Joint Probability Density and Double-Weighted Independent

Article pubs.acs.org/IECR

Joint Probability Density and Double-Weighted Independent Component Analysis for Multimode Non-Gaussian Process Monitoring Qingchao Jiang and Xuefeng Yan* Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology, Shanghai 200237, P. R. China S Supporting Information *

ABSTRACT: A multimode non-Gaussian process monitoring scheme that integrates joint probability density and doubleweighted independent component analysis (DWICA) is proposed. Analyzing each independent component (IC) individually allows estimating the probability density of each IC through a kernel density estimation. Considering that normal and abnormal statuses have different deviations that are captured by each IC, DWICA has been proposed previously for non-Gaussian monitoring performance improvements. The current work aims to extend the DWICA to the multimode version (MDWICA) for multimode non-Gaussian process monitoring. Given that the ICs are as much independent as each other, the joint probability density of the ICs is employed to identify the running-on mode. The MDWICA combines the ability of joint density in mode identification and the superiority of DWICA in fault detection, which is more appropriate for practical application. The benefits of the proposed method are demonstrated by case studies on a numerical process and the Tennessee Eastman challenge process.

1. INTRODUCTION Process monitoring has gained increasing attention with the growing demand for plant safety and product quality.1−3 With the rapid development of data gathering and advanced computing technologies, multivariate statistical process monitoring (MSPM) methods have progressed quickly in recent years.4−12 Among the MSPM methods, principal component analysis (PCA) usually serves as the most fundamental technology and is the most widely used.9,13−18 PCA can effectively handle high-dimensional, highly correlated data, and extract potential process information by projecting the process data into two low-dimensional subspaces. PCA has been extended to solve various process monitoring problems because of its efficiency.19−27 However, PCA-based monitoring methods assume that the monitored variables are Gaussian distributed and only consider the mean and variance information, thereby failing to provide high-order representations for non-Gaussian data. Independent component analysis (ICA)-based methods have been proposed and extended intensively to monitor nonGaussian processes.28−40 ICA decomposes observed data into linear combinations of ICs, which reveal more higher-order statistical information than principal components (PCs). Successful applications of ICA have been reported in both biomedical signal processing and chemical process monitoring fields.28−42 In ICA monitoring, the deviation between normal and abnormal status captured by each IC is different from each other, and the degree of deviation of the ICs cannot be determined by the sequential position.30,43 In some cases, the retained ICs hardly contain the information that is reflected on only one or few ICs, thereby submerging beneficial information for process monitoring. Jiang et al.43 proposed a double-weighted strategy imposed on ICA process monitoring (DWICA) to increase the fault detection sensitivity. DWICA evaluates the importance of each IC online and subsequently sets different © 2014 American Chemical Society

weighting values on the ICs to highlight the deviation information for process monitoring.43 With more online process information and the distribution deviation considered, the monitoring performance has been significantly improved. In DWICA, the estimated probability density is employed to evaluate the importance of ICs.43 In the current work, the joint density of ICs is employed for mode identification, thereby extending the DWICA to the multimode version. In modern chemical industries, processes are often characterized by multiple operating modes caused by product specifications, set-points, or manufacturing strategies.13,44,45 Consequently, the MSPM methods that assume single operation conditions cannot meet the multimodal monitoring requirement, and the incorporation of a multimodal model into an online monitoring scheme has become a new issue.13,44,45 To address this issue, multimode strategies have been intensively researched.46−50 Mixture models have been recently proposed and successfully applied in multimode process monitoring. Chen and Liu51 proposed a mixture PCA model and adopted a heuristic smoothing clustering technology to monitor a process with different operating states. Choi et al.52 developed a Gaussian mixture model (GMM) via PCA and discriminant analysis. Yu and Qin53 proposed a novel multimode process monitoring approach based on finite GMM and Bayesian inference strategy. This approach was demonstrated to be accurate and efficient for various types of faults in multimode processes. Feital et al.13 proposed a maximum likelihood PCA and a component-wise analysis-based multimodal modeling and monitoring approach. Ge and Song44,45 recently proposed a multimode process monitoring method based on Bayesian inference and two-step Received: February 9, 2014 Accepted: December 4, 2014 Published: December 4, 2014 20168

dx.doi.org/10.1021/ie504369x | Ind. Eng. Chem. Res. 2014, 53, 20168−20176

Industrial & Engineering Chemistry Research

Article

Figure 1. Monitoring scheme of the MDWICA.

