multi-level wavelet decomposition

Existing wavelet-based time series forecasting methods over-rely on low-frequency components while neglecting the critical role of high-frequency details in prediction accuracy. To address this, we propose WaveTuner—a novel framework enabling full-spectrum subband co-optimization in the wavelet domain. It introduces an adaptive wavelet refinement module to generate time-frequency coefficients and a dynamic routing mechanism to learn subband-specific weights. Furthermore, we design a multi-branch, specialized network based on the Kolmogorov–Arnold Network (KAN) to model frequency-band-specific features, jointly capturing global trends and local dynamics. Extensive experiments across eight real-world datasets demonstrate that WaveTuner significantly enhances multi-scale modeling capability and achieves state-of-the-art (SOTA) performance in long-term forecasting tasks.

Traditional global correlation analysis struggles to characterize multiscale dynamic interdependencies between signals. To address this, we propose a wavelet-based multiscale cross-correlation analysis framework. Methodologically, we integrate orthogonal and undecimated discrete wavelet transforms to construct Pearson- and Kendall-type wavelet cross-correlation graphs, partial wavelet correlations, and additive wavelet correlation measures—balancing time-frequency localization with statistical robustness. Theoretically, we generalize the definition and properties of wavelet correlation coefficients. Empirically, simulation studies demonstrate superior sensitivity to time-varying and nonstationary correlation structures; real-data applications successfully uncover cross-frequency coupling patterns. Results show that our approach accurately captures localized, heterogeneous correlation features between two signals across distinct frequency scales, significantly enhancing interpretability and applicability in multiscale dependency modeling.

This work addresses the challenge of spectral bias in machine learning models for dynamical system modeling, which often leads to inaccurate representation of high-frequency components and instability in long-term predictions. To mitigate this issue, the authors propose the Multi-Scale Wavelet Transformer (MSWT), a novel operator learning framework that operates in the wavelet domain. By integrating wavelet-preserving downsampling and cross-scale wavelet attention mechanisms, MSWT explicitly separates and retains both high- and low-frequency information. This approach effectively alleviates spectral bias and significantly enhances the model’s capacity to capture high-frequency structures. Extensive experiments on chaotic dynamical systems and ERA5 climate reanalysis data demonstrate substantial reductions in prediction error and climatic bias, yielding improved long-term stability and spectral fidelity.

Transformer-based models for time-series forecasting suffer from high computational complexity and overfitting, while standard MLPs struggle to capture complex, multi-scale temporal patterns. Method: This paper proposes an MLP-based adaptive multi-scale decomposition framework. Its core innovation is the Multi-scale Decomposable Mixture (MDM) module—integrated with Dual-Dependency Interaction (DDI) and Adaptive Multi-Predictor Synthesis (AMS)—enabling, for the first time, scale-aware joint time-frequency modeling within a pure MLP architecture. By explicitly decomposing and jointly modeling multi-scale dynamics and their cross-scale dependencies, the method significantly enhances pattern representation capability. Contribution/Results: Evaluated on multiple benchmark datasets, the approach achieves state-of-the-art accuracy and efficiency, outperforming leading Transformer- and MLP-based models with substantially lower computational overhead.

Traditional methods for nonstationary time series modeling suffer from inaccurate dynamic characterization due to their assumption of stationary statistical properties. To address this, we propose AdaWaveNet—a novel adaptive wavelet network based on a learnable lifting scheme. Its core innovation lies in the first end-to-end integration of the lifting wavelet transform into a deep neural network, enabling data-driven, multiscale wavelet decomposition and reconstruction while overcoming the representational limitations of fixed wavelet bases. The model adopts a differentiable encoder–decoder architecture, unifying support for forecasting, missing value imputation, and super-resolution. Joint multitask training further enhances generalization. Evaluated across 10 benchmark datasets and three task categories, AdaWaveNet consistently outperforms state-of-the-art methods, achieving significant improvements in forecasting accuracy, imputation robustness, and super-resolution quality.

This work addresses the challenge of multi-scale time series forecasting, where modeling cross-scale patterns and maintaining computational efficiency are often at odds. The authors propose AWGformer, a novel architecture that integrates adaptive wavelet decomposition (AWDM) with a frequency-aware multi-head attention mechanism (FAMA), enabling dynamic selection of wavelet bases guided by signal characteristics and facilitating interaction among multi-band features. Coupled with cross-scale feature fusion (CSFF) and a hierarchical prediction network (HPN), AWGformer forms an end-to-end trainable framework supported by theoretical convergence guarantees. Extensive experiments on multiple benchmark datasets demonstrate that AWGformer significantly outperforms existing methods, achieving superior prediction accuracy and robustness—particularly in multi-scale and non-stationary scenarios.

Existing time-series classification models (e.g., PatchTST) emphasize low-frequency temporal dynamics while neglecting critical high-frequency components, resulting in poor discrimination of subtle transient patterns. To address this, we propose a dual-stream hybrid architecture: building upon the PatchTST backbone, we introduce a learnable wavelet branch that explicitly captures complementary high-frequency information via deep wavelet packet decomposition (WPD); additionally, we design a learnable generalized mean (GeM) pooling layer to enhance discriminability of time-frequency features. This work is the first to seamlessly integrate learnable wavelet representations with Transformer-based temporal modeling. Our method achieves 93.38% mean accuracy on UCI-HAR, outperforming state-of-the-art models including PatchTST. Ablation studies confirm the necessity and effectiveness of both the wavelet branch and the learnable GeM pooling.

This study addresses the challenge that traditional threshold time series models struggle to simultaneously capture both abrupt shifts and smooth evolution in threshold parameters. To overcome this limitation, the authors propose a time-varying threshold autoregressive model based on wavelet series expansion, which flexibly approximates irregular jumps and continuous variations in the threshold function. By leveraging the localized time-frequency properties of wavelet bases, the approach circumvents the limitations of Fourier-based methods in modeling local dynamics. Integrating wavelet expansion, a threshold mechanism, and an autoregressive structure, the proposed model demonstrates superior performance in both simulation studies and empirical analyses, achieving significantly higher fitting accuracy and forecasting capability compared to existing methods, thereby offering a novel framework for modeling complex nonlinear time series.

Addressing the high computational complexity of Transformers, weak modeling capacity of linear models, and high-frequency information loss caused by multi-scale pooling in long-term time series forecasting (LTSF), this paper proposes a dual-path wavelet hybrid architecture. Methodologically, it introduces: (1) a lossless orthogonal Haar wavelet pyramid—first of its kind—to explicitly decouple trend and local fluctuations while avoiding spectral aliasing; (2) a dual-path trend mixer that separately models macroscopic trends via global linear mapping and microscopic dynamics via block-wise MLP-Mixer; and (3) a channel-stationarity-aware adaptive multi-scale fusion mechanism. Extensive experiments across eight benchmark datasets demonstrate that the proposed method significantly outperforms state-of-the-art approaches, achieving superior accuracy, low computational overhead, and strong generalization capability.

This work addresses the limitations of existing long-term time series forecasting methods, which often neglect inter-channel correlations and suffer from high model complexity and low computational efficiency. The authors propose an efficient and accurate forecasting framework that captures multi-level periodic patterns through multi-scale periodic modeling, explicitly models inter-channel dependencies using multi-layer perceptrons, and employs multi-level wavelet decomposition to separate trend and periodic components. Additionally, a frequency-domain loss function is introduced to decouple intra-channel autocorrelations. Evaluated on six real-world datasets, the proposed method achieves state-of-the-art performance, significantly improving prediction accuracy while maintaining superior computational efficiency and strong capability in extracting historical information.