Adaptive Selective Ensemble Independent Component Analysis

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Process Systems Engineering

Adaptive Selective Ensemble Independent Component Analysis Models for Process Monitoring Zhichao Li, and Xuefeng Yan Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00591 • Publication Date (Web): 29 May 2018 Downloaded from http://pubs.acs.org on May 29, 2018

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Industrial & Engineering Chemistry Research

Adaptive Selective Ensemble Independent Component Analysis Models for Process Monitoring Zhichao Li, 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

Abstract: Independent component analysis (ICA) has been widely used in non-Gaussian industrial process monitoring. However, the stability of performance and the determination of dominant ICs are still main problems for ICA. Constructing a monitoring model to achieve best performance for different faults is a great challenge owing to the diversity and unknowability of faults. This study develops an adaptive selective ensemble ICA models method to improve the monitoring performance. Ensemble learning based on bagging algorithm is adopted to enhance the stability of ICA. According to the difference of ICs selected for ICA modeling and

*

Corresponding Author: Xuefeng Yan

Email: [email protected] Address: P.O. BOX 293, MeiLong Road NO. 130, Shanghai 200237, P. R. China Tell: 0086-21-64251036

ACS Paragon Plus Environment

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hierarchical clustering, corresponding model sets are constructed for each sample subset. To ensure the accuracy of each sub model, an adaptive method based on just-in-time learning is proposed to model selection. Bayesian inference is applied to determine the final monitoring index. The validity of the proposed approach are attested through a numerical example, TE benchmark process and wastewater treatment plants.

Keywords: Independent component analysis; Process monitoring; Adaptive selective ensemble; Bayesian inference


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1. Introduction Modern industrial processes trend to be more scaled and more complicated with development of science and technology. To ensure the security of the production process and improve the product quality, process monitoring and fault detection in industrial production has become one of the research hotspots of industrial system.1-5 Moreover, the rapid development of computer technology and wide application of distributed control system in the industrial process allow a large of amount of process data to be collected and stored, which makes the data-driven methods attract increasing attention.6-9 Among these methods, principal component analysis (PCA) and partial least squares are considered as two fundamental but practical technologies, and they have been extensively applied to process monitoring.10-13 Both methods extract the main information of high dimensional and variable related process data using dimension reduction strategy so as to analyze the operation status of the process. However, they require the process data to obey the multivariable Gaussian distribution, which is inconsistent with the actual industrial process. Therefore, the monitoring performance may be decreased when they are applied to monitor nonGaussian processes. To address this problem, independent component analysis (ICA) has been introduced into process monitoring.14-19 Owing to the use of non-Gaussian information and high-order statistics when extracting the features from original data, the features extracted by ICA algorithm is of statistically independent non-Gaussian sources. Based on this point, ICA-based monitoring methods have received extensive attention. Albazzaz and Wang suggested a new method for driving statistical process control charts on the basis of ICA and achieved better monitoring performance in batch processes.20 To monitor the non-linear process, kernel-based ICA algorithms were proposed in literatures.21-22 For dynamic process monitoring, P.P. Odiowei et al.


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proposed a state space based ICA method, which took both of the nonlinearity and dynamics of process into account.23 Some researchers combined ICA with support vector machine for better process monitoring.24-25 To further improve the monitoring performance, Rashid et al. proposed to combine multidimensional mutual information with ICA to quantitatively evaluate the variations between the IC sunspaces of the normal operation data and the test sets.26 He et al. combined multikernel independent component analysis with adaptive rank-order morphological filter achieving fault diagnosis of nonlinear processes.27 Although ICA-based process monitoring methods have illustrated their validity and practicability in non-Gaussian process, they still suffer from some drawbacks. First, owing to the random initialization in fastICA algorithm, ICA may obtain different solutions and result in different monitoring effect.16, 19 Aim at this problem, Lee et al. proposed modified ICA method, which utilize the principal components obtained by PCA algorithm as the initial ICs in fastICA algorithm and keep the variance of ICs unchanged.19 The second challenge is determining the proper ICs used to construct ICA model. To solve this problem, the most commonly used dimension reduction criteria in the literature are the following: 1) Lee et al. exploited the Euclidean norm of the rows of separating matrix to sort the ICs;16 2) Back and Weigend suggested to determine the importance of ICs according to the L∞ norm values of ICs;28 3) Hyvärinen thought the importance of ICs is related to the degree of their non-Gaussianity;29 4) Wang et al. selected the appropriate ICs from the perspective of data reconstruction error.30 However, none of them can dominate all the time in the absence of prior knowledge as illustrated by Tong et al.31 In fact, owing to the diversity and complexity of process data, establishing a single deterministic monitoring model based on a certain dimension reduction criterion to achieve an outstanding detection capability for the unknown fault information is difficult.


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Different fault information may be reflected in different ICs. To emphasize the importance degrees of different ICs in different faults, many scholars have done related work. Jiang et al. proposed the adaptively weighted ICA and significantly improved the detection accuracy compared with conventional ICA methods.32 Cai et al. proposed weighted kernel ICA based on Gaussian mixture model and further enhanced the monitoring effect of nonlinear and nonGaussian process.33 GE et al. suggested to utilize the fault information to determine the corresponding dominant ICs.34 Jiang et al. selected the optimal ICs for each fault based on the cumulative change rate of I2 statistic.35 Although the methods in various way, they determine the importance of different ICs or select the optimal ICs to establish the monitoring models according to the different sample information. Therefore, choosing appropriate ICs or monitoring model for the newly collected sample data is very necessary for process monitoring. Ensemble learning is one of the most popular directions in machine learning and has been applied in many areas,36-39 including process monitoring.31, 34, 40-42 Its basic idea is to combine multiple base learners to obtain better predictive performance than could be obtained from any of the constituent base learner alone. Due to the use of multiple learners to predict the data set, the generalization ability of the overall learner is improved. Of course, the base learners must require two conditions: as more accurate as possible, and as more diverse as possible. Therefore, the ensemble learning can be adopted to solve the instability problem of ICA algorithm. Bagging algorithm, which is especially helpful to unstable learning algorithm, is adopted to construct the base models and to improve the overall performance. Moreover, we propose a method of adaptive selective ensemble ICA models (ASEICA) based on fault detection rate (percentage of the samples with monitoring index above the control limit) in a moving window to guarantee the accuracy of each base model. As above analysis, just selecting several ICs to construct a single