expressed as linear combinations of dk(≤mk) unknown independent components sk = [s1,k,s2,k,...,sdk,k]T, and the relationship between the measured data and the independent components is given by35

ICA−PCA strategy. This method has been applied in both continuous and batch multimode process monitoring. Ge and Song45 employed joint probability analysis for mode identification based on three statistics (I2, T2, and SPE) generated by the PCA−ICA model and proved the effectiveness of the method.45 This study aims to develop an efficient monitoring scheme for a non-Gaussian process under different operating conditions by extending the DWICA to the multimode version (MDWICA). First, the ICA model in each operating condition is established with the assumption that the normal operating data in all the modes are available. Second, the probability density of each IC in the ICA model of each operation condition is examined, which has two usages: (1) identifying the running-on mode and (2) evaluating the importance and weighting the ICs. Finally, the process status is evaluated by a statistical weighting strategy. The rest of the article is structured as follows: section 2 demonstrates the proposed method in detail; in section 3, the method is tested in a numerical process and the Tennessee Eastman (TE) process; conclusions are given in section 4.

x k = A ksk + ek

(1)

where Ak = [a1,k, a2,k,...,adk,k] ∈ Rmk×dk is the mixing matrix, and ek is the residual matrix. The mixing matrix Ak and the independent component sk can be estimated by using the FastICA algorithm developed by Hyvarinen and Oja.54,55 Whitening is the initial step, and eigenvalue decomposition is one of the popular methods used to eliminate crosscorrelation between the measurement variables.54,55 Through the FastICA algorithm, the independent components can be calculated as35,54

sk = Wk x k

(2)

where the Wk is the demixing matrix. The ith independent component si,k can be calculated as si , k = wi , kx k (3) where wi,k is the ith projection vector in the matrix Wk. After the ICA transformation, the independent components have means of zero and are as much independent as each other. In this study, the probability density of each IC is calculated from KDE for both mode identification and importance evaluation. The KDE method was introduced by Rosenblatt56 and Parzen.57 The general idea of KDE is to compute the density of the statistical sample in a distribution by using a presumed normal distribution. For a current obtained IC score snew, a univariate kernel estimator with kernel function K is defined by57,58

2. METHODOLOGY This section presents in detail the proposed method, which includes joint probability density for mode identification and DWICA for fault detection. The first step in monitoring a multimode non-Gaussian chemical process is to identify the mode on which the current process is running. Assume that a process has M operation conditions, Mk denotes the kth mode operation, and the process data in normal conditions of all modes are available. First, an ICA model is established in each mode. For the kth mode, the process data are denoted as Xk ∈ Rnk×mk, where nk is the number of samples and mk is the number of measured variables in the kth mode. In the ICA algorithm, the mk measured variables xk = [x1,k, x2,k,...,xmk,k]T are assumed to be

f (s new ) = 20169

1 nh

n

⎧ s new − s j ⎫ ⎬ h ⎩ ⎭

∑ K⎨ j=1

(4)



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where snew is the data point under consideration, sj is an observation value from the data set, h is the window width (also known as the smoothing parameter), and n is the number of observations. The probability density can provide the possibility that the considered point belongs to the reference training data. Larger density indicates that the considered point is more likely to be within the training data, and this property is used for identifying the running-on mode in this study. For a new obtained independent component score snew i,k , the density that concerns the ith IC of the kth mode can be calculated as fi , k (sinew ,k )

1 = nk hk

new ⎧ − sij, k ⎫ ⎪ si , k ⎪ ⎬ ∑ K ⎨⎪ ⎪ hk ⎭ j=1 ⎩ nk

(5)

new si,k

where denotes the ith IC score of the current sample j obtained through the kth ICA model, si,k is the jth observation value from the normal training data set, hk is the window width (also known as the smoothing parameter), and nk is the number of observations.35,57 The kernel function K determines the shape of the bumps, and a number of possible kernel functions are present.58 In practice, the Gaussian kernel function is the most widely used.58 In this study, the Gaussian kernel is employed and the kernel estimator becomes58 fi , k (sinew ,k ) =

