Mathematics in Independent Component Analysis

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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. 19
Chapter 20. Signal Processing 86(3):603-623, 2006 281 median Amari−index 9 8 7 6 5 4 3 2 1 JADE NMF NMF∗ sNMF sNMF∗ SCA 0 −20 −10 0 10 mean SNR (dB) 20 30 40 Fig. 8. Result of the experiment testing the robustness of the different methods against noise. Plotted is the mean over 50 runs. Real s-EMG signal [V] Channel 1 Channel 2 Channel 3 Channel 4 Channel 5 Channel 6 Channel 7 Channel 8 5 0 -5 -10 5 0 -5 -10 5 0 -5 -10 5 0 -5 -10 5 0 -5 -10 5 0 -5 -10 5 0 -5 -10 5 0 -5 -10 100 200 300 400 500 600 700 800 900 1000 Time [ms] Fig. 9. Real s-EMG obtained from a healthy subject performing a sustained contraction at 30% MVC. 20
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Statistical machine learning of bio
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vi Preface
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viii CONTENTS 3 Signal Processing 8
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Chapter 1 Statistical machine learn
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1.1. Introduction 5 auditory cortex
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1.2. Uniqueness issues in independe
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1.2. Uniqueness issues in independe
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S U M M A R Y T his �le contains
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1.3. Dependent component analysis 1
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Table 1.1: BSS algorithms based on
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1.4. Sparseness 25 1.4 Sparseness O
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1.4. Sparseness 27 Theorem 1.4.3 (S
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1.4. Sparseness 29 R 3 R 3 A BSRA R
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1.4. Sparseness 31 Sparse projectio
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Page 306 and 307: 298 BIBLIOGRAPHY Barber, C., Dobkin
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Mathematics in Independent Component Analysis

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