Sparse analysis based fault deviations modeling and its application to fault diagnosis

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1
INTRODUCTION
With rapid development of chemical processes, their large
and complex characteristics drive an increasing interest in
fault detection and diagnosis over the last few decades
[1-4]
.
Multivariate statistical analysis methods, such as partial
least square (PLS)
[5]
, principal component analysis (PCA)
[6]
, fisher discriminant analysis (FDA)
[7]
and etc., have been
widely applied to the field of statistical process monitoring
(SPM). They all share the characteristics of dimensionality
reduction and the ability to deal with highly correlated
variables. For PCA monitoring, two subspaces principal
component subspace (PCS) and residual subspace (RS)
monitored by
2
T
and SPE monitoring statistics respectively
are gained. When the values of monitoring statistics run out
of the desired regions, it can be concluded that some
abnormal or faulty behaviors have occurred. After fault
detection, it is hoped that the type of abnormal conditions
can be quickly confirmed and necessary corrective actions
can be taken to remove them, which means to bring the
out-of-control monitoring statistics back to the normal.
Dunia and Qin
[8]
defined a fault reconstruction concept in
the context of PCA models, which consisted of finding the
reconstruction direction and brought them back to the
normal region along the fault subspace.
FDA
[9-10]
is a popular method for fault diagnosis, in which
fault diagnosis is regarded as a classification problem.
However, it has been reported to have some drawbacks and
limitations (singular problem caused by the within-class
scatter matrix, limitation of the number of discriminant
components by between-class scatter matrix and the
This work is supported by the National Natural Science Foundation of China
(Nos. 61422306 and 61433005).
non-orthogonality of discriminant components). Several
improved methods
[11-13]
have been proposed to solve the
singularity problem of within-class scatter matrix. Zhao et
al.
[14]
proposed a nested-loop fisher discriminant analysis
(NeLFDA) algorithm which performed an inner-loop and
outer-loop calculation to address the three problems of the
conventional FDA algorithm. However, it cannot
comprehensively extract fault features as the variations of
variance from normal data to fault data are not considered.
By comprehensively considering fault types in the process,
Zhao et al. proposed
[15]
a fault degradation oriented Fisher
discriminant analysis (FDFDA) method to bring in the
variance variations between fault data and normal data to
the objective function of FDA.
Nevertheless, these methods treat the whole measurement
variables as a single subject, which do not isolate the
specific faulty variables. Contribution plots method
[16-17]
has been widely used for isolating faulty variables by
comparing the contributions of different variables to the
out-of control monitoring statistics. However, it may lead to
confusing results as the contribution of faulty variables may
disseminate to other variables. Instead of performing on the
context of PCA, Qin et al.
[18]
identified the major
contributing variables based on FDA directions, which has
the drawbacks of not probing into the relationship of
specific variables and still limited by the drawbacks of
traditional FDA method. Zhao et al.
[19]
proposed a faulty
variable isolation method to identified faulty variables on
the NeLFDA direction, which overcame the limitations of
traditional FDA. But variables are evaluated and isolated
one by one in this method, which is inefficient and the
relationship of specific variables is not probed into.
Sparse analysis based fault deviations modeling and its application to fault
diagnosis
Yue Wang
1
, Chunhui Zhao
1∗
, Youxian Sun
1
1. State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou,
310027,China
E-mail: chhzhao@zju.edu.cn
Abstract: In the fault process, some specific variables will be disturbed significantly and cover much fault information,
while some irresponsible variables still keep similar relations with those of normal condition. Therefore, this paper
proposes a sparse relative discriminant fault deviations (SRDFD) modeling algorithm which can extract fault directions
and isolate faulty variables simultaneously to improve fault diagnosis performance. In the proposed algorithm, a sparse
objective function is formulated by bringing an L1 penalization to the objective function of fault degradation oriented
FDA (FDFDA) algorithm, which improved the traditional FDA algorithm by further considering the relative variations
of variance between fault data and normal data. The proposed objective function is not convex, so that the
minorization-maximization approach is used to efficiently optimize it. Then soft threshold operator is performed for
analytic solutions. The extracted sparse directions and the corresponding loadings are used as reconstruction models to
eliminate fault deviations. Online fault diagnosis is then conducted by finding the correct reconstruction models which
can best eliminate the out-of-control monitoring statistics. The performance is verified by the pre-programmed faults of
Tennessee Eastman (TE) benchmark process.