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deterministic monitoring model to achieve best detection performance for all the faults is almost impossible. According to each sample subset obtained by bagging algorithm, this study constructs corresponding model sets for each base model. Take bth sample subset for example, the strategy is described as follow. Numerous models are firstly generated by randomly selecting ICs, and then hierarchical clustering is adopted to model clustering based on the similarity of selected ICs among constructed models. When the number of the clusters meets our requirement, a model that randomly selected from each cluster is retained. All the retained models are used as the model adaptive selection ranges. Because the diversities among the selected ICs of these models have been as large as possible, they have the potential to adapt to all kinds of fault information. After the model adaptive selection is completed (the base learners are determined) Bayesian inference is applied to combine the different results into a final monitoring index to represent process running status. The rest of the paper is arranged as follows. Section 2 briefly introduces conventional ICAbased process monitoring method, and then the necessary of the proposed method is illustrated through a simple numerical example. Section 3 provides a detailed description of the proposed method. The numerical simulation in section 2, the Tennessee Eastman (TE) process and the wastewater treatment plants (WWTPs) are adopted to test the effectiveness of the proposed method, and the results are presented in Section 4. Finally, Section 5 will elaborate the conclusion. 2. Preliminaries This section first briefly reviews the process monitoring method based on ICA, and then a numerical simulation is used to demonstrate the motivation of this paper. 2.1 ICA-based process monitoring


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The basic idea of ICA is to extract and recover the statistically independent non-Gaussian sources from the linear mixed signal. Suppose the ICs needed to be estimated are S=[s1, s2,···, sr]T∈Rr×n (r is the number of the ICs, n is the number of samples), the observed data is X=[x1, x2,···, xm]T∈Rm×n (m is the number of observed variables, n is the number of samples), then16:

X = AS + E ,

(1)

where A∈Rm×r is called mixing matrix and E∈Rm×n is the residual matrix. The purpose of the ICA algorithm is to obtain the ICs s and mixing matrix A when only the training X data is known. Or, from another perspective, it can be equivalent to find the separating matrix W∈Rr×m, so that the the estimated ICs can be gained as follows:

Sˆ = WX .

(2)

The recombinant ICs( Sˆ )obtain the greatest independence. It is assumed that the variance of

ˆ ˆ T = I . Whitening is the first step of ICA algorithm to eliminate correlations Sˆ is unit matrix, SS among observed variables. Based on the singular value decomposition (SVD), the vector after whitening can be represented as follows: -

1

Z = Λ 2U T X = QX ,

(3)

where Λ and U are the eigenvalues and eigenvectors obtained by SVD decomposition of

XX T . Then Eq.(3) can be further transformed into Eq.(4): Z = QX = QAS = BS .

(4)

Define ZZ T = I , then it can be derived that B is an orthogonal matrix through the following formula: ZZ T = BSS T B T = BB T = I .

(5)


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Therefore, we can conclude that W = B T Q . Randomly initialize each column (bi) in B and update iteratively until the non-Gaussianity of each IC ( si = biT z ) reach the maximum (the details can be referred to reference 16). For online process monitoring, two statistics, namely I2 and Q, are calculated to indicate the process operation status after obtaining Sˆ , which can be gained as follows: I 2 ( t ) = skT,t sk ,t ,

(6)

Q ( t ) = et T et , et = xt − Asˆk ,t .

(7)

sk, t represents the selected top k ICs when monitoring the tth sample. Owing to the ICA does

not consider the specific distribution of the data, kernel density estimation (KDE) is applied to calculate the control limits of the statistics.16 2.2 Motivation demonstration It is well-known that the FastICA-based monitoring models can lead to instability of fault detection results thanks to its random initialization, which has been mentioned in34,

40, 42

.

Therefore, to demonstrate the necessary of the proposed method, this part mainly analyzes the influence of selecting different ICs to establish monitoring models on monitoring performance. An numerical example proposed by Ku et al. 43 is employed in this paper, which is also adopted in 16, 32. It can be described as follows:

 0.118 −0.191 0.287  1 2 r ( i ) =  0.847 0.264 0.943  r ( i − 1) +  3 −4 u ( i − 1) ,  −0.333 0.514 −0.217   −2 1 

g (i ) = r (i ) + v (i ) ,

(8)

 0.811 −0.226   0.193 0.689  u (i ) =  u ( i − 1) +    h ( i − 1) ,  0.477 0.415   −0.320 −0.749 


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where h is the input vector, whose elements following the uniform distribution from -2 to 2. v is the noise vector, whose elements following the Gauss distribution with the mean value of 0 and variance of 0.1. The observed variables x are composed of input u and output g, x(i)=[gT(i), uT(i)]T. r and h are not measured. 200 normal samples are collected as the training data. Two

fault conditions (each condition comprises 200 samples) are constructed as follows:

A step change of h1 by 3 is introduced at the 51st sample;

A step change of h2 by 1.3 is introduced at the 51st sample;

