MVFREQZ multivariate frequency response [S,h,PDC,COH,DTF,DC,pCOH,dDTF,ffDTF,pCOH2,PDCF,coh,GGC,Af,GPDC] = mvfreqz(B,A,C,N,Fs) [...] = mvfreqz(B,A,C,f,Fs) INPUT: ======= A, B multivariate polynomials defining the transfer function a0*Y(n) = b0*X(n) + b1*X(n-1) + ... + bq*X(n-q) - a1*Y(n-1) - ... - ap*Y(:,n-p) A=[a0,a1,a2,...,ap] and B=[b0,b1,b2,...,bq] must be matrices of size Mx((p+1)*M) and Mx((q+1)*M), respectively. C is the covariance of the input noise X (i.e. D'*D if D is the mixing matrix) N if scalar, N is the number of frequencies if N is a vector, N are the designated frequencies. Fs sampling rate [default 2*pi] A,B,C and D can by obtained from a multivariate time series through the following commands: [AR,RC,PE] = mvar(Y,P); M = size(AR,1); % number of channels A = [eye(M),-AR]; B = eye(M); C = PE(:,M*P+1:M*(P+1)); N number of frequencies for computing the spectrum Fs sampling rate in [Hz] f vector of frequencies (in [Hz]) OUTPUT: ======= S power spectrum h transfer functions, abs(h.^2) is the non-normalized DTF [11] PDC partial directed coherence [2] DC directed coupling COH coherency (complex coherence) [5] DTF directed transfer function pCOH partial coherence dDTF direct Directed Transfer function ffDTF full frequency Directed Transfer Function pCOH2 partial coherence - alternative method GGC a modified version of Geweke's Granger Causality [Geweke 1982] !!! it uses a Multivariate AR model, and computes the bivariate GGC as in [Bressler et al 2007]. This is not the same as using bivariate AR models and GGC as in [Bressler et al 2007] Af Frequency transform of A(z), abs(Af.^2) is the non-normalized PDC [11] PDCF Partial Directed Coherence Factor [2] GPDC Generalized Partial Directed Coherence [9,10] see also: FREQZ, MVFILTER, MVAR REFERENCE(S): [1] H. Liang et al. Neurocomputing, 32-33, pp.891-896, 2000. [2] L.A. Baccala and K. Samashima, Biol. Cybern. 84,463-474, 2001. [3] A. Korzeniewska, et al. Journal of Neuroscience Methods, 125, 195-207, 2003. [4] Piotr J. Franaszczuk, Ph.D. and Gregory K. Bergey, M.D. Fast Algorithm for Computation of Partial Coherences From Vector Autoregressive Model Coefficients World Congress 2000, Chicago. [5] Nolte G, Bai O, Wheaton L, Mari Z, Vorbach S, Hallett M. Identifying true brain interaction from EEG data using the imaginary part of coherency. Clin Neurophysiol. 2004 Oct;115(10):2292-307. [6] Schlogl A., Supp G. Analyzing event-related EEG data with multivariate autoregressive parameters. (Eds.) C. Neuper and W. Klimesch, Progress in Brain Research: Event-related Dynamics of Brain Oscillations. Analysis of dynamics of brain oscillations: methodological advances. Elsevier. [7] Bressler S.L., Richter C.G., Chen Y., Ding M. (2007) Cortical fuctional network organization from autoregressive modelling of loal field potential oscillations. Statistics in Medicine, doi: 10.1002/sim.2935 [8] Geweke J., 1982 J.Am.Stat.Assoc., 77, 304-313. [9] L.A. Baccala, D.Y. Takahashi, K. Sameshima. (2006) Generalized Partial Directed Coherence. Submitted to XVI Congresso Brasileiro de Automatica, Salvador, Bahia. [10] L.A. Baccala, D.Y. Takahashi, K. Sameshima. Computer Intensive Testing for the Influence Between Time Series, Eds. B. Schelter, M. Winterhalder, J. Timmer: Handbook of Time Series Analysis - Recent Theoretical Developments and Applications Wiley, p.413, 2006. [11] M. Eichler On the evaluation of informatino flow in multivariate systems by the directed transfer function Biol. Cybern. 94: 469-482, 2006.

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