Distributed Gaussian Mixture Model for Monitoring Multimode Plant-wide Process

Transcription

1 istributed Gaussian Mixture Model for Monitoring Multimode Plant-wide Proess Jinlin Zhu, Zhiqiang Ge, Zhihuan Song. State ey Laboratory of Industrial Control Tehnology, Institute of Industrial Proess Control, College of Control Siene and Engineering, Zhejiang University, Hangzhou, 37, P. R. China Abstrat: For large-sale plant-wide proesses with multiple operating onditions, a distributed Gaussian mixture modeling and monitoring mehanism is proposed. To overome the defiient prior modeling knowledge for omplex proess, a two-dimensional probabilisti topi model based tehnique named Bayesian o-lustering method is developed to simultaneously ondut the sub-blok division and operating mode reognition. With the obtained sub-bloks and operating modes, a global Gaussian mixture model is first built and then several loal Gaussian mixture models are extrated and applied for distributed monitoring of plant-wide proesses. By onduting distributed modeling and monitoring, both global and loal proess hanges an be refleted and the fault region an be loalized more easily for further analyses. The feasibility and effetiveness of the proposed method is onfirmed through the Tennessee Eastman benhmark proess. ey Words: Plant-wide proess monitoring; istributed data modeling; Gaussian mixture model; Bayesian o-lustering; Variational Bayesian inferene. Introdutions ue to the ever inreasing development of modern industry, the plant-wide proess monitoring has beome more and more popular in the aademi researh field []. Generally, the plant-wide proess is usually haraterized with multiple operating units, workshops and even plants; moreover, there may also inlude multiple operating onditions in order to produe various produts with different values. To ensure the safe operation and stable prodution quality, automati modeling and monitoring of plant-wide proess are espeially neessary []. Traditional proess modeling and monitoring methods have been designed based on mehanisti model suh as state-spae method. However, the mehanisti analysis and kinemati equation indutions an be sophistiated, time-onsuming and sometimes even unavailable. As a lass of alternatives, the data-based proess monitoring methods have been widely foused and applied [3]. For example, MaGregor proposed a multiblok partial least square (PLS method and monitoring harts are defined for both sub-bloks and the entire proess so as to enhane the monitoring performane [4]. Qin analyzed several multi-blok methods and the deentralized monitoring and diagnosis framework was proposed[5]. However, a ritial issue for these methods is that the deomposition for loal bloks should largely rely on a priori proess knowledge. In pratie, however, a priori proess knowledge is not always available for plant-wide proess. There have been attempts to blok plant-wide data with different automati mehanisms. For example, Ge proposed a distributed prinipal omponent analysis method for plant-wide proess monitoring []. PCA sub-bloks are onstruted through different diretions of prinipal om- This work was supported in part by the ational atural Siene Foundation of China (SFC (67367, ational 973 Projet (CB75, and the Alexander von Humboldt Foundation. ponents. Reently, the Generalized ie s oeffiient is used for measuring similarity among vetors in latent spae, and then PCA sub-bloks are onstruted aording to latent vetors with high similarity [6]. However, both methods are onstruted based on PCA and seldom onsider the multiple operating harateristi of plant-wide proess In this paper, a probabilisti method alled Bayesian o-lustering is developed for analyzing the plant-wide onditions of multiple modes and sub-bloks. Tehnially, the Bayesian o-lustering is based on the latent irihlet alloation (LA and an simultaneously ondut the sub-blok division and operating mode reognition from the two-dimension aspets through topis [7]. The merits of Bayesian o-lustering for plant-wide monitoring an be found from the following three aspets [8]. First, topi models an disover the potential topis and then organize data aording to the disovered theme. otie that in industrial proess, a topi on variable-wise an be viewed as a unit like a reator, a ondenser or even a workshop with a ombination of oupled units, et. While for sample-wise, a topi an be presented with a ertain operating mode. Therefore, this merit is very useful for plant-wide data bloking sine the bloking progress an be regarded as a way to disover the topi of eah blok and then reorganize data with various bloks from different topis. Seond, reent advane show that topi model an be applied to massive olletion of industrial data. With the development of digitized store tehnique, the volume of data is growing, espeially for plant-wide situation. Therefore, probabilisti topi model provides a desirable statisti solution for data mining and information disovery for plant-wide proess issue. Third, topi model an handle different kinds of data. For example, in appliations out of proess ontrol, the topi model has been adapted to other proess system engineering fields like analysis of genetis data, images and soial networks. Atually, with the oming of big data era, the advanement of proess monitoring methods show /6/$3. 6 IEEE 633

