Locally Weighted Canonical Correlation Analysis for Nonlinear

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Process Systems Engineering

Locally Weighted Canonical Correlation Analysis for Nonlinear Process Monitoring Qingchao Jiang, and Xuefeng Yan Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b01796 • Publication Date (Web): 13 Sep 2018 Downloaded from http://pubs.acs.org on September 16, 2018

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Industrial & Engineering Chemistry Research

Locally Weighted Canonical Correlation Analysis for Nonlinear Process Monitoring Qingchao Jiang and Xuefeng Yan* Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology, Shanghai 200237, P. R. China *Corresponding author email: [email protected]

ABSTRACT A locally weighted canonical correlation analysis (LWCCA) method is proposed to achieve efficient nonlinear process monitoring. The basic idea of the LWCCA is to approximate a nonlinear process through several local linear canonical correlation analysis (CCA) models, in which the determination of sample weights is a key step. Slowly decreasing weights will ignore the local behaviors, whereas rapidly decreasing weights will lead to significant false alarms. A randomized algorithm-based approach is proposed to determine the tunable parameter for calculating the weights. Thus, the LWCCA model explores as much local behavior as possible with the false alarm performance guaranteed. When a local CCA model that characterizes the process input and process output correlation is established, optimal fault detection residuals are generated, and monitoring statistics are established. Two experimental studies are conducted through which the efficiency of the LWCCA method is verified. KEYWORDS: Data-driven process monitoring, nonlinear processes, locally weighted canonical correlation analysis, just-in-time learning

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INTRODUCTION Process monitoring is crucial in maintaining the long-term safe operation of a production plant. Rapid advancement of data collecting and transmitting techniques has made abundant of process data that contain meaningful process information available 1-3. Data-driven especially multivariate analysis (MVA) methods play a more important role in process monitoring

4-9

. Generally, a classical MVA monitoring

method follows offline modeling with online monitoring procedures. During the offline modeling, a multivariate data analysis technique is employed to explore the relationship among variables and to construct the feature spaces for monitoring. In the online monitoring procedure, a query sample is projected into the monitoring spaces, and the process status is determined according to monitoring statistics. Principal component analysis (PCA), partial least squares (PLS), and canonical correlation analysis (CCA) are the basic MVA methods. PCA focuses on the variable relationships of the entire process (without discriminating process input and output)

10-12

. It constructs a dominant subspace and a residual

subspace according to the features’ importance for reconstructing the original data. PLS focuses on the quality-related process monitoring

13-15

. It constructs the quality-related subspace and the residual

subspace according to their relationship with difficult-to-measure quality variables. A fault in the quality-related subspace is important, because the fault will generally affect the production quality. CCA explores the correlation between two sets of variables. It is generally used in two different ways for process monitoring, as follows: one way focuses on the relation between the process input and process output; and the other one way focuses on the relation between two coupled units

16-20

. It is proved that

the CCA generates optimal fault detection residual when only one set of variables are affected by a fault


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18


. Given the efficiency, PCA, PLS, and CCA-based monitoring methods have been intensively extended

to solve various monitoring problem. Other MVA-based monitoring methods have also proposed and efficiency has been reported 21-23. However, these methods generally assume that the measured variables are linearly related, which limits their applications in nonlinear processes. For nonlinear process monitoring, the neural networks (NN)-based methods learning methods

26-28

, and the just-in-time learning methods

29, 30

24, 25

, the kernel

are the basic ones. The NN-based

methods extract features through nonlinear mapping, which are relatively sophisticated and generally require a large amount of computation. Moreover, the structure and parameters should be determined in designing an NN for monitoring, which remains a challenge. The kernel-based methods replace the nonlinear mapping through kernel functions and then extract features in the high-dimension space. Although efficiency is shown, the selection of related parameters and kernel functions is subjective, which significantly affects monitoring results. Recently, just-in-time learning (JITL) approaches have been developed for nonlinear process modeling and monitoring

29-31

. The basic idea of a JITL method is to establish several local linear

models to represent a nonlinear process

32

. Although the method is efficient, there are still some

problems that need to be addressed. First, locally weighted PCA (LWPCA) and locally weighted PLS (LWPLS), which are extensions of PCA and PLS, are proposed for modeling nonlinear processes

29, 30

.

