Noninvasive Cardiac Signal Analysis Using Data Decomposition Techniques - Modeling in Computational Biology and Biomedicine - page 96

3.3.2 PCA Solution to BSS

Let us first examine whether the PCA method introduced in Sect. 3.2.2 is suitable
for BSS and under which conditions it can recover the source and the mixing matrix
in model ( 3.17 ). To guarantee the source identity covariance assumption justified by
the scale ambiguity, the principal components in Eq. ( 3.6 ) must be scaled by their
standard deviation to provide the PCA source estimate:

s PCA = D − 2 z = D − 2 U

T

x .

(3.18)

Hence, the separating matrix yielding the estimated sources from the observations
as s PCA = W PCA x is given by W PCA = D − 2 U

T , which amounts to the mixing

matrix estimate:

ˆ

1
2 .

H PCA = UD (3.19)

According to Sect. 3.2.2.4 , the PCA estimates can also be computed from the SVD
of the observed data matrix as

S PCA = √ N V

1
√ N

ˆ ˆ ˆ ˆ

T

H PCA = U D , , (3.20)

where the columns of ˆ S PCA contain the N samples of s PCA corresponding to the

observations in X .
In any case, it can be remarked that the columns of the estimated mixing matrix in
Eqs. ( 3.19 )and( 3.20 ) are always orthogonal due to the orthogonality of the principal
directions. As a result, PCA will be unable to perform the separation whenever the
actual structure of H in model Eq. ( 3.17 ) violates this orthogonality constraint. In
the problem of atrial activity extraction, forcing an orthogonal structure for H seems
a difficult task due to the spatial proximity of the atrial and ventricular sources, and
would most probably require a specific patient-dependent electrode placement. In
general, since both s PCA and s have an identity covariance matrix, they will be
related through an unknown ( M × M ) orthonormal transformation:

s PCA = Qs . (3.21)

The separation can thus be completed by finding Q and applying its transpose to
s PCA . The covariance of s PCA does not depend on Q . Therefore, finding this matrix
requires information other than the second-order statistics exploited by PCA.
As recalled in Sect. 3.2.2.2 , PCA transforms the original data into uncorrelated
components, i.e., into a random vector with diagonal covariance matrix. Hence,
in the context of BSS, PCA tries to recover the sources by diagonalizing the
observation covariance matrix, thus recovering the source diagonal covariance
structure and the source independence at order 2 (uncorrelation). By analogy with
whitening filters that whiten or flatten the frequency spectrum of a given signal
by diagonalizing its correlation matrix, PCA can be seen as a spatial whitening
operation, and the principal components or, rather, their scaled versions in s PCA ,