Incipient fault detection for chemical processes using two-dimensional

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Process Systems Engineering

Incipient fault detection for chemical processes using two-dimensional weighted SLKPCA Xiaogang Deng, and Jiawei Deng Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b04794 • Publication Date (Web): 11 Jan 2019 Downloaded from http://pubs.acs.org on January 15, 2019

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Industrial & Engineering Chemistry Research

Incipient fault detection for chemical processes using two-dimensional weighted SLKPCA Xiaogang Deng ∗ and Jiawei Deng College of Information and Control Engineering, China University of Petroleum, Qingdao China E-mail: [email protected]

Abstract Early detection of incipient faults is a challenging task in chemical process monitoring field. As an effective incipient fault detection tool, statistical local kernel principal component analysis (SLKPCA) has demonstrated its advantage over the traditional kernel principal component analysis (KPCA). However, how to improve its incipient fault detection performance is still a valuable problem. In this paper, an enhanced SLKPCA method, referred to as twodimensional weighted SLKPCA (TWSLKPCA), is proposed by integrating the sample and component weighting strategies. Different to KPCA, SLKPCA monitors the process changes based on the residual vectors computed by the statistical local approach. To highlight the influence of the faulty residual samples, the residual sample weighting strategy is firstly designed based on the distance between the tested samples and the training samples, which puts large weights on the samples with strong fault information. Furthermore, the residual component weighting strategy is developed to assign large weights to the sensitive components, which are judged by computing the mutual information between the sample-weighted residual components and the original measured variables. Based on the two-dimensional weighted residual ∗ To

whom correspondence should be addressed

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

vectors, two monitoring statistics are built to detect the incipient faults. Finally, simulations on a numerical example and the Tennessee Eastman process are used to demonstrate the superiority of the proposed TWSLKPCA method.

1. Introduction As modern chemical processes are large-scale and complicated, fault diagnosis technologies have become indispensable to guarantee safety and increase production. Considering the process complexity, the analytical model-based and process knowledge-based fault diagnosis methods are often difficult to implement in chemical process monitoring field. 1–3 Advanced computer control systems and digital storage technologies have been widely applied in chemical industry and bring massive production data. Therefore, data-based fault diagnosis methods have drawn the most attention from researchers in the last two decades. 4,5 Many data-based methods have been developed by utilizing multivariate statistical analysis tools, 6–8 and one of the most popular methods is principal component analysis (PCA). PCA extracts the un-correlated low-dimensional components from the high-dimensional correlated process variables and constructs two monitoring statistics based on the extracted components for fault detection. PCA and its extended versions have been widely applied to different process monitoring cases. For monitoring complex large-scale processes, Ge et al. 9 proposed a distributed PCA model, which divides the whole process model into several sub-blocks for local information mining. In view of multiscale characteristic of process data, Bakshi 10 presented a modified PCA method by combining wavelet analysis. Considering the process dynamics, a dynamic PCA (DiPCA) method was developed in Dong et al.’s work. 11 As some process data demonstrate the nonGaussian property, Jiang et al. 12 combined independent component analysis with PCA to design a PCA-ICA method. For monitoring multiphase batch processes, a PCA subspace decompositionbased reconstruction model was proposed by Zhao et al. 13 To emphasize the influence of variable selection, a modified PCA method based on the fault-relevant variable selection and Bayesian

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inference was developed by Jiang et al. 14 to improve the monitoring performance. The aforementioned methods have demonstrated their validity in different cases. However, they do not focus on the incipient fault detection task. Incipient faults mean the small changes in the process data and lead to the underlying inapparent damages to process status, which are often difficult to detect by using the basic monitoring methods. In chemical processes, there are many typical incipient faults such as measurement drifting, valve sticking and catalyst deactivation. As time goes by, incipient faults may gradually evolve into serious fault. Timely incipient fault detection is very helpful to carry out the preventive maintenance and avoid the occurrence of serious faults. To conduct the incipient fault monitoring, some works have been developed. Wold et al. 15 firstly developed a modified PCA based fault detection algorithm by combining exponentially weighted moving average (EWMA), which is effective to detect the small shift. Lately, Harrou et al. 16 constructed an improved PCA method for production system monitoring by integrating the multivariate cumulative sum (MCUSUM) control chart. In addition, from the view of probability density function (PDF), the Kullback Leibler divergence-PCA method was implemented by Harmouche et al. 17 because of its high sensitivity for incipient faults. Chen et al. 18 and Chai et al. 19 also discussed the combination of KLD and PCA for incipient fault detection. Considering the extraction of early fault features from process noise, a two-step fault detection method was proposed by Ge et al., 20 which applies wavelet analysis and optimal parity space analysis to distill the characteristics of small amplitude changes. Data filtering is helpful to the detection of weak changes. 21 Ji et al. 22 combined two data filtering methods combined with PCA, which can give better ability to distinguish normal data from fault data. Furthermore, to deal with the incipient fault isolation problem, Ji et al. 23 developed an exponential smoothing reconstruction (ESR) approach. Based on the analysis of PCA and statistics pattern analysis, Shang et al. 24 presented a recursive transformed component statistical analysis (RTCSA). Zhang et al. 25 carried out in-depth study aiming at the incipient fault detection of multistage batch processes. To mine the accurate signal features of electrical drive in high-speed trains, a deep PCA method was designed by Chen et al. 26 by applying multiple data processing layers. Besides, by investigating the changes of data

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distribution as incipient fault occurs, a sparse difference analysis was developed by Zhao et al. 27 Statistical local approach (SLA) is effective to detect the small changes and Kruger et al. 28 integrated SLA with PCA to promote the detection sensitivity to incipient faults. The applications of SLA were also discussed by Chen et al. 29 and Li et al. 30 These aforementioned incipient fault detection methods are all based on the linear PCA model. However, many chemical processes show clear nonlinear characteristic. To monitor the incipient faults in nonlinear processes, some methods have been put forward by combining the nonlinear kernel PCA (KPCA) methods. 31,32 Considering the multi-mode feature of pulse width modulated inverter, Chen et al. 33 proposed a multimode-KPCA method to detect incipient faults of PWM inverter. By incorporating the statistical local approach, Ge et al. 34 constructed an improved statistical local KPCA (SLKPCA) method, which shows superiority on detecting the incipient faults.Although the research on the nonlinear PCA based incipient fault detection has made great progress, there are still some problems deserving further study. When the incipient faults occur, not all the samples and components are affected because of the weak influence of the incipient faults. That is to say, some samples or components may carry clear fault information, while others carry less or no fault information. Traditional methods treat all samples and components equally so that fault information may be concealed. To overcome this problem, some researchers have designed the strategies to explore fault information sufficiently for better detection performance. Cai et al. 35 proposed a novel weighted kernel independent component analysis (KICA) method, which estimates the component probabilities with Gaussian mixture model and assigns the larger weights to the significant components. This method can provide better fault detection performance than the traditional KICA method. However, it only focuses on the component weighting strategy but does not consider the influence of the sample weight. Therefore, how to highlight the fault information in views of the samples weight and the component weight is still a valuable problem, which motivates the study of this paper. To provide better monitoring performance for incipient fault monitoring of chemical processes, this paper is to propose a modified SLKPCA method with two-dimensional weight information,

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which is called Two-dimensional Weighted SLKPCA (TWSLKPCA). In this proposed method, the primary residuals are firstly computed based on SLKPCA modeling. Then the residuals are weighted in view of the sample dimension and the component dimension, respectively. Along the sample dimension, the primary residual samples are weighted to highlight the influence of fault samples. Furthermore, the residual component weighting strategy is constructed to emphasize the sensitive components. With the bidirectional weights, the TWSLKPCA model is developed to detect the process changes. Finally, we apply two case studies to verify the designed methods in this work.

