Frequency-Domain Time Series Modeling

Long-term Time Series Forecasting (LTSF) tasks, which leverage the current data sequence as input to predict the future sequence, have become increasingly crucial in real-world applications such as weather forecasting and planning of electricity consumption. However, state-of-the-art LTSF models often fail to achieve prediction output alignment for the same timestamps across lagged input sequences. Instead, these models exhibit low output alignment, resulting in fluctuation in prediction outputs for the same timestamps, undermining the model's reliability. To address this, we propose AliO (Align Outputs), a novel approach designed to improve the output alignment of LTSF models by reducing the discrepancies between prediction outputs for the same timestamps in both the time and frequency domains. To measure output alignment, we introduce a new metric, TAM (Time Alignment Metric), which quantifies the alignment between prediction outputs, whereas existing metrics such as MSE only capture the distance between prediction outputs and ground truths. Experimental results show that AliO effectively improves the output alignment, i.e., up to 58.2% in TAM, while maintaining or enhancing the forecasting performance (up to 27.5%). This improved output alignment increases the reliability of the LTSF models, making them more applicable in real-world scenarios.

Large Language Models (LLMs) have shown strong potential in multivariate time series forecasting and anomaly detection. Existing studies predominantly inject temporal information into LLMs via direct numerical tokenization or heuristic textual descriptions. However, LLMs still face difficulty in perceiving the underlying structural patterns of numerical time series, particularly the seasonal and trend components obscured by discrete numerical tokens. To bridge this gap, we propose FreSia, a frequency-aware framework that establishes an effective alignment between the semantic space of LLMs and the frequency space of time series. Specifically, FGPrompt, a Frequency-Guided Prompt mechanism within FreSia, distills the frequency-domain structures of time series and projects them into prompts tailored to the semantic space of LLMs. Furthermore, we introduce a Global-driven Context Learning (GCL) component, which uses a global CLS-driven probe to generate global context to bridge the time-frequency domain gap and fuse the multi-modal information. Experiments on eight forecasting benchmarks show that FreSia achieves average improvements of 13.48% and 8.06% in MSE and MAE, respectively.

Real-world time series are inherently non-stationary, with trends, periodic patterns, and uncertainty evolving over time. While the Fourier domain offers a natural lens to model time series, current deep learning approaches do not explicitly model evolution and randomness in the Fourier spectra, which limits their ability to accurately predict both the expected trajectory and its uncertainty in non-stationary time series. Motivated by Evolutionary Spectra (ES) theory, we propose TimeES, a general framework that enables probabilistic and deterministic forecasting via the evolutionary spectra theory. Specifically, we derive a parameterizable evolutionary spectra formulation, recasting non-stationary random process modeling as learning an evolving representation modulated by random variables. Furthermore, we reduce the complexity of the estimated spectra from O(NM) to O(NK), where K << M/2, by exploiting Hermitian symmetry and spectral energy sparsity for frequency selection. Based on a simple linear backbone, our proposed TimeES achieves consistent state-of-the-art performance across both deterministic and probabilistic forecasting tasks, with high efficiency and interpretability. Code is available at: https://github.com/wwy155/TimeES.

Probabilistic time series forecasting remains challenging, largely because modeling distinct trend and seasonal dynamics requires specialized approaches. Existing methods often fail to capture the unique inner properties of these components, lack interpretability, or suffer from heavy memory and runtime overhead. To address these limitations, we propose DecoVAE, a lightweight interpretable trend-seasonal VAE framework that explicitly decomposes time series into trend and seasonal components by applying domain-specific inductive biases. The trend stream enforces structural smoothness using a differential regularizer on the latent trajectory, analogous to the Hodrick-Prescott filter. Concurrently, the seasonal stream operates in the frequency domain via a complex Gaussian VAE, natively capturing the amplitude and phase of periodic patterns. Extensive evaluations across seven real-world benchmarks show that DecoVAE consistently outperforms strong baselines. It achieves reductions of up to 14.96% in CRPS and 23.30% in NMAE for short-term forecasting, and up to 52.68% and 26.51% for long-term horizons. Crucially, DecoVAE yields these accuracy gains while remaining highly efficient, reducing model weight by up to 93% and accelerating speed by up to 74% compared to the second-best method.