2 similar trends that should also ombine information from different kinds of data types, for example, data from spetra measurements, images and sound, et [9]. In this sense, the topi model provides a good alternative. One the bloking information and mode alloation have been obtained, distributed Gaussian mixture models an be onstruted for distributed monitoring. Speifially, a global GMM is first built for apturing the global operating mode of the urrent proess. Based on that, sub-blok GMMs are extrated from the global model so as to monitor the loal behavior of eah proess theme. In ontrast to traditional GMM, the distributed GMM (GMM an be more feasible for refleting loal proess hanges. Meanwhile, the fully probabilisti treatment of bloking, mode alloation, and distributed modeling provides a feasible and effetive solution for the general plant-wide proess monitoring. The rest of this artile is organized as follows. Conventional GMM based monitoring method is first revisited. Then istributed modeling and monitoring with Bayesian o-lustering approah are introdued and developed. In the simulation part, the proposed method is validated on the hallenging Tennessee Eastman proess. Finally, onlusions and future outlooks are made. Proess monitoring with GMM In this part, we give a brief revisit on GMM based proess monitoring. Assume data set = { R n } n= X x is generated from a multimode proess with loal operating onditions, denotes the dimensionality of data and is the total number of sampling data. The probability density funtion of GMM an be defined as follows (, Θ = ( n = ( n p X p z p x θ ( n= = where z n is the latent indiator, Θ= { θ } is the parameter = set, and θ = { μ, Σ }, = { π } = is the weight set for all loal Gaussian omponents and π =. ue to the fixed weights, p( z π = n = is written with π in some ases for simpliity. For parameter estimation, expetation maximization (EM algorithm is ommonly used based on the derivative of log-likelihood funtion, in the E-step, expetations for latent variables are omputed and in the M-step, parameters are updated with the expetations derived in the former step. The expliit steps an be found in []. Yu has also proposed the Bayesian inferene-based probability ( index []. The index is a desirable global probabilisti index designed for fault detetion with GMM. Speifially, for a new oming data x new, the index an be alulated as: = p( xnew, Θ PL ( x new ( = where PL ( x new is defined as the Mahalanobis distane based probability of x new with respet to omponent, and an be omputed by P L ( new = Pr{ M( M( new } T where M ( = ( μ ( Σ + ε I ( μ x x x (3 xnew xnew k x is the new squared Mahalanobis distane. The Mahalanobis distane from the enter of luster follows the χ distribution with degree of freedom. Hene the above equation an be easily solved by a simple integration. Besides, one has new. The upper ontrol limit an be given a priori, suh as.95 or istributed GMM method 3. Bayesian o-lustering struture Latent irihlet alloation or LA is originally used for disrete text modeling, and the goal is to extrat topis from a series of douments so as to provide statisti evidenes for further analysis suh as lassifiation, summarization, novelty detetion, et. It is worth to emphasize that although the samples in the ontext are assumed to be Gaussian distributed whih are not disrete data with tokens, we an still regard that the model as a general form of LA. One an easily see from Eq.( that GMM takes the assumption that mixing weights are fixed for all data items. This is atually a kind of redued desription on a fixed set of omponents. In other words, all data are onstrained to be generated from omponent set with fixed priors. As a Bayesian method, latent irihlet alloation relaxes this limitation by assuming that a separate weight vetor should be employed for eah sampling data. In addition, the weights are further assumed to be sampled from a irihlet distribution with prior, denoted as ir (. Therefore, the probability density funtion of LA an be defined as follows[]: p( X, Θ = ir ( p( z = p( x θ d (4 π n= = n n The basi idea here is that proess data are represented as random mixtures over latent omponents, where eah omponent is haraterized by a distribution over data samples. By doing so, LA assumes that eah data sample should be sampled with the following steps: First, omponent weights are sampled from irihlet distribution; seond, a speifi omponent is