FastICA-based process monitoring method is applied to this process, and the top three ICs determined by L2-ICA criteria are selected to establish monitoring model. Figure 1 shows the monitoring charts for the two faults. Figure 1(a) presents that the IC space can timely and effectively detect the abnormal situation when faults 1 occurs. However, neither IC space nor residual space can achieve significant monitoring performance for fault 2, e.g. high missed detection rates and high detection delays. Figure 2 shows the monitoring charts of ten models for the two faults, which are constructed by randomly selecting three ICs. Table 1 provides the specific ICs selected in each model. The 2nd model, which selects the top three ICs, is equivalent to the L2-ICA model. According to Figure 2, we can conclude that the 2nd, 3rd, 4th, 5th, 8th and 9th models are very suitable for the detection of fault 1; while the 1st, 3rd, 4th, 5th, 7th and 10th models can facilitate the detection of fault 2. By calculating the I2 statistic of ith IC as follows:

Ii2 = xTnew wTi wi xnew ,

(9)

we can gain the degree of response of each IC to different faults, which are shown in Figure 3. Combining Figure 3 and Table 1, the 1st IC that can best reflect the information of fault 1 is selected to construct the 2nd, 3rd, 4th, 5th, 8th and 9th models, and the 4th IC that can best


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reflect the information of fault 2 is selected to construct the 1st, 3rd, 4th, 5th, 7th and 10th models. Consequently, using a single IC selection criterion to establish a determined monitoring model is difficult to effectively detect all faults. Different faults information may be reflected into different IC spaces (different monitoring models). Therefore, constructing different monitoring models based on the difference of the selected ICs to adapt different fault information is necessary. 3. Proposed ASEICA for Process Monitoring This section provides a detailed description of the proposed method in this paper. 3.1 ICA ensemble models based on bagging algorithm and Bayesian inference Bagging algorithm based on the bootstrap sampling is the most famous representative of the parallel ensemble learning method. Given a normal operation data set X∈Rm×n, we firstly randomly take out one sample and put it into the sample set. Then, put the sample back to the initial dataset, so that the sample can still be selected in the next sampling. After N random sampling operations, a sub sample set containing N samples is obtained. Repeat the above operations until we acquire expected T sub sample sets. Therefore, the bth sub sample set can be expressed as Xb∈Rm×N (b=1,2,3,…,T). ICA algorithm is then individually implemented based on these sub sample sets to extract the ICs. Take Xb∈Rm×N for example, sb = W b X b .

(10)

where sb represents all the calculated ICs, Wb is the separating matrix obtained by Xb. The analysis in section 2 shows that different fault information may be reflected in different ICs. In the absence of prior knowledge, we cannot know which ICs can accurately represent the


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fault information. Hence, all the ICs in sb are assumed of equal importance. T model sets are constructed according to each sample subset, and the models in the same model set satisfy certain diversity in the selection of ICs when modeling(the details is provided in Section 3.2). According to an adaptive selection criterion, an appropriate model will be picked up from each model set for the current sample (the details is provided in Section 3.3). In this way, a total of T ICA models are obtained, and each of them is considered of the best model in their own model set. The specific process is shown in Figure 4. For a new sample xnew, the I2 statistic in selected model b (b=1,2,…,T, T is the number of sub models) can be calculated as follows:

sˆb ,k = Wb ,k xnew ,

(11)

Ib2 = sˆbT,k sˆb,k ≤ Ib2，lim ,

(12)

where Wb,k∈Rk×m represents selected k row vectors from Wb, sˆb,k indicates the selected ICs that used to construct model b, I2b,lim is the control limit calculated by KDE. As a method of nonparametric estimation, KDE has been widely applied to calculate the control limits in process monitoring, especially when the distribution of data is unclear. Gaussian kernel function is the most frequently used one. For one-dimensional statistic Ib2, the univariate kernel estimator can be given as bellow:

( )

fˆ Iˆb2

(

 Iˆ 2 − I 2 b b ,i  exp  − = ∑ 2 2h nh 2π i =1   1

n

)

2

  ,  

(13)

2 where Iˆb is the estimated data, Ib2 is the collected data; n is the size of the Ib2 ; h is the

smoothing parameter and is commonly given empirically. The point where the Ib2 value presents a 97.5% confidence level is set as the threshold of the Ib2 statistic.


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After the detection results of each model are gained, a reliable combination method should be applied to fuse these results into a final monitoring index. Bayesian inference has gained extensive attention in process monitoring and tis effectiveness has been also demonstrated 31, 40-42, 44

. It converts traditional monitoring statistics to the corresponding probabilistic information. For

model b, the statistic (Ib2) is transformed to the fault probabilistic information according to the following formula:

PI 2 ( F | xnew ) =

PI 2 ( xnew | F ) PI 2 ( F ) b

b

PI 2 ( xnew )

b

,

(14)

b

where PI 2 ( x new ) = PI 2 ( x new |N ) PI 2 ( N ) + PI 2 ( x new |F ) PI 2 ( F ) , b

b

b

b

(15)

b

N represents the normal operation condition; F represents the fault operation condition; PI 2 ( F ) indicates the probability of failure occurrence; PI 2 ( N ) gives the probability of normal b

b

operation of the process, which is set to α in this paper. The conditional probabilities are defined as follows:

 I2  PI 2 ( xnew |N ) = exp  − 2 b  ,  I  b  b,lim 

(16)

 I b2,lim  PI 2 ( xnew |F ) = exp  − 2  . b  Ib 

(17)

Therefore, the final monitoring index can be calculated through weighted averaged of the PI 2 ( F | x new ) (b=1, 2, …, T) of each model, which is given as follows: b

   PI 2 ( F | xnew ) PI 2 ( F | x new )  b BIC I 2 = ∑  b B  b =1   P F | x ( ) ∑ new I b2   b =1 B

(18)


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When the value of BICI 2 exceeds the control limit (1-α), it means that the process is in fault condition. Otherwise, the process is running properly.

