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ISA Transactions 96 (2020) 24–36
Contents lists available at ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
Research article
Plant-wide process monitoring by using weighted copula–correlation
based multiblock principal component analysis approach and
online-horizon Bayesian method
Ying Tian a,∗, Heng Yao a, Zeqiu Li b
aShanghai Key Lab of Modern Optical System, School of Optical–Electrical and Computer Engineering, University of Shanghai for Science and
Technology, Shanghai 200093, China
bSchool of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
highlights
•The copula-correlation, which describes correlation degree and correlation pattern simultaneously, is used to describe the relationship among
variables.
•The weighted strategy, through assigning different weights to different variables in each sub-block, prevents ‘‘noisy’’ variables involved and reserves
the important information.
•The PCA model is established in each block and the Bayesian inference strategy is adopted for the comprehensive monitoring result.
•An online-horizon based Bayesian fault diagnosis system is established to identify the fault type of the system.
article info
Article history:
Received 19 July 2017
Received in revised form 29 May 2019
Accepted 1 June 2019
Available online 9 July 2019
Keywords:
Plant-wide process monitoring
Copula–correlation analysis
Weighted strategy
Multiblock principal component analysis
(MBPCA)
Bayesian inference
Online-horizon Bayesian fault diagnosis
abstract
Multiblock methods have been proposed to capture the complex characteristics of plant-wide
monitoring due to the enlargement of process industries. These methods based on automatic sub-block
division and copula–correlation, which simultaneously describe the correlation degree and correlation
patterns, are designed for sub-block partition. However, the selection of variables for each sub-
block through copula–correlation analysis requires a pre-defined cutoff parameter which is difficult
to be determined without sufficient prior knowledge, and a ‘‘bad’’ parameter leads to a degraded
performance. Therefore, a weighted copula-correlation-based multiblock principal component analysis
(WCMBPCA) is proposed. First, the variables in each sub-block are obtained through the copula–
correlation analysis-based weighted strategy rather than the cutoff parameter, which highly avoids
information loss and prevents ‘‘noisy’’ information. Second, a PCA model is established in each sub-
block. Third, a Bayesian inference strategy is used to merge the monitoring results of all sub-blocks.
Finally, an online-horizon Bayesian fault diagnosis system is established to identify the fault type
of the system based on the statistics of each sub-block. The average detection rate and the average
diagnosis rate for numerical example are 77.85% and 98.95%, and that for TE example are 80.63% and
89.50%. Comparing with other candidate methods, the proposed method achieves excellent detection
and diagnostic performance.
©2019 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
With the rapid development of distributed control systems,
the collected massive process data have caused the emergence of
multivariate statistical process monitoring (MSPM) methods [1,2].
In MSPM methods, principal component analysis (PCA), indepen-
dent component analysis (ICA), self-organizing map, canonical
correlation analysis, support vector data description, receptor
∗Corresponding author.
E-mail address: tianying@usst.edu.cn (Y. Tian).
density algorithm, and sparse plus low rank models, have been
used over the past few decades to monitor different industrial
processes [3–9].
Considering the enlargement of process industries, the plant-
wide process monitoring has gained considerable attention in
recent years. Decentralized statistical methods have been pro-
posed and extended to capture the complex characteristics, such
as massive monitoring points, strong coupling, and time delay
among plant-wide process variables [10–17]. For example,Wang
https://doi.org/10.1016/j.isatra.2019.06.002
0019-0578/©2019 ISA. Published by Elsevier Ltd. All rights reserved.
Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36 25
proposed a joint-individual monitoring scheme that incorpo-
rates multiset canonical correlation analysis (MCCA) for large-
scale chemical processes with several interconnected operation
units [18]. Jiang proposed randomized algorithm (RA) integrated
with evolutionary optimization-based data-driven distributed lo-
cal fault detection scheme to achieve efficient monitoring of
multiunit chemical processes [19]. Tong developed decentralized
modified autoregressive models for fault detection in dynamic
processes [20]. Moreover, MacGregor developed a multiblock
projection method that established the monitoring charts for
each sub-block and the entire process [10]. Westerhuis compared
different multiblock PCA (MBPCA)/ multiblock partial least square
(MBPLS) methods [11]. Qin used the MBPCA and MBPLS for de-
centralized monitoring and diagnosis [12]. Choi and Lee utilized
the MBPLS method to monitor a complex chemical process and
to model a key process quality variable simultaneously [14].
Kohonen used multiblock methods to divide the mechanical data
from one process and the spectral data from another [16]. Block
division is an important step in these decentralized methods,
and the above mentioned methods usually divide the blocks by
referring to prior knowledge on processes, although such prior
knowledge is inconsistently available for plant-wide processes.
To solve this issue, data-based block division methods have
been introduced and several researchers have investigated these
methods. Grbovic used the decomposition and maximum entropy
decision fusion to form a sparse PCA-based decentralized fault
detection and diagnosis method [21]. Ge constructed sub-blocks
in different principal component directions [22]. Ferrari utilized
overlapping decompositions to monitor the nonlinear dynam-
ical large-scale distributed processes [23]. Jiang proposed mu-
tual information-based MBPCA and mutual information-spectral
clustering-based MB kernel PCA (MBKPCA) for nonlinear large-
scale process monitoring [24,25]. Furthermore, Zhang proposed
a decentralized process monitoring with MBKPLS [26] and MB-
KICA [27] for nonlinear process monitoring. Tong proposed dy-
namic decentralized PCA approach (DDPCA) and weighted DDPCA
approach for dynamic large-scale process monitoring [28,29] and
a modified MBPCA algorithm that considered the specificity in
each block and the relevance among different blocks [30]. Liu
proposed an multiblock concurrent PLS for large-scale continuous
annealing process monitoring [31,32].
These data-based block division methods usually divide the
block based on the relationship among variables, in which the
data with the same correlation degree may have different cor-
relation patterns. The copula function proposed by Sklar is a
multivariate distribution function used to build a bridge between
a multivariate cumulative distribution and the corresponding uni-
variate cumulative marginal distributions [33], which simulta-
neously describes the correlation degree and correlation pat-
terns. Ren proposed a vine copula-based dependence description
for multivariate multi-modes process monitoring [34]. Copula–
correlation analysis uses a copula function among the variables to
estimate the Kendall and Spearman rank correlation coefficients.
Thus, the copula–correlation analysis is introduced for significant
block division.
