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J. W. Choi and K. H. Kim
V x|x,y var(e x,y ) and V y|y,x var(e y,x ) for bivariate AR model.
Here, var(·) denotes the variance.
The Granger causality between two signals, x and y, is calculated as the log-ratio
of variances follows:
GC x→y ln
V y|y
V y|y,x
for the measure of ‘signal x causes signal y’
GC y→x ln
V x|x
V x|x,y
for the measure of ‘signal y causes signal x’
The prediction error of y should not be reduced whether x is considered or not for
the estimation of y, if there exist no causal influence from x to y. This implies that the
variances V y|y and V y|y,x are identical, and thus, GC x→y is close to zero. On the other
hand, causal influence of x to y reduces the prediction error of y when x is considered.
Hence, GC x→y becomes a positive value. The GC measure is directional. If the GCs
of both directions are high, it can be interpreted as a bidirectional connectivity [42].
6.3.7 Partial Directed Coherence (PDC)
PDC is a frequency domain equivalent of the GC, based on multivariate autoregressive (MVAR) modeling of multichannel signals [3]. Let’s assume that the simultaneously recorded m channel signals x(t) [x 1 (t), . . . , x m (t)]
T can be described by
an MVAR model as follows:
x(t)
p
n1
A n x(t − n) + e(t).
(6.11)
here, p is the model order, A n
⎡
⎢
⎢
⎣
a 1,1 (n) · · · a 1,m (n)
. . .
. . .
. . .
a m,1 (n) · · · a m,m (n)
⎤
⎥
⎥
⎦ is the matrix of model
coefficients at time lag n, and e(t) [e 1 (t), . . . , e m (t)]
T is a multivariate Gaussian
white noise with zero mean and covariance matrix . The model coefficients a m,m
indicate the influence among the signals (e.g., a 1,2 (n) is the influence of x 2 (t − n) on
x 1 (t)).
This time domain representation can be transformed into frequency domain by
Fourier transform (FT). ¯
A( f ) I − A( f ) [¯ a 1 ( f )¯ a 2 ( f ) . . . ¯
a m ], where A(f ) is the
FT of the model coefficients and ¯
a i, j ( f ) is the i, jth element of ¯
A( f ). The PDC from
signal x i to signal x j can be calculated as follows:
J. W. Choi and K. H. Kim
V x|x,y var(e x,y ) and V y|y,x var(e y,x ) for bivariate AR model.
Here, var(·) denotes the variance.
The Granger causality between two signals, x and y, is calculated as the log-ratio
of variances follows:
GC x→y ln
V y|y
V y|y,x
for the measure of ‘signal x causes signal y’
GC y→x ln
V x|x
V x|x,y
for the measure of ‘signal y causes signal x’
The prediction error of y should not be reduced whether x is considered or not for
the estimation of y, if there exist no causal influence from x to y. This implies that the
variances V y|y and V y|y,x are identical, and thus, GC x→y is close to zero. On the other
hand, causal influence of x to y reduces the prediction error of y when x is considered.
Hence, GC x→y becomes a positive value. The GC measure is directional. If the GCs
of both directions are high, it can be interpreted as a bidirectional connectivity [42].
6.3.7 Partial Directed Coherence (PDC)
PDC is a frequency domain equivalent of the GC, based on multivariate autoregressive (MVAR) modeling of multichannel signals [3]. Let’s assume that the simultaneously recorded m channel signals x(t) [x 1 (t), . . . , x m (t)]
T can be described by
an MVAR model as follows:
x(t)
p
n1
A n x(t − n) + e(t).
(6.11)
here, p is the model order, A n
⎡
⎢
⎢
⎣
a 1,1 (n) · · · a 1,m (n)
. . .
. . .
. . .
a m,1 (n) · · · a m,m (n)
⎤
⎥
⎥
⎦ is the matrix of model
coefficients at time lag n, and e(t) [e 1 (t), . . . , e m (t)]
T is a multivariate Gaussian
white noise with zero mean and covariance matrix . The model coefficients a m,m
indicate the influence among the signals (e.g., a 1,2 (n) is the influence of x 2 (t − n) on
x 1 (t)).
This time domain representation can be transformed into frequency domain by
Fourier transform (FT). ¯
A( f ) I − A( f ) [¯ a 1 ( f )¯ a 2 ( f ) . . . ¯
a m ], where A(f ) is the
FT of the model coefficients and ¯
a i, j ( f ) is the i, jth element of ¯
A( f ). The PDC from
signal x i to signal x j can be calculated as follows: