Environmental Engineering Reference
In-Depth Information
feature. Using this dimension reduction process, the embedded relation between
the rotational speed and the features can be captured.
A practical method to obtain the matrix T is extracting the main m eigenvectors
that correspond to the largest m eigenvalues k
k
k
¼
1
;
...
;
m
ð
Þ
, where the eigen-
values are written in decreasing order:
k
1
k
2
k
m
k
m
þ
1
k
n
!
0
ð
9
:
9
Þ
In most practical situations, we select m \ n, and the cumulative contribution
k
k
P
P
m
n
k
k
reaches a certain proportion (say, 80 or 95 %) [
45
], which means the
k
¼
1
k
¼
1
m factors have a significant influence on the vibration features. In this work, the
rotational speed will be the only significant factor, which means that m is equal to 1.
In some other practical applications, where the number of environmental factors is
unknown or difficult to determine by observing the eigenvalues, choosing a series of
order m for verification may be considered.
Finally, T is the m eigenvectors corresponding to the m eigenvalues in decreasing
order. Then, we can use T in Eq.
9.8
to project the features into the principal
components space. The loss of information in this process can be assessed by
reconstitution of the projected data back to the original space:
X
¼
T
T
Y
ð
9
:
10
Þ
.
According to the analysis above, Eq.
9.8
is a dimension reduction process;
Eq.
9.10
is a reconstruction process. The PCA method is shown in Fig.
9.2
. The
residual error between the original data and the reconstructed data is estimated as [
46
]:
E
¼
X
X
ð
9
:
11
Þ
The damage index is defined from the prediction error vector E
k
obtained at
time t
k
using the Euclidean norm [
47
]:
NI
k
¼
E
kk
ð
9
:
12
Þ
If it is further assumed that the Euclidean
ind
ices are normally distributed,
statistical analysis may be performed. Defining NI and r as the mean value and
standard deviation of NI for the prediction in the reference state, respectively, the
upper and lower control limits (UL and LL) can be defined as [
23
]:
CL
¼
NI
UL
¼
CL
þ
ar
LL
¼
CL
ar
ð
9
:
13
Þ