hosen from the multinomial distribution of p( z ; in the last step, proess data is generated from the empirial Gaussian distribution. Compared with GMM, suh latent alloation puts more flexibility for modeling. In addition, due to the similar alloation mehanism, data sampled from the same operating mode ould be gathered again after statistial modeling has been ompleted. The above alloation is based on the topi of mode similarity whih is performed on the dimension of sample wise. From the variable wise, the same mehanism an also be introdued. Similar to the sample wise data lustering with mode topi, the variable wise lustering an be viewed as following the topi of blok division. In other words, variables with same impliated topi are gathered in the same luster or blok. As aforementioned, the topi here an th Chinese Control and eision Conferene (CCC

3 refer to a unit like a reator, a ondenser or even a workshop with a ombination of oupled units. Therefore, the entire lustering progress is named as Bayesian o-lustering. For modeling, in order to distinguish between both dimensionalities, we assume two irihlet distributions ( ir for rows and olumns. Besides, mixing weights are defined for eah row u and olumn v and denoted as u and v, u {,,..., } and v {,,..., } ir and (. Latent indiators are also separately annotated as z uv and z uv representing latent row and olumn luster respetively. otie that z uv {,,..., }, zuv {,,..., } where and are total number of row lusters and olumn lusters. The probability density funtion of Bayesian o-lustering model an be defined as follows [7] p ( X,, Θ (5 = ir ( ir ( π π u v u= v= p ( z uv u p( zuv v p( xuv θz uv, z d uv d u= v= zuv = zuv = 3. Bayesian inferene ow turn to the omputational problem where the goal is to estimate the parameters of impliated distributions. However, learning suh a hierarhial model with maximum likelihood is omputationally intratable due to the integral of omplex latent struture. In this work, the variational inferene is used. The ore idea of variational inferene is to approximate the omplex distribution over latent struture with a simpler distribution. efine T { Z, Z,, } for latent variables; the marginal log-likelihood of the model an be given as log p ( X,, Θ = log p( XT,,, Θ dt (6 p ( XT,,, Θ q( T log dt q ( T = F,, Θ, q( T ( otie that F(,, Θ, q( T is usually alled lower bound while q ( T is alled variational posterior. Therefore, maximizing the likelihood turns to the problem of maximization of lower bound. In order to make the alulation more appropriate, variational posterior is further fatorized by mean field approximation as q( T = q( Z φ q( Z φ q( q( (7 where eah variational posterior fator is defined by the same distribution assignation as the true orresponding posterior distribution. Therefore, the lower bound an be further written as: F(,, Θ, q( T = Eq ( log ( + Eq( log ( (8 + Eq( log ( Z + Eq( log ( Z + Eq( log ( X Z, Z, Θ Eq( log q( Z φ Eq( log q( Z φ Eq( log q( Eq( log q( For maximization, take the derivative of F(,, Θ, q( T with respet to variational parameters φ, φ,, and parameters Θ= { θ { ij = μij, σij }, i =,...,, j =,..., } and set to zeros. With some manipulations, one an derive the following updating rules log ( φ vj p xuv θij v j φ ui exp = = ( γ ui γ (9 Ψ Ψ ul + l = m u φ ( ui log p xuv θij u i φvj exp = = ( γvj γ ( Ψ Ψ vl + l = m v γ = α + m φ ( σ ui i u ui γ = α + m φ ( μ vj j v vj φuiφvj xuv u= v= ij = φuiφvj u= v= ( x μ φ φ uv ij ui vj u= v= ij = φuiφvj u= v= (3 (4 otie that Ψ (. is the digamma funtion, φ ui denotes the i th omponent of φ for row u and φ = ui i=, γ ui is the orresponding variational prior, m u and m v are the number of entries in row u and olumn v respetively. Finally, the irihlet parameters and should be updated. In order to estimate suh parameters effiiently, the ewton-rapson optimization method is utilized []. Basially, the formulation is ' = H( g( (5 where H ( and g ( are Hessian matrix and gradient term respetively. Therefore, for eah element α i in, we have ' g i u α i = α i (6 h g i = Ψ α l Ψ α i + Ψ γ ui Ψ γ ul i ( ( (7 l= u= l= h = Ψ '( α (8 i i 6 8th Chinese Control and eision Conferene (CCC 635

