Mathematics in Independent Component Analysis

280 Chapter 20. Signal Processing 86(3):603-623, 2006
divided to give a mean improvement ratio. Taking again mean over the 100
runs yields the following improvements:
JADE NMF NMF∗ sNMF sNMF∗ SCA
mean SIR improvement 4.15 2.54 2.87 0.701 1.90 3.08
This confirms our results; JADE works very well for preprocessing, but also
NMF∗ and, interestingly, SCA.
In order to test the robustness of the methods against noise, we recorded s-
EMG at 0%MVC, that is, when the muscle was totally relaxed; in this way
we obtained a recording of noise only. As expected the resulting recordings
have nearly the same means and variances, and are close to Gaussian. This is
confirmed by a Jarque-Bera test, which asymptotically tests for goodness-of-fit
of an observed signal to a normal distribution. We recorded two different noise
signals, and the test was positive in 11 out of 16 cases (at significance level
α = 5%); the 5 exceptions had p-values not lower than 0.001. Furthermore, the
noise is independent as expected because it has a close to diagonal covariance
matrix, Amari index of 2.1, which is quite low for 8 dimensional signals. The
noise is not fully i.i.d. but exhibits slight non-stationarity. Nonetheless, we
take this findings as confirmation to assume additive Gaussian noise in the
following. We will show mean algorithm performance over 50 runs for varying
noise levels.
Note that due to the Gaussian noise, the models (especially the NMF model
which already uses such a Gaussian error term) hold well given the additive
noise. We multiplied this noise signal progressively by 0, 0.01, 0.05, 0.1, 0.5,
1 and 5 (which corresponds to mean source SNRs of ∞, 36, 22, 16, 2.1, -3.9
and -18 dB) and then added each of the obtained signals to a randomly generated
synthetic s-EMG containing 5 sources as above. The Amari index was
calculated for each method and for each noise level. We thus obtained the
comparative graph shown in Fig. 8. Interestingly, sparse NMF∗ outperforms
JADE in all cases, which indicates that the sNMF model (which already includes
noise), works best in cases of slight to stronger additive noise — which
makes it very well adapted to real applications. Again SCA performs somewhat
problematically, however separates the data well a the noise level of -3.9
dB. We believe that this is due to the involved thresholding parameter in
SCA hyperplane detection; apparently it is necessary to implement an adaptive
choice of this parameter in order to improve separation as in the case of
SNR of -3.9 dB.
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