2. Preliminaries 2.1 Kernel principal component analysis KPCA is a state-of-the-art nonlinear PCA method and has been widely applied to solve nonlinear monitoring task. 32,36–40 In the KPCA modeling procedure, the original variables are firstly transformed to the high-dimensional feature space by an implicit nonlinear mapping. Then in the feature space, linear PCA is performed to obtain the principal components. As the kernel trick is utilized to deal with the nonlinear optimization, this method is called kernel PCA. Its details are given as follows. Given the training dataset X = [xx1 x 2 · · · x n ]T ∈ R n×m , where n is the sample number and m is the variable number, a nonlinear mapping Φ(·) is assumed to map the original data x i to Φ(xxi ). X ) is expressed by The covariance matrix of Φ(X

C=

1 n ∑ Φ(xxi)ΦT(xxi). n i=1

(1)

X ) leads to an eigenvalue problem as: To perform PCA on the data matrix Φ(X

λ jγ j = Cγ j,

5


(2)


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where λ j and γ j denote the j-th eigenvalue and eigenvector of the matrix C , respectively. The eigenvector γ j can be represented by the linear combination of the training samples as 41,42 n

γ j = ∑ αi j Φ(xxi ),

(3)

i=1

where αi j represents the combining coefficient. By integrating Eqs.(1), (2) and (3), we obtain nλ j α j = K α j ,

(4)

where α j = [α1 j , α2 j , · · · , αn j ]T , K represents a kernel matrix and its (i, j)-th element ki j is the inner product of Φ(xxi ) and Φ(xx j ), that means, ki j = ΦT (xxi )Φ(xx j ). By kernel trick, ΦT (xxi )Φ(xx j ) can be computed by one kernel function ker(xxi , x j ). Here, we apply the commonly used Gaussian kernel function ker(xxi , x j ) = e−

||xxi −xx j ||2 c

, where c is the kernel width.

For one testing sample x h at the h-th sampling instant, its projection on the j-th eigenvector γ j leads to the kernel principal component (KPC) th, j , expressed as n

th, j = ΦT (xxh )γ j = ∑ ΦT (xxi )Φ(xxh )αi j = k h α j ,

(5)

i=1

where k h = [ker(xxh , x 1 ) ker(xxh , x 2 ) · · · ker(xxh , x n )] is the kernel vector corresponding to the testing sample x h . In order to monitor the data changes, two statistics are usually used, which are the T 2 and SPE statistics. 41 The T 2 statistic is developed to monitor the principal component subspace, constructed by the first a KPCs corresponding to the first a largest eigenvalues. As to the sample x h , its T 2 statistic is denoted as Th2 , calculated by Th2 = [th,1 th,2 · · · th,a ]Λ−1 [th,1 th,2 · · · th,a ]T ,

6


(6)

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where Λ is a diagonal matrix with its elements as λ1 , λ2 , · · · , λa . The SPE statistic is used to monitor the non-principal component subspace. The sample x h ’s SPEh statistic is computed by

SPEh =

n

a

j=1

j=1

∑ th,2 j − ∑ th,2 j ,

(7)

where n is the dimension of the feature space. In this paper, n is determined to satisfy that the cumulative sum of the first n eigenvalues exceeds 99% of the sum of all eigenvalues. 43 The KPCA-based process monitoring framework can be summarized as Fig.1. The measured process data are transformed by KPCA into kernel principal components, and then two monitoring statistics are developed for fault detection. Measured process data

KPCA

Kernel principal components

Monitoring statistics

Figure 1: KPCA monitoring framework.

2.2 Statistical local kernel principal component analysis Statistical local approach is an effective tool to investigate the changes of model parameters. It was firstly used by Basseville 44 to carry out the on-board incipient fault detection. Furthermore, Ge et al. 34 introduced statistical local analysis into KPCA to construct an enhanced statistical local KPCA (SLKPCA) method and the applications demonstrated that SLKPCA outperforms KPCA in monitoring the incipient faults. The details of SLKPCA are given as follows. Based on the kernel principal components th,k and eigenvalues λk , SLKPCA firstly constructs the primary residual as 2 rh,k = 2th,k − 2λk .

(8)

The n primary residuals construct the primary residual vector r h = [rh,1 rh,2 · · · rh,n ]. Furtherly, the moving window technology is used to develop the improved residual as h 1 rh, j = √ ∑ rl, j , q l=h−q+1

7


(9)


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where q is the moving window length. For the sample x h , its improved residual vector is denoted as r h = [rh,1 rh,2 · · · rh,n ]. Considering that the first a components in the vector r h represents the principal system changes while the other n − a components are the noise part, the residual vector r h can be divided into two parts as (s) r h = [r1 r2 · · · ra ],

(10)

(n) r h = [ra+1 ra+2 · · · rn ].

(11)

Similar to KPCA, SLKPCA builds two monitoring statistics T 2 and SPE, formulated as 34 T Th2 = r h Γ−1 s (rr h ) ,

(12)

T SPEh = r h Γ−1 n (rr h ) ,

(13)

(s)

(s)

(n)

(n)

where Γs , Γn are the residual covariance matrix obtained by the training data primary residual. The monitoring framework of SLKPCA is depicted in Fig.2. Different to KPCA, statistical local analysis (SLA) is introduced before constructing monitoring statistics and benefits the detection of incipient faults. KPCA

Measured process data

SLA Kernel principal components

Residual vectors


Figure 2: SLKPCA monitoring framework. Although SLKPCA has demonstrated its effectiveness, there are some issues worthy of further study. One issue is related to the design of the improved residuals. By Eq. (9), all the residual samples in the sliding window are with the same weights. However, as incipient faults occur, the samples may lead to different influences because of process dynamics. So it is unreasonable to assign the same weights to the samples in the sliding window. Another issue involves the statistics construction. When monitoring statistics are designed in Eqs. (12) and (13), the improved residual components are viewed equally. However, as incipient faults occur, not all the components are 8


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affected equally. Some components may be affected seriously while other components may not. That is also one reason why incipient faults is difficult to detect. Therefore, it is necessary to set different weights for the residual components. Based on these considerations, we will design an improved SLKPCA method.

