Research ArticleMultimode Process Monitoring Based on Sparse PrincipalComponent Selection and Bayesian Inference-Based Probability

Xiaodong Jiang1 Haitao Zhao1 and Bo Jin2

1Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of EducationEast China University of Science and Technology Shanghai 200237 China2Software Engineering Institute East China Normal University Shanghai 200062 China

Correspondence should be addressed to Haitao Zhao htzhaoecustgmailcom

Received 6 May 2015 Revised 27 July 2015 Accepted 28 July 2015

Academic Editor Jean J Loiseau

Copyright copy 2015 Xiaodong Jiang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

According to the demand for diversified products modern industrial processes typically have multiple operating modes At thesame time variables within the same mode often follow a mixture of Gaussian distributions In this paper a novel algorithm basedon sparse principal component selection (SPCS) and Bayesian inference-based probability (BIP) is proposed formultimode processmonitoring SPCS can be formulated as a just-in-time regression between all PCs and each sample SPCS selects PCs according to thenonzero regression coefficients which indicate the compact expression of the sample This expression is necessarily discriminativeamongst all subset of PCs SPCS selects the PCs which most compactly express the sample and rejects all other possible but lesscompact expressions BIP is utilized to compute the posterior probabilities of each monitored sample belonging to the multiplecomponents and derive an integrated global probabilistic index for fault detection of multimode processes Finally to verify itssuperiority the SPCS-BIP algorithm is applied to the Tennessee Eastman (TE) benchmark process and a continuous stirred-tankreactor (CSTR) process

1 Introduction

Over the past two decades with the development of complexchemical processes and the growing demand of plant safetyand stable product quality timely process monitoring isgaining importance Because large amounts of data can begathered by the use of distributed control systems (DCSs)multivariate statistical process monitoring (MSPM) algo-rithms have received great attention Among these algo-rithms principal component analysis (PCA) and partialleast squares (PLS) are the most widely used algorithms[1ndash8] Both algorithms project high-dimensional data ontolower dimensional subspaces Process normal and abnormalconditions can be isolated by the use of Hotellingrsquos 1198792 orsquared predicted error (SPE) [9ndash13] Other complemen-tary MSPM algorithms including independent componentsanalysis (ICA) Fisher discriminant analysis (FDA) andcanonical variate analysis (CVA) are used to overcome somelimitations in PCAPLS-based monitoring schemes [14ndash17]However most of MSPM algorithms rely on the assumption

that the system is in a single operating region and that thedata follow a Gaussian distribution In chemical processesoperating condition shifts are often encountered due tothe changes of various factors such as feedstock productspecification set points and manufacturing strategy Whena process is running under substantially different operatingconditions only a small number of variables actually followGaussian distribution [18] As a result the multimodalityof data distribution might lead to unseemliness for themonitoring of conventional MSPM algorithms To addressthese problems it is necessary to develop new algorithms

In literature multiple models can be built to fit eachindividual operating mode but these are two essential issuesthat need to be addressed One is how to divide the trainingdata into multiple subsets correctly corresponding to dif-ferent operating modes In order to solve this issue manyclustering algorithms are applied In terms of the traditionalapproaches Ge and Song [19] used fuzzy C-means clusteringalgorithm to separate the training data set according to theunique characteristics of each mode He et al [20] applied

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 465372 12 pageshttpdxdoiorg1011552015465372

2 Mathematical Problems in Engineering

the 119896-nearest-neighbor method Srinivasan et al [21 22]identified the different operating modes by evaluating theEuclidean distances between samples in a constructed datawindow and then applied dynamic PCA-based similaritymeasures to cluster the samples Liu andChen [23] developeda method using Bayesian classification for selecting multipleregions from a training data set Zhao et al [24] presenteda multiple principal component analysis (MPCA) algorithmthat selects one suitable model to monitor multimode pro-cesses The other issue is how to determine the final resultsA proper measurement should be employed to determinewhich model is the most suitable one for monitoring atthe current moment Ng and Srinivasan [22] exploited themost suitable PCA model through a minimized distancereflecting both the 1198792 and SPE values Zhao et al [24] closethe local PCAmodel with theminimumSPE value Natarajanand Srinivasan [21] used the distance between the sampleand the center of local models as a criterion Yu and Qin[25] performed Bayesian inference on the postprobabilitiescalculated by the Gaussian mixture model (GMM) or thenonlinear kernel GMM Meanwhile Ge and coworkers [2627] took advantage of Bayesian inference to softly combinethe monitoring results computed by local models built bymeans of probabilistic PCA (PPCA) factor analysis (FA) orsubspace algorithms

To date the problem of how to correctly divide thetraining data into multiple subset can successfully be solvedby many algorithms mentioned in the previous paragraphHowever there are still some issues that need to be resolvedthe most important one is how to select the key principalcomponents (PCs) when using one suitablemodel for processmonitoring Many algorithms for selecting PCs have beenproposed such as cumulative percent variance (CPV) [28]variance of reconstruction error (VRE) [29] and cross vali-dation (CV) [30] Generally most of the classical algorithmsjust take normal operational observations into account andselect the first several PCs with larger variance While PCswith larger variance of normal data cannot guarantee thecapture of the largest variations in fault data online Jolliffe[31] suggested that the last PCs may be as important as thosewith large variance Togkalidou et al [32] noted that the PCswith larger variance do not always containmuch informationfor prediction However this issue is insufficiently discussedin PCA-based process monitoring and the standard PCselection is still not established

Fortunately many researches have been aware of theinherent defects of classical PCA algorithm A lot of work-ers tried to seek a subspace spanned by key PCs whichcontains the most important information for process mon-itoring Peng et al [33] suggested a new feature selectionalgorithm named minimal-redundancy-maximal-relevancecriterion (mRMR) It is based on mutual information andselects the features with highest relevance to the targetclass Jiang et al [34] put forward the sensitive principalcomponent analysis for fault detection and diagnosis inchemical processes They pointed out that PCs selectedby PCA algorithm are not always the key PCs for faultdetectionTheir task was to find the sensitive PCs which have

relationship with fault information Arbel et al [35] proposedthat the process variables that are preponderant in achievingspecific objectives need to be selected

In this paper a process monitoring algorithm usingmultisubspace sparse principal component analysis with theBIP algorithm is put forward First variables are divided intodifferent subblocks corresponding to different units or piecesof equipment to reduce the complexity of process analysis Byusing BIP algorithm multimode data in each subblock aredivided into multiple subgroups BIP can compute the poste-rior probabilities of each monitored sample belonging to themultiple components and derive an integrated global prob-abilistic index for fault detection of multimode processesThe PCs selected by PCA algorithm with larger variances donot always have relationship with fault information Sparseprincipal component selection (SPCS) takes the informationof both normal and abnormal observations into accountThealgorithm is formulated as a just-in-time form that constructsan elastic net regression between all PCs and each sampleSPCS selects PCs corresponding to the nonzero regressioncoefficients which indicate the compact expression of thesampleThis expression is necessarily discriminative amongstall subset of PCs SPCS selects the PCs whichmost compactlyexpress the sample and rejects all other possible but lesscompact expression Third the key PCs are selected by SPCSin each subgroup to solve the problem of fault informationloss It needs to be stressed that the subspace spanned by thekey PCs selected is the feature subspace Finally in order toverify the superiority of the SPCS-BIP algorithm it is appliedto the Tennessee Eastman (TE) benchmark problem and acontinuous stirred-tank reactor (CSTR) process

