We develop new statistical methods for estimating functional connectivity between components of a multivariate time series and for testing differences in functional connectivity across experimental conditions. Here, we characterize functional connectivity by partial coherence, which identifies the frequency band (or bands) that drives the direct linear association between any pair of components of a multivariate time series after removing the linear effects of the other components. Partial coherence can be efficiently estimated using the inverse of the spectral density matrix. However, when the number of components is large and the components of the multivariate time series are highly correlated, the spectral density matrix estimate may be numerically unstable and consequently gives partial coherence estimates that are highly variable. To address the problem of numerical instability, we propose a shrinkage-based estimator which is a weighted average of a smoothed periodogram estimator and a scaled identity matrix with frequency-specific weight computed objectively so that the resulting shrinkage estimator minimizes the mean-squared error criterion. Compared to typical smoothing-based estimators, the shrinkage estimator is more computationally stable and gives a lower mean squared error. In addition, we develop a randomization method for testing differences in functional connectivity networks between experimental conditions. Finally, we report results from numerical experiments and analyze an EEG data set recorded during a visually-guided hand movement task.
Bibliographical noteFunding Information:
The authors would like to thank the handling editor and the referees for their suggestions that helped improve the paper. This research is supported, in part, by NIH R01 EY015451 (J. Sanes), NSF SBE/BCS 0843938 (J. Sanes) and NSF DMS 0806106 (H. Ombao, C. Linkletter and W. Thompson).
- Multivariate time series
- Partial coherence
- Randomization test
- Shrinkage estimator