Noninvasive Cardiac Signal Analysis Using Data Decomposition Techniques - Modeling in Computational Biology and Biomedicine - page 87

wave and a constant wave, represented by the L -dimensional vectors v 1 , v 2 and
1I , respectively. These components are linearly weighted by parameters α i and
β i , which stand for a scaling coefficient and the vertical offset (mean value),
respectively, before adding up to yield the i th observed T wave, corrupted by the
additive noise represented by vector n i . The binary value 0 or 1 for the a variable
will allow us to detect and classify alternans episodes. This model accounts for
a baseline component [ 4 ] that is assumed to be constant in the T-wave interval.
The scaling coefficient α i represents the modulation of the ECG signal during the
recording, which can be due to respiration movements or recording conditions.
According to model ( 3.1 ) the alternans amplitude lies in the range 0-10

Vforthe
particular example of Fig. 3.4 . The alternans phenomenon can be characterized by
estimating the model parameters in Eq. ( 3.1 ) from the observed T-wave data. We
will see next how to do so by using PCA.

μ

3.2.2 Principal Component Analysis

Principal component analysis (PCA) is a classical statistical technique for decom-
posing multivariate observations into uncorrelated variables known as principal
components . A recent review of its application to ECG signal processing can be
found in [ 5 ]. Chapters 5 and 7 of this topic apply PCA to other biomedical data. The
concepts recalled in this section will be useful not only in the design of PCA-based
TWA detectors, but also in the derivation of ICA techniques later in Sect. 3.3 .We
will assume throughout this section that the data to be analyzed are represented by an
L -dimensional random vector x ∈R L . The following mathematical formulations
of this statistical technique focus on the ensemble data covariance matrix, defined as

T

R x =E { xx }, (3.2)

where the data are assumed to have zero mean, E { x } = 0 . The rank of this matrix

R L actually spanned by the observed
data. As will be seen next, the data covariance matrix plays a central role in PCA.
The practical setting where several realizations (typically time samples) of x are
observed will be discussed in Sect. 3.2.2.4 .

yields the dimension of the subspace of

3.2.2.1 PCA as a Variance Maximization Decomposition

Among the various mathematical formulations of PCA, one of the most intuitive
is perhaps in terms of linear combinations or projections with maximum variance.
Let us consider a one-dimensional (scalar) signal or component z ∈R

derived as a

linear combination of the observed data x ∈R L :

T

z = w x .

(3.3)