2 Preliminaries

21 Principal Component Analysis Principal componentanalysis is a multivariate statistical analysis which is widelyused in chemical process monitoring fault detection andso forth [36ndash38] Let x isin R119898 represent an 119898-dimensionalsample vector and X isin R119873times119898 denote a data matrix with zeromean and unit variance where 119873 is the number of samplesand 119898 is the number of variables in the process From thestatistical viewpoint the PCA algorithm could be obtainedby singular value decomposition (SVD) [28 34]

X = TP119879 + E = X + E (1)

where T isin R119873times119896 and P isin R119898times119896 are the score matrix and theloading matrix respectively 119896 is the principal componentsretained number The loading matrix P can be obtained byeigenvalue decomposition on the covariancematrix cov(x) asfollows

cov (x) asymp 1

119873 minus 1X119879X = PΛP119879 (2)

where Λ = diag1205821 1205822 120582

119898 denotes the eigenvalue

matrix and P = [P P] contains the loading matrices ofcomponent subspace and residual subspace respectively

Mathematical Problems in Engineering 3

22 Construction of Finite Gaussian Mixture Model Basedon EM For the process running at multiple operating con-dition owing to the mean shifts or covariance changes theassumption of multivariate Gaussian distribution becomesinvalid [21 22] In this situation the local Gaussian dis-tribution is still appropriate to characterize each subsetof measurement data from the same operating conditionsTherefore the finite Gaussian mixture model is prime suitedto represent the data sources driven by different operatingmodes [13 24 25]

To construct a FGMM given a set of training samplesX isin

R119873times119898 the log-likelihood function can be expressed as

log 119871 (X Θ) =119873

sum

119895=1

log(119871

sum

119894=1

120596119894119892 (119909119895| 120579119894)) (3)

and the parameter estimation problem is formulated as

Θ = argmaxΘ

(log 119871 (X Θ)) (4)

where Θ = 120596119894 120583119894 Σ1 120596

119871 120583119871 Σ119871 120596119894(1 le 119894 le 119871)

is the prior probabilities and 119871 is the number of Gaussiancomponents included in FGMM 120583

119894is the mean vector and

Σ119894is the covariance matrixThere are a lot of learning algorithms such as maximum

likelihood estimation (MLE) EM and the F-J algorithm thathave been put forward formixturemodel estimation [39 40]As amore tractable numerical strategy the EM algorithm hasbeen well used in practice to estimate the maximum like-lihood distribution parameter [39] EM algorithm is imple-mented iteratively by means of repeating the expectation step(E-step) andmaximization step (M-step) to calculate the pos-terior probabilities and then the corresponding distributionparameters until a convergence criterion of the log-likelihoodfunction is satisfied Given the training data X and an initialestimate Θ(0) = 120596

(0)

1 120583(0)

1 Σ(0)

1 120596

(0)

119871 120583(0)

119871 Σ(0)

119871 the

iterative E-step and M-step are expressed as follows(i) E-step

119875(119904)(119862119897| x119895) =

120596(119904)

119897119892 (x119895| 120583(119904)

119897 Σ(119904)

119897)

sum119871

119894=1120596(119904)

119894119892 (x119895| 120583(119904)

119894 Σ(119904)

119894)

(5)

where119875(119904)(119862119897| x119895) denotes the posterior probability of the 119895th

training sample within the 119897th Gaussian component at the 119904thiteration

(ii) M-step

120583(119904+1)

119897=

sum119873

119895=1119875(119904)(119862119897| x119895) x119895

sum119873

119895=1119875(119904) (119862

119897| x119895)

Σ(119904+1)

119897

=

sum119873

119895=1119875(119904)(119862119897| x119895) (x119895minus 120583(119904+1)

119897) (x119895minus 120583(119904+1)

119897)119879

sum119873

119895=1119875(119904) (119862

119897| x119895)

120596(119904+1)

119897=

sum119873

119895=1119875(119904)(119862119897| x119895)

119873

(6)

where 120583(119904+1)119897

Σ(119904+1)119897

and 120596(119904+1)119897

are the mean covariance andprior probability of the 119897thGaussian component at the (119904+1)thiteration respectively

3 Fault Detection with Sparse PrincipalComponent Selection andBayesian Inference-Based Probability

In this section the idea of SPCS-BIP algorithm formultimodeprocess monitoring is demonstrated in detail We first intro-duce the Bayesian inference-based probability which canderive the confidence boundary around the normal operatingregions for process monitoring and fault detection Thenthe sparse principal component selection was introduced forselecting the key Pcs related with fault information Finallythe steps of this algorithm were given

31 Bayesian Inference-Based Probability In the previoussection the FGMM has been constructed and it is essentialto further derive the confidence boundary around the normaloperating regions for process monitoring and fault detectionDue to the multimodality of mixture distribution it is reallydifficult to capture the analytical boundary of the densityfunction 119901(119909 | Θ) in a certain confidence level

In the proposed monitoring approach given an arbitrarymonitored sample 119909

119905belonging to eachGaussian component

Bayesian inference strategy is used to calculate the posteriorprobability as follows

119875 (119909119905isin 119862119897) = 119875 (119862

119897| 119909119905) =

119875 (119862119897) sdot 119901 (119909

119905| 119862119897)

119901 (119909119905)

=119875 (119862119897) sdot 119901 (119909

119905| 119862119897)

sum119871

119894=1119875 (119862119894) sdot 119901 (119909

119905| 119862119894)

(7)

which can also be formulated as

119875 (119909119905isin 119862119897) =

120596119897119892 (119909119905| 120583119897 Σ119897)

sum119871

119894=1120596119894119892 (119909119905| 120583119894 Σ119894)

(119897 = 1 2 119871) (8)

Given that each component 119862119897follows a unimodal Gaus-

sian distribution the squared Mahalanobis distance of 119909119905

from the center of 119862119897follows 1205942 distribution provided that

119909119894belongs to 119862

119897

119863((119909119905 119862119897) | 119909119905isin 119862119897) = (119909

119905minus 120583119897)119879

Σminus1

119897(119909119905minus 120583119897) 1205942

119898 (9)

Under the assumption that 119909119905isin 119862119897and 1205942

119898has 119898 degree

of freedom 119863((119909119905 119862119897) | 119909

119905isin 119862119897) denotes the squared

Mahalanobis distance between 119909119905and the mean center of

119862119897 Owing to colinearity Σ

119897is usually ill-conditioned and

the following regularized Mahalanobis distance is utilizedinstead to avoid too wide confidence regions

119863119903((119909119905 119862119897) | 119909119905isin 119862119897)

= (119909119905minus 120583119897)119879

(Σ119897+ 120576119868)minus1

(119909119905minus 120583119897)

(10)

4 Mathematical Problems in Engineering

where the function of 120576 is to remove the ill condition ofcovariance matrix Σ