3.2 Models construction for each sub sample set The proposed method exhibits difficulty in determining the exact number of ICs that should be retained in each model. Only by choosing the right quantity of ICs, the proposed method can achieve satisfactory results. On the contrary, too many or too few ICs will result in undesirable monitoring performance. On the premise that false alarm rate (FAR) is within the production requirements (for example, FAR≤0.05), the determination of k should make the fault detection rate as high as possible. After numerous experimental tests, the monitoring performance of the ensemble model is found to satisfy our requirements when k is the 25%-50% of the amount of ICs. If k is small, the FAR will be large; if k is too large, the models in the same model set are more similar and the adaptability of the model set will be decreased. For bth sub sample set, the conventional fastICA algorithm is implemented firstly, and all the calculated ICs (Wb) are saved with equal importance. Next, the number of ICs used to construct ICA model is set to k empirically. Then, numerous models are built by randomly selecting k ICs, following with a screening step to obtain several models (model set for bth sub sample set, Mb) with a certain degree of diversity in the choice of ICs. In this way, Mb may have the ability to adapt to various faults. Actually, when k ICs are selected to construct a ICA model, a guiding vector hi∈R1×M (M is the total number of ICs) consisting of “0” and “1” is also generated at the same time. “0” indicates the IC of the corresponding location in sb is not selected, while “1” indicates the IC of the corresponding location in sb is selected. Therefore, a total of k elements are set to 1, and the others are set to 0. Then, the distance of any two models can be represented by the Euclidean


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distance of their corresponding guiding vectors. Suppose that the guiding vectors of any two models (Hi and Hj) are hi and hj, respectively. The distance between Hi and Hj can be calculated as follows, d ( H i , H j ) = dist ( hi , h j )

(19)

where dist(hi, hj) represents the Euclidean distance between the guiding vectors hi and hj. The larger value the d(Hi, Hj), the greater the diversity of ICs selection between Hi and Hj. Mean linkage hierarchical clustering is adopted to cluster the numerous models. It is an algorithm using a bottom-up aggregation strategy. It first treats each model in the model set as an initial cluster. Then at each step of algorithm operation, the two nearest clusters are found to be merged. The process is repeated until the preset number of clusters (C) is reached. In mean hierarchical clustering, the distance between two clusters is represented by the average Euclidean distance between the guiding vectors in clusters. Given two model clusters A and B, the distance can be calculated as follows: d ( A, B ) =

1 A B

∑ ∑ d (A,B ), i

j

(20)

Ai ∈ A B j ∈B

When the clustering step is finished, one model is randomly picked out from each cluster as the representative. Therefore, a total of C models in model set Mb (b=1, 2… T) form the model adaption range for model b.

3.3 Models adaptive selection scheme To ensure the detection accuracy of model b (b=1, 2… T), a model adaptive selection scheme based on just-in-time detection information is proposed in this paper. When monitoring a new sample xnew, finding the best models from each model sets at current time is the first step. Hence, to measure the importance of each model in Mb in real time, we characterize the importance of


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each model by using the number of sample points over the thresholds detected by each model in the time window (set the window width to w, which is suggested to 20-100. If the value is too large, the update of the model will be affected by more old information. If the value is too small, the update of the model may be affected by noise). The larger the value is, the more important the corresponding model. In this study, the model with the largest detection quantity is selected as the optimal monitoring model for the sample at the current moment. Remarkably, in the early 2 stages of the process (runtime is less than w), we use the mean value of C models’ Ib,c,t (c=1,

2, …, C) as the statistic of model b. Subsequently, the monitoring statistic at t moment can be calculated as follows:

I b2,t

 1 C 2 , t≤w  ∑ I b , c ,t =  C c =1 , 2  I , c = index[max( FDN ,L , FDN )], t > w b ,1 b ,C  b , c ,t

(21)

where FDNb,c is the number of the samples that exceed the thresholds when using model c in 2 Mb to monitor the process within the window; Ib,c,t is the statistic of the cth model in model set

Mb; c=index[max(FDNb,1,…, FDNb,C)] represents the model which occupies the most samples over the thresholds. After each model set Mb (b=1, 2, …, T) gain its corresponding statistic or model, Bayesian inference will be applied to combine them into the final monitoring index to represent the status of the process.

3.4 ASEICA-Based Fault detection Scheme Offline modeling: (1) Collect two normal operation data as training data, one for model construction and another for thresholds determination; (2) Apply bagging algorithm to obtain T sub sample sets, and normalized the T sub sample sets;


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(3) For each sub sample set (e.g. Xb), use FastICA algorithm and save the corresponding separation matrix Wb and ICs sb; (4) Construct numerous models (nM) by randomly selecting k ICs from sb; and save the corresponding guiding vectors; (5) Apply KDE to calculate the thresholds of each model; (6) Mean linkage hierarchical clustering is employed to determine the model set Mb (consisting of C models); (7) Save the corresponding thresholds of Mb; (8) Repeat the steps (3)-(7) until the all the relevant model sets Mb (b=1, 2, … , T) are determined according to T sub sample sets. Online monitoring (9) Standardize the current sample xnew based on the means and variances of each sub sample set; (10) According to the model adaptive selection strategy, T models and statistics are determined for the current sample; (11) Combine the statistics into a final monitoring index BICI 2 through Bayesian inference. If the value of BICI 2 exceeds the control limit (1-α), the fault occurs. Otherwise, the process is running properly.