Although the block can be determined by copula–correlation
analysis, the selection of variables in each sub-block still re-
quires a pre-specified cutoff parameter that is determined by
experience and different cutoff parameters lead to apparently
different monitoring results. Therefore, the determination of vari-
ables in each sub-block based on copula–correlation analysis
remains a challenge. Several researchers have proven that allocat-
ing different weights to different variables in each sub-block can
improve the monitoring performance [35–37]. Thus, we allocate
different weights through the copula–correlation analysis on the
variables of each sub-block in this study, and a weighted copula-
correlation-based MBPCA (WCMBPCA) method is proposed. First,
the copula–correlation analysis among the variables is calculated.
Second, the weighted variables in each sub-block are determined
by copula–correlation matrix, which avoids information loss and
prevents ‘‘noisy’’ variables. Then, a PCA model is established in
each sub-block, and a Bayesian inference strategy is used to
merge the monitoring results of all sub-blocks. Since the deter-
mination of the cutoff parameter is not involved in the proposed
WCMBPCA method, the monitoring performance is better than
the copula-correlation-based MBPCA(CMBPCA) without weighted
strategy.
Moreover, the realization of fault diagnosis after fault detec-
tion is a key problem. In recent years, the Bayesian method,
which is an efficient tool in probabilistic inference and performs
fault diagnosis based on fault signature evidence has been de-
veloped [38–40]. Several studies have been conducted in this
framework. For instance, the missing data problem has been
addressed by Zhang [41] and Qi [42], and the mode dependency
and evidence dependency problems have been addressed by Qi
in [43] and [44]. Then, a Bayesian fault diagnosis system with
optimal principal components used as evidence sources has been
established [39], and a Bayesian diagnosis system applied in
distributed process monitoring has been proposed by Jiang [45].
This study proposes an improved Bayesian diagnosis method with
online horizon that can achieve good fault diagnosability by using
the evidence obtained from WCMBPCA.
The remainder of this paper is organized as follows. Section 2
briefly reviews the regular PCA-based process monitoring and
CMBPCA methods. Section 3gives the motivation of the weighted
method and presents the details of the proposed WCMBPCA and
the online-horizon Bayesian diagnosis methods. Section 4illus-
trates the fault detection performance of the WCMBPCA method
and fault diagnosability of the online-horizon Bayesian diagnosis
method by using the numerical example and Tennessee Eastman
(TE) chemical process. Finally, Section 5provides the conclusion.
2. Preliminaries
2.1. PCA
The PCA algorithm, which is a fundamental MSPM method,
decomposes the data matrix X= [x1x2··· xn]T∈Rn×m
consisting of nsamples and mmonitored variables into a low-
dimension space with most process information remained, which
can be expressed as [46]
X=
m
i=1
tipT
i=TPT=ˆ
Tˆ
PT+E(1)
where piis the loading vector for the ith principal direction, tiis
the corresponding uncorrelated score vector, ˆ
Tis the projection
of Xon the principal component subspace (PCS) that is spanned
by the first kloading vectors ˆ
P∈Rm×k, and E=˜
T˜
PTis the
residual matrix in the residual subspace (RS) that is spanned by
the remainder ˜
P∈Rm×(m−k). Parameter kis determined through
a cumulative percentage variance (CPV) rule.
After data transformation, for real-time monitoring, the T2and
SPE statistics are constructed for the online monitoring of PCS and
RS, respectively.
Given the current monitored sample x∈Rm×1, the T2statistic
is
T2=xTˆ
PΛ−1ˆ
PTx,(2)
where Λis a diagonal matrix consists of eigenvalues correspond-
ing to PCS.
The SPE statistic is
SPE =
x−ˆ
Pˆ
PTx
2
=
(I−ˆ
Pˆ
PT)x
2
(3)
26 Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36
Fig. 1. Geometric illustration of T2and SPE. (For interpretation of the references
to colour in this figure legend, the reader is referred to the web version of this
article.)
Proper control limits are usually defined under the assumption
that the data follow multivariate Gaussian distribution. How-
ever, Gaussian assumption is easily violated in practical applica-
tions. To address this issue, a kernel density estimation (KDE) is
introduced for the calculation of control limits.
A univariate KDE is defined as [47,48]
ˆ
f(y)=1
nh
n
i=1
Ky−yi
h(4)
where yis the sample under consideration, yiis the observation
value from the dataset, his the window width that has a crucial
influence on KDE performance and is selected empirically, nis
the number of observations, and Kis the kernel function. In
this study, the Gaussian kernel function is employed and the
univariate KDE becomes
ˆ
f(y)=1
n
1
h√2π
n
i=1
exp −(y−yi)2
2h2(5)
The steps for control limit determination are expressed as
follows. First, T2and SPE values for the normal operating data
are calculated. Second, KDE is performed to estimate the density
function of normal T2and SPE values. Finally, the value within
the 99% area of the density function is used as the control limits
for normal operating data.
According to Yue and Qin, the illustration of T2and SPE is
provided by geometric way [49]. As Fig. 1 shown, the values of T2
and SPE will fall into corresponding areas for normal data, where
the range of T2and SPE is calculated by Eqs. (4)–(5). For abnormal
sample, it is either beyond the range of T2(red point), or beyond
the area of the SPE (the blue point), or even both.
Next, the detailed calculation steps for P,Λand kin PCA is
introduced as follows:
(1) The data Xis zero mean and normalization treated.
(2) Calculate the covariance matrix cov (X)=XTX.
(3) Calculate the eigenvalues λand eigenvectors Pof covariance
matrix.
(4) Arrange the eigenvalues from large to small, and determine
the retained dimension kaccording to CPV rule.
(5) The eigenvectors are arranged into matrices Paccording to the
corresponding eigenvalues, and the first keigenvectors are used
to form matrix ˆ
P=P(:,1:k), and the first keigenvalues is taken
to form diagonal matrix Λ.
(6) T2and SPE statistics is computed according to Eqs. (2)–(3).
2.2. Copula–correlation analysis
Although the regular PCA method can capture
high-dimensional and correlated data, the complex characteris-
tics of plant-wide process and the local behavior are difficult
to interpret. Therefore, the multi-block strategy is introduced to
divide the plant-wide process into several sub-blocks for bet-
ter monitoring. The blocks usually are divided based on the
prior knowledge that are inconsistently available. Therefore, data-
based block division methods have been proposed. However,
these data-based methods only consider the correlation degree
and ignore the correlation patterns among the variables. Thus,
the copula–correlation analysis that simultaneously considers
the correlation degree and correlation pattern is used for block
division.
The copula function, which is the connection between the
marginal cumulative distribution and joint cumulative distribu-
tion of random variables, is defined as [34]
F(x1,x2,...,xd)=C[u1,u2,...,ud](6)
where F(x1,x2,...,xd) is the d-dimensional joint cumulative
distribution function (CDF), which can be decomposed into copula
function Cand dmarginal CDF u1,u2,...,ud,ui(i=1,2,...,d)
are the ith marginal CDF expressed as
ui=Fi(xi)=xi
−∞
fi(xi)dxi,ui∈ [0,1](7)
In Eq. (7),fi(xi) is the marginal probability distribution function
(PDF) of xi.