119897by adding a positive number to all the

diagonal entriesFor the monitored sample 119909

119905 a local Mahalanobis

distance-based probability index relative to each Gaussiancomponent 119862

119897can be defined as

119875(119897)

119871(119909119905)

= Pr 119863 ((119909 119862119897) | 119909 isin 119862

119896) le 119863 ((119909

119894 119862119897) | 119909119905isin 119862119897)

(11)

or

119875(119897)

119871(119909119905) = Pr 119863

119903((119909 119862

119897) | 119909 isin 119862

119897)

le 119863119903((119909119894 119862119897) | 119909119905isin 119862119897)

(12)

Given the appropriate degree of freedom 119875(119897)119871(119909119905) can be

computed by integrating the 1205942 probability density functionUnder a given confidence level this index has the functionof indicating whether the monitored sample is normal orabnormal provided that it belongs to the correspondingGaussian component A global BIP index is proposed tocombine the local probability metrics across all the Gaussianclusters because the random characteristic of eachmonitoredsample may come from multiple Gaussian components withthe corresponding posterior probabilitiesThe formulation ofBIP index for the monitored sample 119909

119905is given by

BIP =119871

sum

119897=1

119875 (119862119897| 119909119905) 119875(119897)

119871(119909119905) (13)

where the posterior probability 119875(119862119897| 119909119905) is used to incor-

porate the contribution of each local Gaussian component tothe overall probabilistic index As 0 le 119875(119897)

119871(119909119905) le 1 we have

0 le BIP le119871

sum

119897=1

119875 (119862119897| 119909119905) = 1 (14)

Under the preset confidence level (1minus120572) 100 the processis determined within normal operation if

BIP le 1 minus 120572 (15)

Otherwise the process operation is treated out of control

32 Sparse Principal Component Selection Sparse represen-tation has proven to be an extremely powerful tool foracquiring representing and compressing high-dimensionaldata [41ndash43] This success is mainly because of the factthat the important reconstruction information of data suchas process data and time series data has naturally sparserepresentations with respect to fixed bases or concatenationsof such bases Qiao et al [44] proposed that the graphsconstructed by the 119871

1-norm have the advantage of greater

robustness to data noise automatic sparsity and adaptiveneighborhood for individual datum What is more anotherimportant advantage is that sparse representation has thepotential discriminative ability since most nonzero elements

are located on the samples in the same class as the representedsample

Given the training sample L = [1198971 1198972 119897119873] isin R119873times119898

a test sample H isin R119898 the solution to the sparse represen-tation problem can be obtained by solving the following ℓ1-minimization problem

(ℓ1) 120573 = argmin 1003817100381710038171003817120573

1003817100381710038171003817

subject to L120573 = H

Card (120573) le 119896

(16)

where 120573 = [1205731 1205732 120573

119873] are the sparse representation

coefficients and Card(120573) denotes the number of nonzeroelements of 120573 From the perspective of statistics formula(16) can be named the Lasso criterion Lasso is a penalizedleast squares algorithm which was originally by quadraticprogramming imposing a constraint on the 119871

1norm of

the regression coefficients Thus the Lasso estimates 120573 areobtained by minimizing the Lasso criterion

120573 = argmin120573

1003817100381710038171003817100381710038171003817100381710038171003817

H minussum

119894=1

L119894120573119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

+ 120582

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

subject to Card (120573) le 119896

(17)

where 120582 is nonnegative However only using the 1198711-norm

penalty in Lasso has its limitation Zou et al [45] proposedthat if there is a group of variables among which the pairwisecorrelations are very high lasso tends to select any variablefrom the group and does not consider which one is selectedFortunately elastic net was put forward by Zou et al whichovercomes the limitation of only using the 119871

1-norm penalty

It is known that combining the 1198711-norm and 119871

2-norm

penalty can result in grouping effectiveness in regression andthus enhance the prediction accuracy For any nonnegative1205821and 120582

2 the elastic net estimates 120573en are given by

120573en = argmin120573

1003817100381710038171003817100381710038171003817100381710038171003817

H minus

119873

sum

119894=1

L119894120573119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

+ 1205821

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

+ 1205822

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

2

subject to Card (120573) le 119896

(18)

In brief it is expected that the elastic net is used to groupa set of sparse coefficients to construct the sparse alignmentmatrices in which the sparse representation informationor the potential discriminative information is encoded toenhance the discriminative ability in an unsupervised man-ner

33 Fault Detection with SPCS and BIP The key problemfor monitoring the multimode process is to select a suitablemodel and choose the subspace spanned by key PCs In theIntroduction we had put forward the fact that the subspace

Mathematical Problems in Engineering 5

Offline modeling

Use EM algorithm to learn the GMM andestimate the model parameters

For each submodel normalize the trainingdata

Obtain the basic PCs using SVDdecomposition

Obtain the sparse PCs of normal data byconstructing elastic net regression

End

Online modeling

Current data normalization

Obtain the sparse PCs of each testingsample by constructing elastic net

regression

Calculate BIP index value of each testingsample

Exceed limit

Next

No

There is a fault in process

YesSpecify a confidence level and constantcontrol limit

Figure 1 The steps of SPCS-BIP algorithm for process monitoring

spanned by the first several PCs with largest explainedvariance does not always have fault information

In the following part a novel multimode process mon-itoring approach based on SPCS and BIP is proposed Thisapproach is in a just-in-time form For each sample an elasticnet regression between all PCs and the sample is constructedand solved The PCs which have nonzero regression coeffi-cients are retained while other PCs are rejected That meansfor each sample we can pick out the most discriminativebases and the others are set to zero Its concrete calculatingsteps are summarized in Figure 1

Offline Modeling

(1) Collect a set of historical training data under allpossible operating conditions

(2) Use the EM algorithm to learn the Gaussian mixturemodel and estimate the model parameter set Θ =

1205831 Σ1 1205961 120583

119896 Σ119896 120596119896 based on the iterative steps

(3) For each submodel get a normal operational obser-vation set X =isin R119873times119898 where 119873 is the number ofsamples and 119898 is the number of variables This set isdenoted as the training set for threshold determiningA testing set Y isin R119878times119898 with both normal andabnormal observations is given for testing

(4) Normalize the training data through the mean valueand variance of each variable

(5) Obtain all principal components using SVD decom-position The training data X is reconstructed by X =

sum119898

119894=1t119894p119879119894 where t

119894is the score vector and p

119894is the

loading vector(6) For training sample x

119895(119895 = 1 2 119873) construct an

elastic net regression between each observation valueof training data and loading vector Pmade of PCs instep (5) according to 120573

119895= argmin

120573x119895minussum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

(7) Corresponding to the nonzero representation coef-ficients 120573

1198951

1205731198952

120573119895119896

construct a new loadingvector P

119895= [p1198951

p1198952

p119895119896

](8) Specify a confidence (1minus120572) 100 and constant control

limit 1 minus 120572

Online Monitoring

(1) Normalize the current time point data by using meanvalues and variance of the training data

