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MVFREQZ multivariate frequency response


function [S,h,PDC,COH,DTF,DC,pCOH,dDTF,ffDTF, pCOH2, PDCF, coh,GGC,Af,GPDC,GGC2]=mvfreqz(B,A,C,N,Fs)


 MVFREQZ multivariate frequency response
 [S,h,PDC,COH,DTF,DC,pCOH,dDTF,ffDTF,pCOH2,PDCF,coh,GGC,Af,GPDC] = mvfreqz(B,A,C,f,Fs)
 [...]  = mvfreqz(B,A,C,N,Fs)
 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)); 

 Fs     sampling rate in [Hz]
 (N     number of frequencies for computing the spectrum, this will become OBSOLETE),  
 f    vector of frequencies (in [Hz])  

 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]


 [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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