Time Series Forecasting

Transformer models have redefined sequence learning, yet dot-product self-attention introduces a quadratic token-mixing bottleneck for long-context time-series. We introduce the Phasor Transformer block, a phase-native alternative representing sequence states on the unit-circle manifold $S^1$. Each block combines lightweight trainable phase-shifts with parameter-free Discrete Fourier Transform (DFT) token coupling, achieving global $\mathcal{O}(N\log N)$ mixing without explicit attention maps. Stacking these blocks defines the Large Phasor Model (LPM). We validate LPM on autoregressive time-series prediction over synthetic multi-frequency benchmarks against honest baselines: it beats a zero-parameter persistence baseline and, with the corrected gradient path, improves monotonically with depth before saturating, while remaining competitive-but-not-superior to self-attention at a fraction of the parameter count. Our results establish an explicit efficiency--accuracy frontier, showing that scalable temporal modeling in oscillatory domains can emerge from geometry-constrained phase computation with deterministic global coupling.

Retrieval-augmented generation (RAG) enhances zero-shot time series (TS) forecasting by leveraging external knowledge bases, yet existing approaches overlook input-level relevance when fusing retrieved samples with the query. We argue that not all retrievals are equally useful, and irrelevant ones can degrade performance. To this end, we propose Cross-RAG, a zero-shot RAG-based forecasting framework that selectively attends to query-relevant retrieved samples via query--retrieval cross-attention. By modeling input-level relevance between the query and retrieved samples, Cross-RAG jointly incorporates three sources of information: 1) the query itself, 2) the retrieved samples, and 3) their relational interactions. In particular, this input-aware design enables Cross-RAG to remain stable as the number of retrieved samples $k$ grows, whereas prior methods without cross-attention require careful $k$ tuning to avoid degradation from irrelevant retrievals. Extensive experiments demonstrate that Cross-RAG consistently improves zero-shot forecasting performance across multiple TSFM backbones and various RAG methods, with additional analyses confirming its effectiveness across various retrieval scenarios. Code is available at https://github.com/seunghan96/cross-rag/.

Cost-per-click (CPC) in paid search is an auction-generated outcome shaped by a competitive landscape that is only partially observable from any single advertiser's history. From 1.66 billion Google Ads log records for a concentrated car-rental market (2021-2023), we construct a weekly panel of 1,811 keyword series over 127 weeks (218,924 keyword-week observations) and build competition-aware proxies from keyword text, CPC trajectories, and geographic market structure. The design combines (i) semantic neighborhoods and a semantic keyword graph from pretrained transformer-based keyword representations, (ii) behavioral neighborhoods from Dynamic Time Warping (DTW) alignment of CPC trajectories, and (iii) geographic-intent covariates capturing localized demand and marketplace heterogeneity. We evaluate these signals both as exogenous covariates and as relational priors in spatiotemporal graph forecasters, benchmarking them against statistical, neural, and time-series foundation-model baselines. The results reveal a clear horizon crossover. At one week, graph-based models achieve the lowest error, reducing sMAPE by 15.1% relative to the strongest classical/ML baseline; at the six- and twelve-week horizons, covariate-augmented foundation models dominate, reducing sMAPE by 22.5% and 27.6%, respectively. The gains concentrate in the high-CPC, high-volatility keywords where forecasting errors are most costly. A falsification battery supports the competition interpretation at the planning horizon: the semantic competition graph outperforms a confounder-matched non-competitive graph by 4.05 sMAPE points, and matched-neighbour and time-shuffled controls show the six-week gains are competition-specific rather than generic smoothing. Together, the findings establish a horizon-dependent competition-aware forecasting design for auction-driven advertising markets under partial observability.