(2) Obtain the loading vector P from offline modeling(3) When a test sample y

119895isin R119898 (119895 = 1 2 119878) is

coming construct an elastic net regression betweenthe sample and loading vector Pmade of PCs in step(2) according to 120573 = argmin

120573y119895minus sum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

6 Mathematical Problems in Engineering

(4) Corresponding to the nonzero representation coef-ficients 120573

1198971

1205731198972

120573119897119896

construct a new loadingvector P

119895= [p1198971

p1198972

p119897119896

](5) Generate theBIP control chartwith the calculatedBIP

index values for all the monitored samples If the BIPindex of a test sample is lower than the control limitwhichmeans the sample is normal go to step (1) Elsethere is a fault in the process

4 Case Studies on the TE and CSTR Process

In this case study the TE benchmark and CSTR processare introduced to verify the effectiveness of the SPCS-BIPalgorithm PCA-GMM is the classic algorithm formultimodeprocessing monitoring And the fault detection index (FDI)is similar to Bayesian inference probability (BIP) So here acomparison was made between SPCS-BIP and PCA-GMMIn addition to verify the improvements of SPCS algorithmwhich can select sparse PCs a comparison was performedbetween the SPCS-BIP algorithm and theMPPCA algorithm

41 Tennessee Eastman Process As a well-known benchmarkprocess the Tennessee Eastman process which was pre-sented by Downs and Vogel has been widely applied toevaluate and compare the efficiency of process monitoringtechniques [46 47] The schematic diagram of the processis illustrated in Figure 2 This process consists of five majorunit operations a reactor a product condenser a vapor-liquidseparator a recycle compressor and a product stripper Inaddition there are six modes of process operation as listedin Table 1 The variables can be divided into three categoriescomposition variables continuous process variables andmanipulated variables In our study only modes 1 and 3were simulated through the Simulink programs developedon the basis of the decentralized control strategy designed byRicker [48]The Simulink programs can be downloaded fromhttpdeptswashingtoneducontrolLARRYTEdownloadhtml The 31 selected monitoring variables contained 9manipulated variables and 22 continuous process variablesThus these variables were divided into five subblocks accord-ing to five units However given that only two variables wereallocated to each the compressor unit and the condenser unitthere were four variables assigned to the other three relatedsubblocks As a result the total of 31 variables was dividedinto three subblocks

There are 20 faults in the multimode TE process whichare listed in Table 2 Among these faults the root causes of thefaults 16ndash20 are unknown [46 47] What is more to simplifyinterpretation the amplitudes of faults 3 9 and 15 are sosmall It is difficult to detect so only the remaining 12 faultswere considered in this study In the modeling stage 2000normal samples which include 1000mode 1 samples and 1000mode 3 samples were collected as the training data set Inthe testing stage 1000 samples of mode 1 were tested firstand then the process switches to mode 3 As a result the testdata set consists of 1000 samples of mode 1 and 1000 samplesof mode 3 And faults occurred from the 1200th sample Aset of 20 faults in multimode TE process which are listed

Table 1 Six process operation modes of TE process

Mode GH mass ratio Production rate1 5050 7038 kghG and 7038 kghH2 1090 1048 kghG and 12669 kghH3 9010 10000 kghG and 1111 kghH4 5050 Maximum5 1090 Maximum6 9010 Maximum

in Table 2 are simulated and the corresponding process dataare collected for testingThe following simulations are run inMATLAB 830 (2014a) environment Here two indicatorswhich are FR (FR) and MR (MR) are often introduced tomeasure the result of process monitoring FR is the rate ofnormal data classified as fault dataMR is the rate of fault dataclassified as normal rate

In the MPPCA algorithm and PCA-GMM algorithmwhen the variance contribution was selected as 85 thedimension of feature space inMPPCAand the number of PCsin PCA-GMM were each selected as 18 In order to comparethemonitoring performances of these algorithms in the samesituation the selected sparse PCs of each mode in SPCS-BIPwere selected as 18 The 99 control limit was assigned to allthree algorithms

First Figure 3 shows that the different submodes can besuccessfully divided by the EM algorithm used in this paperAnd by using other algorithms themodes also can be dividedcorrectly In other words how to correctly divide the trainingdata into multiple subset is not a problem by many relatedalgorithms

The normal process was tested by different algorithm andthe results are shown in Figure 4 In this figure it is hard tofigure out which algorithmrsquos FR is lower In the figure mostsamples of each algorithm are lower than the control limitAnd by calculation the FR of these algorithms are 0333008 025 and 108 respectively corresponding to Fig-ures 4(a) 4(b) 4(c) and 4(d) The monitoring performancesof these three algorithms suggest that the FR are acceptableNext the data sets of 12 faults in mode 3 were tested and theMR of these three algorithms are listed in Table 3 with thesmallest MR shown in bold

From Table 3 we observe that the monitoring perfor-mance of SPCS-BIP is the best compared to the MPPCAand PCA-GMM algorithms for all 12 faults Here we takethe further analysis In comparison with MPPCA and PCA-GMMalgorithms the SPCS-BIP algorithm can exactly dividethe process data into subgroups corresponding to differentmodes by using the E-M algorithm and in each submodeSPCS-BIP can select the most important PCs that have mostrelation with the fault Due to the fact that the subspacespanned by the PCs was monitored by BIP most of the PCsare related to themain process of chemical industrial processand only little PCs are related to the fault process SPCSare discriminative by constructing an elastic net regressionbetween all PCs and each sample So in Table 3 we observe

Mathematical Problems in Engineering 7

Table 2 Process faults for the multimode TE process

Faults number Disturbance state TypeIDV(1) AC feed ratio B composition constant (Stream 4) StepIDV(2) B composition AC ratio constant (Stream 4) StepIDV(3) D feed temperature (Stream 2) StepIDV(4) Reactor cooling water inlet temperature StepIDV(5) Condenser cooling water inlet temperature StepIDV(6) A feed loss (Stream 1) StepIDV(7) C header pressure loss reduced availability (Stream 4) StepIDV(8) A B and C feed composition (Stream 4) Random variationIDV(9) D feed temperature (Stream 2) Random variationIDV(10) C feed temperature (Stream 4) Random variationIDV(11) Reactor cooling water inlet temperature Random variationIDV(12) Condenser cooling water inlet temperature Random variationIDV(13) Reaction kinetics Slow driftIDV(14) Reactor cooling water valve StickingIDV(15) Condenser cooling water valve StickingIDV(16) Unknown UnknownIDV(17) Unknown UnknownIDV(18) Unknown UnknownIDV(19) Unknown UnknownIDV(20) Unknown Unknown

