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General Machine Learning · Sequential, Network, and Time Series Modeling

Andrea Cini, Alexander Jenkins, Danilo Mandic, Cesare Alippi, Filippo Maria Bianchi

We address the problem of uncertainty quantification in time series forecasting by exploiting observations at correlated sequences. Relational deep learning methods leveraging graph representations are among the most effective tools for obtaining point estimates from spatiotemporal data and correlated time series. However, the problem of exploiting relational structures to estimate the uncertainty of such predictions has been largely overlooked in the same context. To this end, we propose a novel distribution-free approach based on the conformal prediction framework and quantile regression. Despite the recent applications of conformal prediction to sequential data, existing methods operate independently on each target time series and do not account for relationships among them when constructing the prediction interval. We fill this void by introducing a novel conformal prediction method based on graph deep learning operators. Our approach, named Conformal Relational Prediction (CoRel), does not require the relational structure (graph) to be known a priori and can be applied on top of any pre-trained predictor. Additionally, CoRel includes an adaptive component to handle non-exchangeable data and changes in the input time series. Our approach provides accurate coverage and achieves state-of-the-art uncertainty quantification in relevant benchmarks.

Deep Learning · Sequential Models, Time series

Peiyuan Liu, Beiliang Wu, Yifan Hu, Naiqi Li, Tao Dai, Jigang Bao, Shutao Xia

Non-stationarity poses significant challenges for multivariate time series forecasting due to the inherent short-term fluctuations and long-term trends that can lead to spurious regressions or obscure essential long-term relationships. Most existing methods either eliminate or retain non-stationarity without adequately addressing its distinct impacts on short-term and long-term modeling. Eliminating non-stationarity is essential for avoiding spurious regressions and capturing local dependencies in short-term modeling, while preserving it is crucial for revealing long-term cointegration across variates. In this paper, we propose TimeBridge, a novel framework designed to bridge the gap between non-stationarity and dependency modeling in long-term time series forecasting. By segmenting input series into smaller patches, TimeBridge applies Integrated Attention to mitigate short-term non-stationarity and capture stable dependencies within each variate, while Cointegrated Attention preserves non-stationarity to model long-term cointegration across variates. Extensive experiments show that TimeBridge consistently achieves state-of-the-art performance in both short-term and long-term forecasting. Additionally, TimeBridge demonstrates exceptional performance in financial forecasting on the CSI 500 and S&P 500 indices, further validating its robustness and effectiveness. Code is available at https://github.com/Hank0626/TimeBridge.

Deep Learning · Sequential Models, Time series

Dilfira Kudrat, Zongxia Xie, Yanru Sun, Tianyu Jia, Qinghua Hu

Time-series forecasting has gained significant attention in machine learning due to its crucial role in various domains. However, most existing forecasting models rely heavily on point-wise loss functions like Mean Squared Error, which treat each time step independently and neglect the structural dependencies inherent in time series data, making it challenging to capture complex temporal patterns accurately. To address these challenges, we propose a novel **P**atch-wise **S**tructural (**PS**) loss, designed to enhance structural alignment by comparing time series at the patch level. Through leveraging local statistical properties, such as correlation, variance, and mean, PS loss captures nuanced structural discrepancies overlooked by traditional point-wise losses. Furthermore, it integrates seamlessly with point-wise loss, simultaneously addressing local structural inconsistencies and individual time-step errors. PS loss establishes a novel benchmark for accurately modeling complex time series data and provides a new perspective on time series loss function design. Extensive experiments demonstrate that PS loss significantly improves the performance of state-of-the-art models across diverse real-world datasets. The data and code are publicly available at: \url{https://github.com/Dilfiraa/PS_Loss}.

Deep Learning · Sequential Models, Time series

Thomas Lee, William Toner, Rajkarn Singh, Artjom Joosen, Martin Asenov

Foundation models (FMs) have emerged as a promising approach for time series forecasting. While effective, FMs typically remain fixed during deployment due to the high computational costs of learning them online. Consequently, deployed FMs fail to adapt their forecasts to current data characteristics, despite the availability of online feedback from newly arriving data. This raises the question of whether FM performance can be enhanced by the *efficient* usage of this feedback. We propose *ELF* to answer this question. ELF is a lightweight mechanism for the online adaption of FM forecasts in response to online feedback. ELF consists of two parts: **a)** the *ELF-Forecaster* which is used to learn the current data distribution; and **b)** the *ELF-Weighter* which is used to combine the forecasts of the FM and the ELF-Forecaster. We evaluate the performance of ELF in conjunction with several recent FMs across a suite of standard time series datasets. In *all* of our experiments we find that using ELF improves performance. This work demonstrates how efficient usage of online feedback can be used to improve FM forecasts.

