Biomedical Engineering Reference
In-Depth Information
For an arbitrary number of channels partial coherence may be defined in terms
of the minors of spectral matrix
S
(
f
)
, which on the diagonal contains spectra and
off-diagonal cross-spectra:
(
)
M
ij
f
κ
ij
(
f
)=
M
ii
(3.11)
(
f
)
M
jj
(
f
)
where
M
ij
is a minor of
S
with the
i
th
row and
j
th
column removed. Its properties
are similar to ordinary coherence, but it is nonzero only for direct relations between
channels. If a signal in a given channel can be explained by a linear combination of
some other signals in the set, the partial coherence between them will be low.
Multiple coherence is defined by:
1
(
)
det
S
f
G
i
(
f
)=
−
(3.12)
S
ii
(
f
)
M
ii
(
f
)
Its value describes the amount of common components in the given channel and the
other channels in the set. If the value of multiple coherence is close to zero then
the channel has no common components with any other channel of the set. The high
value of multiple coherence for a given channel means that a large part of the variance
of that channel is common to all other signals; it points to the strong relation between
the signals. Partial and multiple coherences can be conveniently found by means of
the autoregressive parametric model MVAR.
3.2 Multivariate autoregressive model (MVAR)
3.2.1 Formulation of MVAR model
MVAR model is an extension of the one channel AR model for an arbitrary number
of channels. In the MVAR formalism, sample
x
i
,
t
in channel
i
at time
t
is expressed
not only as a linear combination of
p
previous values of this signal, but also by means
of
p
previous samples of all the other channels of the process. For
k
channels model
of order
p
may be expressed as:
+
E
t
x
t
=
A
1
x
t
−
1
+
A
2
x
t
−
2
+
...
A
p
x
t
−
p
(3.13)
=[
,
,...
]
where :
x
t
x
1
,
t
x
2
,
t
x
k
,
t
is a vector of signal samples at times
t
in channels
.
E
t
{
is a vector of noise process samples at time
t
.
The covariance matrix
V
of a noise process is expressed as:
1
,
2
,...
k
}
=[
E
1
,
t
,
E
2
,
t
,...
E
k
,
t
]
⎛
⎝
⎞
⎠
σ
1
0
...
0
0 σ
2
...
0
E
t
E
t
=
V
=
(3.14)
.
.
0 σ
k
0
...
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