Biomedical Engineering Reference
In-Depth Information
where T denotes transposition.
V
is a matrix representing residual variances of noises assumed in the model. In
the process of model estimation we obtain an estimate of the variance matrix as
V
,
which accounts for the variance not explained by the model coefficients. Matrix
V
usually contains small non-diagonal elements; their value informs us how well the
model fits the data. The MVAR model coefficients for each time lag
l
are
k
×
k
-sized
matrices:
⎛
⎞
A
11
(
l
)
A
12
(
l
)
...
A
1
k
(
l
)
⎝
⎠
A
21
(
l
)
A
22
(
l
)
...
A
2
k
(
l
)
A
t
=
(3.15)
.
.
(
)
...
(
)
(
)
A
k
1
l
A
kk
−
1
l
A
kk
l
Before starting a fitting procedure, certain preprocessing steps are needed. First,
the temporal mean should be subtracted for every channel. Additionally, in most
cases normalization of the data by dividing each channel by its temporal variance is
recommended. This is especially useful when data channels have different amplifi-
cation ratios.
The estimation of model parameters in the case of a multivariate model is similar
to the one channel model. The classical technique of AR model parameters esti-
mation is the Yule-Walker algorithm. It requires calculating the correlation matrix
R
of the system up to lag
p
. The model equation (3.13) is multiplied by
x
t
+
s
,for
s
=
0
,...,
p
and expectations of both sides of each equation are taken:
N
s
t
=
1
x
i
,
t
x
j
,
t
+
s
1
N
s
R
ij
(
s
)=
(3.16)
Assuming that the noise component is not correlated with the signals, we get a set of
linear equations to solve (the Yule-Walker equations):
⎛
⎞
⎛
⎞
⎛
⎞
R
(
0
)
R
(
−
1
)
...
R
(
p
−
1
)
A
(
1
)
R
(
−
1
)
⎝
⎠
⎝
⎠
=
⎝
⎠
R
(
1
)
R
(
0
)
...
R
(
p
−
2
)
A
(
2
)
R
(
−
2
)
(3.17)
.
.
.
.
.
R
(
1
−
p
)
R
(
2
−
p
)
...
R
(
0
)
A
(
p
)
R
(
−
p
)
and
p
j
=
0
A
(
j
)
R
(
j
)
V
=
(3.18)
This set of equations is similar to the formula (2.50) for one channel model,
however the elements
R
k
matrices. Other methods of finding
MVAR coefficients are the Burg (LWR) recursive algorithm and covariance algo-
rithm. The Burg algorithm produces high resolution spectra and is preferred when
closely spaced spectral components are to be distinguished. Sinusoidal components
in a spectrum are better described by the covariance algorithm or its modification.
Recently, a Bayesian approach has been proposed for estimating the optimal model
(
i
)
and
A
(
i
)
are
k
×
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