Given that Cis differentiable, the corresponding ddimensional
joint PDF f(x1,x2,...,xd) can be obtained as
f(x1,x2,...,xd)=c(u1,u2,...,ud)
d
i=1
fi(xi) (8)
where c(u1,u2,...,ud) is defined as
c(u1,u2,...,ud)=∂nC
∂u1∂u2···∂ud
(9)
2D copula is the most commonly used in the copula family,
which defines the joint CDF for 2D random variables x=(x1,x2)T.
2D copula can be divided into Elliptical and Archimedean cat-
egories based on different correlation patterns. The former is
mainly used to describe the variable correlation with symmetrical
tail, such as Gaussian and Student’s t-copulas. The latter is used
to picture the asymmetrical pattern, including Clayton, Gumbel,
and Frank copulas.
To obtain random variables, the copula function is defined
in three steps. (1) Marginal CDF determination. The appropriate
marginal CDF u1=f1(x1)and u2=f2(x2)for x1and x2is an
important prerequisite for copula–correlation analysis that is usu-
ally estimated by a non-parameter KDE method. (2) Appropriate
copula function selection. After the determination of the marginal
CDF, the appropriate copula function is selected based on 2D
histogram of (u1,u2). If the histogram has a symmetrical tail,
then the Gaussian copula or Student’s t-copula should be selected.
Otherwise, the Archimedean category should be selected. (3) Cop-
ula function parameter estimation. Generally, the marginal CDF
and copula function contain unknown parameters. In this study,
the canonical maximum likelihood method is adopted for the
parameter estimation. First, the empirical distribution function
u1=f1(x1)and u2=f2(x2)for random variables x1and x2
are obtained by KDE to replace their marginal CDF f1(x1;θ1) and
f2(x2;θ2). Subsequently, the parameters in the copula function are
estimated as
ˆκ=arg max
n
i=1
ln c(f1(x1i),f2(x2i);κ) (10)
Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36 27
where iis the different sampling time.
After copula function C(u1,u2) defined, the copula function-
based Kendall rank correlation coefficient τand Spearman rank
correlation coefficient ρscan be defined as [33]
τ=41
01
0
C(u1,u2)dC(u1,u2)−1 (11)
ρs=12 1
01
0
u1u2C(u1,u2)−3=12 1
01
0
C(u1,u2)du1du2−3
(12)
where the derivations of Eqs. (11) and (12) are given in the
Appendix and the details can be found in Ref. [33].
Obviously, different copula functions for the same variables
lead to different correlation coefficients. The best block division
can be achieved through the most accurate correlation coeffi-
cient determined by the most suitable copula function. In this
study, the absolute value of copula-based Kendall rank correlation
coefficient is utilized as the copula–correlation coefficient.
2.3. Block division
Suppose that data matrix X= [x1,x2,...,xn]T∈Rn×mis
collected from the plant-wide process. To obtain the copula–
correlation, the marginal CDF of each variable is constructed,
the 2D copula function between two arbitrary variables is se-
lected, and the copula–correlation is estimated. Thus, a copula–
correlation matrix CC ∈m×mis built as
CC =
cc11 cc12 ··· cc1m
cc21 cc22 ··· cc2m
.
.
..
.
.
ccm1ccm2··· ccmm
(13)
where the element ccij indicates the copula–correlation between
the ith and jth variables.
After the copula–correlation matrix is obtained, the jth vari-
ables in matrix Xthat satisfy the following criterion are selected
in the ith sub-block:
ccij > β ·mean(cci) (14)
where mean (cci)is the average value of the ith line of matrix CC
and βis the cutoff parameter that requires to be pre-specified
for block division, which plays an important role in the divi-
sion process. Few variables are selected in each sub-block with
a high value of βand vice versa. Therefore, inappropriate se-
lection of parameter βpotentially deteriorates the monitoring
performance.
2.4. Decentralized monitoring
After the copula-correlation-based block division is completed,
matrix Xis divided into a total of overlapping sub-blocks, which
can be expressed as
X= [X1X2··· Xm](15)
In each sub-block i,Xi∈Rn×mi(i=1,2,...,m) has nsamples
and mivariables. Then, the PCA model is constructed for each sub-
block, the number of PC is obtained through the CPV rule, and the
related control limits of T2and SPE statistics for each sub-block
are also calculated with KDE estimation.
For a new online data sample, matrix Xis divided into msub-
blocks, and msets of T2and SPE statistics are obtained for all
sub-blocks. To render the monitoring indicators easy to read and
analyze, the Bayesian inference strategy is used to combine all
the local results into two probability monitoring indices, namely,
BICT2and BICSPE [50,51].
First, the definition of BICT2is provided. Considering the fault
posterior probability PT2(F/xi) of current sample xiassociated
with T2in one sub-block as [28,30]
PT2(F/xi)=PT2(xi/F)PT2(F)
PT2(xi)(16)
PT2(xi)=PT2(xi/N)PT2(N)+PT2(xi/F)PT2(F) (17)
where PT2(N) and PT2(F) are the prior probabilities of the process
as normal and faulty, which are assigned with αand 1 −αwhen
the confidence limit is α. The conditional probabilities PT2(xi/N)
and PT2(xi/F) are defined as
PT2(xi/N)=e−T2
i/T2
lim (18)
PT2(xi/F)=e−T2
lim/T2
i(19)
where T2
iis the T2statistic of the current sample, and T2
lim is the
corresponding control limit of T2statistic.
Considering that mdifferent sub-blocks exist, the conditional
probabilities PT2(xi/N) and PT2(xi/F) and fault posterior probabil-
ity PT2(F/xi) are calculated in each sub-block, and the final prob-
abilistic index BICT2can be determined by using the weighted
formula,
BICT2=
m
j=1PT2(xi/F)PT2(F/xi)
m
j=1PT2(xi/F)(20)
Similarly, the final probabilistic index BICSPE related to SPE
statistic is combined and calculated in the same manner as BICT2
BICSPE =
m
j=1PSPE (xi/F)PSPE (F/xi)
m
j=1PSPE (xi/F)(21)
The control limits for BICT2and BICSPE are determined by KDE
estimation, as described in Section 2.1, and the value within the
99% area of the density function is used as the control limits of
BICT2and BICSPE . A fault is detected if either of the two statistics
is beyond the control limit.