4. Case Study In this section, the proposed ASEICA is illustrated by the numerical example mentioned in Section 2.2, TE benchmark process and WWTPs benchmark. In addition, the monitoring performance is also compared with several current popular methods.

4.1 Numerical example


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The numerical simulation mentioned in Section 2.2 is used to test the validity of the proposed method. The two faults are considered in this part. The parameters were set as follows: T=10; nM=100; k=3; C=10; α=0.99; w=30. The monitoring results are shown in Figure 5. Compared Figure 1 (a) and Figure 5 (a), both ICA and ASEICA can effectively detect the occurrence of fault 1 with very extremely low non-detection rates (NDRs) and detection delays. However, it is obvious that the traditional ICA performs very badly when referring to fault 2. The IC space can hardly detect the fault. Though the residual space can partly detect the fault, there is still problem of high NDRs. As shown in Figure 5 (b), the NDRs are significantly reduced by ASEICA. This is because the traditional ICA only establishes a single ICA monitoring model based on a certain dimension reduction criterion, which makes it difficult to adapt to the change of fault information. Once the ICs transmitting the fault information are not selected or are dispersed, the effect of the fault detection will be affected. According to the difference of ICs selected in the establishment of ICA models, ASEICA established numerous models, which make it possible select the optimal model in real-time according to the fault information so as to ensure the accuracy of each model (Figure 6 shows the adaptive process of each model). Moreover, bagging algorithm is also used to improve the stability of ICA algorithm.

4.2 TE Benchmark Process TE benchmark process is a platform suggested by Downs and Vogel 45. It has been extensively used to evaluate the performance of process control and monitoring methods 46-47. 22 continuous variables, 11 manipulated variables and 19 compositions are considered in this paper. A set of 21 programmed faults are used to test the validity of ASEICA. Each fault consists of 960 samples, and the fault starts at 161st sample point. The data set adopted for this study is downloaded from http://web.mit.edu/braatzgroup/links.html. 500 normal operation samples are used for model


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construction. Meanwhile, we have divided the testing file for normal operating conditions into two parts: first, 480 samples are randomly selected for parameter evaluation (thresholds) and then the rest 480 samples ,which can be called fault 0, for FAR detection. The parameters were set as follows: T=20; nM=500; k=14; C=10; α=0.99; w=30. To ensure that the normal operation of the process is not affected, using fault 0 to examine the false alarm rate (FAR) is necessary. The FARs of ASEICA, ICA-I2 and Q2 are 0.03, 0.01 and 0.03, which can be negligible in practice. The monitoring performance is compared with several state-of-the-art methods, such as MICA19, KICA21, WKICA-based33, ensemble ICA31, DEMICA42. The monitoring results of above methods are shown in Table 2. As shown in the table, the monitoring performance is evidently improved by ASEICA. ASEICA makes almost all the 21 faults detection rates reach the highest level among these methods (written in bold) except for faults 11, 17 and 19 that achieved by WKICA-based method. Note that WKICA-based method is the solution to the nonlinear problem, but the FDRs are just slight higher than ASEICA’s for the three faults. This further demonstrates the powerful detection capability of the proposed method. The root cause of fault 4 originates from the reactor cooling water inlet temperature occurring as a step change. The monitoring charts of traditional ICA and ASEICA are shown in Figure 7 (a) and (b), from which we can know that ICA and ASEICA can detect the fault in time. However, traditional ICA-based method cannot show the state of failure all the time and is accompanied by high NDRs, especially for IC space. Although the residual space has a better monitoring performance, there is still some points undetected (74% abnormal samples were detected). However, almost all the fault samples were detected by ASEICA and further improved the monitoring performance. Fault 11 is a random variation of reactor cooling water inlet


18

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temperature. Fault 20 is an unknown fault in TE process. Figures 8 and 9 give the monitoring charts of ICA and ASEICA for the two faults. Both the figures draw the conclusion that ASEICA significantly improved the monitoring performance with higher FDRs than ICA.

4.3 WWTPS Benchmark Process Wastewater treatment process (WWTPs) is extremely complex, which not only contains various of physical and chemical reactions, but also biochemical reactions. Moreover, there also exists various uncertainties such as water inflow, water quality and load change, which poses a great challenge to the establishment of monitoring model for WWTPs. Based on this, the benchmark simulation model-1 (BSM1) 48 suggested by international water association has been widely used to assess the validity of the methods in many literatures 49-51. This study also adopts it to evaluate the performance of the proposed ASEICA. The BSM1 model framework is shown in figure 10. It consists of five biological reactors (5999 m3) and one clarifier (6000 m3). Note that the first two reactors are anoxic section (nonaeration) while the last three are aerated section (aeration). The clarifier is modeled as a ten layers (the height of each layer is 0.4 m) non-reactive unit. After flowing through the biological reactors, a part of wastewater circulates through the inner loop (Qa) and the other enters the sixth layer of clarifier (Qf). The top layer of clarifier is treated effluent (Qe), some of the sludge in the bottom is circulated as the carrier of biochemical reaction (Qr) while the rest is drawn out by pump (Qw). The average wastewater treatment flow is 18446 m3/d and an average biodegradable chemical oxygen demand (COD) is 300 g/ m3. The study took a total of 14 days of data, sampling interval of 15 minutes. This paper considers eight variables and the details are shown in Table 3. The influent data in dry weather (available at http://www.benchmarkwwtp.org/) is used to perform the closed-loop