3. The two-dimensional weighted SLKPCA method In order to provide better monitoring performance for incipient faults, this work designs an improved SLKPCA method, called Two-dimensional Weighted SLKPCA (TWSLKPCA). The proposed method applies the weighting strategy at the both sample and component dimension. Considering the difference of the residual samples, the sample weighting strategy is implemented to highlight the roles of the faulty samples. To differentiate the effects of different components, the component weighting strategy is built to emphasize the sensitive components. This is so-called two-dimensional weighting (TDW) strategy, which is depicted in Fig.3.

Ă

vij

w1

r11

r12

Ă

r1 j

r1n

w2

r21

r22

Ă

r2 j

r2n

Ă

Ă

Ă

Ă

wi

ri1

ri 2

Ă

rij

ri n

Ă

Ă

Sample dimension

vi n

Ă

Ă

Ă

vi 2

Ă

vi1

Ă

Sample weighting

Component weighting

Ă

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Component dimension

澳

Figure 3: Two-dimesional weighting strategy.

3.1 Sample weighting strategy For improving the detection sensitivity to incipient faults, the sample weighting strategy is designed to enhance the computation of the improved residuals. In Eq. (9), all samples in the moving window are with the same weight as 1. However, this is unreasonable and cannot benefit the fast incipient fault detection. The roles of the samples should be distinguished by different weights. 9



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Some samples with weak fault information should be given smaller weight, while some samples, reflecting strong fault information, deserve larger weights. To evaluate how much information the tested sample carries, a natural idea is to obtain the distance between the tested sample and the normal data, denoted by dh , defined by

dh =

1 n ∑ dist(rr h, r ∗i ), n i=1

(14)

where r ∗i is the primary residual vector responding to the training sample x i , dist represents the Euclidean distance between two vectors. Considering the randomness of the normal samples, a modified sliding window distance is built as dh =

h 1 ∑ dl . q l=h−q+1

(15)

For normal samples, they have the similar small d h values. If d h is larger than some specified threshold, it means fault information is detected. The larger the d h is, the stronger the fault information is. Based on the distance analysis, a distance weighting factor wh is defined by  d −d   sat(eβ1 hdlimlim ) , d ≥ d h lim wh = ,   1 , otherwise

(16)

where β1 is a tuning parameter, while sat() represents a saturation function expressed as

xout

   xin , i f 0 < xin < xlim    , = sat(xin ) = xlim , i f xin ≥ xlim      0 , i f xin ≤ 0

(17)

where xlim is the upper limit value of the saturation function. In this paper, it is set as 10 empirically. By applying the distance weighting factor, the samples with large d h will be given a weight larger than 1. This can strengthen the influence of incipient faulty samples. However, only considering the h-th sample distance may lead to high false alarm rate because of the existing of process noises. 10


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In practice, we can enforce the large weight when continuous c sample distances d l (h − c + 1 ≤ l ≤ h) exceed the threshold dlim . The threshold dlim is computed by analyzing the distribution of normal samples. By applying the distance weighting factor wh , we build a weighted improved residual as h 1 wrh, j = √ ∑ rl, j wl , q l=h−q+1

(18)

and two monitoring statistics are developed based on the sample-weighted residual vector as wr h )T , Th2 = w r h Γ−1 s (w

(19)

wr h )T , SPEh = w r h Γ−1 n (w

(20)

(s)

(n)

(s)

(s)

(n)

(n)

where w r h = [wrh,1 wrh,2 · · · wrh,a ], w r h = [wrh,a+1 wrh,a+2 · · · wrh,n ].

3.2 Component weighting strategy By investigating the Eqs. (19) and (20), it is observed that the monitoring statistics are developed based on the weighted residual [wr1 wr2 · · · wra ] and [wra+1 wra+2 · · · wrn ], and all the components wri (1 ≤ i ≤ n) are equally treated. However, when incipient fault occurs, maybe only a few components are with significant fault information. By the monitoring statistics in Eqs. (19) and (20), the fault information reflected by a few components may be submerged by other components. In order to detect the incipient faults more effectively, it is necessary to find the sensitive residual components and appoint larger weights to them. Firstly, we give a sensitive component location strategy based on the correlation analysis between the residual components and the process variables. A common nonlinear correlation analysis tool is the mutual information method. 45,46 Mutual information is a measure of the mutual dependence between the two variables. If the mutual information between one residual component and the process variables is large, that means this residual component carries important process infor-

11



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mation and can be viewed as the sensitive component. For two given random variables with n samples, denoted by y 1 = [y11 y12 · · · y1n ] and y 2 = [y21 y22 · · · y2n ], their mutual information is defined by n

im(yy1 , y 2 ) = ∑ p(y1i , y2i )log( i=1

p(y1i , y2i ) ), p(y1i )p(y2i )

(21)

where p(y1i , y2i ) is the joint probability function of y 1 and y 2 , while p(y1i ) and p(y2i ) are the marginal probability distribution function of y 1 and y 2 , respectively. Mutual information can be obtained by the histogram method or the kernel density estimation methods. 47,48 In this paper, the histogram method is applied because of its simplicity. 49,50 In real processes, both linear and nonlinear relations exist among the data. For a comprehensive consideration, the linear correlation coefficient is also integrated to modify the correlation index based on the mutual information. The modified correlation degree index is defined as e y1 , y 2 ) = im(yy1 , y 2 ) + lcr(yy1 , y 2 ) , im(y 2

(22)

where lcr(yy1 , y 2 ) is the measure of linear correlation degree. In this paper, lcr(yy1 , y 2 ) is defined by a modified linear correlation coefficient, expressed by e y2 y T1 e , lcr(yy1 , y 2 ) = ||e y 1 || · ||e y 2 ||

(23)

where e y1, e y 2 are the vector scaled by the mean value of training data. In online monitoring procedure, a data window X h ∈ Rq×m with the q samples and m variables is gathered.



  xh−q+1,1 xh−q+1,2   xh−q+2,1 xh−q+2,2  Xh =   ··· ···   xh,1 xh,2

· · · xh−q+1,m   · · · xh−q+2,m   .  ··· ···   ··· xh,m

(24)

Using statistical local analysis and sample weighting strategy, the corresponding sample-weighted

12


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residual matrix W R h is obtained as 

wrh−q+1,1 wrh−q+1,2 · · · wrh−q+1,n

   wrh−q+2,1 wrh−q+2,2 · · · wrh−q+2,n  W Rh =   ··· ··· ··· ···   wrh,1 wrh,2 ··· wrh,n

     .   

