Current methods for time series forecasting struggle in the online scenario, since it is difficult to preserve long-term dependency while adapting short-term changes when data are arriving sequentially. Although some recent methods solve this problem by controlling the updates of latent states, they cannot disentangle the long/short-term states, leading to the inability to effectively adapt to nonstationary. To tackle this challenge, we propose a general framework to disentangle long/short-term states for online time series forecasting. Our idea is inspired by the observations where short-term changes can be led by unknown interventions like abrupt policies in the stock market. Based on this insight, we formalize a data generation process with unknown interventions on short-term states. Under mild assumptions, we further leverage the independence of short-term states led by unknown interventions to establish the identification theory to achieve the disentanglement of long/short-term states. Built on this theory, we develop a Long Short-Term Disentanglement model (LSTD) to extract the long/short-term states with long/short term encoders, respectively. Furthermore, the LSTD model incorporates a smooth constraint to preserve the long-term dependencies and an interrupted dependency constraint to enforce the forgetting of short-term dependencies, together boosting the disentanglement of long/short-term states. Experimental results on several benchmark datasets show that our LSTD model outperforms existing methods for online time series forecasting, validating its efficacy in real-world applications.

The conventional mean square error for time series forecasting is a point-wise loss function, which ignores the temporal dependency of forecasting data points and results in unstable predictions. Although other methods involve shape information loss functions, such as dynamic time warping, they assign the same weights to each matching pair and result in suboptimal results when a wrong matching pair is chosen. Besides, although the optimal transport can assign different weights for each matching pair, they easily suffer from false alignment due to the time lag. To solve these challenges, we propose a Model-Agnostic loss function via Temporally Sensitive Optimal Transport (MA-TOT) as a differentiable loss function for time series forecasting, which combines temporally sensitive Wasserstein distance for adaptive matching pair chosen and Gromov-Wasserstein distance for multi-level-similarity-measurement. Extensive experiments of several of the latest time series forecasting models with our loss function on seven real-world benchmark datasets reflect the effectiveness of our method.