19


Page 20 of 37

simulation of WWTPs and generates three normal operation data: one for offline modeling, one for control limits determination and the rest for FAR test. A total of 22 faults were designed to test the performance of monitoring methods (details shown in Table 4 and the specific meaning of the parameters refer to [53]). Every fault consists of 1344 samples (14 days) and the fault occurs from the eighth day. The parameters for ASEICA to monitor the WWTPs were set as follows: T=20; nM=100; k=3; C=10; α=0.99; w=30. The monitoring performance is compared with PCA, ICA, KPCA and MICA algorithms, and the detection results are presented in Table 5 (the maximum value of FDRs are written in bold font). The last line gives the FARs of each monitoring method. As the table shown, at the same level of FAR, ASEICA achieved the highest FDRs for all the faults among the compared methods, especially for faults 4, 9, 10, 11, 15, 16, 18-19. For faults 4, 9, 11, 15 and 18, ASEICA has raised the low FDRs obtained by traditional methods to more than 90%, even 100%. For faults 10, 16, 19-21, the FDRs are significantly improved by ASEICA (take fault 10 for example, the maximum FDR obtained by PCA, ICA, KPCA and MICA algorithms is 24%, while ASEICA detects 86% of the fault samples). The faults 4, 9 and 22 are analyzed in detail. The maximum specific growth rate of autotrophic bacteria, which is the critical parameter to characterize the growth of autotrophic bacteria, occurs as a step change in fault 4. Figure 11 gives the monitoring results of ICA and ASEICA for the fault. As the figure shows, ICA can detect 81% of the fault samples in IC space, and the residual space can hardly discover the occurrence of the fault. However, ASEICA further improved the monitoring performance by extreme high FDR (97%) and almost all the fault samples were detected. Fault 9 is a step change of v0, which represents the maximum vesilind settling velocity. Figure 12 (a) indicates that both IC space and residual space of ICA can detect the fault in time, but they


20

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are also accompanied by low FDRs, only 0.15 and 0.51 respectively. However, ASEICA can classify the normal and fault samples more accurately. ASEICA detects the fault with the FDR of 100% and the FAR of 3% Fault 22 is a drift fault occurred in SO5 sensor, which is used to measure the concentration of dissolved oxygen in reactor 5. As figure 13 shows, ICA algorithm detects the fault with high detection delays and high NDRs. The fault is not discovered until since about the 1120th sample. Compared with ICA, ASEICA has detected the fault since about the 770th sample. Therefore, ASEICA further improved the monitoring performance by lower detection delays and high FDR, which reduced the possibility of causing more serious safety accidents.

5. Conclusions A novel monitoring method based on model adaptive selection ensemble ICA is proposed in this paper. Ensemble learning is adopted to further improve the stability of traditional ICA algorithm. Multiple sub sample sets are obtained based on bagging algorithm, then ICA models are established according to each sub sample set. To better adapt to different fault information, several ICA models are constructed and selected, among which the ICs used to construct these models satisfy a certain degree of difference. An adaptive algorithm based on the number of abnormal samples in the window width is proposed to determine the optimal model for current sample monitoring. After each optimal sub model is determined, Bayesian inference is adopted to combine the results into a final probabilistic monitoring index. Cases including a numerical example, the TE and WWTPs benchmark processes have demonstrated the excellent performance of the proposed ASEICA. Meantime, there still exist some defects to be improved. Although numerous models are constructed, for large-scale industrial processes, it is still difficult to guarantee the optimal monitoring models corresponding to various fault information are


21


Page 22 of 37

contained in the model sets. Therefore, how to guarantee the quality of the model sets according to the available fault information is still worth studying.

Acknowledgments The authors are grateful for the support of the Fundamental Research Funds for the Central Universities under Grant of China(222201717006).

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benchmark of wastewater treatment process. Comput. Chem. Eng. 2008, 32, 2849-2856. (51) Chang, K. Y.; Villez, K.; Hulle, S. W. H. V.; Vanrolleghem, P. A. Enhanced process monitoring for wastewater treatment systems. Environmetrics 2008, 19, 602–617.


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Table 1. Specific ICs selected in each model Model 1 2 3 4 5 6 7 8 9 10