(25)

To compute the mutual information between each residual component and the measured variables leads to the following mutual information matrix as 

e e  im1,1 im1,2   im e e  2,1 im2,2 IM =   ··· ···   e m,1 im e m,2 im

 e 1,n im   e 2,n  · · · im  , ··· ···    e m,n · · · im ···

(26)

e i, j = im(X e X h (:, i),W W R h (:, j)) represents the mutual information between the i-th column where im of X h and j-th column of W R h . In Eq.(26), the j-th column gives the mutual information values between the j-th (1 ≤ j ≤ n) component and all process variables. So the sum of the j-th column e is denoted as Im j = ∑m i=1 imi, j , which describes the correlation between the j-th component and the process information. Furthermore, we define the mutual information ratio (MIR) as

Cj =

Im j . n ∑ j=1 Im j

(27)

If one component is sensitive to the process changes, its MIR value C j will be larger than other components. Based on the above analysis, we can find out which components are sensitive. Next, a weighting approach is given to highlight the influence of these components. As to the

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j-th component, its weight is set as    vh, j =

sat{e

β2

wr2 h, j −σ j

σj

  1

} , i f (C j ≥ ε & wr2h, j ≥ σ j )

,

(28)

, otherwise

where ε is the threshold of mutual information ratio, which can be set as

1 n

, while σ j is the 99%

confidence limit of wr2h, j . Considering the component weights, the sample-weighted residual vector is further improved to two-dimensional weighted residual vector as h 1 vwrh, j = √ ∑ rl, j wl vl, j . q l=h−q+1

(29)

With the two-dimensional weights, the process monitoring statistics are built as vw r h )T , Th2 = v w r h Γ−1 s (v

(30)

vw r h )T , SPEh = v w r h Γ−1 n (v

(31)

(s)

(s)

(n)

(n)

(s) where v w r h = [vwrh,1 vwrh,2 · · · vwrh,a ] corresponds to the principal system changes, while , (n) v w r h = [vwrh,a+1 vwrh,a+2 · · · vwrh,n ] represents the noise information.

For the proposed method, its process monitoring framework is shown in Fig.4. It is clear that two-dimensional weighting strategy is inserted after the residual vector computation. This will improve the monitoring performance of SLKPCA. TDW

SLA

KPCA Measured process data

Kernel principal components

Residual vectors

Weighted residual vectors

Figure 4: TWSLKPCA process monitoring framework.

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3.3 The process monitoring procedure based on TWSLKPCA When TWSLKPCA is applied, its process monitoring procedure consists of two stages, which are shown in Fig.5. During the offline modeling stage, normal process data are firstly acquired as the training data and then KPCA modeling is performed. Furtherly, SLA is applied to obtain the residual vectors and the monitoring statistics. The confidence limits of two monitoring statistics are determined by kernel density estimation. At the online monitoring phase, new observation is collected and scaled by the mean and variance of normal data. Primary residual of new data are obtained by projecting on the trained statistical model. Then two-dimensional weighting strategy is carried out on the primary residual vectors. If the statistic based on the weighted residual vectors exceeds confidence limit, that means the system is in abnormal operation.

3.4 Computation complexity analysis In this paper, we propose the TWSLKPCA method for better monitoring performance. However, this method also brings higher computation complexity. The algorithm computation complexity Offline modeling

Online monitoring

Normalize training data X

Normalize testing data

KPCA modeling

KPCA model

Calculate the kernel principal components of the testing data Obtain the primary residual of testing data

Get the primary residual and improved residual

Perform the two-dimensional weighting strategy Compute the monitoring statistic of normal training dataset

Compute the monitoring statistic of testing data

Determine the confidence limit for monitoring statistic using KDE

Judge whether statistic exceeds confidence limit

No

Yes abnormal operation

Figure 5: TWSLKPCA based process monitoring procedure. 15



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analysis is a complicated task. In this paper, we only consider the main computation steps. For the proposed method, its online computation for monitoring one testing sample include five main steps: 1) kernel components computing, 2) primary residual calculating, 3) sample weighting, 4) component weighting, and 5) statistics constructing. At the first step, given a testing sample, its kernel components are obtained by the kernel vector computing in Eq. (5). The corresponding time complexity can be given as

Tkc = O((m + 1)n + nn).

(32)

At second step, the primary residual vector is computed by Eq. (8), and its time complexity is represented by Tpr = O(n).

(33)

At the third step, the sample weighting is performed which includes two sub-stages of the distance computation and the sample weighting factor calculation. The time complexity for this step is Tsw = O(nn + n).

(34)

The fourth step is the implementation of the component weighting strategy, which involves three sub-stages of mutual information matrix computation, component weighting factor calculation and two-dimensional weighted residual vector obtaining. It should be noted that, for simplicity, the computation of mutual information between two vectors are viewed as one single computation step. The time complexity of the fourth step is listed by

Tcw = O(qmn + nq + 2n).

(35)

The last step is the construction of two monitoring statistics, which has the time complexity of

Tst = O(n).

16


(36)

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Table 1: The time complexity of KPCA, SLKPCA, WSLKPCA and TWSLKPCA for monitoring single online sample method time complexity KPCA O(m + n + 1)n + n) SLKPCA O(m + n + 1)n + 2n) WSLKPCA O(m + 2n + 1)n + 3n) TWSLKPCA O(m + 2n + 1)n + (5 + q + qm)n) Table 2: The main stored intermediate variables of KPCA, SLKPCA, WSLKPCA and TWSLKPCA for monitoring single online sample method variables KPCA t h ∈ R 1×n SLKPCA t h ∈ R 1×n , r h ∈ R 1×n , r h ∈ R 1×n WSLKPCA t h ∈ R 1×n ,rr h ∈ R 1×n ,wh ∈ R 1 , w r h ∈ R 1×n TWSLKPCA t h ∈ R 1×n , r h ∈ R 1×n , wh ∈ R 1 ,vvh ∈ R 1×n ,vvw r h ∈ R 1×n

total dimension n 3n 3n + 1 4n + 1

To sum up, the whole computation complexity for the proposed method is obtained by Tall = O((2n + m + 1)n + (5 + q + qm)n). For comparison, the other three methods KPCA, SLKPCA, and WSLKPCA are analyzed similarly. The time complexity results of four methods are listed in the Table 1. It is clear that TWSLKPCA involves the highest time complexity. Besides the time complexity, the storage space required in the algorithm running is also analyzed. When the basic KPCA monitoring statistics are used to monitor the single online sample, only the kernel components t h = [th,1 th,2 · · · th,n ] needs to be pre-computed and stored. However, for the proposed TWSLKPCA method, its monitoring statistics are computed based on more intermediate variables including the kernel components t h , the residual vector r h , the sample weight wh , the component weight v h = [vh,1 vh,2 · · · vh,n ] and the two-dimensional weighted component vector (s) (n) v w r h = [vvw r h v w r h ]. For a straightforward comparison, we list the main intermediate variables

used in computing the monitoring statistics in the Table 2, where the total dimension is the sum of the dimensions of all the used intermediate variables. It can be seen that the total dimension of TWSLKPCA is higher than the other three methods. Based on the above analysis, we observe that the proposed TWSLKPCA method has the higher computation complexity than the traditional KPCA and SLKPCA method. When it is used for process monitoring, engineers can decide if it is applicable according to the process characteristics 17



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such as the scale and the trait.

