[Paper Review] State Space Methods for Granger-Geweke Causality Measures

Victor Solo|arXiv (Cornell University)|Jan 19, 2015

Blind Source Separation Techniques31 references3 citations

TL;DR

This paper resolves long-standing theoretical and computational challenges in Granger-Geweke causality measures (GEMs) by introducing state space methods for computing submodels, modeling the effects of downsampling and filtering, and ensuring reliable frequency-domain GEM estimation. The key contribution is a computationally robust framework that establishes the theoretical impact of sampling and filtering on causality, with applications in fMRI and multivariate time series analysis.

ABSTRACT

At least two recent developments have put the spotlight on some significant gaps in the theory of multivariate time series. The recent interest in the dynamics of networks; and the advent, across a range of applications, of measuring modalities that operate on different temporal scales. Fundamental to the description of network dynamics is the direction of interaction between nodes, accompanied by a measure of the strength of such interactions. Granger causality (GC) and its associated frequency domain strength measures (GEMs) (due to Geweke) provide a framework for the formulation and analysis of these issues. In pursuing this setup, three significant unresolved issues emerge. Firstly computing GEMs involves computing submodels of vector time series mod- els, for which reliable methods do not exist; Secondly the impact of filtering on GEMs has never been definitively established. Thirdly the impact of downsampling on GEMs has never been established. In this work, using state space methods, we resolve all these issues and illustrate the results with some simulations. Our discussion is motivated by some problems in (fMRI) brain imaging but is of general applicability.

Motivation & Objective

• To address unresolved issues in multivariate time series: reliable submodel computation for GEMs, and the impact of downsampling and filtering on causality measures.
• To provide a theoretically sound and computationally feasible framework for computing GEMs in the presence of time-scale mismatches and signal transformations.
• To resolve the forward and reverse causality questions under downsampling and filtering, particularly in neuroimaging contexts where neural dynamics are slower than recorded fMRI signals.
• To establish general conditions under which Granger causality is preserved or distorted under sampling and filtering, filling gaps in econometric and neuroscientific applications.

Proposed method

• Uses state space representations to model vector autoregressive processes and derive stable, causal, and minimum-phase spectral factorizations.
• Applies the discrete algebraic Riccati equation (DARE) to compute optimal state estimators and spectral densities, ensuring stability and detectability.
• Derives a decomposition of non-minimum phase filters into minimum-phase and all-pass components, enabling spectral factorization and GEM computation.
• Models the transformation of time series under downsampling and filtering using state space transitions, preserving the structure of the original process.
• Employs spectral factorization techniques to compute the power spectral density of submodels and joint processes, enabling accurate GEM estimation.
• Validates results through simulations, demonstrating consistency and correctness of GEMs under various filtering and sampling conditions.

Experimental results

Research questions

• RQ1 Does unidirectional Granger causality in a fast time-scale process remain detectable in a downsampled, filtered version of the signal?
• RQ2 Can the reverse causality inference be made—i.e., does observed causality in a downsampled signal imply causality in the original high-rate process?
• RQ3 How do filtering operations, particularly those with variable time delays like the hemodynamic response function, affect the estimation of GEMs?
• RQ4 What are the general conditions under which GEMs remain valid or become distorted under downsampling and filtering?
• RQ5 Can reliable submodel computation be achieved for GEMs without producing negative frequency-domain values?

Key findings

• The paper establishes that the detectability of the state-space system is a sufficient condition for the stability and validity of GEM computations, resolving issues with negative GEM values in prior methods.
• A state space-based method is developed that reliably computes submodels of multivariate time series, eliminating the risk of negative GEMs from inconsistent submodel fitting.
• The impact of downsampling on GEMs is formally characterized: causality can be lost or distorted, and the reverse causality inference is generally invalid without proper modeling.
• Filtering effects on GEMs are analytically resolved using spectral factorization, showing that non-minimum phase filters can be decomposed into minimum-phase and all-pass components to preserve spectral structure.
• The method ensures that the GEMs computed from filtered and downsampled signals are consistent with the underlying true causality when the state space model is properly specified.
• Theoretical conditions are derived under which GEMs remain invariant or transform predictably under sampling and filtering, providing a foundation for reliable inference in fMRI and other applications.