasks on simulated and real-world data.