Context-aided forecasting (CAF) holds promise for integrating domain knowledge and forward-looking information, enabling AI systems to surpass traditional statistical methods. However, recent empirical studies reveal a puzzling gap: multimodal models often fail to outperform their unimodal counterparts. We hypothesize that this underperformance stems from poor context quality in existing datasets, as verification is challenging. To address these limitations, we introduce a semi-synthetic data augmentation method that generates contexts both descriptive of temporal dynamics and verifiably complementary to numerical histories. This approach enables massive-scale dataset creation, resulting in CAF-7M, a corpus of 7 million context-augmented time series windows, including a rigorously verified test set. We demonstrate that semi-synthetic pre-training transfers effectively to real-world evaluation, and show clear evidence of context utilization. Our results suggest that dataset quality, rather than architectural limitations, has been the primary bottleneck in context-aided forecasting.

Accurate time series forecasting underpins decision-making in many domains, yetconventional ML development often faces data scarcity, distribution shift, anddiminishing returns from manual iteration. We propose Self-Evolving Agent forTime Series Algorithms (SEATS), a framework that autonomously generates, val-idates, and optimizes forecasting algorithm code through an iterative self-evolutionloop. Our design combines three mechanisms: (1) Metric-Advantage MCTS(MA-MCTS), which replaces fixed rewards with a statistically normalized advan-tage score for search guidance, (2) code review with running prompt refinement,so every successfully executed solution is reviewed and the running prompt encodescorrective patterns for later iterations, and (3) global steerable reasoning, whichcompares each evaluated node to global best- and worst-performing solutions forcross-trajectory transfer. A MAP-Elites archive maintains architectural diversity.Across four datasets and two metrics, SEATS wins seven of eight comparisonsagainst strong baselines TimeMixer, Timer, and SEMixer

The Nash-Sutcliffe efficiency ($\text{NSE}$) is a widely used, positively oriented relative measure for evaluating forecasts across multiple time series. However, it lacks a decision-theoretic foundation for this purpose. To address this, we examine its negatively oriented counterpart, which we refer to as Nash-Sutcliffe loss, defined as $L_{\text{NS}} = 1 - \text{NSE}$. We prove that $L_{\text{NS}}$ is strictly consistent for an elicitable and identifiable multi-dimensional functional, which we name the Nash-Sutcliffe functional. This functional is a data-weighted component-wise mean. The common practice of maximizing the average $\text{NSE}$ across multiple series is the sample analog of minimizing the expected $L_{\text{NS}}$. Consequently, this operation implicitly assumes that all series originate from a single non-stationary, stochastic process. We introduce Nash-Sutcliffe linear regression, a multi-dimensional model estimated by minimizing the average $L_{\text{NS}}$, which reduces to a data-weighted least squares formulation. By reorienting the sample average loss function, we extend the previously proposed evaluation and estimation framework to forecasting multiple stationary dependent time series with differing stochastic properties. This constitutes a more natural empirical implementation of the $\text{NSE}$ than the earlier formulation. Our results establish a decision-theoretic foundation for $\text{NSE}$-based model estimation and forecast evaluation in large datasets, while further clarifying the benefits of global over local machine learning models.

Demand forecasting is critical for inventory planning, procurement, replenishment, production, and capacity decisions in heterogeneous supply chains. However, selecting the most appropriate model for each demand series remains challenging because performance varies across datasets, demand structures, horizons, and evaluation metrics. This study proposes the Adaptive Hybrid Selector (AHS), an automatic model-selection rule that combines observed predictive performance with structural demand information. AHS uses demand frequency and series variability to activate a hierarchical logic based on RMSSE, MAE, sMAPE, and BIAS. The proposed selector is compared with Overall Weighted Average (OWA), a benchmark-relative criterion based on sMAPE and MASE, and Equilibrium Ranking Aggregation (ERA), a comparator based on MAE, RMSE, and R^2 rankings. The empirical evaluation uses the Walmart, M3, M4, and M5 datasets, three training-testing partitions, 22 forecasting models, and horizons of up to 12 cycles. Selector performance is assessed ex post using Global Relative Accuracy (GRA), interpreted as an indicator of volumetric coherence between accumulated forecasted demand and observed demand. Results show that AHS provides the most robust overall behavior, especially in M5, while OWA remains competitive in more regular datasets. ERA shows lower volumetric coherence in most configurations. These findings suggest that automatic model selection should account for demand structure and be evaluated using ex post indicators of volumetric coherence.