4. Case Study In this section, two examples, including a numerical example and the well-known TE process, are used to demonstrate the monitoring performance of the proposed TWSLKPCA method. Also, three other methods, KPCA, SLKPCA and WSLKPCA, are performed for method comparison. WSLKPCA is the improved SLKPCA method only with the sample weighting strategy. In the following monitoring charts, the statistic is plotted with the solid line while its confidence limit is given by the dashed line. To monitor the changes of each statistic, 99% confidence limit is applied which is estimated by the kernel density estimation technology. 51,52 To evaluate the fault detection performance of different methods, three indices of fault detection time (FDT), fault detection rate (FDR) and false alarm rate (FAR) are used. FDT is defined as the sample number from which the successive six samples exceed the confidence limit. FDR is defined as the percentage of the detected faulty samples over all the faulty samples, while FAR is defined as the percentage of the normal samples exceeding the confidence limit over the total normal data.

4.1 A numerical example A nonlinear dynamic system 53,54 is firstly applied for method testing. Its mathematical expression is given as   x (k)  1   0.118 −0.191 0.287     x (k)  =  0.847 0.264 0.943 2       −0.333 0.514 −0.217 x3 (k) 

  x (k − 1) t(k)    1      x (k − 1)  +  t 2 (k) − 3t(k)    2    −t 3 (k) + 3t 2 (k) x3 (k − 1) 

t(k) = 0.811t(k − 1) + 0.193w(k),

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where x1 , x2 , x3 are the monitored variables, t is the internal variable, ei (i = 1, 2, 3) ∈ N(0, 0.01) are the Gaussian noises , w ∈ [0.01, 2] is the random variable with uniform distribution. By running 18


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the simulation programmes of Eqs. (37) and (38), a normal dataset with 800 samples is generated where the first 400 samples are selected for offline modeling, while other 400 samples are applied as normal testing data. For fault detection algorithm testing, two kinds of faults with 400 samples are simulated, which are list as follows. Both faults involve small changes and can be viewed as incipient faults. • Fault 1: After the 100th sample, a small step bias with the amplitude -0.5 occurs on the variable x2 . • Fault 2: After the 100th sample, a small ramp change with the slop 0.001 occurs on the variable x1 . Four methods of KPCA, SLKPCA, WSLKPCA and TWSLKPCA are used to monitor the simulated system. When KPCA modeling is performed, the Gaussian kernel width c is set to 30 and the retained principal component number a is set as 4 according to the cumulative eigenvalue rule, 55 which requires the principal component subspace explains at least 95% data variance. For SLKPCA, the window width q is selected as 30. The monitoring results on the normal and fault testing data are discussed as follows. Firstly, we present the monitoring charts of normal testing data in Fig. 6. As it can be seen, the monitoring statistics of the four methods are mostly below the confidence limits. The FAR of KPCA SPE chart is 0.5%, while the FARs of the other charts are all 0. Therefore, for the simulated numerical system, all methods give the satisfactory results under the normal case. Then fault F1 is tested and its monitoring results are demonstrated in Fig.7. It is seen that when KPCA is applied, most of fault samples are not detected in Fig.7(a). The FDRs of KPCA T 2 and SPE statistics are 0.33% and 4%, respectively. So KPCA can not detect this fault effectively. By using SLKPCA, which considers the local statistical analysis, its monitoring results in Fig.7(b) show clear improvement compared to KPCA’s. The FDRs of SLKPCA are 23.0% and 27.67% for T 2 and SPE, respectively. With the proposed sample weighting strategy, WSLKPCA further improves the fault detection performance and its monitoring charts are shown in Fig.7(c), 19


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Figure 6: Monitoring charts obtained by the four methods over the normal testing data. where we see that more faulty samples are detected by T 2 and SPE monitoring statistics with the FDRs of 38.67% and 34.00%, respectively. According to Fig.7(d), TWSLKPCA gives the highest FDRs, which are 58.67% and 49.67% for T 2 and SPE, respectively. Among these four methods, TWSLKPCA gives the best monitoring performance, which shows the superiority of the proposed two-dimensional weighting strategy. Then another fault F2 is illustrated. Four methods’ monitoring charts are seen in Fig.8. By KPCA charts in Fig.8(a), we observe that KPCA performs poorly with the very low FDRs, which are 6.67% and 8.33% for T 2 and SPE, respectively. By contrast, SLKPCA charts in Fig.8(b) do better. Its T 2 statistic has a high FDR as 39.33%, while the SPE statistic’s FDR is 45.67%. With the use of WSLKPCA, the monitoring results in Fig.8(c) show further improvements, which give the 47.67% FDR for the T 2 statistic and the 47.33% FDR for the SPE statistic. TWSLKPCA in Fig.8(d) obtains a better result than the WSLKPCA method, and its T 2 and SPE statistics are with the FDRs of 55% and 67.33%, respectively. Therefore, the TWSLKPCA method demonstrates its

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Figure 7: Monitoring charts obtained by the four methods over the fault F1. advantage in the detection of fault F2. To investigate the weighting procedure of TWSLKPCA, the sample and component weights in fault F2 detection procedure are plotted in the Fig.9. By this figure, it is seen that the sample weights in Fig.9(a) are augmented in two areas. One large weight area is between the 345-th and the 400-th sample, which is obviously because that the samples from the 345-th instant depart from the normal data seriously . Another area involves the interval between the 166-th and 180-th sample. In this interval, the samples are slightly away from the normal data so that the SLKPCA method can not detect this clearly. With the sample weights, the WSLKPCA method improves the detection performance on the faulty samples in the interval of the 166-th to 180-th samples. Fig.9(b) gives the weights corresponding to the 8-th residual component, which is a sensitive component at many time instants. This figure indicates that the component weights are increased clearly from the 200th sample. Although the component weights are fluctuating because of random changes, they are still beneficial to monitoring performance improvement.

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Figure 9: Weight charts by TWSLKPCA over the fault F2. For a comprehensive comparison, the FDR indices of monitoring both faults are summed in the Table 3. It can be observed that the WSLKPCA’s average FDRs are 43.17% and 40.67% for T 2 and SPE, respectively, which outperform the KPCA’s and SLKPCA’s, while TWSLKPCA further enhances the FDRs to 56.84% and 58.50%, respectively. Generally, TWSLKPCA gives the best fault monitoring performance for the numerical system.

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Table 3: The fault detection rates (FDRs)(%) of KPCA, SLKPCA, WSLKPCA and TWSLKPCA for faults F1 and F2 Fault KPCA case T 2 SPE F1 0.33 4.00 F2 6.67 8.33 Average 3.50 6.17

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WSLKPCA T2 SPE 38.67 34.00 47.67 47.33 43.17 40.67

TWSLKPCA T2 SPE 58.67 49.67 55.00 67.33 56.84 58.50

4.2 The TE Process The Tennessee Eastman (TE) process is a well-known chemical process, which has been widely applied to evaluate different fault detection and diagnosis methods. 36,56,58 The TE process is established by the Eastman Chemical Company and consists of five basic operating units: reactors, condensers, separators, compressors and strippers. Its flow chart is illustrated in Fig. 10. A simulator of this process provides the simulated process data, which consist of 41 measurement variables and 11 operational variables. Also normal operation and 21 pre-set faults are simulated, whose details can be seen in the literature. 9 Among these faults, Fault IDV-3, IDV-9, and IDV-15 are very famous because of the high detection difficulty. As there are no clear changes observed in the origFC 6

