Biomedical Engineering Reference
In-Depth Information
(a)
(b)
coherence
pdc:2
>1
pdc:1
>2
1
1
0.5
0.5
0
0
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
causality:2
>1
causality:1
>2
dtf:2
>1
dtf:1
>2
1
0.2
0.5
0.1
0
0
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
frequency
frequency
frequency
frequency
Fig. 8.2
Results of computing bivariate measures between
s
1
(
t
)
and
s
2
(
t
)
in the first experiment.
a
Results of computing spectral causality measures and the coherence. Results for the coherence are
shown in the
top panel
. Results for the spectral Geweke causality are shown on the
bottom-left
,and
bottom-right
panels.
b
Results of computing the partial directed coherence (PDC) (
top panels
)and
the directed transfer function (DTF) (
bottom panels
). The abscissas of plots indicate the normalized
frequency. (The frequency is normalized by the sampling frequency in which the value 0.5 indicates
the Nyquist frequency.) The ordinates indicate the relative amplitudes of corresponding measures
can also detect the information flow from the first to the second sources. In this
two-channel experiment, the PDC and DTF provide identical results, because, aside
from the scaling, both measures are equal to
|
A
1
,
2
(
f
)
|
in a bivariate case.
The causality measures were computed between
s
1
(
t
)
and
s
3
(
t
)
, and these results
are shown in Fig.
8.3
. Since there was no interaction between
s
1
(
)
and
s
3
(
)
, all the
causal measures used in the experiment, as well as the coherence, are close to zero
in these results.
t
t
8.8.2 Experiments Using Trivariate Causal Time Series
Next, we conducted experiments using trivariate causal time series. In these exper-
iments, the time series of the three sources are generated using the MVAR process
introduced in [
11
],
⊡
⊤
⊡
⊤
⊡
⊤
s
1
(
t
)
0
.
800
.
4
s
1
(
t
−
1
)
⊣
⊦
=
⊣
⊦
⊣
⊦
s
2
(
t
)
00
90
00
.
s
2
(
t
−
1
)
s
3
(
t
)
.
50
.
5
s
3
(
t
−
1
)
⊡
⊤
⊡
⊤
−
0
.
50
0
s
1
(
t
−
2
)
⊣
⊦
⊣
⊦
+
+
0
80
00
−
0
.
s
2
(
t
−
2
)
e
(
t
).
(8.122)
−
0
.
2
s
3
(
t
−
2
)
This MVAR process represents the causal relationship depicted in Fig.
8.1
b, in which
the second source has a directional causal influence on the third source and the third