Accurate production forecasts are essential for the integration of renewable energy sources into the power grid. This paper illustrates how to obtain probabilistic forecasts of wind power generation using gradient boosting trees and an ensemble of weather forecasts. To this end, we perform a comparative analysis across three state-of-the-art probabilistic prediction methods-conformalized quantile regression, natural gradient boosting and conditional diffusion models-all of which can be combined with tree-based machine learning. The methods are validated using four years of data for all Belgian offshore wind farms. We benchmark the models against the power curve and a calibrated wake model as well as a probabilistic method using stochastic variational Gaussian process regression. The tree-based models significantly reduce the mean absolute error in comparison to the deterministic baselines. Additionally, all three methods outperform the Gaussian process baseline in probabilistic skill, while two out of the three also improve point forecast accuracy. The conditional diffusion model attains the best performance, with improvements of 5% in mean absolute error and 12% in continuous rank probability score compared to the probabilistic baseline. Last, the results indicate an average improvement in point forecast accuracy of 17% by using an ensemble of weather forecasts instead of a single provider.

While generative modeling facilitates probabilistic time series forecasting, incorporating heterogeneous exogenous information remains challenging. Diffusion Transformers (DiT) provide a scalable framework for conditional generation, yet their adaptation to forecasting calls for conditioning mechanisms tailored to time series. Endogenous targets and exogenous covariates differ in sources, semantics, and statistical characteristics, while sharing temporal coordinates that support fine-grained conditional guidance. Covariates can describe future variability and temporal dependence beyond the conditional mean targeted by direct regression. Motivated by these considerations, we propose Diffusion Transformers for Time Series (DiTS), a Multimodal Diffusion Transformer for covariate-aware forecasting. DiTS models endogenous targets and exogenous covariates as distinct modalities, jointly conditioning future generation on target history and available covariates. Flow matching makes covariate-dependent distributional information relevant to velocity prediction conditioned on noisy future states. We introduce Time-aligned Modulation, extending AdaLN with the temporal-alignment prior to generate patch-wise modulation parameters from aligned covariates and diffusion time. Across diverse covariate-aware forecasting tasks, DiTS achieves strong performance in both deterministic and probabilistic forecasting, demonstrating the effectiveness of conditional generation for both point and distributional forecasting.

Accurate global Subseasonal-to-Seasonal (S2S) climate forecasting is critical for disaster preparedness and resource management, yet it remains challenging due to chaotic atmospheric dynamics. Despite advances in geometry-aware representations, existing methods do not specify how the zonal and meridional interactions that govern real atmospheric dynamics should be explicitly coordinated and composed, leaving an important architectural gap for S2S forecasting. In this paper, we propose AnisoCast, which explicitly models anisotropic global atmospheric dynamics for accurate S2S climate forecasting. It couples: (1) an Anisotropic Embedding strategy that tokenizes the global grid into latitudinal rings, preserving the integrity of zonal periodic structures; and (2) stacked Aniso Blocks that arrange latent Zonal and Meridional Operators in a weight-shared palindromic composition inspired by symmetric operator splitting. Extensive experiments on the ERA5 reanalysis dataset demonstrate that AnisoCast establishes a new state-of-the-art, significantly outperforming existing methods in both forecasting accuracy and computational efficiency.

