My implementation of a causal statistic was based on the measure of conditional mutual information proposed by Palus et al.(2001). The individual and joint entropies were estimated in the time domain by computing the corresponding correlation integrals, as proposed in
Prichard and Theiler (1995), and tested using linear and non-linear models, for example, in Chavez et al. (2003) and Vakorin et al. (2009). The significance of the results was tested with surrogate data generated by shuffling the phase in the Fourier domain (Theiler et al, 1992).
[1] Chavez, M. and Martinerie, J. and Le Van Quyen,M., "Statistical assessment of nonlinear causality: application to epileptic EEG signals," J. Neurosci Methods, vol. 124, no. 2, pp.113-128, 2003
[2] Palus, M. and Komarek, V. and Hrncir, Z. and Sterbova, K., "Synchronization as adjustment of infomation rates: Detection from bivariate time series," Phys. Rev. E, vol. 63, pages. 046211, 2001.
[3] Prichard, D. and Theiler, J.,"Generralized redundancies for time series analysis," Physica D, vol. 84, pp. 476--493, 1995.
[4] Theiler,J. and Eubank,S. and Longtin,A. and Galdrikian,B. and Farmer,J.D, "Testing for Nonlinearity in Time Series: The Method of Surrogate Data," Physica D, vol. 58, pp. 77-94, 1992.
[5]Vakorin,V.A. and Krakovska,O.A. and McIntosh?,A.R., "Confounding effects of indirect connections on causality estimation," Journal of Neuroscience Methods, vol. 184, no. 1, pp. 152--160, 2009.
Prichard and Theiler (1995), and tested using linear and non-linear models, for example, in Chavez et al. (2003) and Vakorin et al. (2009). The significance of the results was tested with surrogate data generated by shuffling the phase in the Fourier domain (Theiler et al, 1992).
[1] Chavez, M. and Martinerie, J. and Le Van Quyen,M., "Statistical assessment of nonlinear causality: application to epileptic EEG signals," J. Neurosci Methods, vol. 124, no. 2, pp.113-128, 2003
[2] Palus, M. and Komarek, V. and Hrncir, Z. and Sterbova, K., "Synchronization as adjustment of infomation rates: Detection from bivariate time series," Phys. Rev. E, vol. 63, pages. 046211, 2001.
[3] Prichard, D. and Theiler, J.,"Generralized redundancies for time series analysis," Physica D, vol. 84, pp. 476--493, 1995.
[4] Theiler,J. and Eubank,S. and Longtin,A. and Galdrikian,B. and Farmer,J.D, "Testing for Nonlinearity in Time Series: The Method of Surrogate Data," Physica D, vol. 58, pp. 77-94, 1992.
[5]Vakorin,V.A. and Krakovska,O.A. and McIntosh?,A.R., "Confounding effects of indirect connections on causality estimation," Journal of Neuroscience Methods, vol. 184, no. 1, pp. 152--160, 2009.