3. Process monitoring based on WCMBPCA
3.1. Motivation analysis
Consider the following numerical process, shown in Eq. (22). In
this process, s1and s2follow Gaussian distribution with mean of
60 and 90 and standard deviations of 30 and 15, respectively. e1to
e8also follow Gaussian distribution with zero mean and standard
deviation of 5. x1to x8are the monitored variables.
x1
x2
x3
x4
x5
x6
x7
x8
=AS =
0.5768
0.7382
0.8291
0.6519
0.3766
0.0566
0.4009
0.2070
0.3972
1.7800
0.4600
−0.6100
0.8045
0.7400
0.4500
0.6200
·s1
s2+
e1
e2
e3
e4
e5
e6
e7
e8
(22)
First, 1000 samples are collected under normal condition for
offline modeling. Then, the two following faulty cases are sim-
ulated to test the effects of different cutoff parameter βfor the
monitoring performance: (i) a step change by 90 of the second
variable is introduced from the 351th sample to 1000th sample
28 Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36
Table 1
Detection rates in the numerical process.
Index BICT2BICSPE
β0.8 1 1.1 0.8 1 1.1
Fault1 80.77% 98.15% 6.46% 81.69% 33.38% 99.85%
Fault2 1.65% 1.69% 7.85% 95.54% 99.54% 100.00%
and (ii) a step change from 0.6200 to 0.0100 of the last element in
matrix Ais introduced from the 351th sample to 1000th sample.
The PCA model in sub-block is determined by the CPV >85%
rule, and the detection rates for the two faulty cases are listed in
Table 1, where three different cutoff parameters β={0.8,1.0,
1.1}are used for comparison. As shown in Table 1, the CMBPCA
model with β=1.1 achieves the highest detection rate for faults
1 and 2, whereas the model with β=0.8 obtains the lowest
detection rate. The comparison reveals that cutoff parameter β
has a strong influence on the monitoring performance. Although
the copula–correlation analysis is valuable for block division, the
most suitable cutoff parameter βis still difficult to be determined
and a fixed value of βis not the best choice for monitoring all
conditions.
3.2. Weighted feature selection
As concluded in Section 3.1, the best cutoff parameter is dif-
ficult to be pre-defined. To significantly extract the process in-
formation, the weighted strategy is introduced, which assigns
different weights to different variables to avoid the determination
of cutoff parameter.
Fig. 2 reveals the modeling procedures of the CMBPCA and
WCMBPCA methods. The original CMBPCA method selects the
variables for each sub-block through cutoff parameter βand
constructs the sub-block by allocating the selected variables with
weight 1 and assigning the unselected variables with weight 0. A
high value of βreduces the variables in each sub-block and causes
important information loss. However, a small value of βinvolves
many ‘‘noisy’’ variables in each sub-block. The WCMBPCA method
uses the copula–correlation analysis to weight all the variables in
each sub-block. Considering data matrix X= [x1,x2,...,xn]T∈
Rn×m, the related copula–correlation matrix is shown as Eq. (13).
For the ith sub-block, the weight vector is expressed as
wi= [cci1cci2··· ccim](23)
Then, weighted data Xifor the ith sub-block is expressed as
Xi=
cci1x11 cci2x12 ··· ccimx1m
cci1x21 cci2x22 ··· ccimx2m
.
.
..
.
.
cci1xn1cci2xn2··· ccimxnm
=X·diag(wi) (24)
After the implementation of the proposed weighted strategy,
the variables are weighted rather than selected in each sub-block,
and a total of msub-blocks are constructed. The information
loss is highly avoided, ‘‘noisy’’ information is suppressed due to
their relatively small weights, and the important information are
reserved due to their large weights.
Subsequently, the benefits of weighting in WCMBPCA for de-
tection performance improvement are discussed. Since the overall
monitoring performance of the proposed method depends on the
monitoring performance of each sub-block and the data in each
sub-block are analyzed by PCA after weighted through copula–
correlation analysis, the detection performance of each sub-block
is equivalent to the weighted PCA. So here, the relation be-
tween the fault detection performance and the weighted PCA is
illustrated from the view of fault signal-to-noise(SNR).
Fig. 2. Modeling procedures of the CMBPCA and WCMBPCA methods.
Supposing the observed output with a fault fgi[49]
x=x∗+fgi(25)
where x∗is observation without fault, fis scalar value that shows
the magnitude of the fault and gi=[0···0,1,0···0]Tis a fault
direction vector (the ith component is 1, and the others are 0).
So for the fault data, the T2can be represented by
T2=
Λ−(1/2)ˆ
PTˆ
Pˆ
PTx
2
=
Λ−(1/2)ˆ
PTˆ
Pˆ
PTx∗+ˆ
Pˆ
PTfgi
2
=
Λ−(1/2)ˆ
PTˆ
x∗+fˆ
gi
2
(26)
where ˆ
x∗=ˆ
Pˆ
PTx∗is the projection of x∗on the PCS, ˆ
gi=ˆ
Pˆ
PTgi
is the projection of gion the PCS.
The SPE can be represented by
SPE =
x∗+fgi−ˆ
Pˆ
PTx∗+fgi
2
=
˜
x∗+f˜
gi
2(27)
where ˜
x∗=I−ˆ
Pˆ
PTx∗is the projection of x∗on the RS, ˜
gi=
I−ˆ
Pˆ
PTgiis the projection of gion the RS.
Based on the fact described above, the fault SNR is defined as
follows:
SNRT2=T2
T2
α=
(x∗+fgi)Tˆ
PΛ−1ˆ
PT(x∗+fgi)
2
T2
α
=
Λ−(1/2)ˆ
PTˆ
x∗+fˆ
gi
2
T2
α
(28)
SNRSPE =SPE
SPEα
=
(x∗+fgi)T−(x∗+fgi)Tˆ
Pˆ
PT
2
SPEα
=
˜
x∗+f˜
gi
2
SPEα
(29)
Since the higher fault SNR means the easier fault detection,
so we discuss whether the weighting way can improve the fault
SNR, that is to say, we will discuss whether the weighting way can
improve the fault detection ability. Supposing the original data is
weighted by matrix D, the ratio between the original fault SNR
Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36 29
and weighted fault SNR is shown as Eqs. (30) (31):
KT2=T2new/T2
αnew
T2old/T2
αold
=
((x∗+fgi)D)Tˆ
PnewΛnew−1ˆ
PnewT((x∗+fgi)D)
2
/T2
αnew
(x∗+fgi)Tˆ
PoldΛold −1ˆ
PoldT(x∗+fgi)
2
/T2
αold
(30)
KSPE =SPEnew/SPEαnew
SPEold/SPEαnew
=
((x∗+fgi)D)T−((x∗+fgi)D)Tˆ
Pnewˆ
PnewT
2
/SPEαnew
(x∗+fgi)T−(x∗+fgi)Tˆ
Pold ˆ
PoldT
2
/SPEαold
(31)
Here, KDE is introduced for the calculation of control limits, P
and Λis calculated using Eqs. (32) (33), and CPV rule is used to
determine the number of principal components.