Key Words: faulty variable isolation, relative variations of variance, FDA, minorization-maximization
4509
978-1-5090-4657-7/17/$31.00 c
2017 IEEE

In general, for each case, some specific variables will be
disturbed significantly and cover much fault information. It
is important to consider isolating these faulty variables so as
to improve the diagnosis performance. Therefore, this paper
proposes a sparse relative discriminant fault deviations
(SRDFD) modeling algorithm for comprehensive extracting
fault features and isolating faulty variables at one time. In
this algorithm, we bring in L1 penalization to the objective
function in the work of Zhao et al.
[15]
, which integrates
dispersion and the variations of variance between normal
data and fault data, so that the resulting coefficients of
irresponsible variables are equal to zero for the extracted
directions. A minorization-maximization approach
[20]
and
the soft threshold operator are performed for solving the
proposed objective function. The extracted sparse
directions and the corresponding loadings are used for
reconstruction. Online fault diagnosis is then conducted by
finding correct reconstruction models which can best
eliminate the out-of-control monitoring statistics.
The reminder of the paper is arranged as below. First,
motivation of the proposed algorithm is presented. Then, the
proposed algorithm is mathematically formulated and
verified based on pre-programmed faults from TE process.
At last, conclusions are drawn on the basis of the results of
this study.
2
METHODOLOGY
2.1
Motivation
FDA is a widely used fault diagnosis method, by which data
from different classes are separated well along extracted
directions and then fault diagnosis can be performed.
However, traditional FDA may not extract comprehensive
fault features for the reason that it does not take the
variations of variance between fault data and normal data
into consideration, so that Zhao et al.
[15]
proposed a fault
degradation oriented Fisher discriminant analysis (FDFDA)
method which integrates dispersion and the variations of
variance between normal data and fault data. The objective
function is shown as below,
()
T
T
TT
ȕ
max
f
b
nn
J
§·
=+
¨¸
¨¸
©¹
wSw
wSw
wwSw wSw
(1)
where
b
S
is the between-class scatter matrix;
n
S
and
f
S
are within-class scatter matrices for normal data and fault
data respectively;
ȕ
is a weighting factor.
The three matrices are calculated by the following
expression:
()()
bff f
N
Τ
=−−Sxxxx
(2)
,,
1
()()
f
N
ffiffif
i
Τ
=
=− −
¦
Sxxxx
(3)
,,
1
()()
n
N
nninnin
i
Τ
=
=−−
¦
Sxxxx
(4)
where
f
x
and
n
x
are the mean vectors of fault data and
normal data respectively;
x
is the mean vector of total
samples;
,
fi
x
and
,
ni
x
denote the ith sample in fault data
and normal data respectively.
The weighting factor (
ȕ
) is calculated on the basis of
f
S
and
b
S
:
()
ȕ()
b
f
tr
tr
=
S
S
(5)
where
()tr •
denotes the sum of eigenvalues calculated from
the matrix.
By assuming that
T
=1
n
wSw
and using a Lagrange operator,
a conventional eigenvalue problem can be gained by the
following expression:
()
1
ȕ
nb f
λ
−
+=SS Sw w
(6)
Then a set of fault directions can be obtained one time by
performing singular value decomposition (SVD) on the Eq.
(6). Nevertheless, the work of Zhao et al. did not consider
isolating specific faulty variables for accurately describing
fault characteristics. Moreover, for the traditional solution
which is implemented by SVD, the extracted components
are usually the linear combinations of all the original
variables.
