LOCAL: Learning with Orientation Matrix to Infer Causal Structure from Time Series Data

TL;DR

LOCAL introduces a constraint-free, quasi-maximum likelihood framework for dynamic causal discovery from time-series data. It combines two adaptive modules: Asymptotic Causal Mask Learning (ACML) for DAG-oriented masking and Dynamic Graph Parameter Learning (DGPL) for low-rank, time-varying graph factorization, with optional nonlinear modeling via 1D CNNs. The approach achieves accurate recovery of instantaneous and lagged causal edges, scales to high-dimensional settings, and demonstrates substantial speedups by avoiding matrix exponential operations. Empirical results on synthetic, NetSim, and CausalTime datasets show superior accuracy and efficiency over state-of-the-art baselines, with interpretable decompositions of causal structure. These features position LOCAL as a robust tool for dynamic causal discovery in complex time-series contexts.

Abstract

Discovering the underlying Directed Acyclic Graph (DAG) from time series observational data is highly challenging due to the dynamic nature and complex nonlinear interactions between variables. Existing methods typically search for the optimal DAG by optimizing an objective function but face scalability challenges, as their computational demands grow exponentially with the dimensional expansion of variables. To this end, we propose LOCAL, a highly efficient, easy-to-implement, and constraint-free method for recovering dynamic causal structures. LOCAL is the first attempt to formulate a quasi-maximum likelihood-based score function for learning the dynamic DAG equivalent to the ground truth. Building on this, we introduce two adaptive modules that enhance the algebraic characterization of acyclicity: Asymptotic Causal Mask Learning (ACML) and Dynamic Graph Parameter Learning (DGPL). ACML constructs causal masks using learnable priority vectors and the Gumbel-Sigmoid function, ensuring DAG formation while optimizing computational efficiency. DGPL transforms causal learning into decomposed matrix products, capturing dynamic causal structure in high-dimensional data and improving interpretability. Extensive experiments on synthetic and real-world datasets demonstrate that LOCAL significantly outperforms existing methods and highlight LOCAL's potential as a robust and efficient method for dynamic causal discovery.

Figures (7)

• Figure 1: Illustration of instantaneous dependencies (solid lines) and lagged dependencies (dashed lines) in a dynamic Bayesian network (DBN) with $d = 3$ nodes and an autoregression order of $p = 2$. For clarity, edges that do not influence the variables at time $t$ are shown in a lighter shade.
• Figure 2: Comparison of dependencies learned by various causal discovery models with the ground truth on a synthetic dataset with $d=20$ nodes, $T=1000$ time steps, and a lag order of $p=1$. The results indicate that the dependencies learned by LOCAL closely match the ground truth for both lagged and instantaneous dependencies.
• Figure 3: Comparison of instantaneous dependencies (upper) and lagged dependencies (lower) across various methods on a synthetic dataset with different numbers of nodes, $d = \{5, 10, 20, 50, 100\}$. The performance of each method is evaluated using three metrics: True Positive Rate (TPR), Structural Hamming Distance (SHD), and F1 score. Each row present the results for these three metrics, assessing both intra-slice and inter-slice graphs. The results indicate that LOCAL consistently performe well across different metrics and node configurations.
• Figure 4: Comparison of total runtime and performance across various causal discovery methods. Each method is evaluated on a synthetic dataset with different numbers of nodes, $d = \{5, 10, 20\}$. The results demonstrate that LOCAL achieve a favorable balance between accuracy and efficiency.
• Figure 5: Ablation study of different components in LOCAL. Each component contribute to the learning of instantaneous dependencies (left) and lagged dependencies (right) on a synthetic dataset with $d = \{20, 50, 100\}$ nodes, $T = 1000$ time steps, and a lag order of $p = 1$. The results highlight the impact of removing individual components (DGPL, ACML, and QMLE) on model performance.