Biomedical Engineering Reference
In-Depth Information
Notations
All quantities considered in this chapter belong to the field of real numbers, which
is denoted by
. Scalars, vectors and matrices are represented by lightface (
x
,
X
),
boldface lowercase (
x
) and boldface uppercase (
X
) characters, respectively;
[
X
]
ij
denotes the
(
i,j
)
-entry of matrix
X
. Symbol
E
R
{·}
stands for the mathematical
)
T
for the matrix transpose operator.
expectation, and
(
·
3.2
T-Wave Alternans Detection via Principal Component
Analysis
This section analyzes the TWA phenomenon using PCA. The model definition
proposed in Sect.
3.2.1
points to PCA as a good candidate for the estimation of the
model parameters. The basic concepts behind this statistical technique are recalled
in Sect.
3.2.2
, allowing us to develop PCA-based solutions to TWA detection in
Sect.
3.2.3
. Finally, Sect.
3.2.4
reports some illustrative experimental results.
3.2.1
Mathematical Modeling of T-Wave Alternans
First, we assume that an ECG is recorded at the body surface and that the best lead
for the observation of TWA is selected. An initial detection stage provides the time
epochs of the R wave locations and consequently allows the T-wave segmentation.
We assume each T wave appears at a fixed delay after its corresponding R wave. Let
t
i
denote the time index of the
i
th T wave and
L
the number of samples composing
each T wave, assumed to be constant. If we refer to the signal recorded at the lead
of interest as
x
(
t
)
,the
L
samples of the
i
th T wave can be stored in vector
x
i
as
follows:
1)
T
s
)]
T
∈R
L
,
x
i
=[
x
(
t
i
)
,x
(
t
i
+
T
s
)
,...,x
(
t
i
+(
L −
where
T
s
denotes the sampling period. The concatenated T waves plotted in Fig.
3.4
correspond to indices
330
335
in the dataset used as an example. As
introduced in Sect.
3.1.2.1
, the alternans effect is visible between the two horizontal
bold lines displayed for reference.
Now, we can put forward the following model:
x
i
=
α
i
v
1
+
a
(
≤
i
≤
1)
i
v
2
+
β
i
1I
+
n
i
.
−
(3.1)
The key idea underlying this model is that each observed T wave can be decomposed
as the sum of three components, namely, the average T wave, the alternans
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