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. 1. Introduction Identifying interactions of different temporal scales is a recurrent topic in fields such as neuroscience and meteorological or financial research. The concept of functional connectivity, which is defined as the statistical dependence between different variables, is widely used. However, if activity from one variable directly or indirectly exerts influence on other variables, functional connectivity measures will not identify this dependence [1]. This influence is interpreted as the effective connectivity or causal influence, and one solution for this problem in time series inference (TSI) is the concept of causality introduced by Wiener and formulated by Granger [2], the Granger causality (GC) measure. According to the concept of causality, one stochastic process is causal to a second if the autoregressive predictability of the second process at a given time point is improved by including measurements from the past of the first. GC has shown to be suitable for the study of directionality in neuronal interactions by assessment of neurophysiologic data in both the frequency and time domains [3], as well as of simulated
References
[1]
K. J. Friston, “Functional and effective connectivity in neuroimaging: a synthesis,” Human Brain Mapping, vol. 2, no. 1-2, pp. 56–78, 1994.
[2]
C. W. J. Granger, “Investigating causal relations by econometric models and cross-spectral methods,” Econometrica, vol. 37, no. 3, p. 424, 1969.
[3]
S. L. Bressler and A. K. Seth, “Wiener-Granger Causality: a well established methodology,” NeuroImage, vol. 58, no. 2, pp. 323–329, 2011.
[4]
G. Deshpande, K. Sathian, and X. Hu, “Effect of hemodynamic variability on Granger causality analysis of fMRI,” NeuroImage, vol. 52, no. 3, pp. 884–896, 2010.
[5]
J. Geweke, “Measurement of linear dependence and feedback between multiple time series,” Journal of the American Statistical Association, vol. 77, pp. 304–313, 1982.
[6]
M. Dhamala, G. Rangarajan, and M. Ding, “Analyzing information flow in brain networks with nonparametric Granger causality,” NeuroImage, vol. 41, no. 2, pp. 354–362, 2008.
[7]
N. E. Huang, Z. Shen, S. R. Long et al., “The empirical mode decomposition and the Hubert spectrum for nonlinear and non-stationary time series analysis,” Proceedings of the Royal Society A, vol. 454, no. 1971, pp. 903–995, 1998.
[8]
T. Tanaka and D. P. Mandic, “Complex empirical mode decomposition,” IEEE Signal Processing Letters, vol. 14, no. 2, pp. 101–104, 2007.
[9]
C.-L. Yeh, H.-C. Chang, C.-H. Wu, and P.-L. Lee, “Extraction of single-trial cortical beta oscillatory activities in EEG signals using empirical mode decomposition,” BioMedical Engineering Online, vol. 9, article 25, 2010.
[10]
H. Liang, S. L. Bressler, E. A. Buffalo, R. Desimone, and P. Fries, “Empirical mode decomposition of field potentials from macaque V4 in visual spatial attention,” Biological Cybernetics, vol. 92, no. 6, pp. 380–392, 2005.
[11]
C. M. Sweeney-Reed and S. J. Nasuto, “A novel approach to the detection of synchronisation in EEG based on empirical mode decomposition,” Journal of Computational Neuroscience, vol. 23, no. 1, pp. 79–111, 2007.
[12]
J. K. Kroger and J. Lakey, “Phase locking and empirical mode decompositions in EEG,” Citeseer, no. 1978, 2008.
[13]
D. Liu, X. Yang, G. Wang, et al., “HHT based cardiopulmonary coupling analysis for sleep apnea detection,” Sleep Medicine, vol. 13, no. 5, pp. 503–509, 2012.
[14]
M. Hu and H. Liang, “Noise-assisted instantaneous coherence analysis of brain connectivity,” Computational Intelligence and Neuroscience, vol. 2012, Article ID 275073, 12 pages, 2012.
[15]
J. Rodrigues and A. Andrade, “A New Approach for Granger Causality between Neuronal Signals using the Empirical Mode Decomposition Algorithm,” Biomedical Engineering / 765: Telehealth / 766: Assistive Technologies, 2012.
[16]
M. Kamiński, M. Ding, W. A. Truccolo, and S. L. Bressler, “Evaluating causal relations in neural systems: granger causality, directed transfer function and statistical assessment of significance,” Biological Cybernetics, vol. 85, no. 2, pp. 145–157, 2001.
[17]
J. F. Geweke, “Measures of conditional linear dependence and feedback between time series,” Journal of the American Statistical Association, vol. 79, no. 388, pp. 907–915, 1984.
[18]
M. Eichler, “On the evaluation of information flow in multivariate systems by the directed transfer function,” Biological Cybernetics, vol. 94, no. 6, pp. 469–482, 2006.
[19]
B. Schelter, M. Winterhalder, M. Eichler et al., “Testing for directed influences among neural signals using partial directed coherence,” Journal of Neuroscience Methods, vol. 152, no. 1-2, pp. 210–219, 2006.
[20]
Y. Chen, S. L. Bressler, and M. Ding, “Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data,” Journal of Neuroscience Methods, vol. 150, no. 2, pp. 228–237, 2006.
[21]
G. T. Wilson, “Factorization of matricial spectral densities,” SIAM Journal on Applied Mathematics, vol. 23, no. 4, pp. 420–426, 1972.
[22]
G. Nolte, A. Ziehe, V. V. Nikulin, et al., “Robustly estimating the flow direction of information in complex physical systems,” Physical Review Letters, vol. 100, no. 23, pp. 1–4, 2008.
[23]
G. Nolte, O. Bai, L. Wheaton, Z. Mari, S. Vorbach, and M. Hallett, “Identifying true brain interaction from EEG data using the imaginary part of coherency,” Clinical Neurophysiology, vol. 115, no. 10, pp. 2292–2307, 2004.
[24]
Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise-assisted data analysis method,” Advances in Adaptive Data Analysis, vol. 1, no. 1, pp. 1–41, 2009.
[25]
P. Flandrin, G. Rilling, and P. Gon？alvés, “Empirical mode decomposition as a filter bank,” IEEE Signal Processing Letters, vol. 11, no. 2, pp. 112–114, 2004.
[26]
N. Rehman and D. P. Mandic, “Multivariate empirical mode decomposition,” Proceedings of the Royal Society A, vol. 466, no. 2117, pp. 1291–1302, 2010.
[27]
N. Ur Rehman and D. P. Mandic, “Filter bank property of multivariate empirical mode decomposition,” IEEE Transactions on Signal Processing, vol. 59, no. 5, pp. 2421–2426, 2011.
[28]
C. Park, D. Looney, P. Kidmose, M. Ungstrup, and D. P. Mandic, “Time-frequency analysis of EEG asymmetry using bivariate empirical mode decomposition,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 19, no. 4, pp. 366–373, 2011.
[29]
M. Chávez, M. Le Van Quyen, V. Navarro, M. Baulac, and J. Martinerie, “Spatio-temporal dynamics prior to neocortical seizures: amplitude versus phase couplings,” IEEE Transactions on Bio-Medical Engineering, vol. 50, no. 5, pp. 571–583, 2003.
