Mathematics in Independent Component Analysis

Chapter 20. Signal Processing 86(3):603-623, 2006 277
4
3.5
3
2.5
2
1.5
1
0.5
0
SCA
sNMF *
sNMF
NMF *
NMF
sNMF *
sNMF
(a) matrix comparison by Amari index
NMF *
NMF
JADE
Measure mean [a.u.]
1.5
1
0.5
0
Renyi
QMI
KLD
MuIn
RSDSTW
Measure
Xcor
JADE
Source
SCA NMFNMF Channels
sNMF *
sNMF
*
Signal
(b) comparison of the recovered sources
Fig. 6. Inter-component performance index comparisons. (a) Comparison of the
Amari index of matrices acquired from two different methods each in the case of
synthetic s-EMG decomposition. As the comparison matrix is symmetric, half of
its values have been omitted for clarity. (b) Mean of the inter-components mutual
information for each method estimated by different measures. For comparison, the
measures corresponding to the source signals (minimal indices) and to the channels
of the mixed s-EMG (maximal indices) are also shown.
Finally, we perform SCA using a generalized Hough transform [37]. Note that
there are also other algorithms possible for such extraction, and model generalizations
are possible, see for example [45]. After whitening and dimension
reduction, we perform Hough hyperplane identification with bin-size 180 and
manually identified the maxima in the obtained the Hough accumulator. The
recovered mixing matrix is again visualized in Fig. 4(f). Similar to the previous
results, the three components are roughly most active at locations 2 to 3,
4 and 5 respectively. Multiplication with the pseudoinverse of the recovered
matrices from the above experiments shows that the result coincides quite well
with the matrices from NMF, but differs slightly more from the ICA-recovery.
We calculate and compare the Amari index for each method and as shown in
Fig. 6(a), no major differences can be detected. A comparison of the recovered
sources using the indices from section 2.3 is given in Fig. 6(b), where also
the indices corresponding to the original source signals (minimum mutual
information values) and to the channels of the s-EMG (maximum values) have
been added. All the methods separate the mixtures (improvement in terms of
indices) but the methods yield somewhat different result, with JADE giving
quite different sources than the rest of the methods, and NMF and NMF∗
performing rather similar. In terms of source independence, of course JADE
scores best ratings in the indices, as can be confirmed by calculating the Amari
index of the recovered matrix with the original source matrix:
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