2.2 
The proposed algorithm
For further improving the power of feature extraction for
fault diagnosis, a sparse relative discriminant fault
deviations (SRDFD) modeling algorithm is presented by
rebuilding the objective function in the work of Zhao et al.
In this algorithm, extracting fault directions and isolating
faulty variables can be implemented at one time by bringing
in a sparse constraint, the L1-penalization. Then the
function is solved by the minorization-maximization
approach and the soft threshold operator is used to gain
analytic solutions. At last the extracted directions and
loadings are used as reconstruction models for fault
diagnosis. The specific is described as below:
Two data sets are prepared, normal data set
()
nn
NP×X
where subscript n denotes normal data and one fault data set
()
,,fm fm
NP×X
where subscript f and m denotes fault data
and fault class index respectively. The normal data are
normalized and described by
n
X
for simplicity. Then the
fault data normalized by the data preprocessing information
are described by
,fm
X
for simplicity. Then the within-class
scatter matrix for normal data (
n
S
), within-class scatter
matrix for fault data (
f
S
), between-class scatter matrix
(
b
S
) and weighting factor (
ȕ
) are calculated by Eqs. (2-5).
(1) The proposed objective function
For extracting fault directions and isolating faulty variables
at one time, the proposed objective function is gained by
rebuilding the objective function in the FDFDA algorithm.
The proposed objective function is shown as below:
4510 2017 29th Chinese Control And Decision Conference (CCDC)

()
TT
1
T
ˆ
max +ȕ
ˆ
.=1
P
bf jj
j
n
J
w
st
λσ
=
§·
=−
¨¸
©¹
¦
wwSwwSw
wSw
(7)
where
w
is the extracted direction; P denotes the number
of variables;
•
demotes the one norm;
ˆ
n
S
is the diagonal
estimation of within-class scatter matrix for normal data, in
which
2
ˆ
j
σ
is the jth element and
ˆ
j
σ
is the within-class
standard deviation for variable j for normal data. Including
ˆ
j
σ
in the penalty has the effect that variables that vary more
within each class undergo greater penalization. When
λ
is
large, some elements of the solution
w
will be equal to
zero, so that the resulting direction is sparse. In particular,
the tuning parameter
λ
is calculated as follows:
(0) 1/2 1/2
ˆˆ
=(ȕ)
nb fn
λλ
−−
+SS SS
(8)
where
•
denotes the largest eigenvalue; the value of
penalization factor
(0)
λ
is determined by cross validation.
(2) The minorization-maximization approach
For the proposed objective function, there exists a problem
that it is non-concave, therefore, the minorization-
maximization approach is used to solve it, whose procedure
is briefly introduced here. The minorization-maximization
method is to find a function
()
(| )
m
g
ww
to minorize the
function
()
f
w
at the point
()m
w
if
() () ()
()
()( | )
() ( | )
mmm
m
f
f
=Θ
≥Θ ∀
www
www w
(9)
Then
w
can be calculated by iterating the following
function until convergence:
(1) ()
arg max{ ( | )}
mm+
=Θ
w
www
(10)
For the objective function in Eq.(6), define a matrix
ȕ
s
um b f
=+SS S
first for simplicity, which is obvious to
be positive semi-definite, and define
T
()
sum
f=
wwSw
.
Then it can be gained that
() () ()
() () ()
() () () () ()
() ()
()()( )()
2
()2
mmm
mm m
sum sum
mmmmm
sum sum
mm
sum
ff f
f
Τ
ΤΤ
ΤΤ
Τ
≥+−∇
=−
=−
=
ww ww w
wS w w S w
wwSwwSw
wSw
(11)
Therefore, the objective function in Eq. (6) can be
transferred to be the following form:
()
T()
1
ˆ
max 2
P
m
sum j j
j
J
w
λσ
=
§·
=−
¨¸
©¹
¦
wwSw
(12)
where
()m
w
is a fixed value.