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Table 4: The tested fault list Fault Name IDV-3 IDV-9 IDV-10 IDV-15 IDV-21

Description Step change in D feed temperature (stream 2) Random change in D feed temperature (stream 2) Random change in C feed temperature (stream 4) Condenser cooling water valve sticking Valve position constant(stream 4)

inal data trends, some literatures show that traditional process monitoring methods cannot detect them effectively. 57–59 Fault IDV-10 and IDV-21 are also with low fault detection rates when traditional KPCA methods are applied. 36,38 So these five kinds of faults are utilized to perform method comparison. The detailed fault descriptions are shown in Table 4. By the TE process simulator, we collect the 1460 samples for normal operation and the 960 samples for each fault case. The normal data are used as the training dataset. For each fault case, the fault is introduced after the 160-th sample. Four methods of KPCA, SLKPCA, WSLKPCA and TWSLKPCA are used to monitor these five faults. In the TE monitoring procedure, the Gaussian kernel width c is set to 4000 and the window width q is selected as 140 experientially. The other parameters are determined by the same rules as used in the first case study. Fault IDV-3 is firstly analyzed. When this fault occurs, it is not easy to detect because of its weak influence on the process status. KPCA, SLKPCA, WSLKPCA and TWSLKPCA are performed to detect this fault and the results are shown in Fig.11. For the traditional KPCA method, only a few sample points go beyond the confidence limits in Fig.11(a), where the FDRs of KPCA T 2 and SPE are very low as 5.63% and 1.63%. According to Fig.11(b), the SLKPCA T 2 statistic detects the fault at the 251-th sample with the 16.88% FDR, while SPE can not give an alarm signal during the fault period. Generally, the SLKPCA’s T 2 statistic achieves obvious improvement compared to the KPCA’s. When WSLKPCA is applied, the T 2 detection rate in Fig.11(c) is 32%, which is a very clear improvement, while SPE statistic obtains a slight increased FDR as 2.13%. By analyzing the detection time, the WSLKPCA T 2 statistic detects this fault at the same time as the SLKPCA T 2 statistic does, while the SPE statistic alarms the fault at the 643-th sample, which is earlier than the SLKPCA SPE statistic. This shows that the sample weighting strategy is

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Figure 11: Monitoring charts obtained by the four methods over the TE process fault IDV-3. effective in enhancing the incipient fault detection results. By Fig.11(d), TWSLKPCA detects this fault at the 249-th sample and 585-th sample, respectively, which gives the highest fault detection rates of 39.37% and 14.25%, respectively. To sum up, TWSLKPCA outperforms the other three methods. Fault IDV-9 also injects very weak effect on the process. Similar to the monitoring on the fault IDV-3, KPCA gives a poor result in Fig.12(a), which has the FDRs of 5.25% and 2.00% for T 2 and SPE, respectively. By Fig.12(b), SLKPCA obtains obvious performance increasing. The SLKPCA T 2 FDR is increased to 17.00% , while SPE has a FDR of 8.13%. With the application of WSLKPCA, the monitoring charts in Fig.12(c) give better results. The T 2 and SPE FDRs are improved to 26.50% and 23.62%, respectively, which are 9.5% and 18.37% more than the SLKPCA statistics’. That means WSLKPCA can detect more faulty samples. When TWSLKPCA is applied in Fig.12(d), the FDRs of the T 2 and SPE statistics are 37.87% and 27.12%, respectively. Compared with the basic KPCA method, the TWSLKPCA T 2 and SPE FDRs are increased 25


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Figure 12: Monitoring charts obtained by the four methods over the fault IDV-9. by 32.62% and 25.12%, respectively. The result comparison on the fault IDV-9 proves that the proposed two-dimensional weighting strategy is very effective. The fault detection rates (FDRs), false alarm rates (FARs) and fault detection times (FDTs) of KPCA, SLKPCA, WSLKPCA and TWSLKPCA for five TE process faults are listed in Table 5, Table 6 and Table 7, respectively. For all methods, the FAR is computed on the first 160 normal samples in each faulty dataset. That means, a total of 800 normal samples are used to test FAR. In the Table 6, if the FDT value is 960, it means this fault is not detected as a total of 960 samples are monitored. First thing to note is that SLKPCA can obtain better monitoring results than KPCA for all faults in terms of both FDR and FAR. SLKPCA obtains higher FDRs and lower FARs, which proves the effectiveness of statistical local analysis sufficiently. For fault IDV-15 and IDV-21, WSLKPCA has the similar monitoring performance to SLKPCA, while WSLKPCA performs better than SLKPCA when monitoring the other three faults. Among all these four methods, TWSLKPCA with two-dimensional weighting strategy obtains the highest average F-

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Table 5: The fault detection rates (FDRs) (%) of KPCA, SLKPCA, WSLKPCA and TWSLKPCA for TE process faults Fault KPCA 2 case T SPE IDV-3 5.63 1.63 IDV-9 5.25 2.00 IDV-10 45.25 56.13 IDV-15 6.75 4.75 IDV-21 46.50 45.25 Average 21.88 21.95

SLKPCA T2 SPE 16.88 0.00 17.00 8.13 73.25 95.50 20.75 19.00 63.50 39.37 38.28 32.40

WSLKPCA T2 SPE 32.00 2.13 26.50 23.62 88.75 95.50 20.75 20.13 63.50 39.37 46.30 36.15

TWSLKPCA T2 SPE 39.37 14.25 37.87 27.12 92.13 95.87 21.00 20.50 64.25 66.25 50.92 44.80

Table 6: The false alarm rates (FARs) (%) of KPCA, SLKPCA, WSLKPCA and TWSLKPCA for TE process normal samples Method Statistics FAR

KPCA T 2 SPE 3.00 2.00

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WSLKPCA TWSLKPCA T 2 SPE T2 SPE 0 0 0.37 0.37

Table 7: The fault detection times (Sample No.) of KPCA, SLKPCA, WSLKPCA and TWSLKPCA for TE process faults Fault KPCA SLKPCA 2 case T SPE T2 SPE IDV-3 960 960 251 960 IDV-9 960 960 164 171 IDV-10 257 194 228 197 IDV-15 960 960 795 809 IDV-21 443 427 453 646 Average 716.0 700.2 378.2 556.6

WSLKPCA T2 SPE 251 643 161 167 225 197 795 800 453 646 377.0 490.6

TWSLKPCA T2 SPE 249 585 161 161 224 194 793 797 447 431 374.8 433.6

DRs, which are 50.92% and 44.80%, respectively. However, it should be noted that TWSLKPCA has a little higher FAR than SLKPCA and WSLKPCA. By investigating Table 7, it can be seen that the TWSLKPCA T 2 statistic detects the fault IDV-3, 9, 10 and 15 with the fastest speed. For fault IDV-21, KPCA gives the smallest fault detection time, which is only 4 samples earlier than TWSLKPCA. To summarize the results in Table 5, 6, and 7, TWSLKPCA demonstrates its advantage in the incipient fault detection.