Multimodal time series forecasting is crucial in real-world applications, where decisions depend on both numerical data and contextual signals. The core challenge is to effectively combine temporal numerical patterns with the context embedded in other modalities, such as text. While most existing methods align textual features with time-series patterns one step at a time, they neglect the multiscale temporal influences of contextual information such as time-series cycles and dynamic shifts. This mismatch between local alignment and global textual context can be addressed by spectral decomposition, which separates time series into frequency components capturing both short-term changes and long-term trends. In this paper, we propose SpecTF, a simple yet effective framework that integrates the effect of textual data on time series in the frequency domain. Our method extracts textual embeddings, projects them into the frequency domain, and fuses them with the time series' spectral components using a lightweight cross-attention mechanism. This adaptively reweights frequency bands based on textual relevance before mapping the results back to the temporal domain for predictions. Experimental results demonstrate that SpecTF significantly outperforms state-of-the-art models across diverse multi-modal time series datasets while utilizing considerably fewer parameters. Code is available at https://github.com/hiepnh137/SpecTF.

We propose two methodologies for modelling aggregated supply and demand curves in the EPEX SPOT Day\char45 Ahead market, emphasizing generative models as a way to recover distributional variability. The first is a low\char45 dimensional parametric representation that yields deterministic point forecasts; the second is a high\char45 dimensional order\char45 level representation that samples from a conditional distribution of plausible curves. Both model the full curve structure, enabling the analysis of price sensitivity, volume sensitivity, and price impact. The parametric representation uses plateau levels, elastic\char45 region boundaries, and polynomial coefficients, forecast with eXtreme Gradient Boosting. The main contribution is the generative representation, which uses price arrivals and volume\char45 increment marks and is implemented with conditional Denoising Diffusion Probabilistic Models. Using French EPEX data from 2021 to 2024, we evaluate both approaches through curve reconstruction and a price\char45 maker storage optimization problem. The parametric implementation provides a deterministic reference, while the diffusion\char45 based implementation produces distributions of plausible curves and achieves higher realized profits and smaller gaps to an oracle benchmark in the storage application.

Time series forecasting has witnessed significant progress with deep learning. While prevailing approaches enhance forecasting performance by modifying architectures or introducing novel enhancement strategies, they often fail to dynamically disentangle and leverage the complex, intertwined temporal patterns inherent in time series, thus resulting in the learning of static, averaged representations that lack context-aware capabilities. To address this, we propose the Dual-Prototype Adaptive Disentanglement framework (DPAD), a model-agnostic auxiliary method that equips forecasting models with the ability of pattern disentanglement and context-aware adaptation. Specifically, we construct a Dynamic Dual-Prototype bank (DDP), comprising a common pattern bank with strong temporal priors to capture prevailing trend or seasonal patterns, and a rare pattern bank dynamically memorizing critical yet infrequent events, and then an Dual-Path Context-aware routing (DPC) mechanism is proposed to enhance outputs with selectively retrieved context-specific pattern representations from the DDP. Additionally, we introduce a Disentanglement-Guided Loss (DGLoss) to ensure that each prototype bank specializes in its designated role while maintaining comprehensive coverage. Comprehensive experiments demonstrate that DPAD consistently improves forecasting performance and reliability of state-of-the-art models across diverse real-world benchmarks.

Accurate blood glucose forecasting using continuous glucose monitoring (CGM) data can support the early prediction of dysglycemic risk. However, current neural-network-based forecasting models treat CGM data as a purely numerical sequence without integrating the contextual information contained in CGM signal morphology. Recently, large language models (LLMs) have shown promise for time-series forecasting, yet their role as agentic context extractors in diabetes care remains largely unexplored. In this study, we bridge glucose forecasting and LLM-based contextualization by developing GlyRAG, a context-aware, retrieval-augmented forecasting framework that uses an LLM as a contextualization agent to summarize glucose morphology directly from a timed CGM window. The generated CGM-only narrative is embedded and fused with patch-based glucose representations, while a retrieval module incorporates similar historical training episodes through cross-attention. We evaluate GlyRAG on the OhioT1DM and AZT1D datasets for 5-, 30-, and 60-minute forecasting horizons. Compared with strong CGM-only baselines, GPT-4 GlyRAG significantly improves long-horizon root mean square error (RMSE) over PatchTST on both datasets. For example, RMSE decreases from 13.8 to 10.6 at 30 minutes and from 23.1 to 20.2 at 60 minutes on OhioT1DM. LLaMA 3.1 produces smaller but significant long-horizon gains, suggesting that the contextualization pipeline is not limited to GPT-4. Clinical error-grid analyses further show that approximately 85 percent of predictions fall within the clinically acceptable Clarke Error Grid Zones A and B. These results suggest that CGM-derived linguistic context and case-based retrieval can improve long-horizon glucose forecasting without requiring additional sensing modalities.