So that the solution of
w
is calculated by the following
iteration procedure:
(a) let
(0)
w
be the first eigenvalue of
1
ˆ
nsum
−
SS
(b) for m=1,2,... until convergence: let
(1)m+
w
be the
solution to
T()
1
ˆ
=argmax{2 }
P
m
s
um j j
j
w
λσ
=
−
¦
w
wwSw
(13)
w
denotes the solution at convergence.
(3) Soft threshold operator
To solve the function in Eq. (13), we first consider the
following problem:
()
ˆˆ
min ( 2 )
P
m
nsum jj
j
d
λσ
ΤΤ
−+
¦
d
dSd dS w
(14)
If
=0d
, then
=0w
. Otherwise,
ˆ
=
n
Τ
wd dSd
.
As
ˆ
n
S
is the diagonal estimation of
n
S
, the solution to
function (14) is
(1)
2
ˆ
1(),
ˆ2
j
m
jsumj
j
dS
λσ
σ
−
½
°°
=®¾
°°
¯¿
Sw
(15)
where S is the soft threshold operator, which is defined as
(, ) sgn()( )Sxa x x a
+
=−
(16)
where
•
denotes the absolute value;
()
+
•
outputs the
larger value of the element in the bracket and zero.
For the calculated direction
w
, the extracted component is
calculated for fault data and normal data as follows:
,,
=
=
fm fm
nn
tXw
tXw
(17)
where
,fm
t
and
n
t
denote the components for fault data
and normal data respectively.
The specific procedure of the proposed algorithm is
described as below:
Step A. Fault direction extraction
For the rebuilt objective function in Eq. (7), one sparse fault
direction
w
can be gained by performing the
miniorization-maximization approach and the soft threshold
operator in Eq. (14) and Eq. (15) respectively.
Step B. Data deflation
To guarantee that the extracted components are orthogonal
with each other, data deflation is necessary for fault data and
normal data by removing the information concerning the
extracted component:
1
,,,,,
1
,,,,
()
()
fm fm fm fm fm
nnnnn
fm fm fm fm
nnnn
ΤΤ −Τ
ΤΤ−Τ
Τ
Τ
=
=
=−
=−
ptttX
ptttX
EXtp
EXtp
(18)
where
,fm
Τ
p
and
n
Τ
p
are the loading vectors for fault data
and normal data respectively;
,fm
E
and
n
E
are the
residuals of fault data and normal data without the
information of
,fm
t
and
n
t
respectively.
Step C. Data updating
2017 29th Chinese Control And Decision Conference (CCDC) 4511

Using
,fm
E
and
n
E
to replace fault data and normal data
for updating within-class scatter matrix for fault data (
f
S
)
and between-class scatter matrix (
b
S
) in Eq. (2) and Eq. (3).
Step D. Iterative implementation
Then repeat Steps A to C to extract next directions, loadings
and components until all the needed results are gained.
The outputs are a set of extracted directions (
()PR×W
), a
set of loadings (
,
()
fm
PR×P
) and a set of extracted
components (
,
()
fm f
NR×T
), which are composed of
(1)P×w
,
,
(1)
fm
P×p
and
,
(1)
fm f
N×t
respectively and
are used as reconstruction models to reconstruct fault
deviations of fault class m. R denotes the retained number of
extracted directions, which can be determined by the
diagnosis performance in practical application.
2.3
Fault diagnosis performance evaluation
Two types of fault diagnosis analyses, same-fault analysis
and cross-fault analysis, can be performed for evaluating the
performance of fault reconstruction. For same-fault
analysis, the reconstruction models corresponding to one
fault data are applied to the same fault class and the
in-control monitoring statistics indicate good reconstruction
performance. While for cross-fault analysis, the
reconstruction models corresponding to one fault data are
applied to other fault classes and out-of-control monitoring
statistics indicate correct reconstruction performance.
Correspondingly, two evaluation indexes are defined for
fault reconstruction
[22]
, missing reconstruction ratio
(
MRR
) and false reconstruction ratio (
F
RR
).
2.4
Online fault diagnosis based on reconstruction
strategy
First the variations of normal data are extracted by PCA and
corresponding control limits are established so as to
evaluate the fault samples corrected by fault deviations
concerning each fault class.
