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e
is
i
th eigenvector of
C
, then the covariance matrix in this base becomes:
where
l
0
0
1
0
(9.15)
'
-
1
C TCT
=
=
l
,
i
X
X
0
l
m
λ
is the
i
th eigenvalue that corresponds to
e
.
where
e
are also called principal
components of
X
[
25
].
Before presenting the original data in the new base, the columns of
′
C
are sorted
with respect to their variance, from the largest to the smallest variance. Sorted
C
′
x
CS
′
. After sorting the eigenvalues, the
related eigenvectors are also sorted. If the positions of the
i
th and
j
th eigenvalues in
x
has
λ ≥λ
if
j
>
and is denoted by
i
j
X
C
′
are changed, then the positions of the
i
th and
j
th eigenvectors in
T
have to be
changed accordingly. Sorted
T
is denoted by
TS
. The original data matrix
X
is
expressed in a new base defined by
TS
[
26
] as:
( )
T
−
1
·
T
X
′
=
TS X
.
(9.16)
A new orthogonal coordinate system is chosen in which the covariance matrix
is diagonal with no covariance between columns of
X
. The existing variances are
sorted from the largest to the smallest, with the largest variances in the first columns
of
X
. Systems with a large number of measured variables are often driven with just
a few hidden variables. This results in the
X
matrix having significant variances for
only the first few columns. The first two PCs for input data obtained from a 31-lead
MECG (described in
Section 9.4.1
), and input data projected on the coordinate
system defined by the first two PCs, PC1 and PC2, are shown in Fig.
9.8
.
By utilizing PCA, system variability can be analyzed by analyzing only the first
few significant dimensions of a new base defined by PCs. Often, more than 95% of
a heart's electrical activity can be explained by just the first three PCs (Fig.
9.9
).
9.4.3.1
PCA-Based Reconstruction of 12-Lead ECG
For the purpose of reconstruction, we assume that two test MECG measurements
are available for each person. The first MECG measurement is used for calculating
PCs and selecting optimal unipolar leads. The second MECG measurement is used
first as a data source for a real measurements from the determined optimal unipolar
leads, and second, for the generation of the target 12-lead ECG that will be com-
pared to the reconstructed ECG. We will denote these two MECG measurements as
X
and
X
. The procedure for the reconstruction of a 12-lead ECG from two
MECGs, by using PCA is described in the rest of this section.
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