Noninvasive Cardiac Signal Analysis Using Data Decomposition Techniques - Modeling in Computational Biology and Biomedicine - page 95

Unfortunately, neither the sources nor the mixing matrix are known in practice,
and Eq. ( 3.17 )definesan inverse problem referred to as blind source separation
(BSS) in instantaneous linear mixtures [ 9 ]. Given the observed signals, BSS aims at
estimating the source signals and the mixing matrix in model ( 3.17 ). The separation
is achieved by transforming the observations via a separating matrix W , ideally the
inverse of the mixing matrix, W = H − 1 . Inverse problems arising in confocal
microscopy imaging and brain signal analysis are addressed in Chaps. 4 and 7,
respectively, of this topic.
The BSS model can be considered as a generalization of classical approaches
such as adaptive noise canceling [ 24 ]. This earlier approach estimates the interfer-
ence by suitably filtering some of the observations called reference signals. A crucial
assumption for the success of the noise canceling technique is that the reference
signals must be correlated with the interference but uncorrelated with the signal
of interest. In the atrial activity extraction problem, this assumption constrains the
electrode location, since the reference leads must be free of atrial contributions. By
contrast, the BSS model is more flexible in that the contribution of each source
to each observation (i.e., the coefficients h ij ) can be practically arbitrary provided
the mixing matrix remains left invertible or full column rank, i.e., its columns are
linearly independent. A necessary condition for the left invertibility of H is that
M ≤ L .
Apart from this requirement on the mixing matrix, additional assumptions are
necessary to solve the BSS problem ( 3.17 ). These assumptions concern certain
source properties that are exploited to perform the separation. During AF, atrial
and ventricular activities can be assumed to arise from relatively uncoupled elec-
trophysiological phenomena, since the atrial electrical wavefronts impinge on the
atrio-ventricular node, thus generating a ventricular beat, in a rather random fashion
(see Sect. 3.1.2.2 ; cf. the normal sinus activation described in Sect. 3.1.1 ). Hence, the
atrial and ventricular sources present a certain degree of statistical independence.
Depending on the degree of source independence assumed, BSS can be carried
out by different approaches. The PCA technique reviewed in Sect. 3.2.2 can be
considered as a BSS technique exploiting the source independence assumption up to
second order (uncorrelation), whereas ICA exploits independence at orders higher
than two. The remaining of the chapter will summarize these techniques as well as
their advantages and limitations in the context of BSS.
Before continuing the exposition, remark that a scale factor can be interchanged
between a source and its mixing-matrix column without modifying the observations
nor the source independence. To fix this scale ambiguity , we can assume, without
loss of generality, that the sources have unit variance, E

2
i

{s ( t ) } =1 ,for 1 ≤ i ≤ M ,

T

leading to an identity source covariance matrix: R s =E = I . For analogous

reasons, the exact source ordering cannot be determined without further information
about the sources or the mixture; this is the so-called permutation ambiguity .
The permutation ambiguity makes it difficult to target a specific source without
separating all sources first, unless additional information about the source of interest
is introduced in the separation method; this issue will be further discussed in
Sect. 3.3.4 . In the sequel, we will assume for simplicity that L = M (square mixing)
and the time index t will be dropped for convenience.

{ ss }