Multiple channels (multivariate) signals - Practical Biomedical Signal Analysis Using MATLAB - page 82

3.3.2.2 Granger causality index

For a two channels system GCI is based directly on the principle formulated by
Granger. Namely we check, if the information contributed by the second signal im-
proves the prediction of the first signal. In order to find out, we have to compare
the variance of univariate AR model (equation 3.27) with a variance of the model
accounting for the second variable (equation 3.28).
Granger causality index showing the driving of channel x by channel y is defined
as the logarithm of the ratio of residual variance for a two channel model to the
residual variance of the one channel model:

log ε 1
ε

=

GCI y → x (3.30)

This definition can be extended to the multichannel system by considering how the
inclusion of the given channel changes the residual variance ratios. To quantify di-
rected influence from a channel x j to x i for n -channel autoregressive process in time
domain we consider n -and n 1 dimensional MVAR models. First, the model is fit-

tedtoawhole n -channel system, leading to the residual variance V i , n

−

( t )= var ( E i , n ( t ))

for signal x i .Next,a n − 1 dimensional MVAR model is fitted for n − 1 channels, ex-

cluding channel j , which leads to the residual variance V i , n − 1 ( t )= var ( E i , n − 1 ( t )) .

Then Granger causality is defined as:

V i , n

)
V i , n − 1 (

( t

GCI j → i ( t )= log (3.31)

t )

GCI is smaller or equal to 1, since the variance of the n -dimensional system is lower
than the residual variance of a smaller n

is used to
estimate causality relations in time domain. When the spectral content of the signals
is of interest, which is frequent for biomedical signals, the estimators defined in the
frequency domain: DTF or PDC are appropriate choices.

− 1 dimensional system. GCI ( t )

3.3.2.3 Directed transfer function

Directed transfer function (DTF) introduced by Kaminski and Blinowska, [Kamin-
ski and Blinowska, 1991] is based on the properties of the transfer matrix H

of
MVAR which is asymmetric and contains spectral and phase information concern-
ing relations between channels. DTF describes the causal influence of channel j on
channel i at frequency f :

( f )

2

| ( ) |

H ij f

DTF ij (

f )= (3.32)

∑ k m = 1 |

H im ( f ) |

2

The above equation defines a normalized version of DTF, which takes values from 0
to 1 giving a ratio between the inflow from channel j to channel i to all the inflows
to channel i . Sometimes one can abandon the normalization property and use values
of elements of transfer matrix which are related to causal connection strength. The
non-normalized DTF can be defined as:
θ ij (

2

f )= | H ij ( f ) | (3.33)