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Estimating Granger causality from Fourier and wavelet transforms of time series data
Mukeshwar Dhamala, Govindan Rangarajan, Mingzhou Ding
TL;DR
Existing frequency-domain Granger causality estimation depends on autoregressive modeling, which introduces model-order uncertainty and may not fully capture complex spectral features. This paper derives the required quantities from direct Fourier and wavelet transforms, then uses spectral factorization and Geweke’s decomposition to estimate causality spectra. Synthetic network experiments recover directional influences, including time-varying coupling structure, while additional trials reduce spectral-estimate variance.
Problem
Frequency-domain Granger causality traditionally requires autoregressive modeling, which involves model-order uncertainty and may not fully account for complex spectral features.
Method
The method estimates spectral density matrices from direct Fourier and wavelet transforms, then applies spectral density matrix factorization and Geweke’s variance decomposition.
Results
Synthetic network experiments recover the underlying directional influences, including time-varying coupling structure represented in wavelet-based time-frequency maps.
Takeaways & Limitations
The approach extends nonparametric spectral-analysis tools to estimate Granger causality spectra for investigating information flow in dynamical networks.
Abstract
from arXiv · showhide
Experiments in many fields of science and engineering yield data in the form of time series. The Fourier and wavelet transform-based nonparametric methods are used widely to study the spectral characteristics of these time series data. Here, we extend the framework of nonparametric spectral methods to include the estimation of Granger causality spectra for assessing directional influences. We illustrate the utility of the proposed methods using synthetic data from network models consisting of interacting dynamical systems.
0 A0ATA−T
The proposed Fourier- and wavelet-based nonparametric techniques estimate Granger causality spectra without parametric data modeling. Synthetic network experiments recover directional influences, including changing coupling structure, while spectral estimates can improve with more trials or multitaper and multiwavelet methods.
- Method: Fourier- and wavelet-based nonparametric methods derive transfer functions and noise covariance for Granger causality estimation from transformed time-series data.The approach applies spectral density matrix factorization and Geweke’s variance decomposition.
- Numerical examples: In a fixed-coupling network, NP and P Granger causality spectra agree and correctly recover the underlying directional influence from X2 to X1.The experiment uses 5000 trials, each containing 5000 data points, with coupling C(t)=0.2.
- Numerical examples: Granger causality magnitude increases with coupling strength in the time-varying network experiment.
- Numerical examples: More trials reduce spectral-estimate variance, while multitaper and multiwavelet techniques can improve estimates for shorter datasets with fewer trials.A sufficiently long stationary time series can also be segmented into smaller trials.
- Conclusion: The method eliminates the need for parametric data modeling and extends Fourier- and wavelet-based nonparametric spectral-analysis tools for studying dynamical networks.The authors expect integration into laboratory analysis routines to support deeper insights into dynamical-network organization.
- Numerical examples: Wavelet-based time-frequency maps correctly represent the network’s changing unidirectional coupling structure and recover the correct directional influences.The coupling from X2 to X1 remains 0.25, transitions to zero over [2, 2.25] sec, and then stays zero; the reverse coupling remains zero.