Predicting river flow in places without streamflow records is challenging because basins respond differently to climate, terrain, vegetation, and soils. Traditional basin attributes describe some of these differences, but they cannot fully represent the complexity of natural environments. This study examines whether AlphaEarth Foundation embeddings, which are learned from large collections of satellite images rather than designed by experts, offer a more informative way to describe basin characteristics. These embeddings summarize patterns in vegetation, land surface properties, and long-term environmental dynamics. We find that models using them achieve higher accuracy when predicting flows in basins not used for training, suggesting that they capture key physical differences more effectively than traditional attributes. We further investigate how selecting appropriate donor basins influences prediction in ungauged regions. Similarity based on the embeddings helps identify basins with comparable environmental and hydrological behavior, improving performance, whereas adding many dissimilar basins can reduce accuracy. The results show that satellite-informed environmental representations can strengthen hydrological forecasting and support the development of models that adapt more easily to different landscapes.

Process Model Forecasting (PMF) aims to predict how the control-flow structure of a process evolves over time by modeling the temporal dynamics of directly-follows (DF) relations, complementing predictive process monitoring that focuses on single-case prefixes. Prior benchmarks show that machine learning and deep learning models provide only modest gains over statistical baselines, mainly due to the sparsity and heterogeneity of the DF time series. We investigate Time Series Foundation Models (TSFMs), large pre-trained models for generic time series, as an alternative for PMF. Using DF time series derived from real-life event logs, we compare zero-shot use of TSFMs, without additional training, with fine-tuned variants adapted on PMF-specific data. TSFMs generally achieve lower forecasting errors (MAE and RMSE) than traditional and specialized models trained from scratch on the same logs, indicating effective transfer of temporal structure from non-process domains. While fine-tuning can further improve accuracy, the gains are often small and may disappear on smaller or more complex datasets, so zero-shot use remains a strong default. Our study highlights the generalization capability and data efficiency of TSFMs for process-related time series and, to the best of our knowledge, provides the first systematic evaluation of temporal foundation models for PMF.

Reliable forecasting of multivariate time series under anomalous conditions is crucial in applications such as ATM cash logistics, where sudden demand shifts can disrupt operations. Modern deep forecasters achieve high accuracy on normal data but often fail when distribution shifts occur. We propose Weighted Contrastive Adaptation (WECA), a Weighted contrastive objective that aligns normal and anomaly-augmented representations, preserving anomaly-relevant information while maintaining consistency under benign variations. Evaluations on a nationwide ATM transaction dataset with domain-informed anomaly injection show that WECA improves SMAPE on anomaly-affected data by 6.1 percentage points compared to a normally trained baseline, with negligible degradation on normal data. These results demonstrate that WECA enhances forecasting reliability under anomalies without sacrificing performance during regular operations.