1 1 nk hk 2π

nk

∑ j=1

⎡ (s new − s j )2 ⎤ i,k i,k ⎥ exp⎢ − ⎢⎣ ⎥⎦ 2hk 2

(6)

Extensive studies on the choice of the window width hk have been reported, and finding a proper empirical value for each specific case has been suggested.58−60 The training data set for normal status is easy to obtain with sufficient information. Hence, a satisfactory performance of KDE can be normally guaranteed. If the current sample belongs to the k-th mode, the density f i,k(snew i,k ) would be normally larger; otherwise, the density value would be very small and close to zero. Let the event s•,k ∈ Mk denote that the ICs generated by the kth ICA model are within the kth mode. Thus, every IC is within the kth mode, that is, (s1,k ∈ Mk)∩(s2,k ∈ Mk)∩···∩(sdk,k ∈ Mk). The ICs are as much independent as each other; hence, the joint density of all ICs new f•,k(s•,k ) can be approximately calculated as dk

f•, k (s•new ,k )

=

∏ fi ,k (sinew ,k ) i=1

dk dk ⎡ nk ⎛ (s new − s j )2 ⎞⎤ ⎛1 1 ⎞ i,k i,k ⎟⎟⎥ =⎜ ⎟ ∏ ⎢∑ exp⎜⎜ − 2 ⎢ 2hk ⎝ nk hk 2π ⎠ i = 1 ⎣ j = 1 ⎝ ⎠⎥⎦

Figure 2. Monitoring results of the case 1: (a) joint densities and mode identification results; (b) fault detection results by using MICA, MWICA, and MDWICA.

(7)

The joint probability density provides an indication whether or not the current point is evaluated by its corresponding mode. Larger mode joint density indicates that the current point is more likely to be within the mode. When the point is evaluated in an unmatched mode, the joint probability will be much smaller and very close to zero. Therefore, the current mode on which the process is running can be determined as the mode with the largest mode joint probability density. After identifying the running-on mode, the next step is to identify the presence or absence of a fault in process. ICA can detect faults highly efficiently, and numerous successful applications have been reported. However, Jiang et al.43 showed that

the deviation between normal and abnormal status captured by each IC is different from each other and that constraining the ICs to the same level may hide information that is beneficial for process monitoring as well as submerge the fault information. Therefore, taking the online process information into account and highlighting the beneficial information are important. To achieve this aim, the weighting strategy can be adopted, which can be concluded into two procedures: (1) evaluate the importance of the ICs online and (2) set different weighting values on the ICs. In this study, the importance of 20170



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Figure 3. Monitoring results of case 2: (a) joint densities and mode identification result; (b) fault detection results using MICA, MWICA, and MDWICA; (c) density of each IC in mode 1; (d) density of each IC in mode 2; and (e) density of each IC in mode 3.

each IC is also evaluated based on the obtained probability density of each IC. The detailed procedures of DWICA are presented in the Supporting Information and more details on the DWICA for process monitoring have been presented by

Jiang et al., who introduced both the fault detection and the corresponding contribution plot for fault cause identification.43 The total monitoring procedures of the proposed MDWICA are summarized and illustrated in Figure 1. 20171



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3. ILLUSTRATIONS AND DISCUSSIONS In this section, the proposed MDWICA method is tested by using a numerical process and the TE challenge process. 3.1. Case Study on a Numerical Process. We consider the following modified version of the simple multivariate process suggested by Ku et al.:35,61 ⎡ 0.118 −0.191 0.287 ⎤ ⎡1 2 ⎤ ⎢ ⎥ ⎢ ⎥ z(j) = ⎢ 0.847 0.264 0.943 ⎥z(j − 1) + ⎢ 3 −4 ⎥u(j − 1) ⎢⎣−0.333 0.514 −0.217 ⎥⎦ ⎣− 2 1 ⎦ y(j) = z(j) + v(j) ⎡ 0.193 0.689 ⎤ ⎡ 0.818 −0.226 ⎤ u(j) = ⎢ ⎥g(j − 1) ⎥u(j − 1) + ⎢ ⎣−0.320 0.415 ⎦ ⎣−0.320 −0.749 ⎦

