Rethinking Fourier Transform from A Basis Functions Perspective for Long-term Time Series Forecasting

The interaction between Fourier transform and deep learning opens new avenues for long-term time series forecasting (LTSF). We propose a new perspective to reconsider the Fourier transform from a basis functions perspective. Specifically, the real and imaginary parts of the frequency components can be viewed as the coefficients of cosine and sine basis functions at tiered frequency levels, respectively. We argue existing Fourier-based methods do not involve basis functions thus fail to interpret frequency coefficients precisely and consider the time-frequency relationship sufficiently, leading to inconsistent starting cycles and inconsistent series length issues. Accordingly, a novel Fourier basis mapping (FBM) method addresses these issues by mixing time and frequency domain features through Fourier basis expansion. Differing from existing approaches, FBM (i) embeds the discrete Fourier transform with basis functions, and then (ii) can enable plug-and-play in various types of neural networks for better performance. FBM extracts explicit frequency features while preserving temporal characteristics, enabling the mapping network to capture the time-frequency relationships. By incorporating our unique time-frequency features, the FBM variants can enhance any type of networks like linear, multilayer-perceptron-based, transformer-based, and Fourier-based networks, achieving state-of-the-art LTSF results on diverse real-world datasets with just one or three fully connected layers. The code is available at: https://github.com/runze1223/Fourier-Basis-Mapping.