Biomedical Engineering Reference
In-Depth Information
Ta b l e 1 .
Perfomance index (PI) values for random mixtures: mean and standard deviation for the
four algorithms, with two initializations (whitening and sphering) and using the couple
n/N
(nb.
of channels / data length) given by the heuristic rule derived from figure 6(b).
n
=8
n
=16
n
=24
n
=32
n
=48
N
=2
×
512
N
=3
×
512
N
=4
×
512
N
=5
×
512
N
=7
×
512
W
S
W
S
W
S
W
S
W
S
FastICA
0.048 0.049 0.043 0.043 0.041 0.040 0.040 0.040 0.038 0.038
(0.014) (0.012) (0.006) (0.007) (0.006) (0.006) (0.005) (0.005) (0.004) (0.006)
AMICA
0.024 0.029 0.018 0.019 0.021 0.029 0.036 0.061 0.113 0.170
(0.011) (0.031) (0.004) (0.005) (0.021) (0.051) (0.048) (0.056) (0.050) (0.060)
Extended
0.167
0.199
0.274
0.326
0.288
0.334
0.252
0.300
0.274
0.310
Infomax
(0.091) (0.124) (0.059) (0.055) (0.022) (0.013) (0.019) (0.014) (0.008) (0.007)
JADER
0.044 0.044 0.038 0.038 0.035 0.035 0.035 0.035 0.130 0.132
(0.010) (0.010) (0.005) (0.005) (0.004) (0.004) (0.003) (0.003) (0.029) (0.032)
(Figure 6(a)). The normalization factor
k
has been experimentally set to
8
,takingthe
8
channels mean error
as a reference on which higher number of channels configurations
are scaled.
This rule is reported on the figure 6(c). The proposed rule is rather linear, thus being
in contradiction with current literature suggestions, rather proportional to
n
2
.Ourrule
is then between the bounds given in the literature for low number of channels, but is
increasing much slower and gives lower bounds for number of channels above 24. A
possible way to interpret the figure 6(c) is to use it as a decision rule: for a given number
of channels, one can estimate the minimum number of data points necessary to have a
reliable estimate of the covariance matrix and thus a reliable whitening. This decision
rule leads to data lengths between approximately 2s (
1024
data points) for 8 channels to
7s (
3584
data points) for 48 channels. This range of time length is more compatible with
the stationarity hypothesis than the values obtained using the
30
n
2
rule [12, 11]. Indeed,
with this rule, we get from
1920
(3.75s) to
9720
(2min15s) data points respectively for
8 and 48 channels, which is rather contradictory (at least in a realistic EEG setup) with
the assumption of stationarity on which most of BSS/ICA algorithms are based
3
.
In order to experimentally validate this length rule, we computed the performance
index
PI
(12) for the resulting data length. Mean (over 50 realizations) of the
PI
as
well as its standard deviation for each algorithm (with whitening and sphering initial-
izations) are reported in the Table 1. As it can be seen, FastICA (with either whitening
or sphering) gives rather stable
PI
under
0
.
05
for this given rule (from
0
.
048
for 8
channels to
0
.
038
for
48
channels). It has to be noticed that
PI
values indicate better
performances when the number of channels is increasing with respect to our empirical
rule. This could suggest that our proposed criterion could be relaxed and the number of
points could be reduced further for FastICA. On the other hand, one must take into ac-
count that these tests are performed on simulated stationary random data: if outliers are
present, HOS estimates are more affected than the SOS estimations used to define our
threshold, thus a higher amount of points might be needed for HOS reliable estimation.
3
This observation is important especially for high resolution EEGs, having a high number of
channels.
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