Table of Contents
ISRN Signal Processing
Volume 2013, Article ID 374064, 9 pages
Research Article

Instantaneous Granger Causality with the Hilbert-Huang Transform

Institute of Biophysics and Biomedical Engineering, Faculty of Sciences, University of Lisbon, Campo Grande, 1749-016 Lisboa, Portugal

Received 30 November 2012; Accepted 6 January 2013

Academic Editors: G. A. Tsihrintzis and M. Wicks

Copyright © 2013 João Rodrigues and Alexandre Andrade. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Current measures of causality and temporal precedence have limited frequency and time resolution and therefore may not be viable in the detection of short periods of causality in specific frequencies. In addition, the presence of nonstationarities hinders the causality estimation of current techniques as they are based on Fourier transforms or autoregressive model estimation. In this work we present a combination of techniques to measure causality and temporal precedence between stationary and nonstationary time series, that is sensitive to frequency-specific short episodes of causality. This methodology provides a highly informative time-frequency representation of causality with existing causality measures. This is done by decomposing each time series into intrinsic oscillatory modes with an empirical mode decomposition algorithm and, subsequently, calculating their complex Hilbert spectrum. At each time point the cross-spectrum is calculated between time series and used to measure coherency and compute the transfer function and error covariance matrices using the Wilson-Burg method for spectral factorization. The imaginary part of coherency can then be computed as well as several Granger causality measures in the previous matrices. This work covers the most important theoretical background of these techniques and tries to prove the usefulness of this new approach while pointing out some of its qualities and drawbacks.