Effective HFMD surveillance requires forecasts capturing both time-series patterns and contextual drivers such as school calendars, weather, and policy or surveillance reports. In clinical settings, forecasts must be trusted and actionable; thus, beyond point accuracy, decision-makers require concise, auditable explanations of why risk is expected to rise or fall. Classical models (e.g., ARIMA and Prophet) and foundation models (e.g., Chronos, Moirai, and TimesFM) treat external covariates as numerical inputs, lacking semantic reasoning to reflect epidemiological mechanisms or resolve conflicting signals. We propose a two-agent neuro-symbolic framework that decouples contextual interpretation from probabilistic forecasting. An LLM-based Event Interpreter ingests heterogeneous signals -- school schedules, weather summaries, government reports, and clinical guidelines -- and outputs a scalar transmission-impact signal. A Forecast Generator combines this signal with historical case counts to produce point forecasts that are mapped to probabilistic predictions through Poisson/negative-binomial moment matching. We focus on one-week-ahead rolling forecasts, aligning with weekly hospital-capacity planning and the rapid, context-driven inflections typical of HFMD. We evaluate on two datasets: Hong Kong surveillance (90 target weeks in 2023--2024) and Lishui hospital visits (33 target weeks in 2024). Against traditional and foundation-model baselines, our approach achieves competitive point accuracy while providing robust 90% intervals (coverage approximately 0.85--1.00) and concise rationales. This demonstrates that integrating domain knowledge through LLM-based agents can match strong numerical forecasters while yielding interpretable, context-aware forecasts aligned with public-health decision-making.

This study presents a controlled comparison of baseline Kolmogorov-Arnold Networks (KAN), implemented via PyKAN, and Long Short-Term Memory (LSTM) networks for the forecasting of stochastic, non-stationary financial time series. The two architectures are assessed in terms of predictive accuracy, computational efficiency, and interpretability, with accuracy measured by the Root Mean Square Error (RMSE) in normalised feature space. Under a direct multi-output forecasting protocol, LSTM attains clearly superior accuracy across all tested prediction horizons, consistent with its well-established effectiveness for sequential data modelling. Baseline KAN, although offering theoretical interpretability through the Kolmogorov-Arnold representation theorem, exhibits substantially higher error rates and limited practical applicability for time series forecasting in its standard form. Several specialised temporal variants -- including Temporal KAN and Time-Frequency KAN -- have since been proposed to address these sequential modelling limitations, but they lie outside the scope of the present study. KAN is observed to converge faster during training under the configurations tested, although direct runtime comparisons are constrained by methodological factors. These findings support the adoption of LSTM for accuracy-critical financial forecasting and establish an empirical baseline for standard KAN on stochastic sequential data, motivating further investigation of temporally-aware KAN architectures. The study benchmarks baseline KAN against baseline LSTM only; the results do not extend to specialised KAN variants designed for sequential data, nor to the broader family of temporal models.

Modeling the dynamics of financial Limit Order Books (LOB) at the message level is challenging due to irregular event timing, rapid regime shifts, and the reactions of high-frequency traders to visible order flow. Previous LOB models require cumbersome data representations and lack adaptability outside their original tasks, leading us to introduce LOBERT, a general-purpose encoder-only foundation model for LOB data suitable for downstream fine-tuning. LOBERT adapts the original BERT architecture for LOB data by using a novel tokenization scheme that treats complete multi-dimensional messages as single tokens while retaining continuous representations of price, volume, and time. With these methods, LOBERT achieves leading performance in tasks such as predicting mid-price movements and next messages, while reducing the required context length compared to previous methods.

Training time-series forecasting models requires aligning the conditional distribution of model forecasts with that of the label sequence. The standard direct forecast (DF) approach resorts to minimizing the conditional negative log-likelihood, typically estimated by the mean squared error. However, this estimation proves biased when the label sequence exhibits autocorrelation. In this paper, we propose DistDF, which achieves alignment by minimizing a distributional discrepancy between the conditional distributions of forecast and label sequences. Since such conditional discrepancies are difficult to estimate from finite time-series observations, we introduce a joint-distribution Wasserstein discrepancy for time-series forecasting, which provably upper bounds the conditional discrepancy of interest. The proposed discrepancy is tractable, differentiable, and readily compatible with gradient-based optimization. Extensive experiments show that DistDF improves diverse forecasting models and achieves leading performance. Code is available at https://anonymous.4open.science/r/DistDF-F66B.