1

5

2

3

4

A

D

E

C

FI

FI

FI

FI

FIFI

FI

FI

FI

FI

XA

XB

XC

XD

XE

XF

XG

XH

XD

XE

XF

XG

XH

XA

XB

XC

XD

XE

XF

Compressor

Ana

lyze

rA

naly

zer

Ana

lyze

r

6

7

8

9

10

11

12

SC

PI

PI

PI

JI

TI

Condenser

CWS

CWS

CWR

Reactor

CWR

Stripper

TI

TI

TI

TILI

LI

LI

Purge

Product

Stm

Cond

Vapliqseparator

Figure 2 Control scheme for the TE process

8 Mathematical Problems in Engineering

Sample number0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

11

12

13

14

15

16

17

18

19

2

Labe

l

Figure 3 Different modes of the training data

Table 3 Missed detection rates () of 12 faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 075 025 1 01252 35 85 55 18754 0 0 0 05 0375 0 1125 06 0 0 0 07 0 0 0 08 3375 3875 425 250010 82375 16 89875 787511 225 85 45 112512 1375 1625 15 075013 16125 265 19625 1162514 0 35 0125 0

that the results of SPCS-BIP are better than the results ofMPPCA and PCA-GMM

Figure 5 shows the monitoring performances of fault 10It is easy to see that the FDI of MPPCA-1198792 and the BIP ofPCA-GMM cannot detect the fault effectively in Figures 4(a)and 4(c) In the figure more than half of the fault sampleswere regarded as the normal samples while compared tothe performances of MPPCA-1198792 and PCA-GMM the FDIof MPPCA-SPE shows some improvements However themonitoring performance of MPPCA-SPE does not match theperformance of SPCS-BIP We can find this point both inFigure 4 and Table 3

42 CSTR This study simulated the CSTR process describedby Yoon and MacGregor [49] The diagram of the process ispresented in Figure 6 Due to the fact that the CSTR processconsisted of only one operating unit the number of subblockswas selected as 1

Table 4 Missed detection rates () of two faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 90 100 988 6562 288 996 318 28

In the modeling stage 1000 samples which include 500mode 1 samples and 500mode 2 samples were collected as thetraining data set In the testing stage 1000 samples of mode 2were tested and two faults were introduced to the process asfollows

Case 1 A step of 1 K was added in the cooling watertemperature 119879

119862from the 500th sample

Case 2 A 2 kmol(m3 sdotmin) step was added in the inlet soluteconcentration 119862

119860119860from the 500th sample

In the MMPCA algorithm when the variance contri-bution was selected as 85 the dimension of feature spacein MPPCA is 10 So in order to compare the monitoringperformances of these algorithms in the same situation thenumber of PCs in PCA-GMM and the selected sparse PCs inSPCA-BIPwere both selected as 10The 99 control limit wasassigned to all three algorithms

The same as TEP the FR of these algorithms are 04 012 and 1 respectively In an industry process FR lowerthan 005 is acceptable [28]

The data sets of two faults in mode 2 were tested and theMR were listed in Table 4 In the table the smallest misseddetection rates are shown in bold

As shown in Table 4 the SPCS-BIP algorithm has shownthe best performance for these two faults compared withother algorithms listed in the table It is obvious that neitherMPPCA nor PCA-GMM algorithms can detect the faultbecause their missed detection rates were high In those fouralgorithms only the SPCS-BIP was based on the selectionPCs so the improvements in the proposed sparse principalcomponents selection can be demonstrated through thebetter monitoring performance of the SPCS-BIP algorithm

Fault 1 is a bias in cooling water temperature 119879119862 Due to

the control loop in the CSTR process these would be a biasin outlet temperature 119879 and then the cooling water flow rate119865119862would increase In Figure 7 both the MMPCA and PCA-

GMMalgorithms could not detect fault 1 effectively accordingto the performances of those shown in Figures 7(a) 7(b)and 7(c) In Figure 7(d) it is obvious that the monitoringperformance of SPCS-BIP is much better than the othersThe reason is that the correct classification for each subgroupby using E-M algorithm and the PCs selected by SPCS arediscriminative and could construct the subspace that containsthe important fault information for abnormal data

Fault 2 is a bias in inlet solute concentration 119862119860119860

Thendue to the control loop in the CSTR process there wouldbe biases in outlet concentration 119862 and outlet temperature119879 According to the performances shown in Figure 8(b) theFDI of MPPCA-SPE could not detect fault 2 Comparedto the MPPCA-SPE the FDI of MPPCA-1198792 showed some

Mathematical Problems in Engineering 9

0 500 1000 1500 20000

001

002

003

004

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

0005

001

0015

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 4 Monitoring performance of the normal process

0 500 1000 1500 20000

002

004

006

008

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

05

1

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 5 Monitoring performances of fault 10 in TEP

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the 119896-nearest-neighbor method Srinivasan et al [21 22]identified the different operating modes by evaluating theEuclidean distances between samples in a constructed datawindow and then applied dynamic PCA-based similaritymeasures to cluster the samples Liu andChen [23] developeda method using Bayesian classification for selecting multipleregions from a training data set Zhao et al [24] presenteda multiple principal component analysis (MPCA) algorithmthat selects one suitable model to monitor multimode pro-cesses The other issue is how to determine the final resultsA proper measurement should be employed to determinewhich model is the most suitable one for monitoring atthe current moment Ng and Srinivasan [22] exploited themost suitable PCA model through a minimized distancereflecting both the 1198792 and SPE values Zhao et al [24] closethe local PCAmodel with theminimumSPE value Natarajanand Srinivasan [21] used the distance between the sampleand the center of local models as a criterion Yu and Qin[25] performed Bayesian inference on the postprobabilitiescalculated by the Gaussian mixture model (GMM) or thenonlinear kernel GMM Meanwhile Ge and coworkers [2627] took advantage of Bayesian inference to softly combinethe monitoring results computed by local models built bymeans of probabilistic PCA (PPCA) factor analysis (FA) orsubspace algorithms

To date the problem of how to correctly divide thetraining data into multiple subset can successfully be solvedby many algorithms mentioned in the previous paragraphHowever there are still some issues that need to be resolvedthe most important one is how to select the key principalcomponents (PCs) when using one suitablemodel for processmonitoring Many algorithms for selecting PCs have beenproposed such as cumulative percent variance (CPV) [28]variance of reconstruction error (VRE) [29] and cross vali-dation (CV) [30] Generally most of the classical algorithmsjust take normal operational observations into account andselect the first several PCs with larger variance While PCswith larger variance of normal data cannot guarantee thecapture of the largest variations in fault data online Jolliffe[31] suggested that the last PCs may be as important as thosewith large variance Togkalidou et al [32] noted that the PCswith larger variance do not always containmuch informationfor prediction However this issue is insufficiently discussedin PCA-based process monitoring and the standard PCselection is still not established

Fortunately many researches have been aware of theinherent defects of classical PCA algorithm A lot of work-ers tried to seek a subspace spanned by key PCs whichcontains the most important information for process mon-itoring Peng et al [33] suggested a new feature selectionalgorithm named minimal-redundancy-maximal-relevancecriterion (mRMR) It is based on mutual information andselects the features with highest relevance to the targetclass Jiang et al [34] put forward the sensitive principalcomponent analysis for fault detection and diagnosis inchemical processes They pointed out that PCs selectedby PCA algorithm are not always the key PCs for faultdetectionTheir task was to find the sensitive PCs which have

relationship with fault information Arbel et al [35] proposedthat the process variables that are preponderant in achievingspecific objectives need to be selected

