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# mvfreqz

## PURPOSE

MVFREQZ multivariate frequency response

## SYNOPSIS

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

## DESCRIPTION

``` 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]

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.```

## CROSS-REFERENCE INFORMATION

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