Deep Frequency Derivative Learning for Non-stationary Time Series Forecasting

TL;DR

This paper tackles non-stationary time-series forecasting by arguing that existing normalization toward zero frequency wastes distribution information. It introduces Frequency Derivative Transformation (FDT), a reversible, multi-order frequency-domain transform that yields more stationary representations via the $k$-order Fourier Derivative Operator ${\rm \\mathcal{R}}_k(\mathcal{X}(f))=(j2\\pi f)^k\mathcal{X}(f)$. DERITS combines FDT with an Order-adaptive Fourier Convolution Network (OFCN) in a parallel-stacked architecture to learn frequency dependencies across multiple derivation orders, and uses the inverse transform for time-domain forecasts. Empirical results on seven real-world datasets show consistent improvements over strong baselines and normalization methods, with favorable efficiency and robustness to distribution shifts. The work offers a principled frequency-domain perspective for mitigating non-stationarity and suggests broad applicability of reversible, derivative-based spectral learning for forecasting tasks.

Abstract

While most time series are non-stationary, it is inevitable for models to face the distribution shift issue in time series forecasting. Existing solutions manipulate statistical measures (usually mean and std.) to adjust time series distribution. However, these operations can be theoretically seen as the transformation towards zero frequency component of the spectrum which cannot reveal full distribution information and would further lead to information utilization bottleneck in normalization, thus hindering forecasting performance. To address this problem, we propose to utilize the whole frequency spectrum to transform time series to make full use of data distribution from the frequency perspective. We present a deep frequency derivative learning framework, DERITS, for non-stationary time series forecasting. Specifically, DERITS is built upon a novel reversible transformation, namely Frequency Derivative Transformation (FDT) that makes signals derived in the frequency domain to acquire more stationary frequency representations. Then, we propose the Order-adaptive Fourier Convolution Network to conduct adaptive frequency filtering and learning. Furthermore, we organize DERITS as a parallel-stacked architecture for the multi-order derivation and fusion for forecasting. Finally, we conduct extensive experiments on several datasets which show the consistent superiority in both time series forecasting and shift alleviation.

Figures (8)

• Figure 1: Given one time series and its frequency spectrum, the main comparison between existing works (a) and our method (b).
• Figure 2: The main architecture of DeRiTS.
• Figure 3: The forecasting performance (MAE) comparison between original DeRiTS (multi-order) and its individual-order variant. The lower values indicate the better forecasting performance.
• Figure 4: Impact of lookback length on forecasting. Metrics MAE and RMSE are reported with the length of lookback window prolonged and the prediction length fixed.
• Figure 5: Visualization comparison of original signals and derived signals with Fourier derivative transformation.

Theorems & Definitions (4)

• Definition 1: Fourier Derivative Operator
• Proposition 1
• Proposition 1
• proof