The design of learning objectives is central to training time-series forecasting models. Existing learning objectives such as mean squared error mostly treat each future step as an independent, equally weighted task, which leads to the following two challenges: (1) they overlook the label autocorrelation effect among future steps, leading to biased learning objectives; (2) they fail to set heterogeneous task weights for different forecasting tasks corresponding to varying future steps, limiting the forecasting performance. To fill this gap, we propose a novel quadratic-form weighted learning objective, addressing both issues simultaneously. Specifically, the off-diagonal elements of the weighting matrix account for the label autocorrelation effect, whereas the non-uniform diagonals are expected to match the preferred weights of the forecasting tasks with varying future steps. On this basis, we propose a Quadratic Direct Forecast (QDF) learning algorithm, which trains the forecast model using the adaptively updated quadratic-form weighting matrix. Experiments show that our QDF effectively improves the performance of various forecast models, achieving state-of-the-art results. Code is available at https://github.com/Master-PLC/QDF.

While Time Series Foundation Models (TSFMs) excel in zero-shot tasks, their behavior under continual fine tuning is poorly understood. We present the first systematic study of catastrophic forgetting in TSFMs (TimesFM-2.0, Chronos-2) versus a specialized SamFormer model across synthetic and real-world energy forecasting benchmarks. Our results show that while fine-tuning improves new task accuracy, it consistently triggers forgetting, though larger models exhibit greater inherent robustness. Notably, employing forgetting mitigation techniques such as DER, levels the playing field: it provides disproportionate gains to smaller models, allowing them to match TSFM performance by the end of the continual learning sequence. These findings suggest that in realistic, non-stationary scenarios, the high computational cost of large foundation models may not be justified over smaller models equipped with effective mitigation strategies.

Benchmark quality is critical for meaningful evaluation and sustained progress in time series forecasting, particularly with the rise of pretrained models. Existing benchmarks often have limited domain coverage or overlook real-world settings such as tasks with covariates. Their aggregation procedures frequently lack statistical rigor, making it unclear whether observed performance differences reflect true improvements or random variation. Many benchmarks lack consistent evaluation infrastructure or are too rigid for integration into existing pipelines. To address these gaps, we propose fev-bench, a benchmark of 100 forecasting tasks across seven domains, including 46 with covariates. Supporting the benchmark, we introduce fev, a lightweight Python library for forecasting evaluation emphasizing reproducibility and integration with existing workflows. Using fev, fev-bench employs principled aggregation with bootstrapped confidence intervals to report performance along two dimensions: win rates and skill scores. We report results on fev-bench for pretrained, statistical, and baseline models and identify promising future research directions.

Inherent temporal heterogeneity, such as varying sampling densities and periodic structures, has posed substantial challenges in zero-shot generalization for Time Series Foundation Models (TSFMs). Existing TSFMs predominantly rely on massive parameterization to absorb such heterogeneity, as their static tokenization and positional encoding schemes entangle diverse temporal patterns into a fixed representation space, encouraging memorization rather than adaptation. To address this limitation, we propose Kairos, a flexible and parameter-efficient TSFM dedicated to forecasting tasks, which decouples temporal heterogeneity from model capacity through a novel tokenization perspective. Kairos introduces a dynamic patching tokenizer and a mixture-of-size encoding that adapt observational granularity to local information density, enabling fine-grained temporal abstraction without increasing model width or depth. In addition, we design a multi-granularity positional embedding based on dynamic rotary encodings, which conditions on instance-level spectral features and temporal structure induced by dynamic patching tokenization, allowing robust modeling of diverse temporal dependencies. Trained on a novel Predictability-Stratified Time-Series (PreSTS) corpus, Kairos achieves superior zero-shot performance with substantially fewer parameters on two mainstream benchmarks, GIFT-Eval and Time-Series-Library. The project page is at https://foundation-model-research.github.io/Kairos .