However, the process input and output relationship is not well characterized by the PCA and PLS-based methods. Extensions of CCA to the nonlinear form are necessary. Second, determination of the tunable parameter in calculating the sample weights remains a challenge. The importance of far-away samples is not distinguishable from that of nearer samples when too-slowly decreasing weights are used. This



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situation will ignore the local behaviors. Using too-rapidly decreasing weights will highlight only a few of the nearest samples, which leads to deviation in the estimation of the covariance matrix and create significant false alarms (FAs). In this paper, we proposed a JITL-locally weighted CCA (LWCCA) monitoring scheme for nonlinear processes. First, the tunable parameter for calculating the weights is determined through a randomized algorithm (RA). This allows the model to explore as much local behavior as possible with the false alarm performance protected. Second, LWCCA is established to characterize the process input and process output correlation. Then, fault detection residuals are generated and monitoring statistics are established. The rest of this article is organized as follows. Section 2 introduces the CCA and RA basics. Motivations for establishing a JITL-LWCCA model is presented. Section 3 details the LWCCA monitoring scheme. In Section 4, two experimental studies are presented. Conclusions are provided in Section 5.

2. PRELIMINARIES AND MOTIVATIONS 2.1 CCA fault detection basics CCA is a classical MVA technique which is used to explore the correlation between two sets of random

variables. Let

 x1   x11  x  x 21 X= 2          xn   xn1

x 12 x22 xn 2

x1 p  x2 p    xnp 

(n samples, p variables) and

 y1   y11 y  y 21 Y= 2          yn   yn1

y 12 y22 yn 2

y1q  y2 q    ynq 

(q

variables) be zero-mean process input and output data. CCA tries to find projection vectors J and L such that the J T X T and LT Y T are most correlated, as follows 4, 33:


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 J , L   arg max ( J , L)

J

J T  XY L

T

XJ

1

2

 L  L T

1

(1) 2

Y

where   is the covariance. To obtain the solution to Eq. (1), a matrix K is constructed. Then singular value decomposing is performed on K as follows 4, 33 K   X1/2  XY  Y1/2  R  V T  diag (1 , 0 

where   

,  l ) 0  pq 0

(2)

and l  rank    . Then, we have 4, 33

J   X1/2 R, L   Y1/2 V .

(3)

For samples x   p1 and y  q1 , the following residual vector is generated 16, 18

r  J T x   LT y

(4)

E  r   J T E  x    LT E  y   0 ,

(5)

where

r  Ip    T .

(6)

Then, T 2 statistic for the residual is established as 16, 18 T 2  r T  r1 r

(7)

With Gaussian assumption, the threshold of the T 2 is determined as Tcl2  2  mr  , where mr  rank ( r )

, given a level of significance  . We assume that the x and y are related as

A  x     By ,

where  is process noise. Under the Gaussian assumption, these T 2 statistic is optimal

for detecting a fault that affects only x

16, 18

. Similarly, the T 2 statistic for detecting a fault that affects

variables in y can be established. 2.2 RA



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Let  be a random variable with density D   . Let D be the support of  . Let  be the accuracy level. RA delivers an estimate pˆ    of probability of at least 1  

p     prob( J     )

, which satisfies p    pˆ     with

. The pˆ    is calculated based on the generated N independent identical

34, 35

distributed random samples  (1) , ,  ( N )  D as follows 34, 35:

pˆ    

1 N

1,if  i   D i i   ,     D    0,otherwise ,  D  i 1  N

(8)

where D is the support of  . N determines the reliability of the estimate pˆ    , as stated in the following Theorems 1 and 2 34, 35. Theorem 1. (Hoeffding inequality) If we let zi   ai , bi  ( i  1, , N ) be an independent random variable, then for any 

 0 , we

have 34, 35:   2   2   , prob( zi  E   zi    )  exp   N  2  i 1  i 1    (bi  ai )   i 1  N

N

    N 2 2  N  . prob( zi  E   zi    )  exp   N  2  i 1  i 1    (bi  ai )   i 1 

(9)

We further assume  ai , bi   [0,1] . The following Theorem 2 is obtained: Theorem 2. For any    0,1 and    0,1 , the N satisfies one-side Chernoff bound 34, 35

N

1 2 ln , 2 2 

(10)

derives p    pˆ     with probability prob( p    pˆ     )  1   . Theorem 2 provides guidance on the selection of a proper N to obtain a reliable estimate pˆ    of a real probability p    . 2.3 Motivations