Multi-modality transportation refers to urban systems composed of multiple transportation modes, such as traffic flow and public transit, whose dynamics are coupled by shared temporal patterns. Accurate multi-modality transportation forecasting remains challenging because (1) different modalities exhibit distinct spectral characteristics and (2) interact unevenly across frequencies, whereas most existing methods operate primarily in the time domain or rely on coarse feature fusion. To address these limitations, we propose a lightweight yet effective Frequency-Domain Multi-Modality modeling (FreMo) that explicitly exploits the frequency domain to enable adaptive and selective cross-modality synergy. FreMo disentangles modality-wise spectral refinement from cross-modality synergy and supports plug-and-play integration with general time series backbones. Specifically, FreMo introduces a Modality-Wise Frequency Filter (MFF) to adaptively refine spectral components within each modality, emphasizing informative frequencies while suppressing noise. FreMo further incorporates a Frequency-Guided Synergy Integrator (FSI) that selectively aggregates information across modalities based on their relative contribution at each frequency, facilitating effective cross-modality knowledge sharing while mitigating negative transfer. Extensive experiments on real-world datasets show that FreMo consistently outperforms state-of-the-art baselines, with superior performance and generalization across diverse forecasting scenarios. The code is available at https://github.com/beginner-sketch/FreMo.

Time series forecasting leverages historical patterns to predict future values, but traditional methods face challenges when dealing with complex, non-stationary patterns that are difficult to memorize during training. Retrieval-augmented approaches have emerged as promising solutions by retrieving similar historical patterns to enhance predictions. However, existing retrieval methods suffer from two fundamental limitations: spectral blindness, which overlooks critical frequency-domain characteristics that capture underlying periodic structures, and temporal recency, which treats all historical data equally without emphasizing recent, more relevant patterns. In this paper, we propose SpecReTF, a novel retrieval method that addresses these issues by converting time series into windowed frequency representations, measuring similarity with a combined metric that captures both amplitude and phase information. To balance recency and historical context, we apply an exponential moving average weighting scheme that emphasizes recent windows. Extensive experiments on benchmark datasets demonstrate that SpecReTF outperforms time-domain retrieval methods, achieving superior forecasting accuracy across diverse, non-stationary time series.

Diffusion models have demonstrated strong performance in time series modeling due to their ability to progressively capture complex data distributions through iterative denoising. However, existing approaches struggle with frequency-sensitive denoising, high-frequency reconstruction and balancing global trends with local dynamics. To address these limitations, we propose HyFAD, a Hybrid time-frequency Diffusion model with Frequency-Aware embedding for time series imputation. Built upon the DDPM paradigm, HyFAD adopts a coupled time-frequency diffusion framework, in which the reverse denoising proceeds sequentially from the time domain to the frequency domain, enabling coarse-to-fine generation. Specifically, the time-domain diffusion process captures low-frequency global trends, while the frequency-domain diffusion process refines high-frequency spectral components. We further introduce a frequency-aware step embedding that exploits the relationship between diffusion steps and spectral components, providing step-dependent spectral guidance and facilitates more accurate band-wise reconstruction. Extensive experiments on multiple benchmark datasets demonstrate that HyFAD achieves state-of-the-art performance. Our source code is available at https://github.com/hongfangao/HyFAD.

Long-term time-series forecasting needs models that are accurate yet efficient enough for commodity hardware. Lightweight linear forecasters are remarkably strong in this regime, yet they leave two openings: reversible instance normalization (RevIN) de-normalizes the entire horizon with a single lookback statistic, which is inaccurate under non-stationarity, and time-domain trend/seasonal decomposition relies on a fixed, non-adaptive filter. We present FreqLite, an ultra-lightweight, channel-independent frequency-decomposed linear forecaster: a learnable, lossless, partition-of-unity spectral filter splits the input into bands that are forecast by per-band linear heads and, unlike low-pass-truncation approaches, the high-frequency band is retained and modeled. FreqLite is the best lightweight model on the standard long-term forecasting benchmarks and, at long lookback (L=336), attains a lower average error than a PatchTST Transformer (0.3244 vs. 0.3587 MSE) while using 4x fewer parameters, 2.2x less memory, and 2.2x less time per epoch on a single 4 GB laptop GPU; although modest in magnitude, its improvements are statistically significant under paired Wilcoxon tests across all matched cells (p < 1e-5). We further introduce Adaptive Reversible Instance Normalization (A-RevIN), a regime-adaptive reversible normalization that strictly generalizes RevIN (recovered exactly when its gate is closed), engages under non-stationarity, and reduces to RevIN without harm on stationary data. We validate this on both a real strongly non-stationary dataset (ILI, up to ~5% MSE reduction) and a controlled synthetic drift sweep in which A-RevIN's benefit and its learned gate both rise monotonically with injected non-stationarity. Every component is independently ablatable (Linear and RLinear are special cases of FreqLite), and all results are reproducible on commodity hardware.

