TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting

TL;DR

TimeBridge tackles the divergent effects of non-stationarity in multivariate time series forecasting by decoupling short-term stabilization from long-term dependency modeling. It introduces a patch-based architecture with Integrated Attention to mitigate short-term non-stationarity within variates and Cointegrated Attention to preserve non-stationarity for capturing cross-variate cointegration, aided by patch embedding and downsampling to enrich long-horizon information. Empirical results show state-of-the-art performance across long-term, short-term, and financial forecasting tasks, including CSI 500 and S&P 500 datasets, underscoring the method's robustness to volatility and inter-variable relationships. The work provides theoretical and empirical support for balancing non-stationarity in modeling and offers a practical, open-source solution for complex real-world forecasting problems.

Abstract

Non-stationarity poses significant challenges for multivariate time series forecasting due to the inherent short-term fluctuations and long-term trends that can lead to spurious regressions or obscure essential long-term relationships. Most existing methods either eliminate or retain non-stationarity without adequately addressing its distinct impacts on short-term and long-term modeling. Eliminating non-stationarity is essential for avoiding spurious regressions and capturing local dependencies in short-term modeling, while preserving it is crucial for revealing long-term cointegration across variates. In this paper, we propose TimeBridge, a novel framework designed to bridge the gap between non-stationarity and dependency modeling in long-term time series forecasting. By segmenting input series into smaller patches, TimeBridge applies Integrated Attention to mitigate short-term non-stationarity and capture stable dependencies within each variate, while Cointegrated Attention preserves non-stationarity to model long-term cointegration across variates. Extensive experiments show that TimeBridge consistently achieves state-of-the-art performance in both short-term and long-term forecasting. Additionally, TimeBridge demonstrates exceptional performance in financial forecasting on the CSI 500 and S&P 500 indices, further validating its robustness and effectiveness. Code is available at https://github.com/Hank0626/TimeBridge.

Figures (13)

• Figure 1: Visualization of the impact of non-stationarity on short-term and long-term modeling. The goal is to forecast two cointegrated sequences, $X_t$ and $Y_t$, where $X_t$ exhibits two random fluctuations. (a) Retaining non-stationarity preserves long-term cointegration between variates but leads to spurious regressions in short-term modeling (orange line). (b) Removing non-stationarity avoids short-term spurious regressions but disrupts long-term similar trends (green line).
• Figure 2: Time series forecasting methods categorized by normalization and dependency modeling.
• Figure 3: Overall architecture of TimeBridge: (a) Patch Embedding divides the input sequence into non-overlapping patches and embeds each as a token; (b) Integrated Attention models temporal dependencies within each variate by mitigating short-term non-stationarity; (c) Patch Downsampling reduce patches to aggregates long-term information and lower complexity; (d) Cointegrated Attention captures long-term relationships across variates while keeping non-stationarity.
• Figure 4: (a) Comparison of intra-variate attention maps under stationary and non-stationary conditions for different patches in the Electricity dataset. (b) Comparison of inter-variate attention maps between different variates under stationary and non-stationary conditions in the Solar dataset. (c) Impact of varying the number of downsampled patches $M$ on forecasting performance across different datasets. See \ref{['tab:app_abla_patch_number']} for full results.
• Figure 5: Visualization of the effect of retaining or removing non-stationarity in Integrated Attention and Cointegrated Attention on the Weather dataset for temperature ($T$) and dew point temperature ($T_{\text{dew}}$). (a) Both Integrated and Cointegrated Attention retain non-stationarity. (b) Both remove non-stationarity. (c) Only Integrated Attention retains non-stationarity. (d) Only Cointegrated Attention retains non-stationarity.