Perform PCA on the normal data
n
X
to gain monitoring
system:
,
,
,
,,
ˆ
()
ˆ
no n o
no n o o
no n o o
nnono
Τ
Τ
=
=
=−
=+
TXP
XXPP
EXIPP
XX E
(19)
where
,
()
no n
NL
×T
are the principal components and
,
()
no n
NP
×E
are the residuals; L denotes the number of
principal components retained;
,
ˆ
no
X
are the data
reconstructed by the principal loadings
()
o
PL
×P
. The
subscript o denotes that these results are used for original
monitoring system establishing
Then
2
T
and SPE statistics are calculated as follows:
21
,, ,,
,,
()()
no no o no n o
no no
T
SPE
Τ−
Τ
=− Σ −
=
tt tt
ee
(20)
where
,no
t
is the row vector in
,no
T
;
,no
t
denotes the mean
vector of
,no
T
;
o
Σ
is the variance-covariance matrix of
principal components from normal data;
,no
e
is the row
vector in
,no
E
.The control limits
2
T
Ctr
and
SPE
Ctr
are
calculated by an F-distribution and a weighted
Chi-distribution respectively for
2
T
and SPE.
Whenever a new fault observation,
(1)
new
xP
×
is available,
it is first corrected by fault deviations concerning fault class
m:
,new new new f m
∗Τ Τ Τ Τ
=−xxxWP
(21)
where
new
∗
x
denotes the corrected fault sample with the fault
deviations concerning fault class m removed.
Then the original monitoring system are performed on the
corrected fault sample
new
∗
x
to gain corrected monitoring
statistics:
21
,,
()
()()
new new o
new new o o
new new no o new no
new new new
T
SPE
∗Τ ∗Τ
∗Τ ∗Τ Τ
∗∗ Τ−∗
∗∗Τ∗
=
=−
=− Σ −
=
txP
exIPP
tt tt
ee
(22)
If the corrected monitoring statistics
2
new
T
∗
and
new
SPE
∗
are
in their control limits respectively, it means that fault
deviations in the new fault sample can be well corrected by
the models from fault class m, which indicates that the fault
sample belongs to the class m.
3
ILLUSTRATION RESULTS
In this section, the performance of the proposed algorithm is
illustrated by Tennessee Eastman benchmark process,
which contains 41 measured variables and 11 manipulated
variables. The specific description can be found in the work
of Downs and Vogel
[21]
. Four sets of fault data (#2, #4, #7,
#8) and one normal data are used here and each involve 480
samples for training data and 100 samples for testing data.
For these four fault classes, the sparse reconstruction
models are established by the proposed algorithm.
010 20 30 40 50 60
-0.02 5
-0.02
-0.01 5
-0.01
-0.00 5
0
0.005
0.01
0.015
0.02
0.025
values of coeffi cients
variable No.
Fig 1. The values of coefficients in the first fault direction for fault #2
4512 2017 29th Chinese Control And Decision Conference (CCDC)

020 40 60 80 100
0
20
40
60
model #2
reconstructed SPE
020 40 60 80 100
0
20
40
60
reconstructed SPE
(a)
020 40 60 80 100
0
20
40
60
model #4
reconstructed SPE
020 40 60 80 100
0
5
10
15
reconstructed SPE
(b)
020 40 60 80 100
0
20
40
60
model #7
reconstructed SPE
020 40 60 80 100
0
10
20
30
reconstructed SPE
(c)
020 40 60 80 100
0
20
40
60
model #8
reconstructed SPE
020 40 60 80 100
0
20
40
60
reconstructed SPE
(d)
Fig 2. Reconstruction results for fault #4 by models developed from (a)
fault #2, (b) fault #4, (c) fault #7 and (d) fault #8 (the dotted line represents
the control limit)
Taking fault #2 for example, the coefficient for each
variable of the first fault direction is displayed in Figure 1,
which shows that the coefficients for some variables are
zero, such as the fifth and sixth variables. It indicates that
some variables that do not influence the process of fault #2
are not contained in the models, which agrees with the actual
situation. Therefore, the proposed algorithm can extract
fault directions and isolate faulty variables at the same time.