Accurate and efficient long-term multivariate time series forecasting requires capturing recurring temporal structure while keeping inference cheap across many variables and horizons. Frequency-space models represent long-range and periodic variation compactly, but they typically process the real and imaginary spectral components as weakly coupled streams and treat periodic cues as ordinary input features, even when such cues are unreliable. This paper proposes FRWKV-Plus, a lightweight periodic-aware frequency-space forecasting model built on the efficient FRWKV backbone. FRWKV-Plus introduces a cross-branch spectral gate that reweights each spectral branch using a summary of its sibling branch, and a trust-gated residual correction that converts compact within-period context into a bounded, sign-flexible adjustment of these gates under a learned, data-dependent trust score. By construction, the correction is identity-preserving at initialization and strictly bounded, so periodic evidence can refine but never dominate or invert the base interaction. On seven standard benchmarks, FRWKV-Plus is consistently competitive with strong linear, frequency-domain, recurrent-style, and Transformer-based forecasters while preserving the lightweight profile of the backbone. Controlled three-seed ablations show that each component contributes, that the benefit is modest on strongly periodic data and pronounced on the harder Exchange and ILI datasets, and that the within-period context is the most influential single component. The implementation is publicly available at https://github.com/yangqingyuan-byte/FRWKV-plus.

This paper presents a preliminary analysis of the ability of Chronos foundation model to process and internally represent frequency domain information. Foundation models that process time-series data offer practitioners a unified architecture capable of learning generic temporal representations across diverse tasks and domains, reducing the need for task-specific feature engineering and enabling transfer across signal modalities. Despite their growing adoption, the extent to which such models encode fundamental signal properties remains insufficiently characterised. We address this gap by analysing Chronos under controlled conditions, starting from the simplest class of signals: discrete sinusoids generated at fixed frequencies. Using lightweight online minimum description length probes applied to the decoder architecture, we test for the presence and separability of frequency information in the model's internal representations. The results provide insight into how frequential content is captured across the frequency spectrum and highlight regimes in which representation quality may degrade or require particular care. These findings offer practical guidance for users of Chronos in signal processing and information fusion contexts, and contribute to ongoing efforts to improve the interpretability and evaluation of foundation models for temporal data.

Accurate long-term time series forecasting (LTSF) requires the capture of complex long-range dependencies and dynamic periodic patterns. Recent advances in frequency-domain analysis offer a global perspective for uncovering temporal characteristics. However, real-world time series often exhibit pronounced cross-domain heterogeneity where variables that appear synchronized in the time domain can differ substantially in the frequency domain. Existing frequency-based LTSF methods often rely on implicit assumptions of cross-domain homogeneity, which limits their ability to adapt to such intricate variability. To effectively integrate frequency-domain analysis with temporal dependency learning, we propose AdaMamba, a novel framework that endogenizes adaptive and context-aware frequency analysis within the Mamba state-space update process. Specifically, AdaMamba introduces an interactive patch encoding module to capture inter-variable interaction dynamics. Then, we develop an adaptive frequency-gated state-space module that generates input-dependent frequency bases, and generalizes the conventional temporal forgetting gate into a unified time-frequency forgetting gate. This allows dynamic calibration of state transitions based on learned frequency-domain importance, while preserving Mamba's capability in modeling long-range dependencies. Extensive experiments on seven public LTSF benchmarks and two domain-specific datasets demonstrate that AdaMamba consistently outperforms state-of-the-art methods in forecasting accu racy while maintaining competitive computational efficiency. The code of AdaMamba is available at https://github.com/XDjiang25/AdaMamba.

Time series foundation models (TSFMs) have recently achieved remarkable success in universal forecasting by leveraging large-scale pretraining on diverse time series data. Complementing this progress, incorporating frequency-domain information yields promising performance in enhancing the modeling of complex temporal patterns, such as periodicity and localized high-frequency dynamics, which are prevalent in real-world time series. To advance this direction, we propose a new perspective that integrates explicit frequency-domain representations into scalable foundation models, and introduce WaveMoE, a wavelet-enhanced mixture-of-experts foundation model for time series forecasting. WaveMoE adopts a dual-path architecture that jointly processes time series tokens and wavelet tokens aligned along a unified temporal axis, and coordinates them through a shared expert routing mechanism that enables consistent expert specialization while efficiently scaling model capacity. Preliminary experimental results on 16 diverse benchmark datasets indicate that WaveMoE has the potential to further improve forecasting performance by incorporating wavelet-domain corpora.