Deep Learning · Sequential Models, Time series

Siru Zhong, Weilin Ruan, Ming Jin, Huan Li, Qingsong Wen, Yuxuan Liang

Recent advancements in time series forecasting have explored augmenting models with text or vision modalities to improve accuracy. While text provides contextual understanding, it often lacks fine-grained temporal details. Conversely, vision captures intricate temporal patterns but lacks semantic context, limiting the complementary potential of these modalities. To address this, we propose Time-VLM, a novel multimodal framework that leverages pre-trained Vision-Language Models (VLMs) to bridge temporal, visual, and textual modalities for enhanced forecasting. Our framework comprises three key components: (1) a Retrieval-Augmented Learner, which extracts enriched temporal features through memory bank interactions; (2) a Vision-Augmented Learner, which encodes time series as informative images; and (3) a Text-Augmented Learner, which generates contextual textual descriptions. These components collaborate with frozen pre-trained VLMs to produce multimodal embeddings, which are then fused with temporal features for final prediction. Extensive experiments demonstrate that Time-VLM achieves superior performance, particularly in few-shot and zero-shot scenarios, thereby establishing a new direction for multimodal time series forecasting. Code is available at https://github.com/CityMind-Lab/ICML25-TimeVLM.

Deep Learning · Sequential Models, Time series

Leon Götz, Marcel Kollovieh, Stephan Günnemann, Leo Schwinn

Despite recent advances in subquadratic attention mechanisms or state-space models, processing long token sequences still imposes significant computational requirements. Token merging has emerged as a solution to increase computational efficiency in computer vision architectures. In this work, we perform the first investigations of token merging in time series analysis on both transformers and state-space models. We further introduce local merging, a domain-specific token merging algorithm that selectively combines tokens within a local neighborhood, achieving two major benefits: a) Local merging can adjust its computational complexity from quadratic to linear based on the neighborhood size to effectively scale to long sequences; b) Local merging is the first causal merging scheme enabling token merging in transformer decoders. Further, we identify spectral properties of the input data that reliably predict the potential benefits of local merging without requiring evaluation on downstream tasks. Our comprehensive empirical evaluation demonstrates that local merging offers substantial efficiency gains with minimal impact on accuracy, achieving up to 5400% acceleration on the recently proposed Chronos foundation model.

Deep Learning · Sequential Models, Time series

Elizabeth M Fons Etcheverry, Alejandro Sztrajman, Yousef El-Laham, Luciana Ferrer, Svitlana Vyetrenko, Manuela Veloso

Time series with missing or irregularly sampled data are a persistent challenge in machine learning. Many methods operate on the frequency-domain, relying on the Fast Fourier Transform (FFT) which assumes uniform sampling, therefore requiring prior interpolation that can distort the spectra. To address this limitation, we introduce a differentiable Lomb--Scargle layer that enables a reliable computation of the power spectrum of irregularly sampled data.We integrate this layer into a novel score-based diffusion model (LSCD) for time series imputation conditioned on the entire signal spectrum. Experiments on synthetic and real-world benchmarks demonstrate that our method recovers missing data more accurately than purely time-domain baselines, while simultaneously producing consistent frequency estimates. Crucially, our method can be easily integrated into learning frameworks, enabling broader adoption of spectral guidance in machine learning approaches involving incomplete or irregular data.

Deep Learning · Sequential Models, Time series

Sungwon Han, Seungeon Lee, MEEYOUNG CHA, Sercan Arik, Jinsung Yoon

Time series forecasting uses historical data to predict future trends, leveraging the relationships between past observations and available features. In this paper, we propose RAFT, a retrieval-augmented time series forecasting method to provide sufficient inductive biases and complement the model's learning capacity. When forecasting the subsequent time frames, we directly retrieve historical data candidates from the training dataset with patterns most similar to the input, and utilize the future values of these candidates alongside the inputs to obtain predictions. This simple approach augments the model's capacity by externally providing information about past patterns via retrieval modules. Our empirical evaluations on ten benchmark datasets show that RAFT consistently outperforms contemporary baselines with an average win ratio of 86%.