The results of fault diagnosis are shown in Figure 2, which
are gained by performing the data from fault #4 on different
reconstruction models. It can be seen that the fault can be
correctly diagnosed, since only the models developed from
fault #4 can well bring the alarming monitoring statistics
back to the normal. While for reconstruction models
developed from other fault classes, at least one monitoring
statistic is still out-of control.
To evaluate the performance of the reconstruction models,
missing reconstruction ratio (MRR) in same-fault analysis
and false reconstruction ratio (FRR) in cross-fault analysis
are used in this case. The penalization factor
(0)
λ
and the
number of extracted directions in the models for each fault
case are determined by making MRR for training data just
smaller than 5%. The results for the proposed algorithm
concerning testing data are displayed in Table 1. For
same-fault analysis, MRR results for fault #2, #4 and #8 are
larger than 5%, which means that fault characteristics for
these three faults have changed more or less. For cross-fault
analysis, results of FRR for
2
T
are much larger but those
for SPE are small, which indicates that the diagnosis power
for
2
T
is less than that of SPE. What's more, fault #2 cannot
be correctly diagnosed, as results for FRR corresponding to
the reconstructed fault data from fault #2 corrected by
models developed from fault #8 are extremely large, which
may have the reason that characteristics of fault #2 have
changed to be close to those of fault #8.
Table 1
Fault reconstruction results for 2
T
and SPE evaluated
by MRR in same-fault analysis and FRR in cross-fault analysis
using RC algorithm and the proposed algorithm (the shaded
values indicate the same-class fault analysis results evaluated by
MRR% and the unshaded values indicate the cross-class fault
analysis results evaluated by FRR%)
Model #
Fault #
2 4 7 8
2
T
2 0 0 0 85
4 80 0 82 77
7 0 0 0 31
8 33 1 55 0
SPE 2 7.5 0 0 44
4 0 13.5 6.5 0
7 0 0 4 3.5
8 2.5 1 19.5 24
Then the comparative analysis is conducted between the
proposed algorithm and the FDFDA method. For Fault #8,
the results of comparative analysis are displayed in Figure 3.
By correcting the fault deviations along the directions
extracted by FDFDA method, the reconstructed fault data
still appear alarming signals, which means that fault
information cannot be comprehensively removed by
FDFDA method. While for the proposed algorithm, the
alarming monitoring statistics are both bring back to the
normal region, which shows the effectiveness.
2017 29th Chinese Control And Decision Conference (CCDC) 4513

020 40 60 80 100
0
20
40
60
model #8
reconstructed SPE
020 40 60 80 100
0
5
10
15
20
reconstructed SPE
(a)
020 40 60 80 100
0
200
400
600
model #8
reconstructed SPE
020 40 60 80 100
0
50
100
150
200
reconstructed SPE
(b)
Fig 3. Reconstruction results for fault #8 by models developed from the
same fault data using (a) the proposed algorithm and (b) FDFDA
algorithm (the dotted line represents the control limit)
4
CONCLUSION
In the present work, a sparse relative discriminant fault
deviations (SRDFD) modeling algorithm is proposed. The
proposed algorithm rebuilds the objective function of
FDFDA algorithm, which added relative variations of
variance between fault data and normal data to FDA, by
bringing in an L1 penalization. Therefore, the extraction of
comprehensive fault deviations and the isolation of specific
variables can be accomplished at the same time. For solving
the proposed algorithm, which is non-cave , minorization-
maximization approach is adopted and soft threshold
operator is performed for analytic solutions. Then the
reconstruction models are established based on the
extracted fault deviations. Its application to fault diagnosis
is illustrated and shows superiority to the FDFDA method.
The feasibility and performance of the proposed method
have been verified by pre-programmed faults of TE process.
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4514 2017 29th Chinese Control And Decision Conference (CCDC)