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The above CCA fault detection method handles the linear process well but may not function well for a nonlinear process. Let a nonlinear process model be expressed as y = f  x    . Here, we employ the single input single output process as an example, as presented in Figure 1. In Figure 1, the blue stars denote the process samples generated by the real nonlinear process model (the blue curve). The red bold line denotes a global linear CCA model using all samples, and the dashed red lines represent the control limits for the linear CCA model. The bold green lines denote local CCA models, which are used to approximate the nonlinear model, and the green dashed lines represent the control limits for the local CCA models. The points A and B denote two query samples. For the sample A, the global CCA model has significantly relaxed the control limits, which fails to identify A as a faulty point. For the sample B, different local linear models can be established by using different number of samples. When using too low amount of samples, the covariance matrix cannot be properly calculated. When using too much samples, nonlinearity becomes involved, and the local behavior cannot be well characterized. The number of involved functional samples is directly determined by the tunable parameter  , as presented in the subfigure of Figure 1. Given the abovementioned analysis, when the real nonlinear process model is not available, it is first important to represent the nonlinear process with local linear models. A global model will generally ignore many local behaviors and fail to efficiently monitor a nonlinear process. Second, the weights on samples directly determine the importance of a sample in establishing a local model. It is important to assign proper weight values to samples to establish a local model. Using too-rapidly decreasing weights will highlight only a few of the nearest samples, whereas using too-slowly decreasing weights will not make the importance of samples distinguishable.



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Figure 1. Motivations for establishing LWCCA models

3. JITL-LWCCA PROCESS MONITORING JITL method aims to represent a nonlinear function with several local CCA models. Here, we present the proposed JITL-LWCCA monitoring scheme in a step-by-step manner. Let input and output vectors, as follows:

   x T

yT    T

p  q 1



be a sample with both

.

Step 1: Historical data normalization. First, the historical process data are scaled to the same level through mean-variance normalization. This is to avoid the domination of variables with larger variance in the neighboring sample selection. Step 2: Similarity selection and weight determination. Let

q

be a mean-variance scaled query

sample and  i be the i-th sample in the training dataset. Then the similarity between the

q

and the

 i is represented by the Euclidean distance, as follows:

di 



 q   i  q  T

i

 i  1,

Then, the weight wi assigned to the  i is as follows:


, n ,

(11)

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wi  exp  di2 /  2  ,

(12)

where  is a tunable parameter. The  determines the decreasing speed of the weights. A smaller  makes the weight decrease rapidly, whereas a larger  makes the weight decrease slowly (illustrated in the Figure 1). A sample near the query sample is assigned a larger weight, whereas a sample far away from the query sample will be assigned a smaller weight. When the weights for all samples are determined, the weighting matrix is obtained as W = diag  w1

wn   nn .

Step 3: Locally weighted CCA modeling. After weighting, the training samples become  w1 x11  X W  WX    wn xn1 

w1 x1 p    wn xnp 

 w1 y11

and YW  WY  

 wn yn1 

w1 y1 p    wn ynp 

. The importance of nearer samples is

highlighted, whereas the effect of a far-away sample is suppressed. The mean of the local linear model relies more on the nearest samples. Frist, the mean of the weighted data is removed. Then, the CCA is performed between X W and YW to obtain the local canonical correlation vectors J and L . Step 4: Residual generation. A query sample is the first mean removed according to the mean of the weighted training samples. Then, the following residuals are generated: rx  J T x   LT y, ry  LT y   T J T x.

(13)

The residual rx reflects the variations of x while considering the correlation information from y, whereas the residual ry reflects the variations of y while considering the correlation information from x. Step 5: Monitoring statistics construction. The process data are assumed as linearly distributed locally. The T 2 statistics are constructed for monitoring, which are as follows:



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Tx2  rxT  rx1rx

(14)

Ty2  ryT  ry1ry

where  rx and

 ry

can be calculated from Eq. (6).

Step 6: Threshold determination. Based on the locally Gaussian assumption, the thresholds Tx2,cl and Ty2,cl of the statistics can be directly obtained using the Chi-squared distribution, as follows: Tx2,cl  2  mx  ,

(15)

Ty2,cl  2  m y  .

where mx  rank ( rx ) and my  rank ( ry ) . Step 7: Process status identification. The process status can be identified through the following decision logic: Tx2  Tx2,cl orTy2  Ty2,cl  faulty,

(16)

Tx2  Tx2,cl and Ty2  Ty2,cl  fault  free.