Multivariate time series forecasting often struggles to capture long-range dependencies due to fixed lookback windows. Retrieval-augmented forecasting addresses this by retrieving historical segments from memory, but existing approaches rely on a channel-agnostic strategy that applies the same references to all variables. This neglects inter-variable heterogeneity, where different channels exhibit distinct periodicities and spectral profiles. We propose CRAFT (Channel-wise retrieval-augmented forecasting), a novel framework that performs retrieval independently for each channel. To ensure efficiency, CRAFT adopts a two-stage pipeline: a sparse relation graph constructed in the time domain prunes irrelevant candidates, and spectral similarity in the frequency domain ranks references, emphasizing dominant periodic components while suppressing noise. Experiments on seven public benchmarks demonstrate that CRAFT outperforms state-of-the-art forecasting baselines, achieving superior accuracy with practical inference efficiency.

Text-to-time-series generation is particularly important in meteorology, where natural language offers intuitive control over complex, multi-scale atmospheric dynamics. Existing approaches are constrained by the lack of large-scale, physically grounded multimodal datasets and by architectures that overlook the spectral-temporal structure of weather signals. We address these challenges with a unified framework for text-guided meteorological time-series generation. First, we introduce MeteoCap-3B, a billion-scale weather dataset paired with expert-level captions constructed via a Multi-agent Collaborative Captioning (MACC) pipeline, yielding information-dense and physically consistent annotations. Building on this dataset, we propose MTransformer, a diffusion-based model that enables precise semantic control by mapping textual descriptions into multi-band spectral priors through a Spectral Prompt Generator, which guides generation via frequency-aware attention. Extensive experiments on real-world benchmarks demonstrate state-of-the-art generation quality, accurate cross-modal alignment, strong semantic controllability, and substantial gains in downstream forecasting under data-sparse and zero-shot settings. Additional results on general time-series benchmarks indicate that the proposed framework generalizes beyond meteorology.

Separating multiple effects in time series is fundamental yet challenging for time-series forecasting (TSF). However, existing TSF models cannot effectively learn interpretable multi-effect decomposition by their smoothing-based temporal techniques. Here, a new interpretable frequency-based decomposition pipeline MLOW captures the insight: a time series can be represented as a magnitude spectrum multiplied by the corresponding phase-aware basis functions, and the magnitude spectrum distribution of a time series always exhibits observable patterns for different effects. MLOW learns a low-rank representation of the magnitude spectrum to capture dominant trending and seasonal effects. We explore low-rank methods, including PCA, NMF, and Semi-NMF, and find that none can simultaneously achieve interpretable, efficient and generalizable decomposition. Thus, we propose hyperplane-nonnegative matrix factorization (Hyperplane-NMF). Further, to address the frequency (spectral) leakage restricting high-quality low-rank decomposition, MLOW enables a flexible selection of input horizons and frequency levels via a mathematical mechanism. Visual analysis demonstrates that MLOW enables interpretable and hierarchical multiple-effect decomposition, robust to noises. It can also enable plug-and-play in existing TSF backbones with remarkable performance improvement but minimal architectural modifications.

Mining time-frequency features is critical for time series forecasting. Existing research has predominantly focused on modeling low-frequency patterns, where most time series energy is concentrated. The overlooking of mid to high frequency continues to limit further performance gains in deep learning models. We propose FreqCycle, a novel framework integrating: (i) a Filter-Enhanced Cycle Forecasting (FECF) module to extract low-frequency features by explicitly learning shared periodic patterns in the time domain, and (ii) a Segmented Frequency-domain Pattern Learning (SFPL) module to enhance mid to high frequency energy proportion via learnable filters and adaptive weighting. Furthermore, time series data often exhibit coupled multi-periodicity, such as intertwined weekly and daily cycles. To address coupled multi-periodicity as well as long lookback window challenges, we extend FreqCycle hierarchically into MFreqCycle, which decouples nested periodic features through cross-scale interactions. Extensive experiments on seven diverse domain benchmarks demonstrate that FreqCycle achieves state-of-the-art accuracy while maintaining faster inference speeds, striking an optimal balance between performance and efficiency.

Time Series forecasting (TSF) in the modern era faces significant computational and storage cost challenges due to the massive scale of real-world data. Dataset Distillation (DD), a paradigm that synthesizes a small, compact dataset to achieve training performance comparable to that of the original dataset, has emerged as a promising solution. However, conventional DD methods are not tailored for time series and suffer from architectural overfitting and limited scalability. To address these issues, we propose Harmonic Dataset Distillation for Time Series Forecasting (HDT). HDT decomposes the time series into its sinusoidal basis through the FFT and aligns the core periodic structure by Harmonic Matching. Since this process operates in the frequency domain, all updates during distillation are applied globally without disrupting temporal dependencies of time series. Extensive experiments demonstrate that HDT achieves strong cross-architecture generalization and scalability, validating its practicality for large-scale, real-world applications.