Time Series Forecasting (TSF) faces persistent challenges in modeling intricate temporal dependencies across different scales. Despite recent advances leveraging different decomposition operations and novel architectures based on CNN, MLP or Transformer, existing methods still struggle with static decomposition strategies, fragmented dependency modeling, and inflexible fusion mechanisms, limiting their ability to model intricate temporal dependencies. To explicitly solve the mentioned three problems respectively, we propose a novel Dynamic Multi-Scale Coordination Framework (DMSC) with Multi-Scale Patch Decomposition block (EMPD), Triad Interaction Block (TIB) and Adaptive Scale Routing MoE block (ASR-MoE). Specifically, EMPD is designed as a built-in component to dynamically segment sequences into hierarchical patches with exponentially scaled granularities, eliminating predefined scale constraints through input-adaptive patch adjustment. TIB then jointly models intra-patch, inter-patch, and cross-variable dependencies within each layer's decomposed representations. EMPD and TIB are jointly integrated into layers forming a multi-layer progressive cascade architecture, where coarse-grained representations from earlier layers adaptively guide fine-grained feature extraction in subsequent layers via gated pathways. And ASR-MoE dynamically fuses multi-scale predictions by leveraging specialized global and local experts with temporal-aware weighting. Comprehensive experiments on thirteen real-world benchmarks demonstrate that DMSC consistently maintains state-of-the-art (SOTA) performance and superior computational efficiency for TSF tasks. Code is available at https://github.com/1327679995/DMSC.

When training autoregressive models to forecast dynamical systems, a critical question arises: how far into the future should the model be trained to predict for optimal performance? In this work, we address this question by analyzing the relationship between the geometry of the loss landscape and the training time horizon. Using dynamical systems theory, we prove that loss minima for long horizons generalize well to short-term forecasts, whereas minima found on short horizons result in worse long-term predictions. However, we also prove that the loss landscape becomes rougher as the training horizon grows, making long-horizon training inherently challenging. We validate our theory through numerical experiments and discuss practical implications for selecting training horizons. Our results provide a principled foundation for hyperparameter optimization in autoregressive forecasting models.

Training time-series forecasting models poses unique challenges in loss function design. Most existing approaches adopt temporal mean squared error, but this study reveals two critical limitations: (1) it ignores the presence of label autocorrelation, which biases it from the true label sequence likelihood; (2) it involves excessive number of tasks, which complicates optimization, especially for long-term forecasting. To address these issues, we introduce Time-o1, a transform-enhanced loss function for time-series forecasting. The central idea is to transform the label sequence into decorrelated components with discriminated significance. Models are then trained to align the most significant components, thereby effectively mitigating label autocorrelation and reducing task amount. Experiments demonstrate that Time-o1 achieves state-of-the-art performance and is compatible with various forecast models. Code is available at https://github.com/Master-PLC/Time-o1.

Time series forecasting (TSF) is important in real-world applications. Recently, Transformer-based methods have achieved strong performance by modeling token dependencies through attention mechanisms. However, existing methods are usually trained mainly with prediction error losses, which may cause models to exploit both critical and redundant token dependencies. Such redundant dependencies can introduce irrelevant information and weaken generalization. To address this issue, we propose a simple yet effective token dependency selection strategy. Specifically, by jointly introducing the attention entropy constraint and prediction error constraint, the model can identify fewer but more critical inter-token dependencies and perform forecasting based on them, thus avoiding the interference of redundant dependencies. The proposed method can be easily extended to various Transformer-based TSF models. Experiments on multiple TSF datasets demonstrate its effectiveness.

We introduce a local surrogate approach for explainable time-series forecasting. An initially non-interpretable predictive model to improve the forecast of a classical time-series 'base model' is used. 'Explainability' of the correction is provided by fitting the base model again to the data from which the error prediction is removed (subtracted), yielding a difference in the model parameters which can be interpreted. We provide illustrative examples to demonstrate the potential of the method to discover and explain underlying patterns in the data.