In this paper a process monitoring algorithm usingmultisubspace sparse principal component analysis with theBIP algorithm is put forward First variables are divided intodifferent subblocks corresponding to different units or piecesof equipment to reduce the complexity of process analysis Byusing BIP algorithm multimode data in each subblock aredivided into multiple subgroups BIP can compute the poste-rior probabilities of each monitored sample belonging to themultiple components and derive an integrated global prob-abilistic index for fault detection of multimode processesThe PCs selected by PCA algorithm with larger variances donot always have relationship with fault information Sparseprincipal component selection (SPCS) takes the informationof both normal and abnormal observations into accountThealgorithm is formulated as a just-in-time form that constructsan elastic net regression between all PCs and each sampleSPCS selects PCs corresponding to the nonzero regressioncoefficients which indicate the compact expression of thesampleThis expression is necessarily discriminative amongstall subset of PCs SPCS selects the PCs whichmost compactlyexpress the sample and rejects all other possible but lesscompact expression Third the key PCs are selected by SPCSin each subgroup to solve the problem of fault informationloss It needs to be stressed that the subspace spanned by thekey PCs selected is the feature subspace Finally in order toverify the superiority of the SPCS-BIP algorithm it is appliedto the Tennessee Eastman (TE) benchmark problem and acontinuous stirred-tank reactor (CSTR) process

2 Preliminaries

21 Principal Component Analysis Principal componentanalysis is a multivariate statistical analysis which is widelyused in chemical process monitoring fault detection andso forth [36ndash38] Let x isin R119898 represent an 119898-dimensionalsample vector and X isin R119873times119898 denote a data matrix with zeromean and unit variance where 119873 is the number of samplesand 119898 is the number of variables in the process From thestatistical viewpoint the PCA algorithm could be obtainedby singular value decomposition (SVD) [28 34]

X = TP119879 + E = X + E (1)

where T isin R119873times119896 and P isin R119898times119896 are the score matrix and theloading matrix respectively 119896 is the principal componentsretained number The loading matrix P can be obtained byeigenvalue decomposition on the covariancematrix cov(x) asfollows

cov (x) asymp 1

119873 minus 1X119879X = PΛP119879 (2)

where Λ = diag1205821 1205822 120582

119898 denotes the eigenvalue

matrix and P = [P P] contains the loading matrices ofcomponent subspace and residual subspace respectively

Mathematical Problems in Engineering 3

22 Construction of Finite Gaussian Mixture Model Basedon EM For the process running at multiple operating con-dition owing to the mean shifts or covariance changes theassumption of multivariate Gaussian distribution becomesinvalid [21 22] In this situation the local Gaussian dis-tribution is still appropriate to characterize each subsetof measurement data from the same operating conditionsTherefore the finite Gaussian mixture model is prime suitedto represent the data sources driven by different operatingmodes [13 24 25]

To construct a FGMM given a set of training samplesX isin

R119873times119898 the log-likelihood function can be expressed as

log 119871 (X Θ) =119873

sum

119895=1

log(119871

sum

119894=1

120596119894119892 (119909119895| 120579119894)) (3)

and the parameter estimation problem is formulated as

Θ = argmaxΘ

(log 119871 (X Θ)) (4)

where Θ = 120596119894 120583119894 Σ1 120596

119871 120583119871 Σ119871 120596119894(1 le 119894 le 119871)

is the prior probabilities and 119871 is the number of Gaussiancomponents included in FGMM 120583

119894is the mean vector and

Σ119894is the covariance matrixThere are a lot of learning algorithms such as maximum

likelihood estimation (MLE) EM and the F-J algorithm thathave been put forward formixturemodel estimation [39 40]As amore tractable numerical strategy the EM algorithm hasbeen well used in practice to estimate the maximum like-lihood distribution parameter [39] EM algorithm is imple-mented iteratively by means of repeating the expectation step(E-step) andmaximization step (M-step) to calculate the pos-terior probabilities and then the corresponding distributionparameters until a convergence criterion of the log-likelihoodfunction is satisfied Given the training data X and an initialestimate Θ(0) = 120596

(0)

1 120583(0)

1 Σ(0)

1 120596

(0)

119871 120583(0)

119871 Σ(0)

119871 the

iterative E-step and M-step are expressed as follows(i) E-step

119875(119904)(119862119897| x119895) =

120596(119904)

119897119892 (x119895| 120583(119904)

119897 Σ(119904)

119897)

sum119871

119894=1120596(119904)

119894119892 (x119895| 120583(119904)

119894 Σ(119904)

119894)

(5)

where119875(119904)(119862119897| x119895) denotes the posterior probability of the 119895th

training sample within the 119897th Gaussian component at the 119904thiteration

(ii) M-step

120583(119904+1)

119897=

sum119873

119895=1119875(119904)(119862119897| x119895) x119895

sum119873

119895=1119875(119904) (119862

119897| x119895)

Σ(119904+1)

119897

=

sum119873

119895=1119875(119904)(119862119897| x119895) (x119895minus 120583(119904+1)

119897) (x119895minus 120583(119904+1)

119897)119879

sum119873

119895=1119875(119904) (119862

119897| x119895)

120596(119904+1)

119897=

sum119873

119895=1119875(119904)(119862119897| x119895)

119873

(6)

where 120583(119904+1)119897

Σ(119904+1)119897

and 120596(119904+1)119897

are the mean covariance andprior probability of the 119897thGaussian component at the (119904+1)thiteration respectively

3 Fault Detection with Sparse PrincipalComponent Selection andBayesian Inference-Based Probability

In this section the idea of SPCS-BIP algorithm formultimodeprocess monitoring is demonstrated in detail We first intro-duce the Bayesian inference-based probability which canderive the confidence boundary around the normal operatingregions for process monitoring and fault detection Thenthe sparse principal component selection was introduced forselecting the key Pcs related with fault information Finallythe steps of this algorithm were given

31 Bayesian Inference-Based Probability In the previoussection the FGMM has been constructed and it is essentialto further derive the confidence boundary around the normaloperating regions for process monitoring and fault detectionDue to the multimodality of mixture distribution it is reallydifficult to capture the analytical boundary of the densityfunction 119901(119909 | Θ) in a certain confidence level

In the proposed monitoring approach given an arbitrarymonitored sample 119909

119905belonging to eachGaussian component

Bayesian inference strategy is used to calculate the posteriorprobability as follows

119875 (119909119905isin 119862119897) = 119875 (119862

119897| 119909119905) =

119875 (119862119897) sdot 119901 (119909

119905| 119862119897)

119901 (119909119905)

=119875 (119862119897) sdot 119901 (119909

119905| 119862119897)

sum119871

119894=1119875 (119862119894) sdot 119901 (119909

119905| 119862119894)

(7)

which can also be formulated as

119875 (119909119905isin 119862119897) =

120596119897119892 (119909119905| 120583119897 Σ119897)

sum119871

119894=1120596119894119892 (119909119905| 120583119894 Σ119894)

(119897 = 1 2 119871) (8)

Given that each component 119862119897follows a unimodal Gaus-

sian distribution the squared Mahalanobis distance of 119909119905

from the center of 119862119897follows 1205942 distribution provided that

119909119894belongs to 119862

119897

119863((119909119905 119862119897) | 119909119905isin 119862119897) = (119909

119905minus 120583119897)119879

Σminus1

119897(119909119905minus 120583119897) 1205942

119898 (9)