Deep Learning · Sequential Models, Time series

Min-Yeong Park, Won-Jeong Lee, Seong Tae Kim, Gyeong-Moon Park

Recently, forecasting future abnormal events has emerged as an important scenario to tackle realworld necessities. However, the solution of predicting specific future time points when anomalies will occur, known as Anomaly Prediction (AP), remains under-explored. Existing methods dealing with time series data fail in AP, focusing only on immediate anomalies or failing to provide precise predictions for future anomalies. To address AP, we propose a novel framework called Anomaly to Prompt (A2P), comprised of Anomaly-Aware Forecasting (AAF) and Synthetic Anomaly Prompting (SAP). To enable the forecasting model to forecast abnormal time points, we adopt a strategy to learn the relationships of anomalies. For the robust detection of anomalies, our proposed SAP introduces a learnable Anomaly Prompt Pool (APP) that simulates diverse anomaly patterns using signal-adaptive prompt. Comprehensive experiments on multiple real-world datasets demonstrate the superiority of A2P over state-of-the-art methods, showcasing its ability to predict future anomalies.

General Machine Learning · Sequential, Network, and Time Series Modeling

Shikang Liu, Chuyang Wei, Xiren Zhou, Huanhuan Chen

Analyzing inherent temporal dynamics is a critical pathway for time series classification, where Reservoir Computing (RC) exhibits effectiveness and high efficiency. However, typical RC considers recursive updates from adjacent states, struggling with long-term dependencies. In response, this paper proposes a Spectral-Aware Reservoir Computing framework (SARC), incorporating spectral insights to enhance long-term dependency modeling. Prominent frequencies are initially extracted to reveal explicit or implicit cyclical patterns. For each prominent frequency, SARC further integrates a Frequency-informed Reservoir Network (FreqRes) to adequately capture both sequential and cyclical dynamics, thereby deriving effective dynamic features. Synthesizing these features across various frequencies, SARC offers a multi-scale analysis of temporal dynamics and improves the modeling of long-term dependencies. Experiments on public datasets demonstrate that SARC achieves state-of-the-art results, while maintaining high efficiency compared to existing methods.

General Machine Learning · Sequential, Network, and Time Series Modeling

Mouxiang Chen, Lefei Shen, Zhuo Li, Xiaoyun Wang, Jianling Sun, Chenghao Liu

Foundation models have emerged as a promising approach in time series forecasting (TSF). Existing approaches either repurpose large language models (LLMs) or build large-scale time series datasets to develop TSF foundation models for universal forecasting. However, these methods face challenges due to the severe cross-domain gap or in-domain heterogeneity. This paper explores a new road to building a TSF foundation model from rich, high-quality natural images. Our key insight is that a visual masked autoencoder, pre-trained on the ImageNet dataset, can naturally be a numeric series forecaster. By reformulating TSF as an image reconstruction task, we bridge the gap between image pre-training and TSF downstream tasks. Surprisingly, without further adaptation in the time series domain, the proposed VisionTS could achieve better zero-shot forecast performance than existing TSF foundation models. With fine-tuning for one epoch, VisionTS could further improve the forecasting and achieve state-of-the-art performance in most cases. Extensive experiments reveal intrinsic similarities between images and real-world time series, suggesting that visual models may offer a "free lunch" for TSF and highlight the potential for future cross-modality research. Our code is available in the https://github.com/Keytoyze/VisionTS.

Deep Learning · Sequential Models, Time series

Adrien Cortes, Remi Rehm, Victor Letzelter

We introduce $\texttt{TimeMCL}$, a method leveraging the Multiple Choice Learning (MCL) paradigm to forecast multiple plausible time series futures. Our approach employs a neural network with multiple heads and utilizes the Winner-Takes-All (WTA) loss to promote diversity among predictions. MCL has recently gained attention due to its simplicity and ability to address ill-posed and ambiguous tasks. We propose an adaptation of this framework for time-series forecasting, presenting it as an efficient method to predict diverse futures, which we relate to its implicit *quantization* objective. We provide insights into our approach using synthetic data and evaluate it on real-world time series, demonstrating its promising performance at a light computational cost.

