Biomedical Engineering Reference
In-Depth Information
order and model parameters. In most cases, however, the spectra produced by dif-
ferent algorithms are very similar to each other. MVAR model and methods of de-
termination of its coefficients are described in: [Priestley, 1981, Kay, 1988, Marple,
1987, Lutkepohl, 1993].
Several criteria of the determination of the MVAR model order were proposed
in [Lutkepohl, 1993]. Similarly to the one channel case we seek the minimum of the
function consisting of two terms: the first one is a reward for minimizing the residual
variance, the second one is a punishment for a too high model order. The first term
depends on the estimated residual variance
V
(
)
for a given
p
, the second one is a
function of model order, number of channels
k
, and number of data points
N
.The
criteria presented below differ in respect to the second term:
p
AIC
criterion:
2
pk
2
N
V
AIC
(
p
)=
log
[
det
(
)] +
(3.19)
Hannan-Quin
criterion:
pk
2
N
V
HQ
(
p
)=
log
[
det
(
)] +
2log
(
log
(
N
))
(3.20)
Schwartz
criterion:
pk
2
N
V
SC
(
p
)=
log
[
det
(
)] +
log
(
N
)
(3.21)
AIC criterion is the one which is mostly used, but some authors [Kay, 1988, Marple,
1987] claim that it sometimes gives a too high model order.
3.2.2 MVAR in the frequency domain
In analogy to the procedure described in Sect. 2.3.2.2.3 equation (3.13) can be
easily transformed to describe relations in the frequency domain:
E
(
f
)=
A
(
f
)
X
(
f
)
(3.22)
A
−
1
(
)=
(
)
(
)=
(
)
(
)
X
f
f
E
f
H
f
E
f
(3.23)
where
m
=
0
A
(
m
)
e
−
2π
im f
Δ
t
−
1
p
H
(
f
)=
(3.24)
1
and Δ
t
F
s
;
F
s
is the sampling frequency
From the form of that equation we see that the model can be considered as a
linear filter with white noises
E
=
(
f
)
on its input (flat dependence on frequency) and
the signals
X
contains information about
all relations between channels of a process. From the transfer matrix, spectra and
cross-spectra may be calculated:
(
f
)
on its output. The transfer matrix
H
(
f
)
X
∗
(
E
∗
(
H
∗
(
VH
∗
(
S
(
f
)=
X
(
f
)
f
)=
H
(
f
)
E
(
f
)
f
)
f
)=
H
(
f
)
f
)
(3.25)
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