Biomedical Engineering Reference
In-Depth Information
Denoting the number of selected voxels
q
, we express the
q
-channel voxel time
series using the vector
y
T
(
)
(
)
=[
y
1
(
),...,
y
q
(
)
]
where
y
j
(
)
is the time series
of the
j
th selected voxel at time
t
. Here the time
t
is expressed using a unit-less value.
We impose multivariate vector autoregressive (MVAR) modeling on the time series
y
t
:
y
t
t
t
t
(
t
)
, such that
P
y
(
t
)
=
A
(
p
)
y
(
t
−
p
)
+
e
(
t
).
(8.1)
p
=
1
Here,
A
is the residual
vector. The MVAR process is expressed in the frequency domain. By computing the
Fourier transform of Eq. (
8.1
), we get
(
p
)
is the AR coefficient matrix,
P
is the model order, and
e
(
t
)
P
e
−
2
ˀ
ipf
y
y
(
f
)
=
A
(
p
)
(
f
)
+
e
(
f
),
(8.2)
p
=
1
where the Fourier transforms of
y
(
t
)
and
e
(
t
)
are expressed in
y
(
f
)
and
e
(
f
)
.We
here use the relationship,
e
−
2
ˀ
ipf
y
y
(
t
−
p
)
exp
(
−
2
ˀ
ift
)
d
t
=
(
f
).
(8.3)
Equation (
8.2
) is also expressed as
⊡
⊣
I
e
−
2
ˀ
ipf
⊤
P
⊦
y
−
A
(
p
)
(
f
)
=
e
(
f
).
(8.4)
p
=
1
¯
A
Defining a
q
×
q
matrix
(
f
)
such that
P
¯
A
e
−
2
ˀ
ipf
(
f
)
=
I
−
A
(
p
)
,
(8.5)
p
=
1
we can obtain
¯
A
(
f
)
y
(
f
)
=
e
(
f
).
(8.6)
)
=
¯
A
)
−
1
, the relationship
Also, defining
H
(
f
(
f
y
(
f
)
=
H
(
f
)
e
(
f
)
(8.7)
can be obtained. According to the equation above,
e
can be interpreted
as the input and the output of a linear-system whose transfer function is
H
(
f
)
and
y
(
f
)
(
f
)
.
