Information Technology Reference
In-Depth Information
series analysis, the measurements from a single lead are called a univariate time
series (UTS), while measurements from more than one lead are referred to as
multivariate time series (MTS) [
19
].
For the purpose of defining measures of similarity between MECGs (between
MTS), two MECG measurements are represented as:
{
}
X X Xi Xm
X X Xi Xm
=
(1),
……
,
( ),
,
(
)
(9.6)
1
1
1
1
{
}
=
(1),
……
,
( ),
,
(
) ,
2
2
2
2
Xi
are
i
th leads (i.e., UTS) of the first and second MECG mea-
surements.
m
is the total number of leads, i.e., the length of the MTS.
The data from a lead are UTS, defined as a series of observations (measurements
or data samples) made sequentially through time:
Xi
and
1
()
2
()
where
Xi Xi
() { (), , ()},
n
=
…
Xi
(9.7)
1
where
n
is the total number of observations.
In order to evaluate the difference between reconstructed and target ECG, which
are both MTS, we have used various similarity measures, described in the rest of
this section.
The
Euclidean distance
between two MTS
X
and
X
of equal lengths can be
defined as the mean distance of the corresponding UTSs [
20
]:
m
k
dXkXk
(
)
=
∑
(), ()
(9.8)
( )
1
2
=
1
ED X X
,
,
12
m
where
dX k X k
is the Euclidean distance between the two UTS defined by:
(
( ),
( ))
1
2
n
(9.9)
(
)
(
)
2
=
∑
dXkXk
(), ()
Xk Xk
()- () .
1
2
1
l
2
l
l
=
1
Correlation
is a measure of statistical dependence of one MTS on another.
Values of correlation near 0 often indicate that the variables are uncorrelated, while
values near 1 or −1 indicate a strong positive or negative correlation.
The correlation between two MTS
X
and
X
can be defined as:
m
k
(
)
=
∑
corr X k X k
(), ()
(9.10)
( )
1
2
CORR X X
,
=
1
,
12
m
where
corr X k X k
(
( ),
( ))
is the correlation between two UTS defined as [
21
]:
1
2
(
(
)(
)
)
∑
n
Xk Xk Xk Xk
()
−
()· ()
−
()
(9.11)
1
l
1
2
l
2
(
)
l
=
1
corr X k X k
(), ()
=
,
1
2
( )
1
ns
−
1
s
Xk Xk
() ()
2
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