Biomedical Engineering Reference
In-Depth Information
Algorithm 1
PCA based on the SVD of the observed data matrix
1: Store the
N
observed data samples into matrix
X
=[
x
1
,
x
2
,...,
x
N
]
.
2: Compute the SVD of the data matrix as
X
=
UDV
T
.
3: Recover the principal directions in the columns of matrix
U
.
4: Recover the principal component samples in the columns of matrix
Z
=
DV
T
.
Dimensionality reduction
can be performed by discarding the singular vectors associated with
negligible singular values in Steps 3-4 (see Sect.
3.2.2.3
for details).
using an observation model well adapted to the PCA approach. Indeed, model (
3.1
)
can be written as
x
i
=
M
θ
i
+
n
i
,
(3.12)
θ
i
=
α
i
,α
i
a
(
−
1)
i
,β
i
]
T
. Thus, the information
contained in the current T wave is summarized by a few parameters, represented
by vector
where
M
=
v
1
,
v
2
,
1I
]
and
θ
i
, using some global knowledge, condensed into matrix
M
, over the
total amount of data. In general, the columns of
M
are not orthogonal, i.e.,
M
T
M
is not a diagonal matrix, because there is no evidence that vectors
v
1
,
v
2
and
1I
are mutually orthogonal. As a result, the principal directions of
x
i
are unlikely to
coincide with these vectors.
To overcome this problem, we will first remove the contribution of the offset
vector from the original T-wave data by making use of the dimensionality reduction
capabilities of PCA recalled in Sect.
3.2.2.3
. To this end, we minimize the MSE (
3.7
)
but fixing the projection vector
h
=
1I
. This leads to the minimization of function
E
{
x
i
−
1I
z
2
T
}
with respect to the linear combination vector
w
, with
z
=
w
x
i
.
2
and the offset-corrected
From Eqs. (
3.7
)and(
3.8
), it turns out that
w
=
1I
/
1I
data are thus given by
2
x
i
=
T
T
1I1I
1I1I
x
i
=
x
i
−
I
−
x
i
.
(3.13)
1I
1I
2
This is the orthogonal projection of the original T-wave data
x
i
onto the orthogonal
complement of vector
1I
, as could be intuitively expected. Now the overall variance
of the offset-corrected observations is mainly due to the scaled T and alternans
waves only, so we can write
˜
x
i
=
˜
M
θ
i
+
n
i
,
(3.14)
where
˜
M
=[
v
1
,
v
2
]
,
˜
1)
i
]
T
,and
n
i
is the noise term
n
i
in Eq. (
3.12
)
after projection (
3.13
). As a result, PCA should now produce dominant principal
directions
u
1
and
u
2
related to
v
1
and
v
2
, so that vector
˜
θ
i
=[
α
i
,α
i
a
(
−
θ
i
can be estimated in the
least square sense by
θ
i
=
˜
˜
T
x
i
.
M
(3.15)
Assuming that the alternans effect is now condensed in
˜
θ
i
, the detection is
performed over this set of data. An alternative but equivalent development is given
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