Deep Learning · Sequential Models, Time series

Yifan Hu, Guibin Zhang, Peiyuan Liu, Disen Lan, Naiqi Li, Dawei Cheng, Tao Dai, Shutao Xia, Shirui Pan

Time series forecasting methods generally fall into two main categories: Channel Independent (CI) and Channel Dependent (CD) strategies. While CI overlooks important covariate relationships, CD captures all dependencies without distinction, introducing noise and reducing generalization. Recent advances in Channel Clustering (CC) aim to refine dependency modeling by grouping channels with similar characteristics and applying tailored modeling techniques. However, coarse-grained clustering struggles to capture complex, time-varying interactions effectively. To address these challenges, we propose TimeFilter, a GNN-based framework for adaptive and fine-grained dependency modeling. After constructing the graph from the input sequence, TimeFilter refines the learned spatial-temporal dependencies by filtering out irrelevant correlations while preserving the most critical ones in a patch-specific manner. Extensive experiments on 13 real-world datasets from diverse application domains demonstrate the state-of-the-art performance of TimeFilter. The code is available at https://github.com/TROUBADOUR000/TimeFilter.

Deep Learning · Sequential Models, Time series

Rom N. Parnichkun, Neehal Tumma, Armin Thomas, Alessandro Moro, Qi An, Taiji Suzuki, Atsushi Yamashita, Michael Poli, Stefano Massaroli

As the space of causal sequence modeling architectures continues to grow, the need to develop a general framework for their analysis becomes increasingly important. With this aim, we draw insights from classical signal processing and control theory, to develop a quantitative measure of *memory utilization*: the internal mechanisms through which a model stores past information to produce future outputs. This metric, which we call ***effective state-size*** (ESS), is tailored to the fundamental class of systems with *input-invariant* and *input-varying linear operators*, encompassing a variety of computational units such as variants of attention, convolutions, and recurrences. Unlike prior work on memory utilization, which either relies on raw operator visualizations (e.g. attention maps), or simply the total *memory capacity* (i.e. cache size) of a model, our metrics provide highly interpretable and actionable measurements. In particular, we show how ESS can be leveraged to improve initialization strategies, inform novel regularizers and advance the performance-efficiency frontier through model distillation. Furthermore, we demonstrate that the effect of context delimiters (such as end-of-speech tokens) on ESS highlights cross-architectural differences in how large language models utilize their available memory to recall information. Overall, we find that ESS provides valuable insights into the dynamics that dictate memory utilization, enabling the design of more efficient and effective sequence models.

Deep Learning · Sequential Models, Time series

Jiecheng Lu, Shihao Yang

Autoregressive attention-based time series forecasting (TSF) has drawn increasing interest, with mechanisms like linear attention often outperforming vanilla attention. However, deeper Transformer architectures frequently misalign with autoregressive objectives, obscuring the underlying VAR structure embedded within linear attention and hindering their ability to capture the data generative processes in TSF. In this work, we first show that a single linear attention layer can be interpreted as a dynamic vector autoregressive (VAR) structure. We then explain that existing multi-layer Transformers have structural mismatches with the autoregressive forecasting objective, which impair interpretability and generalization ability. To address this, we show that by rearranging the MLP, attention, and input-output flow, multi-layer linear attention can also be aligned as a VAR model. Then, we propose Structural Aligned Mixture of VAR (SAMoVAR), a linear Transformer variant that integrates interpretable dynamic VAR weights for multivariate TSF. By aligning the Transformer architecture with autoregressive objectives, SAMoVAR delivers improved performance, interpretability, and computational efficiency, comparing to SOTA TSF models.

Deep Learning · Sequential Models, Time series

Jiecheng Lu, Xu Han, Yan Sun, Shihao Yang

We propose a Weighted Autoregressive Varying gatE (WAVE) attention mechanism equipped with both Autoregressive (AR) and Moving-average (MA) components. It can adapt to various attention mechanisms, enhancing and decoupling their ability to capture long-range and local temporal patterns in time series data. In this paper, we first demonstrate that, for the time series forecasting (TSF) task, the previously overlooked decoder-only autoregressive Transformer model can achieve results comparable to the best baselines when appropriate tokenization and training methods are applied. Moreover, inspired by the ARMA model from statistics and recent advances in linear attention, we introduce the full ARMA structure into existing autoregressive attention mechanisms. By using an indirect MA weight generation method, we incorporate the MA term while maintaining the time complexity and parameter size of the underlying efficient attention models. We further explore how indirect parameter generation can produce implicit MA weights that align with the modeling requirements for local temporal impacts. Experimental results show that WAVE attention that incorporates the ARMA structure consistently improves the performance of various AR attentions on TSF tasks, achieving state-of-the-art results.

