Multi-Band Variable-Lag Granger Causality: A Unified Framework for Causal Time Series Inference across Frequencies

2025-08-01

AI Summary

Traditional Granger causality analysis assumes a fixed time lag, limiting its ability to characterize the time–frequency dynamics of causal relationships in complex systems. To address this, we propose the Multi-Band Variable-Lag Granger Causality (MB-VLGC) model—the first framework to extend variable-lag mechanisms into the frequency domain, enabling separate modeling of time-varying causal delays across distinct frequency bands (e.g., alpha, delta). MB-VLGC overcomes the dual constraints of fixed lag and single-band filtering inherent in conventional approaches. It integrates spectral decomposition with an optimization-based procedure for optimal time-lag alignment, ensuring computationally efficient causal inference. Extensive experiments on synthetic and real-world datasets from neuroscience and economics demonstrate that MB-VLGC significantly outperforms state-of-the-art methods in accurately recovering dynamic, cross-frequency, and cross-lag causal structures. The implementation code and benchmark datasets are publicly available.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Understanding causal relationships in time series is fundamental to many domains, including neuroscience, economics, and behavioral science. Granger causality is one of the well-known techniques for inferring causality in time series. Typically, Granger causality frameworks have a strong fix-lag assumption between cause and effect, which is often unrealistic in complex systems. While recent work on variable-lag Granger causality (VLGC) addresses this limitation by allowing a cause to influence an effect with different time lags at each time point, it fails to account for the fact that causal interactions may vary not only in time delay but also across frequency bands. For example, in brain signals, alpha-band activity may influence another region with a shorter delay than slower delta-band oscillations. In this work, we formalize Multi-Band Variable-Lag Granger Causality (MB-VLGC) and propose a novel framework that generalizes traditional VLGC by explicitly modeling frequency-dependent causal delays. We provide a formal definition of MB-VLGC, demonstrate its theoretical soundness, and propose an efficient inference pipeline. Extensive experiments across multiple domains demonstrate that our framework significantly outperforms existing methods on both synthetic and real-world datasets, confirming its broad applicability to any type of time series data. Code and datasets are publicly available.

Problem

• Modeling frequency-dependent causal delays in time series
• Overcoming fixed-lag limitations in Granger causality analysis
• Unifying causal inference across multiple frequency bands

Innovation

• Multi-Band Variable-Lag Granger Causality framework
• Models frequency-dependent causal delays explicitly
• Efficient inference pipeline for time series