Biomedical Engineering Reference
In-Depth Information
Algorithm 3 RobustICA algorithm for extracting all independent sources from a
linear mixture
1: Compute the PCA of x and let x 0 = s PCA be the whitened observation vector of dimension
M ,where s PCA is given by Eq. ( 3.18 ).
2: for k =1 ( M − 1) do
3:
Extract s k from x k− 1 with the help of Algorithm 2 (Sect. 3.3.3.4 ).
4:
Subtract the contribution of s k from x k− 1 by using the regression formulas ( 3.27 )-( 3.28 ):
x k = x k− 1 E {s k x k− 1 }
.
E {s k }
5:
Compute the standardized observation vector x k
with reduced dimensionality as defined
in ( 3.29 ): x k = P k x k .
6: end for
7: Set s M = x M .
The deflation algorithm explained earlier in this section corresponds, up to di-
mensionality reduction, to the approximation E
T
k }≈
1 }, E
2 },...,
{
s k s
diag(E
{
s
{
s
2
E
) in Eq. ( 3.30 ), that is, to the case where the estimated source covariance
matrix is diagonal. By taking into account the whole covariance matrix of the
estimated sources, this alternative deflation scheme achieves improved robustness
to residual source correlations that may remain when dealing with short sample
sizes.
{
s
k }
3.3.3.6
Remarks
An interesting advantage of the RobustICA algorithm of Sect. 3.3.3.4 is that
it does not require prewhitening, which improves its asymptotic (large sample)
performance [ 26 ]. Hence, the whitening stage (Algorithm 3 , line 1) can just be
omitted, and x 0 can simply be initialized with the observed data x . Nevertheless,
running PCA before extracting the sources improves numerical conditioning due to
the diagonal covariance matrix and unit variance estimates. It can hence still be used
as a preprocessing step before deflation, even if it limits asymptotic performance.
All the results stated in the present framework for the full separation of M sources
hold in two situations: (1) in the presence of at least ( M −
1) non-Gaussian sources
without noise, or (2) in the presence of M non-Gaussian sources with additive
Gaussian noise. In the presence of non-Gaussian noise, results become approximate
and can become erroneous if the non-Gaussian noise is too strong. Finally, although
ICA performs independent source separation under linear mixing model ( 3.17 ), it
can also be employed to find interesting alternative data representations (useful, e.g.,
for classification) even if this observation model is not fulfilled.
3.3.4
Refining ICA for Improved Atrial Signal Extraction
ICA was first applied to atrial signal estimation in AF ECG episodes in [ 20 , 21 ],
yielding satisfactory results when the amplitude of the atrial sources presents
 
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