Deep Learning · Sequential Models, Time series

Weiwei Ye, Zhuopeng Xu, Ning Gui

Due to the dynamics of underlying physics and external influences, the uncertainty of time series often varies over time. However, existing Denoising Diffusion Probabilistic Models (DDPMs) often fail to capture this non-stationary nature, constrained by their constant variance assumption from the additive noise model (ANM). In this paper, we innovatively utilize the Location-Scale Noise Model (LSNM) to relax the fixed uncertainty assumption of ANM. A diffusion-based probabilistic forecasting framework, termed Non-stationary Diffusion (NsDiff), is designed based on LSNM that is capable of modeling the changing pattern of uncertainty. Specifically, NsDiff combines a denoising diffusion-based conditional generative model with a pre-trained conditional mean and variance estimator, enabling adaptive endpoint distribution modeling. Furthermore, we propose an uncertainty-aware noise schedule, which dynamically adjusts the noise levels to accurately reflect the data uncertainty at each step and integrates the time-varying variances into the diffusion process. Extensive experiments conducted on nine real-world and synthetic datasets demonstrate the superior performance of NsDiff compared to existing approaches. Code is available at https://github.com/wwy155/NsDiff.

Deep Learning · Sequential Models, Time series

Marten Lienen, Abdullah Saydemir, Stephan Günnemann

State space models are emerging as a dominant model class for sequence problems with many relying on the HiPPO framework to initialize their dynamics. However, HiPPO fundamentally assumes data to be noise-free; an assumption often violated in practice. We extend the HiPPO theory with measurement noise and derive an uncertainty-aware initialization for state space model dynamics. In our analysis, we interpret HiPPO as a linear stochastic control problem where the data enters as a noise-free control signal. We then reformulate the problem so that the data become noisy outputs of a latent system and arrive at an alternative dynamics initialization that infers the posterior of this latent system from the data without increasing runtime. Our experiments show that our initialization improves the resistance of state-space models to noise both at training and inference time.

Deep Learning · Sequential Models, Time series

Shubham Gupta, Thibaut Durand, Graham Taylor, Lilian Bialokozowicz

We present a novel prompt design for Large Language Models (LLMs) tailored to **Asynchronous Time Series**. Unlike regular time series, which assume values at evenly spaced time points, asynchronous time series consist of timestamped events occurring at irregular intervals, each described in natural language. Our approach effectively utilizes the rich natural language of event descriptions, allowing LLMs to benefit from their broad world knowledge for reasoning across different domains and tasks. This allows us to extend the scope of asynchronous time series analysis beyond forecasting to include tasks like anomaly detection and data imputation. We further introduce **Stochastic Soft Prompting**, a novel prompt-tuning mechanism that significantly improves model performance, outperforming existing finetuning methods such as QLORA. Through extensive experiments on real-world datasets, we demonstrate that our approach achieves state-of-the-art performance across different tasks and datasets.

Deep Learning · Sequential Models, Time series

Seunghan Lee, Taeyoung Park, Kibok Lee

Channel identifiability (CID) refers to the ability to distinguish among individual channels in time series (TS) modeling. The absence of CID often results in producing identical outputs for identical inputs, disregarding channel-specific characteristics. In this paper, we highlight the importance of CID and propose Channel Normalization (CN), a simple yet effective normalization strategy that enhances CID by assigning distinct affine transformation parameters to each channel. We further extend CN in two ways: 1) Adaptive CN (ACN) dynamically adjusts parameters based on the input TS, improving adaptability in TS models, and 2) Prototypical CN (PCN) introduces a set of learnable prototypes instead of per-channel parameters, enabling applicability to datasets with unknown or varying number of channels and facilitating use in TS foundation models. We demonstrate the effectiveness of CN and its variants by applying them to various TS models, achieving significant performance gains for both non-CID and CID models. In addition, we analyze the success of our approach from an information theory perspective. Code is available at [https://github.com/seunghan96/CN](https://github.com/seunghan96/CN).