Biomedical Engineering Reference
In-Depth Information
We have just proved an important result that can be generalized as follows: the
orthogonal projection of the data onto their first
k
principal directions (
k<L
),
given by
U
k
U
T
k
x
, with
U
k
=[
u
1
,
u
2
,...,
u
k
]
, is the rank-
k
approximation best
resembling the data in the MSE sense; its minimum MSE is given by
i
=
k
+1
λ
i
.
The original data are thus approximated by the
k
principal directions and the
realizations of the corresponding principal components. If
N
data realizations are
available, this approximation reduces the storage requirements from
O
(
LN
)
to
O
(
kN
)
in order of magnitude. The popularity of PCA as a data compression
technique hinges upon this property. In particular, discarding principal components
associated with negligible eigenvalues allows us to reduce the dimensionality of the
data with little loss of information. No information loss is actually incurred when
the covariance matrix is rank-deficient with
(
L − k
)
null eigenvalues, as the
L
-
dimensional data lie in a subspace of dimension
k<L
in that case.
3.2.2.4
PCA in Practice
As we have seen in the preceding sections, PCA relies on the computation of the
data covariance matrix
R
x
. In practice,
N
data samples or realizations
{
x
n
}
n
=1
are
observed, which can be stored in the observation matrix
X
=[
x
1
,
x
2
,...,
x
N
]
∈
R
L×N
. Then, the covariance matrix can be estimated from the available data by
sample averaging:
N
1
N
1
N
XX
ˆ
T
n
T
R
x
=
n
=1
x
n
x
=
.
(3.10)
PCA can then be obtained from the EVD of this covariance matrix estimate.
Nevertheless, a computationally more robust yet theoretically equivalent alternative
to compute PCA is the singular value decomposition (SVD) of the observed data
matrix
X
=
ˆ
ˆ
T
U
DV
,
(3.11)
where the columns of unitary matrices
ˆ
U
and
V
contain, respectively, the left
and right singular vectors, and diagonal matrix
ˆ
D
contains the singular values
of
X
. Plugging Eq. (
3.11
) into Eq. (
3.10
) and comparing the result with Eq. (
3.5
),
it turns out that the left singular matrix
ˆ
U
provides an estimat
e o
f
t
he principal
directions whereas the singular-value matrix
ˆ
D
is an estimate of
√
N
D
1
2
. According
to Eq. (
3.6
) the realizations of the principal components are stored in the columns
matrix
Z
=
ˆ
T
. Algorithm
1
summarizes how to compute the SVD-based PCA.
DV
3.2.3
PCA-Based Solution to T-Wave Alternans Detection
Now let us come back to the TWA detection problem. Although estimating the
parameters in model (
3.1
) is directly feasible [
16
], we can relax some constraints by
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