Fourier Basis Mapping: A Time-Frequency Learning Framework for Time Series Forecasting

2025-07-12

AI Summary

Existing Fourier-transform-based time series forecasting methods suffer from several limitations: misalignment of periodic phase origins, sensitivity to sequence length, ambiguous frequency interpretation, and neglect of temporal dynamics. To address these issues, this paper proposes Fourier Basis Mapping (FBM), a novel frequency-domain representation reformulated from the perspective of basis functions. FBM explicitly models frequency components as hierarchical sine-cosine basis coefficients and jointly encodes time-frequency features. Furthermore, we design FBM-S—an enhanced architecture incorporating seasonal-trend-interaction decomposition, interaction masking, centering, and multi-scale downsampling. FBM is plug-and-play compatible with diverse backbone models, including linear models, MLPs, and Transformers. Extensive experiments on multiple real-world benchmarks demonstrate that FBM significantly improves both long-term and short-term forecasting accuracy and robustness, achieving state-of-the-art performance.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Planning, Routing, and Scheduling: Temporal Planning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The integration of Fourier transform and deep learning opens new avenues for time series forecasting. We reconsider the Fourier transform from a basis functions perspective. Specifically, the real and imaginary parts of the frequency components can be regarded as the coefficients of cosine and sine basis functions at tiered frequency levels, respectively. We find that existing Fourier-based methods face inconsistent starting cycles and inconsistent series length issues. They fail to interpret frequency components precisely and overlook temporal information. Accordingly, the novel Fourier Basis Mapping (FBM) method addresses these issues by integrating time-frequency features through Fourier basis expansion and mapping in the time-frequency space. Our approach extracts explicit frequency features while preserving temporal characteristics. FBM supports plug-and-play integration with various types of neural networks by only adjusting the first initial projection layer for better performance. First, we propose FBM-L, FBM-NL, and FBM-NP to enhance linear, MLP-based, and Transformer-based models, respectively, demonstrating the effectiveness of time-frequency features. Next, we propose a synergetic model architecture, termed FBM-S, which decomposes the seasonal, trend, and interaction effects into three separate blocks, each designed to model time-frequency features in a specialized manner. Finally, we introduce several techniques tailored for time-frequency features, including interaction masking, centralization, patching, rolling window projection, and multi-scale down-sampling. The results are validated on diverse real-world datasets for both long-term and short-term forecasting tasks with SOTA performance.

Problem

• Inconsistent starting cycles and series lengths in Fourier-based methods
• Precise interpretation of frequency components lacking temporal information
• Integration of time-frequency features for improved forecasting performance

Innovation

• Integrates Fourier transform with deep learning
• Proposes Fourier Basis Mapping for time-frequency features
• Enhances models with specialized time-frequency techniques