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and d
j
L are the composition of c
j
and the detail scales D
j
, j
¼
1
; ...;
for every
m variables of the data matrix X:
c
1
;
c
2
; ...;
c
j
m
;
A
j
¼½
j
¼
1
; ...;
L
ð
36
Þ
d
1
;
d
2
; ...;
d
j
m
:
D
j
¼½
To select the wavelet decomposition level L it is considered the minimum
number of decomposition levels, and used to obtain an approximation signal A
L
so
that the upper limit of its associated frequency band is under the fundamental
frequency f, as described by the following condition Antonino-Daviu et al. (
2006
),
Bouzida et al. (
2011
):
2
ð
L
þ
1
Þ
f
s
\
f
:
ð
37
Þ
where f
s
is the sampling frequency of the signals and f is the fundamental frequency
of the machine. From this condition, the decomposition level of the approximation
signal is the integer L given by:
¼
log
2
ð
b
f
s
=
Þ
c:
ð
38
Þ
L
f
1
4.2 MSPCA Formulation
WT and PCA can be combined to extract maximum information from multivariate
sensor data. MSPCA can be used as a tool for fault detection and diagnosis by
means of statistical indexes. In particular, faults are detected by using Eqs.
16
and
18
and the isolation is conducted by the contribution method (Eq.
19
). In this way it
is possible to detect which sensor is most affected by fault (see Misra et al.
2002
).
Two fundamental theorems exist for the MSPCA formulation, they assess that PCA
assumptions remain unchanged under the Wavelet transformation. These theorems
are useful to apply MSPCA methodology, as stated in Bakshi (
1998
).
0
2 R
N
N
the orthonormal matrix
representing the orthonormal wavelet transformation operator containing the
H
0
L
;
G
0
L
;
G
0
L
1
; ...;
G
0
1
Theorem 4.1 Let W
¼
filter
coef
cients, the principal component loadings obtained by the PCA of X and WX
are identical, whereas the principal component scores of WX are the wavelet
transform of the scores of X.
Theorem 4.2 MSPCA reduces to conventional PCA if neither the principal com-
ponents nor the wavelet coef
cients at any scale are eliminated.
The developed FDD MSPCA based procedure consists of two stages:
in
the
first step, the faultless data are processed and a model of this data is built.
MSPCA training steps are summarized below:
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