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The predictor
B
is multidimensional because it is composed of 3 variables, as are
the functions
f
that form the terms of the model. For a three-dimensional predictor
, terms for the model could include
fB
=
(or for example,
1
(
)
(1)
BBBB
=
{ (1),
(2),
(3)}
fB
=
) (linear term),
1
(
)
(2)
fB
=
(quadratic term), and
2
(
)
(1)
2
fB B B
3
( ) (1)• (2)
=
(pairwise interaction term). Typically, the function
()1
fB
=
is included among
f
,
so that the design matrix
M
contains a column of ones and the model contains a
constant, y-intercept, term.
We have explored the use of linear additive models. They have terms
()1
fB
=
and
f B B
=
In the case of three bipolar leads, the design matrix
M
of the linear
additive model is:
(
)
( ).
1
B
(1)
B
(2)
B
(3)
1
1
1
M
=
,
(9.32)
1
B
(1)
B
(2)
B
(3)
n
n
n
with four coefficients in the vector
αααα=
that are obtained after
the system solution. If we denote the solution coefficients by
[
]
1
2
3
4
re
α
, then the result of
the
•
res
M
α
is the best approximation of
ECG
12 in the sense of the least-square
solution, obtained from the three selected differential leads
B
.
For reconstruction purposes, only the best combination of differential leads is
required.
Section 9.4.4.1
explains how such a combination of three differential leads
is selected from the set
tota
B
. If the system (9.31) is solved for the three optimal
differential leads, then
•
res
M
α
is considered to be the best approximation to the target
12-lead ECG. If the wireless electrodes, which implement differential leads, are posi-
tioned on the same places on the body surface as optimal differential leads, the
calculated coefficients vectors can be applied on the measurements from wireless
electrode to reconstruct a 12-lead ECG.
9.4.4.1
Optimal Differential Leads Selection for MLR
In the previous section the set of all possible combinations of three differential leads
tota
B
was introduced. In order to find an optimal combination of three differential
leads, a “brute force attack” type of algorithm was used: MLR is conducted for every
element of the set
tota
B
. The optimal combination of three differential leads is
considered to be the one that reconstructs a 12-lead ECG that has the strongest
correlation with the target 12-lead ECG. This procedure is illustrated in Fig.
9.11
.
9.4.5
Personalization
The optimal lead set and the transformation matrix for reconstructing the 12-lead
ECG are personalized, in the sense that a different optimal lead set is found for
every patient and a different transformation matrix is calculated for every patient.
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