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4.3 Some discussions 4.3.1 Online computation efficiency A noteworthy problem about the proposed TWSLKPCA method has higher computation complexity than the traditional KPCA and SLKPCA methods. In order to evaluate the implementation efficiency of the four methods of KPCA, SLKPCA, WSLKPCA and TWSLKPCA, we introduce an index called the online computation time per sample (CTime), which means the computer program running time for monitoring each sample online. We run the online monitoring programs of each method 20 times on the same computer, which is configured with Intel CoreTM i7-6700 CPU (3.4GHz) and 8G RAM memory. We list the average CTimes per sample in Table 8. By investigating this table, KPCA needs the least online CTime of 0.0035s. SLKPCA has a little higher time cost, which has the CTime of 0.0043s. This is because SLKPCA further applies the statistical local analysis, which leads to more computations. When WSLKPCA is applied, its CTime is 0.0081s, higher than the SLKPCA method. WSLKPCA improves the monitoring performance by utilizing the sample weighting strategy, which needs a lot of sample distance computations. That is the reason for the increased CTime. In contrast to these mentioned methods, TWSLKPCA has the highest computation complexity, which carries out the both sample and component weighting strategies. Especially, the component weighting strategy involves the sensitive component determination based on the mutual information analysis, which brings an obvious CTime increasing. However, the CTime of TWSLKPCA is 0.1543s, which is still far from 1s. Considering that most of the chemical process DCS systems collect the data at a period of 1s and the industrial computer usually has a higher computation configuration, it is practicable to implement the proposed method for online monitoring. Table 8: The average computation time per sample in online monitoring procedure by the four methods Method KPCA SLKPCA WSLKPCA CTime(s) 0.0035 0.0043 0.0081

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4.3.2 Parameter selection In the proposed method, there are many important parameters including the tuning parameter β1 in Eq. (16), the tuning parameter β2 in Eq. (28), the upper limit xlim in Eq. (17), and the moving window length q. It is difficult to determine their optimal values theoretically. In fact, the selection of these parameters is usually problem-dependent. In this paper, we analyze their influences by experimental testing and give some empirical values. In the following discussions, three indices of the average fault detection time (AFDT), the average fault detection rate (AFDR), and the average fault false rates (AFAR) are used for performance analysis. AFDT is the average of the T 2 and SPE fault detection times for all the five tested TE process datasets. Similarly, AFDR is the average of T 2 and SPE fault detection rates, while AFAR is the average of the false alarming rates for all the tested datasets. The tuning parameter β1 : This parameter is used in Eq.(16) to tune the influence of the distance d h on the weight wh . Considering that the fault sample’s distance d h at the incipient stage is not very significant in contrast to the normal samples’ distances, β1 is set to a value bigger than 1 to give a zoom-in investigation. We change β1 to 1, 5, 10, 15, 20, and list the WSLKPCA and TWSLKPCA monitoring results in the Table 9, where S-No. represents sample number. As β1 is designed for the sample weighting strategy, its influence to the WSLKPCA monitoring results is very clear. We observe that when β1 is increased, the AFDTs are reduced and the AFDRs are increased, which means the fault detection results are improved. However, the large β1 may lead to more false alarms. When the β1 is larger than 10, there will be more false alarming samples. The TWSLKPCA monitoring results give the similar conclusions. When β1 is set 1, the TWSLKPCA is with a high AFDT of 404.6 and a low AFDR of 45.2%, while the false alarming index AFAR is also low as 0.25%. For the bigger β1 values, the TWSLKPCA’s fault detection index AFDR is clearly increased while the false alarming index AFAR is also increased on the whole. Based on the above discussions, it is reasonable to set the β1 value between 10 and 15. In this paper, we select its value as 10. The tuning parameter β2 : The parameter β2 is used in the component weighting strategy, 29



Table 9: The WSLKPCA and TWSLKPCA monitoring results with different β1 values

β1 1 5 10 15 20

WSLKPCA TWSLKPCA AFDT AFDR AFAR AFDT AFDR AFAR (S-No.) (%) (%) (S-No.) (%) (%) 467.00 36.42 0 404.60 45.20 0.25 466.20 38.48 0 404.30 46.38 0.37 433.80 41.23 0 404.20 47.86 0.37 430.00 42.64 0.06 404.10 51.11 0.37 429.70 43.59 0.13 403.90 51.56 0.50

which only affects the TWSLKPCA method. In the component weighting procedure, the single sensitive component wrh, j has a large fluctuation and may lead to the high false alarm rate. Therefore, β2 should be set to a value smaller than 1. We test different β2 values including 0.1, 0.3, 0.5, 0.7, and 0.9. The corresponding results are demonstrated in the Table 10. This table indicates that the large β2 values lead to the small AFDTs and the high AFDRs, which mean good fault monitoring performance. However, too large β2 values also brings the high AFARs. When β2 is set to 0.9, the AFAR is up to 5.38%. Considering that 99% confidence limit is applied, this is very high false alarm rate. That means many normal samples are mistaken as the faulty samples. By balancing the fault detection performance and the normal monitoring performance, the β2 is selected as 0.5 in this paper. Table 10: The TWSLKPCA monitoring results with different β2 values

β2 0.1 0.3 0.5 0.7 0.9

AFDT AFDR (S-No.) (%) 412.40 44.88 406.90 45.80 404.20 47.86 401.70 49.84 365.70 52.25

AFAR (%) 0 0.06 0.37 1.38 5.38

The upper limit xlim : This parameter is to limit the maximal value of weights. If xlim is set to a small value, the weights computed by the sample and component weighting strategies can not highlight the faulty samples. However, a very large xlim is also not necessary, because it leads to a large weight and further bring a huge statistic. For testing its influence, we change its value to 1, 5, 10, 15, and 20, respectively, and the monitoring results of WSLKPCA and TWSLKPCA are 30


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listed in the Table 11. When xlim = 1 , the weighting strategy is disabled. So WSLKPCA and TWSLKPCA has the same results. An increase of xlim can bring a slight performance improvement. However, the performances for xlim = 5, 10, 15, 20 are very similar generally. Therefore, the proposed methods are not very sensitive to the parameter xlim . In this paper, we set xlim to 10. Table 11: The WSLKPCA and TWSLKPCA monitoring results with different xlim values xlim 1 5 10 15 20

WSLKPCA TWSLKPCA AFDT AFDR AFAR AFDT AFDR AFAR (S-No.) (%) (%) (S-No.) (%) (%) 467.40 35.34 0 467.40 35.34 0 433.80 41.18 0 405.10 46.53 0.37 433.80 41.23 0 404.20 47.86 0.37 433.80 41.24 0 404.00 48.71 0.37 433.80 41.24 0 403.90 49.36 0.37