Under the assumption that 119909119905isin 119862119897and 1205942

119898has 119898 degree

of freedom 119863((119909119905 119862119897) | 119909

119905isin 119862119897) denotes the squared

Mahalanobis distance between 119909119905and the mean center of

119862119897 Owing to colinearity Σ

119897is usually ill-conditioned and

the following regularized Mahalanobis distance is utilizedinstead to avoid too wide confidence regions

119863119903((119909119905 119862119897) | 119909119905isin 119862119897)

= (119909119905minus 120583119897)119879

(Σ119897+ 120576119868)minus1

(119909119905minus 120583119897)

(10)

4 Mathematical Problems in Engineering

where the function of 120576 is to remove the ill condition ofcovariance matrix Σ

119897by adding a positive number to all the

diagonal entriesFor the monitored sample 119909

119905 a local Mahalanobis

distance-based probability index relative to each Gaussiancomponent 119862

119897can be defined as

119875(119897)

119871(119909119905)

= Pr 119863 ((119909 119862119897) | 119909 isin 119862

119896) le 119863 ((119909

119894 119862119897) | 119909119905isin 119862119897)

(11)

or

119875(119897)

119871(119909119905) = Pr 119863

119903((119909 119862

119897) | 119909 isin 119862

119897)

le 119863119903((119909119894 119862119897) | 119909119905isin 119862119897)

(12)

Given the appropriate degree of freedom 119875(119897)119871(119909119905) can be

computed by integrating the 1205942 probability density functionUnder a given confidence level this index has the functionof indicating whether the monitored sample is normal orabnormal provided that it belongs to the correspondingGaussian component A global BIP index is proposed tocombine the local probability metrics across all the Gaussianclusters because the random characteristic of eachmonitoredsample may come from multiple Gaussian components withthe corresponding posterior probabilitiesThe formulation ofBIP index for the monitored sample 119909

119905is given by

BIP =119871

sum

119897=1

119875 (119862119897| 119909119905) 119875(119897)

119871(119909119905) (13)

where the posterior probability 119875(119862119897| 119909119905) is used to incor-

porate the contribution of each local Gaussian component tothe overall probabilistic index As 0 le 119875(119897)

119871(119909119905) le 1 we have

0 le BIP le119871

sum

119897=1

119875 (119862119897| 119909119905) = 1 (14)

Under the preset confidence level (1minus120572) 100 the processis determined within normal operation if

BIP le 1 minus 120572 (15)

Otherwise the process operation is treated out of control

32 Sparse Principal Component Selection Sparse represen-tation has proven to be an extremely powerful tool foracquiring representing and compressing high-dimensionaldata [41ndash43] This success is mainly because of the factthat the important reconstruction information of data suchas process data and time series data has naturally sparserepresentations with respect to fixed bases or concatenationsof such bases Qiao et al [44] proposed that the graphsconstructed by the 119871

1-norm have the advantage of greater

robustness to data noise automatic sparsity and adaptiveneighborhood for individual datum What is more anotherimportant advantage is that sparse representation has thepotential discriminative ability since most nonzero elements

are located on the samples in the same class as the representedsample

Given the training sample L = [1198971 1198972 119897119873] isin R119873times119898

a test sample H isin R119898 the solution to the sparse represen-tation problem can be obtained by solving the following ℓ1-minimization problem

(ℓ1) 120573 = argmin 1003817100381710038171003817120573

1003817100381710038171003817

subject to L120573 = H

Card (120573) le 119896

(16)

where 120573 = [1205731 1205732 120573

119873] are the sparse representation

coefficients and Card(120573) denotes the number of nonzeroelements of 120573 From the perspective of statistics formula(16) can be named the Lasso criterion Lasso is a penalizedleast squares algorithm which was originally by quadraticprogramming imposing a constraint on the 119871

1norm of

the regression coefficients Thus the Lasso estimates 120573 areobtained by minimizing the Lasso criterion

120573 = argmin120573

1003817100381710038171003817100381710038171003817100381710038171003817

H minussum

119894=1

L119894120573119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

+ 120582

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

subject to Card (120573) le 119896

(17)

where 120582 is nonnegative However only using the 1198711-norm

penalty in Lasso has its limitation Zou et al [45] proposedthat if there is a group of variables among which the pairwisecorrelations are very high lasso tends to select any variablefrom the group and does not consider which one is selectedFortunately elastic net was put forward by Zou et al whichovercomes the limitation of only using the 119871

1-norm penalty

It is known that combining the 1198711-norm and 119871

2-norm

penalty can result in grouping effectiveness in regression andthus enhance the prediction accuracy For any nonnegative1205821and 120582

2 the elastic net estimates 120573en are given by

120573en = argmin120573

1003817100381710038171003817100381710038171003817100381710038171003817

H minus

119873

sum

119894=1

L119894120573119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

+ 1205821

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

+ 1205822

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

2

subject to Card (120573) le 119896

(18)

In brief it is expected that the elastic net is used to groupa set of sparse coefficients to construct the sparse alignmentmatrices in which the sparse representation informationor the potential discriminative information is encoded toenhance the discriminative ability in an unsupervised man-ner

33 Fault Detection with SPCS and BIP The key problemfor monitoring the multimode process is to select a suitablemodel and choose the subspace spanned by key PCs In theIntroduction we had put forward the fact that the subspace

Mathematical Problems in Engineering 5

Offline modeling

Use EM algorithm to learn the GMM andestimate the model parameters

For each submodel normalize the trainingdata

Obtain the basic PCs using SVDdecomposition

Obtain the sparse PCs of normal data byconstructing elastic net regression

End

Online modeling

Current data normalization

Obtain the sparse PCs of each testingsample by constructing elastic net

regression

Calculate BIP index value of each testingsample

Exceed limit

Next

No

There is a fault in process

YesSpecify a confidence level and constantcontrol limit

Figure 1 The steps of SPCS-BIP algorithm for process monitoring

spanned by the first several PCs with largest explainedvariance does not always have fault information

In the following part a novel multimode process mon-itoring approach based on SPCS and BIP is proposed Thisapproach is in a just-in-time form For each sample an elasticnet regression between all PCs and the sample is constructedand solved The PCs which have nonzero regression coeffi-cients are retained while other PCs are rejected That meansfor each sample we can pick out the most discriminativebases and the others are set to zero Its concrete calculatingsteps are summarized in Figure 1

Offline Modeling

(1) Collect a set of historical training data under allpossible operating conditions

(2) Use the EM algorithm to learn the Gaussian mixturemodel and estimate the model parameter set Θ =

1205831 Σ1 1205961 120583

119896 Σ119896 120596119896 based on the iterative steps

(3) For each submodel get a normal operational obser-vation set X =isin R119873times119898 where 119873 is the number ofsamples and 119898 is the number of variables This set isdenoted as the training set for threshold determiningA testing set Y isin R119878times119898 with both normal andabnormal observations is given for testing

(4) Normalize the training data through the mean valueand variance of each variable

