Beyond Time: Cross-Dimensional Frequency Supervision for Time Series Forecasting

Time series forecasting plays a crucial role in various fields, and the methods based on frequency domain analysis have become an important branch. However, most existing studies focus on the design of elaborate model architectures and are often tailored for limited datasets, still lacking universality. Besides, the assumption of independent and identically distributed (IID) data also contradicts the strong correlation of the time domain labels. To address these issues, abandoning time domain supervision, we propose a purely frequency domain supervision approach named cross-dimensional frequency (X-Freq) loss. Specifically, based on a statistical phenomenon, we first prove that the information entropy of the time series is higher than its spectral entropy, which implies higher certainty in frequency domain and thus can provide better supervision. Secondly, the Fourier Transform and the Wavelet Transform are applied to the time dimension and the channel dimension of the time series respectively, to capture the long-term and short-term frequency variations as well as the spatial configuration features. Thirdly, the loss between predictions and targets is uniformly computed in the frequency domain. Moreover, we plug-and-play incorporate X-Freq into multiple advanced forecasting models and compare on 14 real-world datasets. The experimental results demonstrate that, without making any modification to the original architectures or hyperparameters, X-Freq can improve the forecasting performance by an average of 3.3% on long-term forecasting datasets and 27.7% on short-term ones, showcasing superior generality and practicality. The code will be released publicly.