Mathematics in Independent Component Analysis

Chapter 7. Proc. ISCAS 2005, pages 5878-5881 127
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(a) ECG recordings
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(b) extracted sources
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(c) MECG part
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(d) FECG part
Fig. 4. Fetal ECG example. (a) shows the ECG recordings. The underlying FECG (4 heartbeats) is partially visible in the dominating MECG (3 heartbeats).
Figure (b) gives the extracted sources using MHICA with k =2and 500 Hessians. In (c) and (d) the projections of the mother sources (components 1 and
2 in (b)) respectively the fetal sources (component 3) onto the mixture space (a) are plotted.
Our goal is to extract an MECG and an FECG component;
however it cannot be expected to find an only one-dimensional
MECG due to the fact that projections of a three-dimensional
vector (electric) field are measured. Hence modelling the data
by a multidimensional BSS problem with k =2(but allowing
for an additional one-dimensional component) makes sense.
Application of MHICA (with 500 Hessians) and MSOBI
(with 50 autocorrelation matrices) extracts a two-dimensional
MECG component and a one-dimensional FECG component.
After block-permutation we get the following estimated mixing
matrices (A using MHICA and A ′ using MSOBI)
� �
0.37 0.42 −0.81
A = −0.75 0.89 −0.16 A
0.55 −0.16 0.57
′ � �
0.22 0.91 −0.40
= −0.84 0.23 −0.33 .
0.50 0.34 0.85
The thus estimated sources using MHICA are plotted in
figure 4(b). In order to compare the two mixing matrices,
calculation of
�
.
A −1 A ′ �
0.85 1.02 0.64
= −0.23 1.11 0.35
−0.01 −0.08 0.98
yields a somewhat visible block structure; the performance
index is E (2) (A−1A ′ ) = 1.12. The block structure is not
very dominant, which indicates that the two models — block
independence versus time-block-decorrelation — are not fully
equivalent.
A (scaling invariant) decomposition of the observed ECG
data can be achieved by composing the extracted sources using
only the relevant mixing columns. For example for the MECG
part this means applying the projection ΠM := (a1, a2, 0)A−1 to the observations. This yields the projection matrices
�
�
0.52 0.38 0.84
ΠM = −0.10 1.08 0.17
0.34 −0.27 0.41
ΠF =
� 0.48 −0.38 −0.84
0.10 −0.08 −0.17
−0.34 0.27 0.59
onto the mother respectively the fetal ECG using MHICA and
Π ′ � �
0.78 0.21 0.45
M = −0.18 1.17 0.36 Π
0.47 −0.44 0.05
′ � �
0.22 −0.21 −0.45
F = 0.18 −0.17 −0.36 .
−0.47 0.44 0.95
using MSOBI. The results of the first algorithm are plotted in
figures 4 (c) and (d). The fetal ECG is most active at sensor
1 (as visual inspection of the observation confirms). When
comparing the projection matrices with the results from [1], we
get quite high similarity of the ICA-based results, and a modest
difference with the projections of the time-based algorithm.
Other one-dimensional ICA-based results on this data set are
reported for example in [14].
�
VI. CONCLUSION
We have shown how the idea of joint block diagonalization
as extension of joint diagonalization helps us to generalize
ICA and time-structure based algorithms such as HICA and
SOBI to the multidimensional ICA case. The thus defined
algorithms are able to robustly decompose signals into groups
of independent signals. In future work, besides more extensive
experiments and tests with noise and outliers, we want to
extend this result to a version of JADE using moments instead
of cumulants, which preserve the block structure.
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