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where (
)
Xk Xk
()
()
Xk ,
1 () l
is t he deviation of an individual UTS observation
1
l
1
from the UTS mean
Xk , and
s
is the standard deviation of
Xk UTS, and
1 ()
1 ()
Xk
1 ()
the same for
Xk .
The PCA similarity factor (S PCA ) is the measure of distance between the two
MTS. It requires matrix representation of data, so that every column represents a
single UTS from the MTS in question. To calculate S PCA it is necessary to obtain
the principal components (PCs) for each matrix, i.e., each MTS, and to choose the
first k PCs using one of the heuristic approaches. For example, the PCs can be
sorted by their variances and the first of them are selected whose sum of variances
represent 95% of the total variance. S PCA reflects the similarity of the first k PCs.
The S PCA of two MTS
2 ()
X and
X is defined as follows [ 22 ]:
kk
= ∑∑
( )
(9.12)
2
S XX
,
cos (
θ
),
PCA
1
2
pq
pq
==
11
where
p θ is the angle between the p th PC of
X and q th PC of
X .
9.4.3
Principal Component Analysis
PCA is a multivariate statistical technique whose purpose is to condense the informa-
tion of a large set of correlated variables into a few variables called principal compo-
nents (PCs), while not losing the variability present in the original data set. The PCs
are derived as a linear combination of the variables of the data set, with weights
chosen so that the PCs become mutually uncorrelated. Each component contains new
information about the data set, and is ordered so that the first few components account
for most of the variability. In signal processing applications, PCA is performed on a
set of time samples rather than on a data set of variables [ 23 ].
If we represent a MECG measurement by a matrix
, where n is the
number of time samples and m is the number of leads, then a covariance matrix is
defined as:
X nm
[
]
c
c
11
1
m
1
(9.13)
T
C
=
XX
=
c
,
X
C
C
ij
n
1
c
c
m
1
mm
where X is centralized MECG measurement X , obtained by subtracting the lead's
mean from every lead, and
c = is the covariance between the i th and j th leads.
The covariance matrix is diagonal in a coordinate system defined by its eigen-
vectors [ 24 ], so if the base transformation matrix is defined as:
ij
ji
(9.14)
[
]
Teee
= ……
,
i m
1

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