ICs 4 2 3 2 4 3 4 3 2 3

2 1 1 1 1 2 3 1 1 2

5 3 4 4 5 5 5 5 5 4

Table 2 Fault detection rates of the methods

ICA

MICA

WKICABASED

KICA

Fault

Ensemble ICA

DEMICA

ASE ICA

I2

Q

I2

Q

I2

Q

WI2

WQ2

BIC_ I2

BIC_ Q

BIC_ T2

BIC_ Q

BIC_ I2

1

1.00

1.00

1.00

1.00

1.00

1.00

0.99

1.00

1.00

1.00

1.00

1.00

1.00

2

0.99

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.99

3

0.02

0.04

0.01

0.01

0.01

0.03

/

/

0.02

0.01

0.02

0.08

0.08

4

0.46

0.74

0.65

0.95

0.81

1.00

0.92

1.00

0.99

0.71

1.00

0.99

1.00

5

1.00

1.00

0.24

0.02

0.25

0.28

0.26

0.28

1.00

1.00

1.00

1.00

1.00

6

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

7

0.98

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

8

0.98

0.98

0.97

0.98

0.97

0.98

0.97

0.98

0.98

0.97

0.98

0.98

0.98

9

0.02

0.06

0.01

0.02

0.01

0.03

/

/

0.01

0.01

0.03

0.05

0.07

10

0.80

0.76

0.70

0.64

0.81

0.78

0.79

0.84

0.84

0.75

0.88

0.85

0.91

11

0.44

0.62

0.43

0.66

0.58

0.77

0.51

0.82

0.67

0.56

0.75

0.71

0.76

12

1.00

1.00

0.98

0.97

0.99

0.99

0.99

0.99

1.00

1.00

1.00

1.00

1.00

13

0.95

0.95

0.95

0.94

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.96

14

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

15

0.03

0.12

0.01

0.02

0.03

0.05

/

/

0.04

0.05

0.05

0.12

0.15

16

0.77

0.70

0.76

0.73

0.77

0.87

0.79

0.90

0.87

0.74

0.91

0.87

0.93

17

0.88

0.89

0.87

0.94

0.91

0.97

0.92

0.97

0.95

0.91

0.96

0.94

0.96

18

0.91

0.90

0.90

0.90

0.89

0.91

0.90

0.91

0.90

0.90

0.90

0.90

0.91

19

0.60

0.19

0.25

0.29

0.70

0.85

0.54

0.89

0.80

0.44

0.87

0.78

0.88

20

0.79

0.71

0.70

0.66

0.50

0.65

0.55

0.68

0.90

0.76

0.91

0.83

0.91

21

0.47

0.45

0.54

0.19

/

/

0.40

0.44

0.48

0.44

0.54

0.56

0.60


29


Page 30 of 37

Table 3. Specific meanings of the considered eight variables Variable

Description

xCOD xS_S5 xS_O5 xS_NO5 xS_NH5 xS_ALK5 xQ_f xQ_a

The concentration of COD in effluent flow The concentration of readily biodegradable substrate in 5th reactor The concentration of dissolved oxygen in 5th reactor The concentration of nitrate and nitrite nitrogen in 5th reactor The concentration of NH4++ NH3 nitrogen in 5th reactor The alkalinity in 5th reactor The flow rate entering the clarifier The flow rate of internal reflux

Table 4. Description of the designed faults

Fault 1 2 3 4 5 6 7 8

Description

Fault type

Bioreactor parameters change

9

mu_H K_S b_H mu_A b_A K_X ny_h i_XB

step：4.0 to 2.0 step：10.0 to 20.0 step：0.3 to 0.62 step：0.5 to 0.8 step：0.05 to 0.2 step：0.1 to 0.03 step：0.8 to 0.4 step：0.08 to 0.086

v0

step：474 to 300

r_p

step：0.00286 to 0.005

11

f_ns

step：0.00228 to 0

12

KLA 3

step：240 to 120

KLA 4

step：240 to 120 fixed at 240

10

13

Clarifier parameters change Variation of aeration rate in biochemical reactors

14 15 16 17 18 19 20 21 22

KLA 5 external carbon flow rate change sludge discharge flow rate change External return flow change Internal return flow change Sensor fault

Reactor 2 Reactor 5

random disturbance by [0,1] Influence of Gaussian noise by N(1， 0.1)

Qw

step：385 to 100

Qr

increased to 140%

Qa

decreased to 12%

Qa

fixed at 20000

SNO2_sensor

fixed deviation (+0.5)

SO5_sensor

drift fault (slope is 0.1)


30

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Table 5. Fault Detection Rates of PCA, ICA, KPCA and MICA for WWTPs

