Biomedical Engineering Reference
In-Depth Information
AR model is defined by the equation:
p
i
=
1
a
i
x
t
−
i
+
ε
t
x
t
=
(2.46)
where ε
t
is a white noise. It has been shown [Marple, 1987] that the AR model of
sufficiently high model order can approximate the ARMA model well. The deter-
mination of the model order and the coefficients for the AR model is much simpler
than for the ARMA model which requires non-linear algorithms for the estimation
of parameters. Many efficient algorithms of AR model fitting exist; therefore in the
following we shall concentrate on AR model. More details on ARMA may be found
in [Marple, 1987]. One more reason for application of the AR model in practice is
the fact that in transfer function
H
(equation 2.6), which describes the spectral
properties of the system, AR model coefficients determine the poles of transfer func-
tion, which correspond to the spectral peaks. MA coefficients determine the zeros
of
H
(
z
)
, which correspond to the dips of spectrum. In the practice of biomedical
signals processing, we are usually more interested in the spectral peaks, since they
correspond to the rhythms present in the time series.
(
z
)
2.3.2.2.1 AR model parameter estimation
There is a number of algorithms for
estimation of the AR model parameters. Here we present the Yule-Walker algorithm.
First, equation (2.46) is multiplied by the sample at time
t
−
m
:
p
i
=
1
a
i
x
t
−
i
x
t
−
m
+
ε
t
x
t
−
m
x
t
x
t
−
m
=
(2.47)
then we calculate expected values of the left and right sides of the equation taking
advantage of the linearity of the expected value operator:
p
i
=
1
E
{
a
i
x
t
−
i
x
t
−
m
}
+
E
{
ε
t
x
t
−
m
}
R
(
m
)=
E
{
x
t
x
t
−
m
}
=
(2.48)
It is easy to see that the expected value
E
{
x
t
x
t
−
m
}
is the autocorrelation function
R
(
m
)
,and
E
{
ε
t
x
t
−
m
}
is nonzero only for
t
=
m
so:
p
i
=
1
a
i
R
(
m
−
i
)+
σ
2
ε
δ
(
m
)
R
(
m
)=
(2.49)
where
m
=
0
,...,
p
.
For
m
>
0 we can write a set of equations:
⎡
⎤
⎡
⎤
⎡
⎤
R
(
0
)
R
(
−
1
)
...
R
(
1
−
p
)
a
1
a
2
.
a
p
R
(
1
)
⎣
⎦
⎣
⎦
=
⎣
⎦
R
(
1
)
R
(
0
)
R
(
−
1
)
R
(
2
)
(2.50)
.
.
.
R
(
p
−
1
)
...
...
R
(
0
)
R
(
p
)
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