As detailed in Section 2.3, determining the value  is important. A larger  makes the calculation of covariance more accurate. However, a too-large  involves more nonlinearity and ignores the local behavior. Too-small  is also problematic, because it will significantly deviate the estimation of a covariance matrix and cause significant false alarms. Here, we determine the  using a RA-based approach, which leads to the model’s exploration of as much local behaviors as possible with the false alarm performance guaranteed. The RA-based tunable parameter determination is as follows: Step 1: Determine the number N according to the RA; determine a FAR according to practical application. Step 2: Collect N samples from normal process condition.




that is allowed

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Step 3: Set  as a smaller value. Step 4: Establish LWCCA monitors using the  ; test the N samples under the normal condition; Step 5: If FAR<  , increase the  as     1 , where 1 is the increasing step length. Repeat steps 4 and 5 until FAR   . Step 6: Decrease  as      2   2  1  and go to step 4. Calculate the FAR. Repeat steps 6 and 4 until FAR   . These procedures will lead to the proper  , which would lead to the model’s exploration of most of the local behavior and guarantees the false alarm performance. The framework of the proposed JITL-LWCCA modeling and monitoring scheme is summarized in Figure 2. offline

Mean-variance normalized historical data

Initial parameter

Mean-variance scaled query samples

online

Initial LWCCA model

RA optimized parameter

JITL-LWCCA models

Process status

Decision logic

Statistic calculation

Figure 2. Schematic of the proposed JITL-CCA

4. EXPERIMENTAL STUDIES 4.1 A literature numerical example Considering a four variable system with three input and one output as follows 36:

x1  t 2  t  1  e1 x2  sin 0.5t  e2 x3  t 3  t  e3 y  x  x1 x2  3cos x3  e4 2 1


(17)


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where t follows a uniform distribution U  1,1 and e1 to e4 are zero-mean Gaussian noise with a standard deviation of 0.01. Under normal operation conditions, 500 samples are collected as training data. Two faulty sets, each consisting of 200 samples, are constructed as follows: Fault 1: x2  x2  0.3 from the 51st to the 150th points; Fault 2: y  y  0.01 (k  50) (k is the sample instant) from the 51st to the 150th points. Here, we present the monitoring results of PCA, CCA, LWPCA, and the proposed LWCCA. First, based on RA, the  is determined as 3.5 to guarantee that the false alarm rate (FAR) is less than 0.05. The monitoring results using PCA, CCA, LWPCA, and LWCCA for fault 1 are provided in Figures 3(a)-(d), respectively. The LWCCA

Tx2

which has the least non-detection (ND) points performs the best.

The fault detection results using PCA, CCA, LWPCA, and LWCCA for fault 2 are provided in Figure 4, which show that the LWCCA Ty2 detects the fault at the earliest time and has the least ND points. Here, we also presents the monitoring results of LWCCA using different  in Figure 5, which shows that the non-detection rate (NDR) will generally increase with  . Carlo tests of 100 times are carried out and the average NDRs and FARs using different methods are provided in Figure 6. The results in Figure 6 show that the LWCCA has the lowest NDRs for the two faults. Compared with CCA, the LWCCA performs better for nonlinear processes, because more local process behaviors are explored. From the abovementioned results and analysis, the efficiency of the LWCCA method is demonstrated.


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(a)

(b)

(c)

(d)

Figure 3. Monitoring results for the fault 1: (a) PCA, (b) CCA, (c) LWPCA, and (d) LWCCA

(b)

(a)



(c)

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(d)

Figure 4. Monitoring results for the fault 2: (a) PCA, (b) CCA, (c) LWPCA, and (d) LWCCA

Figure 5. Influence of  on the monitoring performance

Figure 6. Monte Carlo tests for the two faults: (a) fault 1, (b) fault 2 4.2 The Tennessee Eastman Process (TEP) TEP provides a benchmark platform for monitoring performance assessment 37, 38. There are five typical


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units, and the process flow chart is presented in Figure 7. For process monitoring, 11 control variables (XMV) and 22 measured variables (XMEAS), as listed in Supporting Information Table S1, are generally used

18

. Here, the 11 control variables are regarded as process input, and the 22 measured

variables are regarded as process output. There are 21 programmed faults in the simulator. The latest simulation code is presented at http://depts.washington.edu/control/LARRY/TE/download.html. A benchmark dataset is available at http://web.mit.edu/braatzgroup/links.html 3. We limit the section that describes the TEP because details on the process can be easily obtained from references the benchmark dataset is directly used to provide comparable results.