cov (X)=XTX=PoldΛold PT
old (32)
cov (XDi)=DT
iXTXDi=PnewΛnew PT
new (33)
Since the data and the ˆ
P,Λ,SPEα,T2
αchange together, so it
is difficult to judge whether the weighted fault SNR is larger by
mathematical deduction. Therefore, a numerical example is used
to illustrate the effect of weighting to fault SNR. Suppose that
X=
2.5 2.4 2.1 2.0
0.5 0.7 0.3 0.6
2.2 2.9 2.8 2.3
1.9 2.2 1.7 2.0
3.1 3.0 3.5 2.9
2.3 2.7 2.4 2.6
2 1.6 1.8 1.7
1 1.1 0.9 0.8
1.5 1.6 1.7 1.4
1.1 1.0 0.9 1.2
D=
0.8 0 0 0
0 0.6 0 0
0 0 0.5 0
0 0 0 0.7
(34)
When a fault gi=[1,0,0,0],f=10 occurs in the 2nd
sampling, the difference between the original fault SNR and the
weighted fault SNR is calculated. For the original data,
T2old (2)=12.1564,T2
αold =3.3982
SPEold (2)=296.2162,SPEαold =0.3652
For the weighted fault data,
T2new(2)=81.7500,T2
αnew=3.3765
SPEnew(2)=255.8138,SPEαnew=0.2254
Using Eqs. (30) (31), the ratio of fault SNR between the original
data and the weighted data is as follows:
KSPE =SPEnew/SPEαnew
SPEold/SPEαnew=1.40
KT2=T2new/T2
αnew
T2old/T2
αold =5.93
That is to say, after weighted, both the T2fault SNR and SPE
fault SNR increased, so faults can be more easily detected.
However, we should also note that not for all weights and all
faults, the value of fault SNR increased. This situation can be seen
from the TE example (Section 4.2), for some faults, the detection
rate is improved, but for other faults, the detection rate has not
improved.
3.3. Online-horizon Bayesian fault diagnosis
A diagnosis system identifies the underlying fault status once
a fault is detected, which is important for eliminating the fault.
However, the real process usually is a multivariable system, so the
determination of fault type is difficult. Recently, a Bayesian fault
diagnosis method has been developed for distributed monitor-
ing [45]. In this study, an online-horizon based Bayesian diagnosis
system based on Ref. [45] is proposed, which is suitable for the
evidence obtained through the WCMBPCA method.
Before the introduction of the online-horizon based Bayesian
method, the following concepts should be clarified:
(1) Evidence E: Evidence E, which is the input of the Bayesian
fault diagnosis system, reveals the monitor readings or outputs
of the sources. An evidence of a process with Bmonitors can
be denoted as E={π1, π2...,πB}, where πiis the ith source,
which has qidiscrete values. The collection of all evidence is
ε={e1,e2,...,eK}, where K=B
i=1qi. For instance, the values
of πiindicate the faulty status with regard to the ith sub-block,
and the value is discretized as ‘‘0’’ and ‘‘1’’. The value ‘‘1’’ indicates
that an abnormal event is detected in the related sub-block, and
the value ‘‘0’’ indicates that no abnormality is detected. For the
proposed WCMBPCA monitoring method, the discrete evidence
of current sample can be generated as
πi=0if T 2
i≤T2
i,lim and SPEi≤SPE2
i,lim
πi=1if T 2
i>T2
i,lim or SPEi>SPE2
i,lim
(35)
where i=1,2,...,mdenotes the block number. K=2m
possible evidence values are observed in total.
(2) Process status F: This status reveals the actual internal
state of the system, which can be a normal condition or any
abnormal situation. Assuming that the number of all possible
internal states is G, then the collection on all states of the system
is marked as F={F1,F2. . . , FG}. The use of Bayesian for fault
diagnosis determines the internal state of the system.
(3) Historical dataset D: The historical training data Dcontains
the evidence from all process statuses, which can be denoted as
D=d1,d2,...,dN, where Nis the number of historical training
samples. Sample dtcontains etand process status Ftat time t:
dt=et,Ft. Specifically, historical data are tagged and assumed
to be independent.
The Bayesian fault diagnosis strategy determines the cause of
the failure according to Bayes’ rule. Assuming current evidence ec
and historical evidence data D, then the posterior probability of
each possible fault status Fjcan be calculated as
pFj|ec,D=p(ec|Fj,D)p(Fj|D)
Fjpec|Fj,Dp(Fj|D)(36)
where p(ec|Fj,D) is the likelihood probability, p(Fj|D) is the prior
probability of fault Fj, and pFj|ec,Dis the posteriori probability
of Fjunder current evidence ecand historical database D. The
process status with a large posterior probability is considered to
be the probable cause of failure. Usually, the prior probability
is known. Thus, the key problem of Bayesian fault diagnosis
is the likelihood probability estimation. A marginalization-based
solution involving prior knowledge is provided as [40,42]
pec|Fj,D=nec|DFj+αec|DFj
K
l=1nel|DFj+K
l=1αel|DFj(37)
where nec|DFjis the number of historical samples that satisfy
evidence ecand fault status Fj, and αec|DFjis the number of
prior samples on evidence ecunder fault status Fj. Specifically,
the likelihood probability is determined by prior knowledge and
historical data.
30 Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36
Fig. 3. Monitoring procedures of the WCMBPCA method.
In this study, since the state of the process is continuously
changing, so the normal and fault states persist in multiple sam-
plings. Therefore, a single sampling on the control limit is as-
sumed to be an outlier caused by measurement error. To elim-
inate the influence of outliers and enhance the diagnosis ef-
fect, the online-horizon-based Bayesian fault diagnosis method is
proposed, and the detail is shown as follows:
Online horizon His defined as H=h0,h1,h2,...,hr−1,
where hlrefers to the forward sampling from the current sam-
pling, and ris the length of the horizon. Data hlconsists of an
evidence vector eland posteriori probability of Fjunder evidence
el, namely hl=el,pF1|el,D,...,pFG|el,D. Then, the final
posteriori probability of Fjfor current evidence ecis determined
by the historical and online horizon data, which can be calculated
as
pFj|ec,D,H=
c
k=c−r+1
pFj|ec,D(38)
where Fjwith the maximum probability is identified as the cause
of failure.