(8)

The input g is a random vector of which the element is uniformly distributed, and the ranges of g vary per mode status. The output y is equal to z plus a random noise vector v. Each element of v has zero mean and a standard variance of 0.1. Both input u and output y are measured, but z and g are not, and the data vector consists of xj = [yT(j)uT(j)]T. To simulate a multimode process, three modes of the process are considered, which are listed as follows: Mode 1: g ∼ uniform (−1, 1) Mode 2: g ∼ uniform (6, 7) Mode 3: g ∼ uniform (−9, −8) Normal data with 400 samples in each mode are generated to build the ICA mode. Two cases are designed to test the performance of multimode process monitoring (Supporting Information Table S1). The monitoring results of the two cases are presented in Figures 2 and 3. The density threshold CLDk , the weighting values α, β, and the statistical weighting window width n are empirically set as 0.05, 10, 1, and 4, respectively. The joint probability densities for the three modes and the identification results of case 1 are illustrated in Figure 2a. Only the largest joint densities in each point are plotted because the joint densities of the other modes are much too small. The runningon modes are clearly shown in Figure 2a). The monitoring results of multimode ICA (MICA), multimode WICA (MWICA), and MDWICA are presented in Figure 2b, which shows that the monitoring performance is not damaged by using the multimode and weighting strategy. The average false alarm rates of MICA, MWICA, and MDWICA are 0.020, 0.022, and 0.022, respectively, which are acceptable in practical applications. The joint densities and the mode identification results of case 2 are presented in Figure 3a. The figure shows that the running-on modes have been successfully identified by the largest densities. The monitoring results of case 2 obtained by using MICA, MWICA, and MDWICA are illustrated in Figure 3b. Multimode WICA and multimode DWICA perform better than MICA in detecting the first and third faults with lower nondetection rates. The densities along each IC are presented in Figure 3 panles c−e to further analyze the monitoring performance. Figure 3c shows the density of each IC on mode 1. Only the first IC deviates significantly from the density. Figure 3d shows the densities on mode 2, and Figure 3e shows the densities on mode 3. These figures illustrate that the ICs perform differently in reflecting the deviation information. 3.2. Case Study on the TE Benchmark Process. The TE process is presented by Downs and Vogel based on an industrial process used in the Eastman Chemical Company in

Figure 4. Monitoring results of case 1 in the TE process: (a) joint densities and mode identification results and (b) fault detection results by using MICA, MWICA, and MDWICA.

Tennessee and has been widely used for monitoring scheme testing.62 As shown in Supporting Information Figure S1, the process consists of five major unit operations: a reactor, a condenser, a compressor, a separator, and a stripper.4,8,62 More details on the TE process are presented by Down and Vogel,62 Lyman and Georgakis,63 Ricker,64 and Chiang et al.4 Various control strategies have been developed to provide a stable closed-loop operation. This study adopts the decentralized 20172



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Figure 5. Monitoring results of case 2 in TE process: (a) joint densities and mode identification results; (b) fault detection results by using MICA, MWICA, and MDWICA; (c) density of each IC on mode 1.

control structure developed by Ricker.64 The simulation codes can be downloaded at http://depts.washington.edu/control/ LARRY/TE/download.html. According to the G/H mass ratios, the process can simulate six operating modes, which are listed in Supporting Information Table S2. Operating modes 2 and 3 are selected for the simulation of the multimode processes. The control scheme by Lyman and Georgakis63 is employed to simulate mode 1 and compare the performance of MICA, MWICA, and MDWICA. For the monitoring, 22 continuous measurements and 9 manipulated variables (the steady-state values of the recycle valve and stem valve, the agitator rates are not included because they remained constant in modes 2 and 3) are selected and listed in Supporting Information Table S3. A set of 20 programmed faults that can be simulated in each mode are listed in Supporting Information Table S4. To establish the monitoring model, 500 samples under each mode are collected with a sampling time of 0.05 h. Four typical cases are designed to evaluate the efficiency of the proposed monitoring scheme (Supporting Information Table S5).