Figure 7. Flow chart of the TEP 37, 38


18, 37, 38

. Here,


Page 16 of 25

Figure 8. Influence of  on the FAR First, the FAR changing with  values is provided in Figure 8. At about   45 the FARs are lower than 0.03. Thus,   45 is employed in the current work. All 21 programmed faults are tested, among which two faults, i.e., faults 5 and 10, are analyzed in detail. Fault 5 induces a step change in the condenser cooling water inlet temperature 3, 37. The fault effect is significant at the beginning, but most of the fault effects are compensated by the control loop, which makes detecting the fault difficult with time. The monitoring results using PCA, CCA, LWPCA, and LWCCA are provided in the Figure 9. The Figure 9 shows the fault is detected by all four methods. However, the PCA and LWPCA methods fail to detect the fault after the 400th point, whereas the CCA and LWCCA methods keep indicating the fault. The CCA and LWCCA similarly provide the same best monitoring performance. Fault 10 introduces a random change of the temperature in the C feed. The monitoring results of the four methods are presented in Figure 10, which shows that the CCA and LWCCA outperforms the PCA and LWPCA-based methods, with much lower NDRs. Both the CCA and LWCCA provide good monitoring performance, and the difference is not significant. The nonlinearity in the TE process is not


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significant, and the CCA can explore most of the local structure. The NDRs of all 21 faults are provided in Table 1. The results in the Table 1 show that the CCA and LWCCA perform the best for most faults. In some cases, e.g., faults 8, 10, 11, 15, 16, 17, and 21, the LWCCA performs slightly better than the CCA, because more local behaviors are explored in LWCCA. The abovementioned results and analysis verify the superiority of the LWCCA method. Table 1. Monitoring results for all the faults in TEP Fault No.

PCA

CCA

LWPCA

Tx2

Ty2

0.00

0

0.01

0

0.01

0.02

0.00

0.03

0.01

0.94

0.90

0.85

0.85

0.93

0.88

0

0

0.04

0.11

0

0

0.03

0.67

0.53

0

0

0.64

0.49

0

0

6

0.01

0

0

0

0.00

0

0

0

7

0

0.01

0

0.05

0

0.01

0

0.02

8

0.02

0.04

0.06

0.02

0.02

0.03

0.04

0.01

9

0.90

0.86

0.97

0.93

0.87

0.85

0.96

0.90

10

0.48

0.24

0.17

0.11

0.45

0.22

0.16

0.10

11

0.32

0.25

0.30

0.31

0.29

0.23

0.28

0.27

12

0.01

0.03

0.01

0.00

0.01

0.02

0.01

0.00

13

0.05

0.04

0.05

0.05

0.05

0.04

0.05

0.05

14

0

0.07

0

0

0

0.04

0

0

15

0.85

0.81

0.91

0.82

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Page 18 of 25

(a)

(b)

(c)

(d)

Figure 9. Monitoring results for the TEP fault 5: (a) PCA, (b) CCA, (c) LWPCA, and (d) LWCCA

(a)

(b)


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(c)

(d)

Figure 10. Monitoring results for the TEP fault 10: (a) PCA, (b) CCA, (c) LWPCA, and (d) LWCCA

5 CONCLUSIONS A novel JITL-LWCCA method is proposed for nonlinear process monitoring. The basic idea is to represent a nonlinear process through several local CCA models. A RA-based method is proposed to determine the parameter for calculating sample weights, which leads to LWCCA’s exploration of as much local behavior as possible with the restriction that the fault alarm performance is not destroyed. Compared with the PCA and LWPCA methods, the LWCCA focuses more on the relation between input data and output data. Compared with the CCA method, the LWCCA explores more local process behavior and is more suitable for nonlinear processes. The efficiency of the JITL-LWCCA method is verified through experimental studies on a numerical example and the TEP. This paper mainly focuses on the fault detection issue of nonlinear processes. Since a nonlinear process is approximated by several linear models, classical linear process fault isolation and diagnosis methods can be integrated within the JITL framework. Future efforts could also be devoted to more complex nonlinear process characters, such as non-Gaussian, dynamic, and multiple operation modes.

ASSOCIATED CONTENT ACS Paragon Plus Environment


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Supporting Information The input and output variables in the TEP are listed in the Supporting Information Table S1. This information is available free of charge via the Internet at http://pubs.acs.org/.

AUTHOR INFORMATION Corresponding Author *E-mail address: [email protected]. Mailing address: East China University of Science and Technology, P.O. Box 293, MeiLong Road no. 130, Shanghai 200237, P. R. China. Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT The authors gratefully acknowledge the support from the following foundations: National Natural Science Foundation of China (61603138), Shanghai Pujiang Program (17PJD009), Fundamental Research Funds for the Central Universities (222201717006, 222201714027), and the Programme of Introducing Talents of Discipline to Universities (the 111 Project, B17017).

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Table Of Contents (TOC) graphic

Table Of Contents (TOC) graphic: Motivations for establishing LWCCA models


Locally Weighted Canonical Correlation Analysis for Nonlinear

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