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the set formed by a selected electrode and its nearest neighbors on the body surface.
In the case of the MECG with 31 electrodes, this set contains just 81differential
=
81
leads; however, there are
possible combinations of three reconstruction
85320
3
leads denoted by
tota
B
.
Let a set of three arbitrary bipolar differential leads be denoted as:
BBBB
=
{ (1),
(2),
(3)}.
(9.27)
As explained in
Section 9.2.1
, 12-lead ECG can be represented as a set of
12-leads:
ECG
12
=
{ ,
I II III aVR aVL aVF V V V V V V
,
,
,
,
, 1, 2, 3, 4, 5, 6}.
(9.28)
As already noted, every MECG contains enough leads to generate the measured
standard 12-lead ECG, so
ECG
is extracted from
X
(Eq. (9.25)) and represents
a target ECG for reconstruction.
Generally, a linear regression model represents the relationship between a
response (i.e., criterion variable)
12
ECG
and a predictor
B
[
28
]:
12
ECG
12
=
a
f B
(
)
+…+
a
f B
(
)
+…+
a
f B
(
)
+
e
(9.29)
11
jj
pp
The response is modeled as a linear combination of functions (themselves not
necessarily linear) of the predictor, plus a random error
ε
. The expressions
(
α =…
are the coef-
ficients. Errors
ε
are assumed to be uncorrelated and distributed with mean 0, and
constant, but unknown, variance. Our problem can be covered by the multivariate
regression due to the fact that the response variable
=…
are the terms of the model while
, (
j
1,
,
p
)
j
fB j p
) , (
1,
,
)
j
ECG
is multidimensional,
12
i.e., it is composed of 12 leads (variables) [
29
].
Given
n
independent observations (samples),
(
)
B ECG
,
12 ,
…
, (
B ECG
,
12 )
,
1
1
n
n
of the predictor
B
and the response
ECG
, the linear regression model becomes
12
an n-by-p system of equations:
fB
()
…
fB
()
ECG
12
ae
11
p
1
1
1
1
(9.30)
=
·
+
,
ECG
12
f B
(
)
…
f B
(
)
ae
n
1
n pn p n
or
ECG
12
= +
Mae
·
,
(9.31)
where
M
is the design matrix of the system. The columns of
M
are the terms of the
model evaluated at the predictors. To fit the model to an input data, the above system
must be solved for the
p
coefficient vectors:
T
…
. We solved the system
by applying the least-squares solution separately for every lead of
α
α
1
p
ECG
and
therefore reducing the multivariate regression model to 12 multiple regressions
[
30
], applying the MATLAB
regress
function [
29
].
12
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