The monitoring results of the cases are further analyzed below. The density threshold CLDk , the weighting values α and β, and the statistics weighting window width n are empirically set as 0.01, 10, 1, and 4, respectively. Fault 0 in the TE process represents the normal condition of the process and is usually used for testing the false alarm performance of the monitoring schemes. In this study, normal conditions of different modes are analyzed to test the mode identification performance and the monitoring performance from the aspect of false alarms. The monitoring results of case 1 are presented in Figure 4. Figure 4a presents the joint probabilities of the modes and indicates that the process runs on mode 1 from the first to 500th points, on mode 2 from the 501st to 1000th points, on mode 3 from the 1001st to 1500th points, and back to mode 2 after the 1501st point. This identification result is in accordance with the actual situation. The monitoring results of MICA, MWICA, and MDWICA are presented in Figure 4b. This figure shows that the false alarm rates in all the three monitoring schemes are as low as 0.019. 20173



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Figure 7. Monitoring results of case 4 in the TE process: (a) joint densities and mode identification results; (b) fault detection results by using MICA, MWICA, and MDWICA.

Figure 5b. The figures show that the faults have been detected successfully. However, in the detection of fault 4 on mode 1, the nondetection rate of MICA is large. The nondetections have been significantly reduced in the MWICA and MDWICA methods. The density of each IC on mode 1 is analyzed and presented in Figure 5c to further analyze the monitoring performance. This figure shows that the densities along the 10th and 11th ICs are changed significantly when the fault occurs. This result indicates that most fault information is reflected on the two ICs. The monitoring results of cases 3 and 4 are illustrated in Figures 6 and 7, respectively. The joint densities and the mode identification results of case 3 are presented in Figure 6a, in which the running-on modes can be identified successfully. The fault detection results of the faults in case 3 are depicted in Figure 6b, in which the faults can be detected successfully. The proposed MDWICA performs the best with the lowest nondetections. The monitoring results of case 4 are illustrated

Figure 6. Monitoring results of case 3 in the TE process: (a) joint densities and mode identification results; (b) fault detection results by using MICA, MWICA, and MDWICA.

From a practical engineering aspect, these low false alarms rates can be neglected. Fault 4 in the TE process increases the reactor temperature. The monitoring results of case 2 are illustrated in Figure 5. Figure 5a shows the joint densities and the mode identification results of case 2, from which we can see that the running-on modes have been identified successfully. The fault detection results of MICA, MWICA, and MDWICA are illustrated in 20174



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in Figure 7 panels a and b, which is similar to those of case 3. MDWICA has the best performance among the three methods. The monitoring performance of MDWICA for the other faults in the three modes is also tested, and the nondetection rates are listed in Supporting Information Table S6. Fault 6 is not considered because the process is damaged by the fault on modes 2 and 3. Monitoring result comparisons of DWICA with those of other state-of-the-art methods are provided by Jiang et al.,43 who demonstrated that the fault detection and diagnosis abilities of DWICA are among the best of the monitoring methods.

4. CONCLUSIONS This study proposes a joint probability density-based multimode process monitoring scheme and extends the DWICA to the multimode version. In DWICA, the probability density of each IC is examined for importance evaluation. In MDWICA, the densities are also used for mode identification. The ICA model in each mode is established on the basis of the normal training data of a multimode process. During online monitoring, the probability density of the IC score in each mode is examined. The joint probability density is used for mode identification, and the densities on the running-on mode are used to construct the double-weighted monitoring model. Case studies on both a numerical process and the TE benchmark process indicate that the proposed method is effective.

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ASSOCIATED CONTENT

* Supporting Information S

Double-weighted ICA for fault detection and the tables and figures as mentioned in the article. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Address: East China University of Science and Technology, P.O. Box 293, MeiLong Road No. 130, Shanghai 200237, P. R. China. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the support from the following foundations: 973 Project of China (2013CB733600), National Natural Science Foundation of China (21176073), and the Fundamental Research Funds for the Central Universities.

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REFERENCES

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Joint Probability Density and Double-Weighted Independent

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