3.4. WCMBPCA-based process monitoring
The monitoring procedures are shown in Fig. 3, and the en-
tire process can be divided into offline modeling and online
monitoring phases. The details are also provided as follows:
Offline modeling phase:
(1) Normal dataset Xis obtained and is normalized to zero
mean and unit variance.
(2) The copula–correlation analysis between two arbitrary
variables is executed to obtain the copula–correlation matrix.
(3) Each sub-block dataset is constructed based on the copula–
correlation matrix and Eqs. (23) (24), so msub-blocks are ob-
tained.
(4) For the msub-block established through step (1)–(3), the
mlocal PCA models are built. Then the T2and SPE statistics as
well as the corresponding control limit of each local PCA model
are obtained.
(5) Bayesian inference decision fusion strategy is used to ob-
tain the two final monitoring indices BICT2and BICSPE for normal
dataset, and the control limits of BICT2and BICSPE are calculated
by KDE estimation.
(6) The normalized historical data are obtained from the nor-
mal and different fault conditions, the evidence for all samples
are constructed with Eq. (35), and the likelihood probability is
estimated by Eq. (37).
Online monitoring phase:
(1) Monitored sample xis normalized by using the mean and
variance of normal dataset X.
(2) Sample xis weighed into msub-blocks based on the
copula–correlation matrix which is obtained in the offline mod-
eling phase.
(3) For the msub-block of the online data xwhich is obtained
through step (1)–(2), the real-time T2and SPE statistics are cal-
culated by the mlocal PCA models which are established in the
offline modeling phase. In other words, and msets of T2and SPE
statistics are obtained.
(4) The Bayesian inference decision fusion strategy is imple-
mented to obtain the two final monitoring indices BICT2and
BICSPE .
(5) Determine whether BICT2and BICSPE is over the related
control limit, if yes, then the monitored sample is in faulty state.
(6) The evidence for current online sample is constructed
based on to Eq. (35), the posteriori probability of Fjis calculated
by Eqs. (36) (38), and Fjwith the maximum posteriori probability
selected as the cause of failure.
4. Case study
In this section, the proposed WCMBPCA-based process moni-
toring approach, PCA, MBPCA, and CMBPCA methods are applied
to monitor the numerical example and TE process.
4.1. Numerical example
For the numerical process given in Section 3.1, the compar-
isons on the average detection rates of T2and SPE achieved by
Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36 31
Table 2
Correct diagnostic rate comparison for numerical example.
Bayesian fault
diagnosis
Online-horizon
Bayesian fault diagnosis
Normal condition 91.14% 100%
Fault 1 97.43% 98.29%
Fault 2 93.71% 98.57%
Ave 94.09% 98.95%
Fig. 4. Average detection rates achieved for fault 2.
different methods in monitoring the previously defined fault 2
are displayed in Fig. 4. For all methods, the principal compo-
nents are determined by the CPV >85% rule, and the control
limits are calculated with α=99%. The sub-blocks are con-
structed at each PC direction for MBPCA. As shown in Fig. 4,
the MBPCA approach presents superior performance over the
regular PCA method, whereas CMBPCA method achieves better
detection rates with different βvalues that lead to different
detection rates. For example, the detection rates are 48.60% for
β=0.8 and 53.93% for β=1.1. Although the CMBPCA approach
with β=1.1 achieves the best monitoring result among the
three different cutoff parameters, determining the best parameter
without enough prior knowledge is still difficult. However, the
WCMBPCA method does not require parameter βand has the best
monitoring performance among all methods.
The online-horizon based Bayesian fault diagnosis results are
presented in Table 2 and Fig. 5. As shown in Fig. 5, several
misclassified points are observed at the beginning of fault 1 and
the beginning of fault 2. This condition is because posteriori prob-
ability of Fjfor current evidence is calculated through the moving
Table 4
False alarm rates in the TE benchmark process (%).
Method Index False alarm rates
PCA T20.42
SPE 1.04
MBPCA BICT20.21
BICSPE 4.38
CMBPCA β=0.8BICT20.37
BICSPE 0.5
β=1.0BICT20.31
BICSPE 2.08
β=1.1BICT20.73
BICSPE 1.15
WCMBPCA BICT21.04
BICSPE 1.04
Fig. 5. Bayesian diagnosis results for numerical example.
online horizon. Thus, the initial samplings after the change on
internal process status are mistakenly identified as the former
process state. The fault statuses are generally correctly detected
over time. Table 2 also reveals that the online horizon-based
Bayesian fault diagnosis has better diagnostic performance com-
pared with the original Bayesian fault diagnosis for the evidence
produced from WCMBPCA.
Table 3
Process faults.
Variable Description Type
IDV (1) A/C feed ratio, B composition constant (stream 4) Step
IDV (2) B composition, A/C ratio constant (stream 4) Step
IDV (3) D feed temperature (stream 2) Step
IDV (4) Reactor cooling water inlet temperatures Step
IDV (5) Condenser cooling water inlet temperature Step
IDV (6) A feed loss (stream 1) Step
IDV (7) C header pressure loss-reduced availability (stream 4) Step
IDV (8) A, B, C feed composition (stream 4) Random variation
IDV (9) D feed temperature (stream 2) Random variation
IDV (10) C feed temperature (stream 4) Random variation
IDV (11) Reactor cooling water inlet temperature Random variation
IDV (12) Condenser cooling water inlet temperature Random variation
IDV (13) Reaction kinetics Slow drift
IDV (14) Reactor cooling water valve Sticking
IDV (15) Condenser cooling water valve Sticking
IDV (16) Unknown Unknown
IDV (17) Unknown Unknown
IDV (18) Unknown Unknown
IDV (19) Unknown Unknown
IDV (20) Unknown Unknown
IDV (21) The valve for stream 4 was fixed at the steady-state position Constant position
32 Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36
Table 5
Detection rates in the TE benchmark process (%).
