Biomedical Engineering Reference
In-Depth Information
P
P
y
(
t
)
=
A
(
p
)
y
(
t
−
p
)
+
B
(
p
)
x
(
t
−
p
)
+
p
=
1
p
=
1
=
(
)
+
(
)
+
=
+
.
A
y
t
B
x
t
C
z
(8.91)
=
(
),...,
(
)
=[
,
]
where
B
is a residual vector.
The Granger causality from the time series
x
to
y
,
[
B
1
B
P
],
C
A
B
, and
G
x
ₒ
y
is defined as
log
|
ʣ
e
|
G
x
ₒ
y
=
|
ʣ
|
,
(8.92)
where
ʣ
e
is the covariance matrix of the residual
e
in Eq. (
8.90
), and
ʣ
is the
covariance matrix of the residual
in Eq. (
8.91
). Using Eq. (C.57), we have
|
ʣ
e
|=|
ʣ
yy
−
ʣ
y y
ʣ
−
1
T
y
y
ʣ
y
|
,
(8.93)
˜
y
˜
˜
and
|
ʣ
|=|
ʣ
yy
−
ʣ
y z
ʣ
−
1
T
y
z
ʣ
z
|
.
(8.94)
z
˜
˜
˜
Substituting Eqs. (
8.93
) and (
8.94
)into(
8.92
), we get
log
|
ʣ
yy
−
ʣ
y y
ʣ
−
1
T
y y
ʣ
y y
|
G
x
ₒ
y
=
z
|
.
(8.95)
|
ʣ
yy
−
ʣ
y z
ʣ
−
1
T
y
z
ʣ
z
˜
˜
˜
The right-hand side of the equation above is exactly equal to that of Eq. (
8.89
) except
for the multiplicative constant 1
/
2, indicating that these two measures are equivalent.
8.6.4 Computation of Transfer Entropy
In Sect. C.3.2 in the Appendix, we show that, assuming the Gaussian processes for
the
r
1 real random vector
y
, the mutual
information between these two vectors is expressed as
×
1 real random vector
x
and the
q
×
d
1
2
log
1
1
2
1
I(
,
)
=
yx
=
−
ʻ
j
,
x
y
log
(8.96)
1
−
ʣ
−
1
yy
ʣ
yx
ʣ
−
1
I
xx
ʣ
j
=
1
where we assume
d
=
min
{
q
,
r
}
, and
ʻ
j
is the
j
th eigenvalue of the matrix,
ʣ
−
1
yy
ʣ
yx
ʣ
−
1
yx
ʣ
.
xx