4 u = w g / h i i i= + h l l = (9 w= Ψ α i ( l = For, one an obtain the similar derivations. In the E-step, the optimizations of variational parameters are onduted for Eq. (9-(, while in the M-step, the goal is to maximize the lower bound on the log-likelihood with respet to model prarameters Eq.(3 -( (and also with equations for based on the suffiient statistis omputed in E-step. The two steps iterates until the lower bound onverges. 3.3 istributed proess monitoring One the Bayesian o-lustering has been implemented, the blok division and mode reognition have also been simultaneously aomplished. For variable bloking, variables with the same latent topi are lustered together and an be viewed as generated from the same blok. Therefore, the entire plant-wide proess is divided into several sub-blok proesses. In the meantime, samples have also been automatially lustered with respet to the orresponding mode topi. It should be notied that in this work, the atual number of variable bloks and operating modes are both assumed to be unknown a priori. In that ase, some overestimated initializations ould be assigned. In fat, as an be seen in the following ase study, the Bayesian o-luster method with probabilisti assignation is self-driven and those redundant lusters an be removed one the o-lustering model has been established. With all the above well-paved steps, the last step whih is to build the distributed Gaussian mixture models an be intuitional results. Sine the loal bloks may not always indiate the global multimode harateristis, a global GMM is first onstruted aording to the obtained mode information. The global GMM model aptures and traks the entire operating onditions of the plant-wide proess. Moreover, it an be used as an inferene mode indiator for monitoring the urrent status of the entire plant. As mentioned above, suh monitoring results an only be served as the refletion of global proess operating ondition. In order to loalize the monitoring results, the distributed method is utilized. On the basis of o-lustered variables, all sub-models of distributed GMM an be diretly extrated from the global mixture model. In pratial, it is worth to notie that as a Bayesian method whih is based on probabilisti mehanism, the variable bloking results may also show some unertainties. Still use the above five variable proess as an example. In some ases, variables may be divided into more bloks than expeted. Moreover, the divided bloks may show inonsistent results by different running. In order to deal with suh logial but not so aurate results, a reasonable alternative is to ondut Monte Carlo methods and ondut statistial analysis from multiple random Bayesian lustering results. Speifially, assume that a total of t multiple experiments have been onduted. Furthermore, let b denotes the initial total number of bloks. A blok b matrix BM i R is defined for experiment ii, =,,..., t. Eah element b pq in BM i denotes the ownership probability of variable p belongs to blok luster q. For simpliity, in this work we assume that eah variable belongs to the luster with the highest assigning probability, that is: bpq =, q = argmaxk bpk ( bpq =, others After the definition of the single blok matrix, we have to make a statisti analysis on the overall bloking results. In other words, the goal is to alulate the total times that ertain variables are lustered in the same blok. In this paper, a onfusion luster matrix C R is further defined for statistial analysis of multiple results as C = BM BM BM ( t where the general operator denotes the aumulation proedure of the same alloated luster variables. After the aumulation of all blok matrix, eah element mn in the onfusion matrix denotes the total times of variable n whih is lustered together with variable m, while the diagonal element mm in C denotes the times that variable m should be alloated alone with a single blok. Assume that all variables have been labeled with a positive integer number. Then C = A B an be alulated with the following algorithm: Algorithm Input: Blok matrix A, B. Output: Confusion matrix C.. Find the non-zero olumns (e.g. non-zero variable entries in a olumn an be viewed as a single variable blok of both matrixes;. Initialize a token set Ts with zero entries, this vetor indiates the manipulated variables; 3. For eah non-zero olumn, find the variable with the smallest label; 3. Chek if the urrent variable with smallest label has already been manipulated in the token set, if so, go bak to step 3 for the next non-zero olumn; 3. If the blok ontains more than one variable then the entry with respet to the smallest label should be negleted and set as zero; 3.3 Add the transpose of eah olumn to the orresponding row of C (the row number equals to the smallest label for eah blok; 3.4 Add the variable token of non-zero entries to the token set Ts. In order to make the multiple results more reasonable with unertainties, blok alloation probability (BAP index is further introdued by normalizing eah element of C with th Chinese Control and eision Conferene (CCC

