Information Technology Reference
In-Depth Information
Given the signal represented formally as a multiple time series
X
(
t
)
, one can
extract the instantaneous phases
of the signal
as shown in Section 18.4 (either via a convolution with the Morlet wavelet or by
applying the Hilbert transform). The phase extraction procedure produces a new
multiple time series
φ
X
i
(
t
)
from each 1D component
X
i
(
t
)
of the correspondent phases.
Next, we derive new measures of phase synchrony of the signal based on the
concepts introduced in Section 18.3. Let us observe that the left-hand side of Equa-
tion (18.23) represents the linear combination of the respective phases
φ
X
(
t
)
φ
X
1
(
t
)
and
φ
X
2
(
with integer coefficients. Also recall that condition (18.23), which defines
phase locking between two signals
X
1
(
t
)
, needs to be modified in practice
to account for the noise in the signal. Taking into account presence of the stochastic
noise in the phase series, let us introduce a
modified
concept of the phase synchrony
between two signals
by relaxing the integrality condition on the coefficients in the
linear combination
as follows.
Two signals
X
1
(
t
)
and
X
2
(
t
)
are considered to be
generally phase synchronized
,
if the correspondent instantaneous phases
t
)
and
X
2
(
t
)
φ
X
1
(
t
)
and
φ
X
2
(
t
)
satisfy the condition
below:
∃
,
(
)+
(
)=
,
c
1
c
2
:
c
1
φ
t
c
2
φ
t
z
t
(18.28)
X
1
X
2
where
z
t
is a stochastic variable that represents the deviation from
the constant level
C
as a result of the noise. Notice that in the contrast to condi-
tion (18.23) in the classic definition of phase synchronization, the coefficients
c
1
and
c
2
in the definition of generalized phase synchrony (18.28) do not need to be
integer.
Furthermore, it is straightforward that the new condition (18.28) means that a
2D process
X
∼
N
(
C
,
σ
)
))
is cointegrated. Based on this observation, we can
extend our modified concept of phase synchronization between two signals to the
multivariate case in the following manner.
The multichannel signal
X
(
t
)=(
X
1
(
t
)
,
X
2
(
t
(
t
)=(
X
1
(
t
)
,...,
X
K
(
t
))
is considered to be
phase-
synchronized of rank r
, if the process
φ
X
(
t
)
composed of the correspondent instan-
taneous phases
K
is cointegrated of rank
r
.
In the subsequent subsections, we first discuss the role of the cointegration rank
in the framework of multivariate phase synchronization, and then apply this ap-
proach to multichannel EEG data collected from the patients with absence epilepsy.
φ
X
i
(
t
)
,
i
=
1
,...,
18.5.1 Cointegration Rank as a Measure of Synchronization
among Different EEG Channels
(
)
Note that integrated autoregressive processes
I
are shown to exhibit behavior
similar to that of a random walk. In a short paper [21], Michael Murray used an
example of drunkard and her dog to illustrate the concept of the cointegration. To
explain our reasoning behind the rank of cointegration as a measure of synchrony,
we briefly summarize and then further extend his analogy.
d