The moving window length q: This is an important parameter influencing the method performance. Generally speaking, the larger the window width q is, the stronger the role of historical information is. It can accumulate fault information and help to detect incipient faults effectively. However, if the window width is too large, the current data may be concealed by the historical data, and it will be not conducive to the initial detection of the faults especially the significant faults. That means the larger window width results in the large detection time delay. If the window width is too small, the data cumulation effect will be weakened, which does not benefit the detection of incipient faults and leads to the low fault detection rate of the incipient faults. Moreover, due to the existence of process noises, the small window width may also lead to the increase of false alarm rate. In this paper, the influence of the window width q is analyzed by experiment testing. We set the different q values from 40 to 180 with the interval of 20, and the monitoring results of SLKPCA, WSLKPCA and TWSLKPCA are tabulated in Table 12. We firstly investigate the SLKPCA monitoring results with different q values. When q is set to 40, SLKPCA has a high AFDT of 615 and a low AFDR of 27.01%. That means the faults are not quickly detected with a lot of missing alarming fault samples. When q is increased to 60, the AFDT is reduced to 422.3 and the AFDR is promoted to 30.07%. This indicates that a larger 31



window width q is beneficial to incipient fault detection. However, when q is increased from 80, both the AFDT and the AFDR are increased gradually, which means that too large q leads to a large fault detection delay although more fault samples are detected. When q is increased from 60 to 180, the AFAR is decreased correspondingly. This show that a large q is helpful to reduce the false alarming sample. The results validate our above analysis on the influence of the window width. The WSLKPCA’s performance change trend is similar to the SLKPCA’s. Compared with the SLKPCA method, WSLKPCA obtains the better fault monitoring performance with the smaller AFDTs and the larger AFDRs. However, WSLKPCA’s normal sample monitoring results are worsen with a high false alarming index AFAR. With the two-dimensional weighting strategy, TWSLKPCA obtains the highest AFDR and the smallest AFDT among these three methods, but it also leads to the higher AFAR. It should be noted that even with a large q, the fault may be earlier detected by TWSLKPCA with a small AFDT. That means the two-dimensional weighting strategy is helpful to weaken the influence of q. Taking into consideration of all these things, we set the parameter q to 140 for the TE process monitoring in this paper. Table 12: The average fault detection times (AFDTs), average fault detection rates (AFDRs), and the average fault false rates (AFARs) of SLKPCA, WSLKPCA, and TWSLKPCA for the five tested TE process faults q 40 60 80 100 120 140 160 180

SLKPCA WSLKPCA TWSLKPCA AFDT AFDR AFAR AFDT AFDR AFAR AFDT AFDR AFAR (S-No.) (%) (%) (S-No.) (%) (%) (S-No.) (%) (%) 615.00 27.01 2.25 380.60 36.01 9.06 352.50 44.97 13.69 422.30 30.07 3.19 380.20 40.50 5.38 336.60 49.59 5.56 419.80 33.11 2.25 385.00 41.30 4.56 338.50 50.38 4.81 427.40 34.74 1.81 422.90 40.66 3.75 397.30 48.81 4.00 464.90 35.60 0.38 428.10 41.18 1.00 402.10 47.83 5.31 467.40 35.34 0.00 433.80 41.23 0.00 404.20 47.86 0.37 468.30 36.84 0.00 435.80 41.44 0.00 372.80 48.85 0.94 469.60 37.81 0.00 432.70 42.64 0.00 369.10 54.15 1.00

Usually the parameters are determined orderly but not simultaneously. The SLKPCA-related parameter, that is the moving window length q, is firstly determined. Then, the WSLKPCA-related

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parameters, including the tuning parameter β1 and the upper limit xlim , are selected. Lastly, the parameters used in TWSLKPCA are analyzed which include the parameter β2 . Regarding the parameters determination problem, this paper only provides some empirical rules based on the simulations, and authors would like to point out that the optimal selection of the parameters is an open problem and deserves the further study.

4.3.2 Detection of significant fault Although the proposed method is designed for the incipient fault detection, it still has a good fault detection capability when the significant faults occur. To demonstrate this, we give a monitoring case on one step fault. The fault IDV-1 of the TE process is the step change of feed ratio, which is a typical step-type fault. Its monitoring charts are listed in Fig. 13. The corresponding monitoring results are listed in Table 13. By this figure, it can be easily detected by traditional KPCA method. The FDTs of KPCA T 2 and SPE is the 167-th and 163-th sample, respectively, which can give the

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Table 13: The fault detection times (FDTs)(Sample No.) and fault detection rates (FDRs) (%) of KPCA, SLKPCA, WSLKPCA and TWSLKPCA for TE process IDV-1 fault Fault

KPCA SPE 167 163 99.38 99.75 T2

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SLKPCA T2 SPE 173 170 98.50 98.88

WSLKPCA TWSLKPCA T2 SPE T2 SPE 173 170 169 165 98.50 98.88 99.00 99.50

alarm signal timely. The KPCA FDRs are 99.38% and 99.75% respectively. When SLKPCA is applied in Fig. 13(b), the FDRs are 98.5% and 98.88%, respectively, which are a little lower than the KPCA method. The detection times of SLKPCA are delayed compared with KPCA. SLKPCA’s T 2 and SPE statistics detects the fault at the 173-th and 170-th sample, respectively. This is because the improved residual vector in the SLKPCA is computed based on the moving window technology, which monitors each sample with a data window. This will lead to the increase of the detection delay. WSLKPCA has the same monitorig performance to SLKPCA. This is because the sample weighting strategy in WSLKPCA does not work on the step faults. When TWSLKPCA is applied to this fault, its T 2 and SPE FDTs are the 169-th and 165-th sample, respectively, and the FDRs are 99% and 99.5%, respectively. The component weighting strategy brings a slight performance boost in comparison to the WSLKPCA method. To compare the TWSLKPCA and KPCA methods, KPCA performs a little better than TWSLKPCA for the significant step faults. On the whole, KPCA does a little better in the case of step fault detection while TWSLKPCA performs clearly better in the case of incipient fault detection. In the real applications, we can consider the combination of different monitoring to ensure the comprehensive monitoring of different kinds of faults.

5. Conclusion A novel two-dimensional weighting strategy based on the enhance SLKPCA method has been proposed for incipient fault detection. Based on the basic SLKPCA method, which utilizes the kernel components to construct the residual vector for better incipient fault monitoring , this paper aims at reinforcing its performance by mining the local sample and residual component information. 34


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Specifically, the contribution lies in three aspects. Firstly, a two-dimensional weighting framework is designed to build an improved SLKPCA method, which considers the weight information at the sample and component dimension. Secondly, to mine the local information hidden in the samples, a sample weighting strategy is developed based on the distance analysis. Thirdly, considering the difference of residual components, a component weighting strategy is put forward by analyzing the correlation degree between the components and the monitored variables. Simulations on a simulated numerical system and the benchmark TE process have confirmed that the proposed method has a superior performance over the existing KPCA and SLKPCA methods.

Acknowledgements This work is supported by the Shandong Provincial Key Program of Research and Development under Grant 2018GGX101025, the Fundamental Research Funds for the Central Universities of China under Grant 17CX02054, the National Natural Science Foundation of China under Grant 61403418 & 21606256, and the Natural Science Foundation of Shandong Province under Grant ZR2016FQ21 & ZR2016BQ14.

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Table of Contents (TOC) Graphic is listed as follows.

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Incipient fault detection for chemical processes using two-dimensional

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