5 t and C C t. With the defined BAP index, X i is alloated in the same blok with X j if ( j ji, arg max k ik j = max arg max C C (3 Based on the Eq.(3, bloks an be easily obtained. Assume one has finally derive f bloks, then all bloks an be denoted as B, B, B f. With the final blok information, distributed GMM an be onstruted. The entire distributed modeling proess is summarized as follows. Algorithm Input: Proess data X and Labeled variables,,,. Output: Bloks B, B, B f and the orresponding distributed GMM.. Condut multiple Bayesian o-lustering, derive mode information and multiple blok matrixes BM, BM,, BM t ;. Transform eah blok matrix into - entries with(; 3. Calulate the onfusion matrix C with (3 and then normalize eah element; 4. Separate bloks B, B, B f from the onfusion matrix C ; 5. Construt the general GMM modeling based on proess data X and the derived multimode information; 6. Extrat distributed GMMs based on the separated variable bloks. The extrated distributed GMMs an be readily applied for plant-wide proess monitoring. Speifially, for eah new oming sample x new, first alulate the potential mode. Then ompute the orresponding index defined in Eq.( for eah GMM blok. otie that the trends in eah blok an finally serve as a meaningful referene for distributed monitoring. In some ases, one an determine the impliated topi for eah blok. Although this work is foused on fault detetion, the topi information by various bloks an provide important guidane for further fault identifiation and diagnosis. Therefore, if any sub-bloks trigger alarms, the operators an first determine the orresponding topis and then make further measurements. Finally, the monitoring performane of the entire distributed model an be denoted by an integrated index with - entries ( for normal and for fault. Basially, it is assumed that the proess is in faulty ondition if any sub-bloks alarm faults. For a new sample x new, let,,, f denote the monitoring results from all bloks and CL is the defined upper ontrol limit. The integrated index is then defined as if any index exeeds CL. 4 Case studies In this setion, the TE benhmark proess is onsidered as a simulated plant-wide proess. For losed loop simulation, the TE proess is equipped with deentralized ontrol strategy as desribed in []. There are 4 measured proess variables olleted for monitoring. ata ases are employed by samples and all faults in the testing dataset are introdued in the TE proess at sample 5. As a plant-wide proess, the TE benhmark onsists of six different operating modes with different G/H mass ratios (stream. In this work, mode and mode 3 are used for simulation. For Bayesian modeling, a total of samples are generated from eah mode. After that, the initial guesses for modes and variable bloks are randomly seleted as 4 and respetively. The Bayesian o-lustering is then applied for bi-lustering of both modes and bloks. As above, the derived variable bloks through the normalized onfusion matrix for variable bloks with BAP index after 5 experiments of the TE proess are given in Table. For samples-wise lustering, a typial derived result for all samples is depited in Figure. One an infer from Table that, five bloks an be finally derived aording to the lustering results. It should be notied that the variables are lustered aording to the potential topis. Therefore, variables whih belong to different units are lustered in the same blok due to the impliated oupled units. In other words, topi an be foused on multiple units and produtions. For example, it an be speulated that the topi of blok should be foused on omponents B and G; the seond blok mainly pays attention to impliated variables with regard to the input and prodution proedure of omponent A and C; the topi of blok 3 should be relevant with levels of reator/separator/stripper as well as the prodution of omponent ; the potential theme of blok 4 should be mainly about temperature of reator/separator/stripper/separator water, the underflow of separator/stripper and the prodution of omponent E, F and H; and the last blok is foused on omponent C. Besides topi related variables, some bloks are also involved with attahed misellaneous variables after probabilisti dividing. For example, blok is attahed with separator pressure and omponent B of reator feed analysis; blok is attahed with three misellaneous variables inluding purge rate, reator water temperature and omponent F; blok 3 has three misellaneous variables inluding reator pressure, stripper steam flow and ompressor work; blok 4 is also attahed with three misellaneous variables ontaining reator feed rate, stripper pressure and reyle flow. One an learly see that the divided bloks an effetively reflet the essential parts of the plant-wide proess. Hene, it is validated that the lustering result is onsistent with the atual modeling data. Table : Obtained variable bloks of the TE proess Bloks Variable Topi,3,4, 6,3,35,4 Components B and G,4,,, 3,9,3,39 Component A and C 7,8,,5, 9,,3,37 Levels of reator/separator/str ipper; Component 3,5,6,9,,4,6,7, 8,,7,8,33,34,36, 38,4 Temperature of reator/separator/stripper/se parator water; the underflow of separator/stripper; Component E, F and H One the Bayesian o-lustering has been ompleted, distributed GMM an be easily onstruted. Follow the diagram given in algorithm, a global GMM is first established based on the mode reognition result. ext, five separate GMMs an be extrated from the global model on the basis of variable bloking result. After then, the distributed GMM is applied for monitoring the multimode 5 Component C 6 8th Chinese Control and eision Conferene (CCC 637