No. PCA MBPCA CMBPCA WCMBPCA
β=0.8β=1.0β=1.1
T2SPE BICT2BICSPE BICT2BICSPE BICT2BICSPE BICT2BICSPE BICT2BICSPE
1 99.25 99.75 99.25 99.88 99.5 99.75 99.62 100 99.62 99.88 99.75 99.75
2 98.5 98.62 98.75 98.38 98.62 98.5 98.5 98.38 98.75 98.5 98.5 98.62
4 28.13 100 35.75 100 52 100 65.38 100 64.25 100 100 97.88
5 23.75 33.25 23.38 100 24 100 24.75 100 25.75 100 28 100
6 98.75 100 99.25 100 99.62 100 100 100 99.38 100 99.5 100
7 100 100 100 100 100 100 100 78.13 100 100 95.5 100
8 97.25 96.88 97.25 97.38 97.5 97.88 97.62 97.75 97.62 98 97.75 97.75
10 25.62 44.5 28.38 57.5 37.5 46.75 39 55.37 39.37 52.5 49.25 75.75
11 45.12 69.13 52.62 81.87 54.13 67.37 58 70.87 57.63 71.38 72.88 57.13
12 98.38 95.13 98.75 99.75 98.75 99.62 98.88 99.62 98.88 99.75 98.88 99.38
13 94.5 95.13 94.63 95.63 95 95.37 95 95.5 95 95.75 95 95
14 99 99.88 100 99.88 100 99.88 100 99.88 100 100 100 99.88
16 11.13 43.25 10.88 84.88 18 44.62 19.25 52 20.75 48.5 29.63 78
17 76.12 95.63 86.12 97.25 87.88 97 94.37 97.12 88.88 97.38 94.5 91.87
18 89 90.13 89.25 91.25 89.38 90.38 89.88 90.63 89.98 90.63 90.25 89.62
19 6.25 20.37 10.5 29 11 14.87 13.63 30.13 8.13 18.25 4 52.38
20 28 55.75 32.37 68.5 42.88 69.63 45.87 73.88 46.25 72.5 55.25 65.13
21 39.5 49.75 38.87 48.75 40.88 50 40.5 54.63 43.13 52.12 44 52
Ave 70.71 76.28 75.51 77.06 76.90 80.63
4.2. TE process
The TE process contains a reactor, a condenser, a compressor,
a separator, and a stripper, and can be used as a benchmark
for evaluating process monitoring methods. A set of 52 pro-
cess variables in TE process is used for monitoring, including 22
continuous process measurements (XMEAS (1–22)), 19 composi-
tion measurements (XMEAS (23–41)), 11 manipulated variables
(XMV(1–11)), and the XMEAS and XMV notations are from Downs
and Vogel [52]. A total of 21 pre-programmed faults (IDV (1–21),
shown in Table 3) are introduced for testing. The simulated data
sampled every 3 min can be downloaded from http://web.mit.
edu/braatzgroup.
The normal dataset consists of 960 samples is used for model
development, and 21 kinds of faulty datasets with fault intro-
duced from 161th sample are used for testing purpose. For all
methods, the principal components are retained by the CPV >90%
rule, and the confidence limit is set to 0.99. For the MBPCA
method, the sub-blocks are constructed at different principal
component directions. For the CMBPCA method, three different
cutoff parameters β= {0.8,1.0,1.1}is adopted, 52 local blocks
are defined with different variables in each sub-block for each
different cutoff parameters. For the WCMBPCA method, 52 sub-
blocks are defined with 52 different weighted variables in each
sub-block.
The false alarm rates comparison study for normal TE process
among the four different methods are listed in Table 4. False
alarm rates should be compared because several methods have
obtained good detection rates with high false alarm rates. All the
above-mentioned methods have reasonable false alarm rates.
The detection rate comparisons for all 18 faults (excluding
faults 3, 9, and 15) are listed in Table 5. As shown in the ta-
ble, the detection rates of the four methods for faults 1, 2, 6,
8, 12, 13, and 14 are close to 100% because these faults have
significant magnitude and are easy to be detected. In the case
of fault 7, all methods, except the BICSPE of CMBPCA with β=
1.0 achieve excellent monitoring performance. For fault 5, all
decentralized methods achieve good detection rates because they
can exactly extract the local behavior. For fault 11, all methods
provide similar and medium detection rates. For faults 16, 17,
and 20, the WCMBPCA method provides the best monitoring
performance, and other one or two methods also achieve the best
detection rates. For example, the CMBPCA with β=1.0 and
WCMBPCA have similar effect for fault 17 because β=1.0 is
a ‘‘good’’ parameter for this fault. However, as previously men-
tioned, determining which parameter is the best one is difficult
without sufficient prior knowledge. For faults 4, 10, and 19, the
proposed CMBPCA method can detect the fault with remarkable
high index of BICT2or BICSPE . Considering the average detection
rates, shown in the last row of Table 5, the regular PCA-based
monitoring method has the lowest average detection rate with
70.71%, whereas the decentralized MBPCA provides 76.26% detec-
tion rate. For the CMBPCA, the monitoring performance depends
on different conditions of β. The best effect is obtained by the
proposed WCMBPCA.
Fault 4 is caused by the step change of reactor cooling water
inlet temperature. The detection performance is shown in Fig. 6.
The SPE index of all methods can detect the fault well. However,
only the proposed WCMBPCA achieves good monitoring effect for
T2index.
As shown in Tables 4 and 5and Fig. 6, the proposed WCMBPCA
achieves the best monitoring results. This finding is because the
parameter selection is unrequired and the weighted strategy is
adopted for WCMBPCA. Important information loss is avoided,
‘‘noisy’’ variables are prevented, and necessary knowledge is re-
served through the weighted strategy. Therefore, the WCMBPCA
approach is a reliable and efficient tool in monitoring plant-wide
processes.
For the online-horizon Bayesian fault diagnosis of TE process,
the system is divided into 52 sub-blocks because it has 52 moni-
toring variables based on the proposed WCMBPCA method. If the
statistics of all sub-blocks are selected as evidence source, then
the number of possible evidence is 252 =4.5×1015, which
is beyond the calculation capability of the computer. Therefore,
this study selects the statistics of the first 22 sub-blocks as the
evidence source. The online-horizon based Bayesian fault diag-
nosis results are provided in Table 6 and Fig. 7. The error rate
analysis is also provided in Fig. 8. From the figures and table,
we can obtained that the average error rate for Bayesian fault
diagnosis method is more than 30%, while that for the online-
horizon Bayesian fault diagnosis method is only about 10%. Nearly
for all faults except fault 14, the online-horizon Bayesian fault
diagnosis is better. All in all, the improved diagnosis method has
better performance compared with the original Bayesian fault
diagnosis because it is more suitable to the evidence produced
from the WCMBPCA.
Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36 33
Fig. 6. Monitoring results of fault 4 with (a) PCA; (b) MBPCA; (c) CMBPCA with β=0.8; (d) CMBPCA with β=1.0; (e) CMBPCA with β=1.1; and (f) WCMBPCA.
5. Conclusion
To capture the complex characteristics in plant-wide moni-
toring and achieve fault diagnosis, a novel monitoring and fault
identification model based on WCMBPCA approach and online-
horizon Bayesian method are proposed. In this method, the
weighted strategy based on copula–correlation analysis is intro-
duced to handle block division, which avoids the cutoff param-
eter selection by assigning different weights to different vari-
ables in each sub-block. After the construction of sub-blocks,
the PCA model is established in each block, and the Bayesian
inference strategy is adopted for comprehensive monitoring re-
sult. Finally, the online-horizon-based Bayesian fault diagnosis
system is established to identify the fault type of the system
based on the statistics of each sub-block. Compared with regular-
PCA, MBPCA and CMBPCA, the numerical example and TE pro-
cess evidently demonstrate the effectiveness of the WCMBPCA
method. Compared with the original Bayesian diagnosis method,
the improved Bayesian method can better handle the collected
evidence through the WCMBPCA method and realize superior
fault diagnosability.
