Biomedical Engineering Reference
In-Depth Information
8.7 Estimation of MVAR Coefficients
8.7.1 Least-Squares Algorithm
The Granger-causality and related measures rely on the modeling of the multivariate
vector auto-regressive (MVAR) process of the source time series. Use of these mea-
sures requires estimating the MVAR coefficient matrices. This section deals with the
estimation of MVAR coefficients. A
q
×
1 random vector
y
(
t
)
is modeled using the
MVAR process, such that
P
y
(
t
)
=
A
(
p
)
y
(
t
−
p
)
+
e
(
t
).
(8.106)
p
=
1
We can estimate the MVAR coefficients
A
i
,
j
(
p
)
where
i
,
j
=
1
,...,
q
and
p
=
1
P
based on the least-squares principle. To derive the least-squares equation,
let us explicitly write the MVAR process for the
,...,
th component
y
(
t
)
as
q
q
y
(
t
)
=
A
,
j
(
1
)
y
j
(
t
−
1
)
+
A
,
j
(
2
)
y
j
(
t
−
2
)
j
=
1
j
=
1
q
+···+
A
(
P
)
y
j
(
t
−
P
)
+
e
(
t
).
(8.107)
,
j
j
=
1
We assume that the source time series are obtained at
t
=
1
,...,
K
where
K
q
×
P
. Then, since Eq. (
8.107
) holds for
t
=
(
P
+
1
), . . . ,
K
, a total of
K
−
P
linear
=
(
+
), . . . ,
equations are obtained by setting
t
P
1
K
in Eq. (
8.107
).
These equations are formulated in a matrix form,
y
=
Gx
+
e
.
(8.108)
Here, the
(
K
−
P
)
×
1 column vector
y
is defined as
]
T
y
=
[
y
(
P
+
1
),
y
(
P
+
2
),...,
y
(
K
)
.
(8.109)
In Eq. (
8.108
),
G
is a
(
K
−
P
)
×
Pq
matrix expressed as
⊡
⊣
⊤
⊦
.
y
1
(
P
)
···
y
q
(
P
)
···
y
1
(
1
)
···
y
q
(
1
)
y
1
(
P
+
1
)
···
y
q
(
P
+
1
)
···
y
1
(
2
)
···
y
q
(
2
)
G
=
(8.110)
.
y
1
(
K
−
1
)
···
y
q
(
K
−
1
)
···
y
1
(
K
−
P
)
···
y
q
(
K
−
P
)
