DAG: A Dual Causal Network for Time Series Forecasting with Exogenous Variables

TL;DR

The paper tackles time series forecasting with exogenous covariates, arguing that future covariates and causal interactions between variables are underutilized. It introduces DAG, a dual-causal-network framework with Temporal Causality Modules and Channel Causality Modules that discover and inject causal relationships along temporal and channel dimensions, respectively. The model optimizes a joint objective with temporal, channel, and forecasting losses, using patch embeddings and Transformer-based blocks with a gating mechanism to fuse attention streams. Extensive experiments on eight standard TSF-X datasets and 12 real-world TSF-X datasets show DAG achieves state-of-the-art performance, and the authors provide open-source datasets and code for reproducibility.

Abstract

Time series forecasting is crucial in various fields such as economics, traffic, and AIOps. However, in real-world applications, focusing solely on the endogenous variables (i.e., target variables), is often insufficient to ensure accurate predictions. Considering exogenous variables (i.e., covariates) provides additional predictive information, thereby improving forecasting accuracy. However, existing methods for time series forecasting with exogenous variables (TSF-X) have the following shortcomings: 1) they do not leverage future exogenous variables, 2) they fail to account for the causal relationships between endogenous and exogenous variables. As a result, their performance is suboptimal. In this study, to better leverage exogenous variables, especially future exogenous variable, we propose a general framework DAG, which utilizes dual causal network along both the temporal and channel dimensions for time series forecasting with exogenous variables. Specifically, we first introduce the Temporal Causal Module, which includes a causal discovery module to capture how historical exogenous variables affect future exogenous variables. Following this, we construct a causal injection module that incorporates the discovered causal relationships into the process of forecasting future endogenous variables based on historical endogenous variables. Next, we propose the Channel Causal Module, which follows a similar design principle. It features a causal discovery module models how historical exogenous variables influence historical endogenous variables, and a causal injection module incorporates the discovered relationships to enhance the prediction of future endogenous variables based on future exogenous variables.

Figures (5)

• Figure 1: Time series forecasting algorithms can be classified as follows: (a) univariate/multivariate algorithms without exogenous variables, e.g., PatchTST patchtst and DUET qiu2025duet; (b) algorithms considering only historical exogenous variables, e.g., TimeXer wang2024timexer and CrossLinear zhou2025crosslinear; (c) algorithms that account for both historical and future exogenous variables, e.g., TiDE das2023tide and TFT lim2021temporal; and (d) algorithms that consider both exogenous variables and causal relationships.
• Figure 2: Diagram of Temporal Causality and Channel Causality. Granger causality is used to measure the correlation between historical data and future data. Pearson correlation is used to measure the correlation between exogenous and endogenous variables.
• Figure 3: The architecture of DAG. (a) Overview of the DAG framework, which comprises Temporal Causal Modules ($\mathcal{F}_{\theta_1}$ and $\mathcal{G}_{\theta_2}$) and Channel Causal Modules ($\mathcal{F}_{\theta_3}$ and $\mathcal{G}_{\theta_4}$). (b) Detailed structure of the Temporal Causal Module. (c) Detailed structure of the Channel Causal Module. (c1) The standard Transformer block (Trmblock). (c2) The Causal Trimblock, which injects learned causality into the Trmblock. (d) The loss function. Note that the Gating, Trimblock, and Causal Trimblock used in (b) and (c) share the same architecture.
• Figure 4: Parameter sensitivity studies of main hyper-parameters in DAG.
• Figure 5: Forecasting performance (MSE) with varying look-back windows on 2 datasets: Electricity and Energy. The look-back windows are selected to be H = 48, 96, 192, 336 and 720, and the forecasting horizons are F = 720 and 360.

Theorems & Definitions (3)

• definition 1: Time Series
• definition 2: Endogenous Time Series
• definition 3: Exogenous Time Series