This study provides an effective solution for plant-wide pro-
cess monitoring, which is not restricted to PCA. Moreover, this
study may be extended to ICA, SVDD, or other statistic methods
to handle the nonlinear, non-Gaussian, and dynamical plant-wide
process monitoring.
34 Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36
Fig. 7. Online-horizon Bayesian diagnosis results for TE process.
Fig. 8. Error rate analysis.
Acknowledgment
This work was sponsored by the Shanghai Sailing Program,
China (No. 17YF1428300, 17YF1413100) and the Funding Pro-
grams for Youth Teachers of Shanghai Colleges and Universities,
China (ZZslg16009).
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Appendix
(1) The derivation of Kendall rank correlation coefficient τ
is given.
Let (X1,Y1)and (X2,Y2)be the independent vectors of con-
tinuous random variables. Then, the Kendall’s τis defined as the
probability of concordance minus the probability of discordance:
τ=P[(X1−X2) (Y1−Y2)>0]
−P[(X1−X2) (Y1−Y2)<0](A.1)
Considering that the random variables are continuous, then
P[(X1−X2) (Y1−Y2)<0]
=1−P[(X1−X2) (Y1−Y2)>0](A.2)
and
τ=2P[(X1−X2) (Y1−Y2)>0]−1 (A.3)
P[(X1−X2) (Y1−Y2)>0]
=P[X1>X2,Y1>Y2]+P[X1<X2,Y1<Y2](A.4)
Assuming that the joint distribution functions for (X1,Y1)and
(X2,Y2)are H1and H2, respectively, the margins of X1and X2
are the same and defined as F, and the margins of Y1and Y2are
the same and defined as G. Thus, H1(x,y)=C1(F(x),G(y)) and
H2(x,y)=C2(F(x),G(y)), where C1and C2denote the copulas
of (X1,Y1)and (X2,Y2).
Then, the probabilities P[X1>X2,Y1>Y2]and P[X1<X2,
Y1<Y2]can be evaluated by integrating over the distribution of
Y. Tian, H. Yao and Z. Li / ISA Transactions 96 (2020) 24–36 35
Table 6
Correct diagnostic rate comparison for TE process.
Bayesian fault
diagnosis
Online-horizon
Bayesian fault diagnosis
Normal condition 78.50% 98.50%
Fault 1 48.00% 68.00%
Fault 2 67.50% 79.50%
Fault 4 88.50% 98.50%
Fault 5 47.50% 80.50%
Fault 6 77.00% 83.00%
Fault 7 61.50% 76.50%
Fault 8 52.00% 70.50%
Fault 10 81.50% 97.50%
Fault 11 83.00% 97.50%
Fault 12 52.50% 96.00%
Fault 13 48.00% 86.50%
Fault 14 99.00% 98.00%
Fault 16 73.50% 97.00%
Fault 17 75.50% 98.00%
Fault 18 48.00% 83.50%
Fault 19 76.50% 98.00%
Fault 20 55.50% 95.50%
Fault 21 73.00% 98.00%
Ave 67.71% 89.50%
the vectors (X1,Y1)or (X2,Y2). In this study, (X1,Y1)is used.
P[X1>X2,Y1>Y2]
=P[X2<X1,Y2<Y1]
=R2
P[X2<x,Y2<y]dC1(F(x),G(y))
=R2
C2(F(x),G(y)) dC1(F(x),G(y))
(A.5)
where Ris the ordinary real line (−∞,+∞).
Let u1=F(x)and u2=G(y), then
P[X1>X2,Y1>Y2]=I2
C2(u1,u2)dC1(u1,u2)(A.6)
where I=[0,1].
Similarly,
P[X1<X2,Y1<Y2]
=P[X2>X1,Y2>Y1]
=R2
P[X2>x,Y2>y]dC1(F(x),G(y))
=R2
[1−F(x)−G(y)+C2(F(x),G(y))]
·dC1(F(x),G(y))
=I2
[1−u1−u2+C2(u1,u2)]dC1(u1,u2)
(A.7)
Moreover, E(u1)=E(u2)=1/2. Therefore,
P[X1<X2,Y1<Y2]=1−1
2−1
2
+I2
C2(u1,u2)dC1(u1,u2)=I2
C2(u1,u2)dC1(u1,u2)
(A.8)
Thus,
P[(X1−X2) (Y1−Y2)>0]=2I2
C2(u1,u2)dC1(u1,u2)(A.9)
Namely,
τ=2P[(X1−X2) (Y1−Y2)>0]−1
=4I2
C2(u1,u2)dC1(u1,u2)−1(A.10)
Furthermore, assume that (X1,Y1)and (X2,Y2)are the inde-
pendent vectors of continuous random variables with common
joint distribution functions Hrather than the different distribu-
tion functions H1and H2, which indicate that (X1,Y1)and (X2,Y2)
come from the same (x,y), the popular version of Kendall’s τis
given based on the derivation as
τ=4I2
C(u1,u2)dC (u1,u2)−1 (A.11)
(2) Second, Spearman’s ρsis deduced as follows:
Let xand ybe continuous random variables in which their
copulas are C,(X1,Y1),(X2,Y2), and (X3,Y3)are three indepen-
dent random vectors from (x,y). Thus, the three vectors have
common joint distribution function H, common margins F(of X1,
X2,X3) and G(of Y1,Y2,Y3), and same copula Cbecause H(x,y)=
C(F(x),G(y)) =C(u1,u2). The Spearman’s ρsis defined to
be proportional to the probability of concordance minus the
probability of discordance for the two vectors (X1,Y1) (X2,Y3):.
ρs=3(P[(X1−X2) (Y1−Y3)>0]
−P[(X1−X2) (Y1−Y3)<0])(A.12)
On the basis of the above assumption, the joint distribution
function of (X1,Y1)is H(x,y)=C(u1,u2). Considering that X2
and Y3are independent, then the joint distribution function of
(X2,Y3)is F(x)G(y)=u1u2. Thus, the copulas of X2and Y3are
F(x)G(y). By using the definition of Kendall’s τ, we have the
following:
ρs=34I2
C2(u1,u2)dC1(u1,u2)−1
=12 I2
u1u2dC (u1,u2)−3
=12 I2
C(u1,u2)du1du2−3
(A.13)
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