Time series forecasters are widely used across various domains. Among them, MLP (multi-layer perceptron)-based forecasters have been proven to be more robust to noise compared to Transformer-based forecasters. However, MLP struggles to capture complex features, resulting in limitations on capturing long-range dependencies. To address this challenge, we propose XLinear, an MLP-based forecaster for long-range forecasting. Firstly, we decompose the time series into trend and seasonal components. For the trend component which contains long-range characteristics, we design Enhanced Frequency Attention (EFA) to capture long-term dependencies by leveraging frequency-domain operations. Additionally, a CrossFilter Block is proposed for the seasonal component to maintain the model's robustness to noise, avoiding the problems of low robustness often caused by attention mechanisms. Experimental results demonstrate that XLinear achieves state-of-the-art performance on test datasets. While keeping the lightweight architecture and high robustness of MLP-based models, our forecaster outperforms other MLP-based forecasters in capturing long-range dependencies.

Nonstationary time series forecasting suffers from the distribution shift issue due to the different distributions that produce the training and test data. Existing methods attempt to alleviate the dependence by, e.g., removing low-order moments from each individual sample. These solutions fail to capture the underlying time-evolving structure across samples and do not model the complex time structure. In this paper, we aim to address the distribution shift in the frequency space by considering all possible time structures. To this end, we propose a Time-Invariant Frequency Operator (TIFO), which learns stationarity-aware weights over the frequency spectrum across the entire dataset. The weight representation highlights stationary frequency components while suppressing non-stationary ones, thereby mitigating the distribution shift issue in time series. To justify our method, we show that the Fourier transform of time series data implicitly induces eigen-decomposition in the frequency space. TIFO is a plug-and-play approach that can be seamlessly integrated into various forecasting models. Experiments demonstrate our method achieves 18 top-1 and 6 top-2 results out of 28 forecasting settings. Notably, it yields 33.3% and 55.3% improvements in average MSE on the ETTm2 dataset. In addition, TIFO reduces computational costs by 60% -70% compared to baseline methods, demonstrating strong scalability across diverse forecasting models.

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.

Multivariate time series imputation is fundamental in applications such as healthcare, traffic forecasting, and biological modeling, where sensor failures and irregular sampling lead to pervasive missing values. Existing Transformer- and diffusion-based imputers achieve strong performance, but they often rely mainly on time-domain modeling and lack adaptive spectral bias for recovering structured temporal gaps. We propose FADTI, a Fourier- and attention-driven diffusion framework for multivariate time series imputation. FADTI introduces a Fourier Bias Projection (FBP) module that injects learnable frequency-aware bias into intermediate hidden states during denoising. It projects intermediate hidden states onto Fourier bases, avoiding direct spectral estimation from masked or zero-filled inputs. With DFT, STFT, and FSST instantiations, FBP captures global periodicity, localized time--frequency variations, and non-stationary oscillatory patterns. By coupling FBP with self-attention and gated convolution, FADTI integrates frequency-domain guidance, temporal dependency modeling, and probabilistic denoising in a unified framework. Experiments on multiple benchmarks, including a new biological imputation benchmark, show that FADTI improves accuracy, uncertainty estimation, and sampling efficiency, especially under high missing rates and structured missing patterns. Code is available at https://github.com/RazeenLI/FADTI

Long-term time-series forecasting supports a wide range of applications, including weather prediction and electricity demand planning. Frequency-domain methods address this task by decomposing observations into components that describe temporal variations at different scales. However, separate representations do not by themselves provide an explicit mechanism for adjusting the training emphasis across components. Under a shared forecasting objective, the frequency-specific networks can retain different levels of coefficient prediction error, motivating an error-dependent adjustment to their gradients. To this end, we propose BEAT (Balanced frEquency Adaptive Tuning), a framework that combines frequency-specific error monitoring with adaptive gradient modulation. We design a Frequency-Specific Monitor that compares predicted and target wavelet coefficients in a common normalized space and expresses each discrepancy relative to a reference error computed from the detail components. We further introduce a Dynamical Gradient Balancer that converts these ratios into positive, bounded coefficients. Components with higher relative errors receive larger gradient weights, whereas those with lower relative errors receive smaller weights. A shared modulation-strength parameter controls the departure from unmodulated training, and the monitoring and balancing operations are used only during training. Experiments on seven real-world datasets show that BEAT achieves competitive performance against state-of-the-art forecasting methods.