6 plant-wide proess. Atually, one an effetively monitor the plant-wide proess by monitoring multiple distributed models with different topis. As an illustration, we take Fault (IV in mode as a study ase. To evaluate the effetiveness of the proposed method, the distributed GMM is ompared with traditional GMM. IV is ativated with random variations in C feed temperature (stream 4. The monitoring results of GMM and distributed GMM are given in Figure. The ontrol limits denoted with red lines are set as.95. It an be seen that the Mode is orretly reognized by the global GMM. In the meanwhile, one an infer that the global GMM an only tell the ourrene of the faulty event. On the ontrast, it an be seen that distributed GMM an further indiate that only blok 4 is responsible for this fault. Mode Posterior Blok Blok Blok 3 Blok 4 Blok 5.5 Probability.5 Samples 3 Mode Figure : Mode lustering result for TE samples Time.5 (a 4 Mode Mode Time (b Samples ( Figure : Monitoring results of Fault with (a Mode reognition by global GMM (b trends of global GMM ( trends of distributed GMM Atually, as an be inferred from the TE working diagram, C feed flows into the stripper and the random variations in C feed an also result in orresponding variations of stripper temperature. However, under the ontrol mehanism, although the variations in temperature keep on existing, impats from IV have been rejeted by the slave temperature ontrol loops. Therefore, as an be validated from Table and Figure, only the 4 th distributed GMM blok alarms faults while other models indiate not being affeted. On the basis of distributed modeling, operators an loalize the root ause of fault aording to the topi of blok 4. Monitoring results by traditional GMM an hardly loalize the fault area whih may result in shutdown of the entire plant-wide proess. However, by bloking and distributed monitoring, one may diretly determine the operating ondition (mode and also derive the general topi of fault one abnormal event happens. In other words, the Bayesian o-lustering method provides an effetive alternate for bloking and mode reognition of large sale plant-wide proess in spite of the little proess knowledge that an be obtained a priori. 5 Conlusions In this work, a distributed Gaussian mixture model has been proposed for modeling and monitoring multimode pant-wide proesses. Given the fat that proess knowledge is not always available, the Bayesian o-lustering approah is developed to extrat topi information simultaneously from both sample-wise and variable-wise diretions. Based on the o-lustering mehanism, variables with same potential topis are gathered together in the same blok. Meanwhile, samples are also lustered aording to the topi of underlying operating modes. In order to deal with the modeling unertainties with probabilisti alloation, multiple experiment and analysis diagram have been onstruted. Simulation results on a numerial study ase and the TE benhmark have demonstrated the effetiveness of the developed approah. Inferenes [] S. J. Qin, "Survey on data-driven industrial proess monitoring and diagnosis," Annual Reviews in Control, vol. 36, pp. -34,. [] Z. Ge and Z. Song, "istributed PCA Model for Plant-Wide Proess Monitoring," Industrial & Engineering Chemistry Researh, vol. 5, pp , 3. [3] Z. Ge, Z. Song, and F. Gao, "Review of Reent Researh on ata-based Proess Monitoring," Industrial & Engineering Chemistry Researh, vol. 5, pp , 3. [4] J. F. MaGregor, C. Jaekle, C. iparissides, and M. outoudi, "Proess monitoring and diagnosis by multiblok PLS methods," AIChE Journal, vol. 4, pp , 994. [5] S. J. Qin, S. Valle, and M. J. Piovoso, "On unifying multiblok analysis with appliation to deentralized proess monitoring," Journal of hemometris, vol. 5, pp ,. [6] B. Wang, X. Yan, Q. Jiang, and Z. Lv, "Generalized ie's oeffiientbased multiblok prinipal omponent analysis with Bayesian inferene for plantwide proess monitoring," Journal of Chemometris, vol. 9, pp , 5. [7] H. Shan and A. Banerjee, "Bayesian o-lustering," in ata Mining, 8. ICM'8. Eighth IEEE International Conferene on, 8, pp [8]. M. Blei, "Probabilisti topi models," Communiations of the ACM, vol. 55, pp ,. [9] S. J. Qin, "Proess data analytis in the era of big data," AIChE Journal, vol. 6, pp. 39-3, 4. [] J. Yu and S. J. Qin, "Multimode proess monitoring with Bayesian inferenebased finite Gaussian mixture models," AIChE Journal, vol. 54, pp. 8-89, 8. []. M. Blei, A. Y. g, and M. I. Jordan, "Latent dirihlet alloation," the Journal of mahine Learning researh, vol. 3, pp. 993-, 3. []. Lawrene Riker, "eentralized ontrol of the Tennessee Eastman hallenge proess," Journal of Proess Control, vol. 6, pp. 5-, th Chinese Control and eision Conferene (CCC