Coarse-to-Fine Learning of Dynamic Causal Structures

TL;DR

DyCausal is introduced, a dynamic causal structure learning framework that leverages convolutional networks to capture causal patterns within coarse-grained time windows, and then applies linear interpolation to refine causal structures at each time step, thereby recovering fine-grained and time-varying causal graphs.

Abstract

Learning the dynamic causal structure of time series is a challenging problem. Most existing approaches rely on distributional or structural invariance to uncover underlying causal dynamics, assuming stationary or partially stationary causality. However, these assumptions often conflict with the complex, time-varying causal relationships observed in real-world systems. This motivates the need for methods that address fully dynamic causality, where both instantaneous and lagged dependencies evolve over time. Such a setting poses significant challenges for the efficiency and stability of causal discovery. To address these challenges, we introduce DyCausal, a dynamic causal structure learning framework. DyCausal leverages convolutional networks to capture causal patterns within coarse-grained time windows, and then applies linear interpolation to refine causal structures at each time step, thereby recovering fine-grained and time-varying causal graphs. In addition, we propose an acyclic constraint based on matrix norm scaling, which improves efficiency while effectively constraining loops in evolving causal structures. Comprehensive evaluations on both synthetic and real-world datasets demonstrate that DyCausal achieves superior performance compared to existing methods, offering a stable and efficient approach for identifying fully dynamic causal structures from coarse to fine.

Figures (17)

• Figure 1: Conceptual overview of DyCausal. DyCausal uses a sliding convolutional window to traverse temporal series and encodes coarse-grained time-varying causal structures. It then defines a linear interpolation strategy inspired by Taylor expansion to refine the causal structures at each time step $\{\mathbf{W}_t\}^T_{t=1}$. Based on the acyclic constraint $h_{norm}$ enforced by matrix $1$-norm, DyCausal reconstructs time series to optimize causal structures and obtains dynamic causal graphs $\mathcal{G}$ that conform to the dynamic distribution of time series.
• Figure 2: The decoding process. Vector $\mathbf{Z}_t$ is cut and decoded in parallel to obtain vectors $\{\tilde{\mathbf{z}}_{t,i}\}^m_{i=1}$, which are again cut, recombined and decoded in parallel to obtain vectors $\{\tilde{\mathbf{u}}_{t,j}\in\mathbb{R}^{dm}\}^{d\tau}_{j=1}$. The combination of $\{\tilde{\mathbf{u}}_{t,j}\in\mathbb{R}^{dm}\}^{d\tau}_{j=1}$ forms the matrix $\mathbf{W}_t$.
• Figure 3: Results on time series with dynamic causality. $\{\mathbf{W}_1,\cdots,\mathbf{W}_{10}\}$ represent the causal graphs estimated at each time step.
• Figure 4: Comparison of acyclic constraints and their gradients, and runtime.
• Figure 5: Estimated dynamic causal structures on traffic subset. We observed that: (i) causality $x_6\rightarrow x_4$, $x_4\rightarrow x_{19}$, $x_6\rightarrow x_{19}$, $x_{15}\rightarrow x_7$ and $x_{19}\rightarrow x_6$ are dynamic up to the time $t=20$ and static after;(ii) causality $x_4\rightarrow x_8$, $x_{19}\rightarrow x_4$ and $x_{16}\rightarrow x_{10}$ are static before and after $t=20$, but with different causal strengths. In particular, the strength of $x_4\rightarrow x_8$ is almost zero before $t=20$, which means that $x_4\rightarrow x_8$ can only be identified from time series after $t=20$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2