Multiple channels (multivariate) signals - Practical Biomedical Signal Analysis Using MATLAB - page 85

cascade flows are also present, e.g., from channel 1 to 3 and 5.

3.3.2.3.2 SDTF In biomedical applications it is often of interest to grasp the dy-
namical changes of signal propagation, which means that we have to apply a very
short measurement window. However, it deteriorates strongly the statistical proper-
ties of the estimate. In order to fit the MVAR model properly the number of data
points in the window should be bigger than the number of model parameters. For
MVAR, number of parameters is pk 2 (where k number of channels, p model or-
der), number of data points is kN , so the condition for proper model fitting is that k N
should be small, preferably smaller than 0.1. The same rule holds for all estimators
based on MVAR. In case of time varying processes, when we would like to follow
the dynamics of the transmission the data window has to be short.
To increase the number of the data points we may use multiple repetitions of the
experiment. We may treat the data from each repetition as a realization of the same
stochastic process. Then the number of data points is kN S N T (where N T is number of
realizations, N S is number of data points in the window), and their ratio to the number
of parameters ( pk 2 ) effectively increases. Based on this observation, we can divide
a non-stationary recording into shorter time windows, short enough to treat the data
within a window as quasi-stationary. We calculate the correlation matrix between
channels for each trial separately. The resulting model coefficients are based on the
correlation matrix averaged over trials. The correlation matrix has a form:

N T
r = 1 R ( r )

N T
r = 1

N S
t = 1 X ( r )

1
N T

1
N T

1
N S

t X ( r )

R ij

( s )= ( s )= (3.36)

ij i , j , t + s

R ( r )

ij is the correlation matrix for short windows of N S points, and r is the index of the
repetition.
The averaging concerns correlation matrices for short data windows—data are not
averaged in the process. The choice of thewindowsizeisalwaysacompromise
between quality of the fit (depending on the ratio between number of data points and
number of model parameters) and time resolution. By means of the above described
procedure for each short data window the MVAR coefficients are obtained and then
the estimators characterizing the signals: power spectra, coherences, DTFs are found.
By application of the sliding window, multivariate estimators may be expressed as
functions of time and frequency and their evolution in time may be followed.
The tutorial and software for calculation of DTF can be found at http://eeg.pl.

3.3.2.4 Partial directed coherence

The partial directed coherence (PDC) was introduced by Baccal´aandSameshima
[Baccala and Sameshima, 2001] in the following form:

A ij ( f )

a j (

P ij ( f )= (3.37)

f ) a j ( f )