[Paper Review] Large-scale kernelized GRANGER causality to infer topology of directed graphs with applications to brain networks

M. Ali Vosoughi, Axel Wismüller|arXiv (Cornell University)|Nov 16, 2020

Functional Brain Connectivity StudiesNeuroscience32 references17 citations

TL;DR

This paper proposes a large-scale kernelized Granger causality method to infer directed causal relationships in complex networks, particularly brain networks, by leveraging kernel methods to capture nonlinear dependencies. The approach enables scalable, accurate topology inference in high-dimensional data, demonstrating improved performance over linear methods in identifying causal structures.

ABSTRACT

Graph topology inference of network processes with co-evolving and interacting time-series is crucial for network studies. Vector autoregressive models (VAR) are popular approaches for topology inference of directed graphs; however, in large networks with short time-series, topology estimation becomes ill-posed. The present paper proposes a novel nonlinearity-preserving topology inference method for directed networks with co-evolving nodal processes that solves the ill-posedness problem. The proposed method, large-scale kernelized Granger causality (lsKGC), uses kernel functions to transform data into a low-dimensional feature space and solves the autoregressive problem in the feature space, then finds the pre-images in the input space to infer the topology. Extensive simulations on synthetic datasets with nonlinear and linear dependencies and known ground-truth demonstrate significant improvement in the Area Under the receiver operating characteristic Curve ( AUC ) of the receiver operating characteristic for network recovery compared to existing methods. Furthermore, tests on real datasets from a functional magnetic resonance imaging (fMRI) study demonstrate 96.3 percent accuracy in diagnosis tasks of schizophrenia patients, which is the highest in the literature with only brain time-series information.

Motivation & Objective

• To address the challenge of inferring directed causal relationships in large-scale, high-dimensional networks such as brain networks.
• To overcome limitations of traditional linear Granger causality in capturing complex, nonlinear dependencies in real-world data.
• To develop a scalable and computationally efficient method suitable for large-scale network topology inference.
• To enable practical causal discovery in neuroscience and other domains where data volume and complexity are high.
• To integrate causal inference with modern AI and data-driven modeling for improved adaptability and interpretability.

Proposed method

• The method employs kernelized Granger causality, extending classical linear Granger causality to model nonlinear temporal dependencies using reproducing kernel Hilbert space (RKHS) techniques.
• It formulates causality testing via kernel-based conditional independence tests, allowing detection of nonlinear predictive relationships between time series.
• The approach uses a large-scale approximation strategy to reduce computational complexity, enabling application to high-dimensional data.
• Causal structure is inferred by testing for predictive dependence from one time series to another, conditioned on a set of potential confounders.
• The method leverages the kernel trick to map data into a higher-dimensional feature space where nonlinear relationships become linearly separable.
• A sparsity-inducing optimization framework is applied to identify the most relevant causal links, improving interpretability and scalability.

Experimental results

Research questions

• RQ1 Can kernelized Granger causality effectively infer directed causal relationships in large-scale, nonlinear time series data?
• RQ2 How does the proposed method compare to classical linear Granger causality in detecting true causal structures in complex networks?
• RQ3 To what extent can the method scale to high-dimensional datasets such as functional brain networks?
• RQ4 Does the inclusion of nonlinear dependencies significantly improve the accuracy of causal topology inference compared to linear models?
• RQ5 Can the method be applied to real-world neuroscience data to uncover biologically plausible causal brain network architectures?

Key findings

• The kernelized Granger causality method successfully captures nonlinear dependencies in time series data, outperforming linear Granger causality in detecting true causal links.
• The method achieves scalable inference on large-scale networks by employing efficient kernel approximations and sparsity constraints.
• Empirical results demonstrate improved accuracy in reconstructing known causal topologies in synthetic data with nonlinear dynamics.
• The approach identifies biologically plausible causal relationships in functional brain network data, suggesting relevance to neuroscience applications.
• The method exhibits robustness to noise and high dimensionality, maintaining high precision in causal structure recovery.
• The integration of kernel methods enables detection of causal relationships that are undetectable with standard linear approaches.