Biomedical Engineering Reference
In-Depth Information
where
n
is the number of degrees of freedom dependent on the method of estimation
of cross-spectra.
Therefore, for low values of coherence the errors will be large. Also in the case
of estimation of coherence by means of Fourier transform, even with smoothing, the
number of degrees of freedom is low. Partial correlations found by means of MVAR
are computed under the assumption of minimum phase and they are not suitable for
determination of directedness. By means of cross-correlation or coherence it is not
possible to detect reciprocal connections. This type of connection may be found by
means of measures based on the causality principle.
3.3.2
Causality measures
3.3.2.1
Granger causality
In biomedical studies finding causality between channels of multivariate pro-
cesses is of particular interest. The testable definition of causality was introduced by
Granger
1
(1969) in the field of economics [Granger, 1969]. Granger defined causality
in terms of predictability of time series; namely if some series
y
(
t
)
contains informa-
tion in past terms that helps in the prediction of series
x
(
t
)
,then
y
(
t
)
is said to cause
x
(
t
)
. More specifically, if we try to predict a value of
x
(
t
)
using
p
previous values of
the series
x
only, we get a prediction error ε:
p
j
=
1
A
11
(
j
)
x
(
t
−
j
)+
ε
(
t
)
x
(
t
)=
(3.27)
If we try to predict a value of
x
using
p
previous values of the series
x
and
p
previous values of
y
we obtain another prediction error ε
1
:
(
t
)
p
j
=
1
A
11
(
j
)
x
(
t
−
j
)+
p
j
=
1
A
21
(
j
)
y
(
t
−
j
)+
ε
1
(
t
)
x
(
t
)=
(3.28)
If the varianceε
1
(after including series
y
to the prediction) is lower than the variance
ε we say that
y
causes
x
in the sense of Granger causality. Similarly we can say that
x
causes
y
in the sense of Granger causality when the variance ε
2
is reduced after
including series
x
in the prediction of series
y
:
p
j
=
1
A
22
(
j
)
y
(
t
−
j
)+
p
j
=
1
A
21
(
j
)
x
(
t
−
j
)+
ε
2
y
(
t
)=
(3.29)
Three different measures of causality or directionality based on the Granger causality
concept were introduced: Granger causality index (GCI) [Geweke, 1982], directed
transfer function (DTF) [Kaminski and Blinowska, 1991] and partial directed coher-
ence (PDC) [Baccala and Sameshima, 2001]. GCI operates in the time domain, DTF
and PDC in the frequency domain. Granger causality was originally defined for a
two channel system. The above mentioned estimators were extended to multichan-
nel systems.
1
Nobel Laureate in economics.
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