PCA

ICA

KPCA

MICA

ASEICA

Fault T_2

SPE

I_2

SPE

T_2

SPE

I_2

SPE

BIC_I2

1

0.06

0.99

0.91

0.52

0.01

1.00

0.86

1.00

1.00

2

0.05

0.94

0.84

0.23

0.02

0.99

0.72

1.00

1.00

3

0.02

1.00

0.84

0.66

0.01

1.00

0.96

1.00

1.00

4

0.04

0.26

0.81

0.00

0.06

0.36

0.03

0.37

0.97

5

1.00

1.00

1.00

1.00

0.00

1.00

1.00

1.00

1.00

6

0.01

0.59

1.00

0.23

0.03

0.80

0.78

0.99

1.00

7

0.02

0.82

1.00

0.47

0.03

0.94

1.00

1.00

1.00

8

0.01

0.02

0.12

0.02

0.02

0.07

0.01

0.01

0.22

9

0.04

0.74

0.15

0.51

0.03

0.84

0.30

0.67

1.00

10

0.01

0.12

0.13

0.03

0.02

0.24

0.04

0.14

0.86

11

0.01

0.38

0.15

0.41

0.02

0.64

0.18

0.31

1.00

12

0.21

0.52

0.86

0.26

0.06

0.62

0.55

0.79

0.95

13

0.06

0.34

0.70

0.05

0.03

0.53

0.43

0.48

0.80

14

1.00

0.98

0.94

0.88

0.51

1.00

0.57

1.00

1.00

15

0.01

0.03

0.87

0.00

0.03

0.11

0.02

0.21

0.96

16

0.01

0.02

0.51

0.00

0.02

0.11

0.06

0.01

0.81

17

0.05

0.80

0.91

0.08

0.05

0.92

0.03

0.80

1.00

18

0.04

0.27

0.74

0.07

0.06

0.40

0.01

0.06

0.95

19

0.02

0.26

0.76

0.24

0.04

0.38

0.46

0.60

0.87

20

0.01

0.12

0.30

0.16

0.02

0.18

0.14

0.13

0.63

21

0.02

0.09

0.28

0.11

0.04

0.15

0.10

0.17

0.70

22

0.48

0.46

0.30

0.07

0.50

0.68

0.06

0.59

0.69

FARs

0.01

0.02

0.02

0.03

0.04

0.05

0.01

0.01

0.03


31


I2

I2

150

15

100

10

50

5

0

0

50

Page 32 of 37

100

150

200

0

0

50

100

150

200

100 150 (b) fault 2

200

Q2

Q2

20

60

15

40

10 20

5 0

0

50

100 150 (a) fault 1

200

0

0

50

Figure 1. Monitoring charts of L2-ICA for the two faults I2 of m1

I2 of m2

I2 of m3

I2 of m4

I2 of m5

2.5

10

10

10

10

2

8

8

8

8

1.5

6

6

6

6

I2 of m1 4

I2 of m2

I2 of m3

1.5

3

1

2

0.5

1

I2 of m4 2.5

I2 of m5 3

2

3

2

1.5

2 1

4

4

4

4

0.5

2

2

2

2

0

0

100 200

0

0

I2 of m6

100 200

0

0

I2 of m7

2

2

1.5

1.5

1

1

0.5

0.5

0

0

100 200

100 200

0

100 200

0

100 200

0

I2 of m9 10

2.5

8

8

2

6

6

1.5

4 2

0.5

0

0

0

100 200

0

100 200

0

100 200

0

0

I2 of m6

100 200

0

100 200

0

0

I2 of m7

2

4

1.5

3

1

1

2

0.5

1

0

0

100 200

0

0

I2 of m8 1.5

1

1

2 0

0

1 0.5

I2 of m10

10

4

0

0

I2 of m8

1

100 200

0

I2 of m9

0

100 200

I2 of m10

2

4

1.5

3

1

2

0.5

1

0.5

0

100 200

0

100 200

0

100 200

0

0

100 200

0

0

100 200

0

0

100 200

(a)(b)

Figure 2. The monitoring charts of ten models for the two faults: (a) fault 1 (b) fault 2


32

Page 33 of 37 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


I 21

100

I 22

10

50

I 23

6

I 24

15

4

10

2

5

6

5

0

4

0 0

100 200

I 21

10

0 0

100 200

I 22

10

5

100 200

100 200

I 23

0 0

100 200

I 24

30

0

I 25

15

20

10

10

5

100 200

5

0 0

2

0 0

10

5

0

I 25

8

0 0

100 200

0 0

100 200

0 0

100 200

0

100 200

Figure 3. Ii2 for two faults when only ith IC was selected to establish the model

Figure 4. Specific process of the proposed ASEICA EI2 0.2 0.15 0.1 0.05 20

40

60

80

100 120 (a) fault 1

140

160

180

200

140

160

180

200

2

EI 0.25 0.2 0.15 0.1 0.05 20

40

60

80

100 120 (b) fault 2

Figure 5. The monitoring charts of ASEICA


33


model 1

model 2

10

8

model 3

model 4

10

6 5

4

5

2 0

0

100 200

0

0

model 6

100 200

0

0

model 7

10

8

100 200

4

0

100 200

0

5

0

100 200

8

6

6

4

4

2

2

0

0

0

0

0

100 200 model 9

2 0

model 5

8

model 8 10

6 5

Page 34 of 37

100 200

8

6

6

4

4

2

2 0

100 200

model 10

8

0

0

0

100 200

0

100 200

Figure 6. The adaptive process of each base model (Ordinates represent the numbering of each model in the b th model sets) EI2

I2

0.18

100

0.16 50

0.14 0.12

0

0.1

0

100

200

300

400

500

600

700

800

900

1000

600

700

800

900

1000

Q2

0.08 300 0.06 200 0.04 100

0.02 0

0

100

200

300

400

500

600

700

800

900

1000

0

0

100

200

(a)

300

400

500

(b)

Figure 7. Monitoring charts of (a) ASEICA and (b) ICA for fault 4


34

Page 35 of 37 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


EI2

I2

0.8

150

0.7

100

0.6

50

0.5 0

0

100

200

300

400

500

600

700

800

900

1000

600

700

800

900

1000

0.4 Q2 600

0.3

400

0.2

200

0.1 0

0

100

200

300

400

500

600

700

800

900

1000

0

0

100

200

(a)

300

400

500

(b)

Figure 8. Monitoring charts of (a) ASEICA and (b) ICA for fault 11 EI2

I2

1.2

4000 3000

1 2000 1000

0.8

0

0

100

200

300

400

500

600

700

800

900

1000

600

700

800

900

1000

0.6 Q2 4000

0.4

3000 2000

0.2 1000

0

0

100

200

300

400

500

600

700

800

900

1000

0

0

100

200

(a)

300

400

500

(b)

Figure 9. Monitoring charts of (a) ASEICA and (b) ICA for fault 20

Figure 10. General overview of the BSM1 plant


35


EI2

I2 150

0.7

100

0.6

50

0.5

0

0

200

400

600

800

1000

1200

Q2

0.4

0.3

60

40

0.2

20

0.1

0

0

200

400

600

Page 36 of 37

800

1000

1200

0

200

400

(a)

600

800

1000

1200

(b)

Figure 11. Monitoring results of WWTPs fault 4: (a) ICA (b) ASEICA EI2

I2 60 0.9 40 0.8 20 0

0.7 0.6 0

200

400

600

800

1000

1200 0.5

Q2 300

0.4 0.3

200

0.2 100 0

0.1 0

200

400

600

800

1000

1200

200

400

(a)

600

800

1000

1200

(b)

Figure 12. Monitoring results of WWTPs fault 9: (a) ICA (b) ASEICA EI2

I2 60

0.9

40

0.8

0.6 0

EI2

0.7

20

0

200

400

600

800

1000

1200

0.5

0.04 0.03 0.02

Q2 60

0.4

0.01

0.3

0

40

700

750

800

850

900

950

0.2 20 0

0.1

0

200

400

600

800

1000

1200

200

(a)

400

600

800

1000

1200

(b)

Figure 13. Monitoring results of WWTPs fault 22: (a) ICA (b) ASEICA


36

Page 37 of 37 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


Table of Contents and Abstract Graphics


37

Adaptive Selective Ensemble Independent Component Analysis

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