(5) Obtain all principal components using SVD decom-position The training data X is reconstructed by X =

sum119898

119894=1t119894p119879119894 where t

119894is the score vector and p

119894is the

loading vector(6) For training sample x

119895(119895 = 1 2 119873) construct an

elastic net regression between each observation valueof training data and loading vector Pmade of PCs instep (5) according to 120573

119895= argmin

120573x119895minussum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

(7) Corresponding to the nonzero representation coef-ficients 120573

1198951

1205731198952

120573119895119896

construct a new loadingvector P

119895= [p1198951

p1198952

p119895119896

](8) Specify a confidence (1minus120572) 100 and constant control

limit 1 minus 120572

Online Monitoring

(1) Normalize the current time point data by using meanvalues and variance of the training data

(2) Obtain the loading vector P from offline modeling(3) When a test sample y

119895isin R119898 (119895 = 1 2 119878) is

coming construct an elastic net regression betweenthe sample and loading vector Pmade of PCs in step(2) according to 120573 = argmin

120573y119895minus sum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

6 Mathematical Problems in Engineering

(4) Corresponding to the nonzero representation coef-ficients 120573

1198971

1205731198972

120573119897119896

construct a new loadingvector P

119895= [p1198971

p1198972

p119897119896

](5) Generate theBIP control chartwith the calculatedBIP

index values for all the monitored samples If the BIPindex of a test sample is lower than the control limitwhichmeans the sample is normal go to step (1) Elsethere is a fault in the process

4 Case Studies on the TE and CSTR Process

In this case study the TE benchmark and CSTR processare introduced to verify the effectiveness of the SPCS-BIPalgorithm PCA-GMM is the classic algorithm formultimodeprocessing monitoring And the fault detection index (FDI)is similar to Bayesian inference probability (BIP) So here acomparison was made between SPCS-BIP and PCA-GMMIn addition to verify the improvements of SPCS algorithmwhich can select sparse PCs a comparison was performedbetween the SPCS-BIP algorithm and theMPPCA algorithm

41 Tennessee Eastman Process As a well-known benchmarkprocess the Tennessee Eastman process which was pre-sented by Downs and Vogel has been widely applied toevaluate and compare the efficiency of process monitoringtechniques [46 47] The schematic diagram of the processis illustrated in Figure 2 This process consists of five majorunit operations a reactor a product condenser a vapor-liquidseparator a recycle compressor and a product stripper Inaddition there are six modes of process operation as listedin Table 1 The variables can be divided into three categoriescomposition variables continuous process variables andmanipulated variables In our study only modes 1 and 3were simulated through the Simulink programs developedon the basis of the decentralized control strategy designed byRicker [48]The Simulink programs can be downloaded fromhttpdeptswashingtoneducontrolLARRYTEdownloadhtml The 31 selected monitoring variables contained 9manipulated variables and 22 continuous process variablesThus these variables were divided into five subblocks accord-ing to five units However given that only two variables wereallocated to each the compressor unit and the condenser unitthere were four variables assigned to the other three relatedsubblocks As a result the total of 31 variables was dividedinto three subblocks

There are 20 faults in the multimode TE process whichare listed in Table 2 Among these faults the root causes of thefaults 16ndash20 are unknown [46 47] What is more to simplifyinterpretation the amplitudes of faults 3 9 and 15 are sosmall It is difficult to detect so only the remaining 12 faultswere considered in this study In the modeling stage 2000normal samples which include 1000mode 1 samples and 1000mode 3 samples were collected as the training data set Inthe testing stage 1000 samples of mode 1 were tested firstand then the process switches to mode 3 As a result the testdata set consists of 1000 samples of mode 1 and 1000 samplesof mode 3 And faults occurred from the 1200th sample Aset of 20 faults in multimode TE process which are listed

Table 1 Six process operation modes of TE process

Mode GH mass ratio Production rate1 5050 7038 kghG and 7038 kghH2 1090 1048 kghG and 12669 kghH3 9010 10000 kghG and 1111 kghH4 5050 Maximum5 1090 Maximum6 9010 Maximum

in Table 2 are simulated and the corresponding process dataare collected for testingThe following simulations are run inMATLAB 830 (2014a) environment Here two indicatorswhich are FR (FR) and MR (MR) are often introduced tomeasure the result of process monitoring FR is the rate ofnormal data classified as fault dataMR is the rate of fault dataclassified as normal rate

In the MPPCA algorithm and PCA-GMM algorithmwhen the variance contribution was selected as 85 thedimension of feature space inMPPCAand the number of PCsin PCA-GMM were each selected as 18 In order to comparethemonitoring performances of these algorithms in the samesituation the selected sparse PCs of each mode in SPCS-BIPwere selected as 18 The 99 control limit was assigned to allthree algorithms

First Figure 3 shows that the different submodes can besuccessfully divided by the EM algorithm used in this paperAnd by using other algorithms themodes also can be dividedcorrectly In other words how to correctly divide the trainingdata into multiple subset is not a problem by many relatedalgorithms

The normal process was tested by different algorithm andthe results are shown in Figure 4 In this figure it is hard tofigure out which algorithmrsquos FR is lower In the figure mostsamples of each algorithm are lower than the control limitAnd by calculation the FR of these algorithms are 0333008 025 and 108 respectively corresponding to Fig-ures 4(a) 4(b) 4(c) and 4(d) The monitoring performancesof these three algorithms suggest that the FR are acceptableNext the data sets of 12 faults in mode 3 were tested and theMR of these three algorithms are listed in Table 3 with thesmallest MR shown in bold

From Table 3 we observe that the monitoring perfor-mance of SPCS-BIP is the best compared to the MPPCAand PCA-GMM algorithms for all 12 faults Here we takethe further analysis In comparison with MPPCA and PCA-GMMalgorithms the SPCS-BIP algorithm can exactly dividethe process data into subgroups corresponding to differentmodes by using the E-M algorithm and in each submodeSPCS-BIP can select the most important PCs that have mostrelation with the fault Due to the fact that the subspacespanned by the PCs was monitored by BIP most of the PCsare related to themain process of chemical industrial processand only little PCs are related to the fault process SPCSare discriminative by constructing an elastic net regressionbetween all PCs and each sample So in Table 3 we observe

Mathematical Problems in Engineering 7

Table 2 Process faults for the multimode TE process

Faults number Disturbance state TypeIDV(1) AC feed ratio B composition constant (Stream 4) StepIDV(2) B composition AC ratio constant (Stream 4) StepIDV(3) D feed temperature (Stream 2) StepIDV(4) Reactor cooling water inlet temperature StepIDV(5) Condenser cooling water inlet temperature StepIDV(6) A feed loss (Stream 1) StepIDV(7) C header pressure loss reduced availability (Stream 4) StepIDV(8) A B and C feed composition (Stream 4) Random variationIDV(9) D feed temperature (Stream 2) Random variationIDV(10) C feed temperature (Stream 4) Random variationIDV(11) Reactor cooling water inlet temperature Random variationIDV(12) Condenser cooling water inlet temperature Random variationIDV(13) Reaction kinetics Slow driftIDV(14) Reactor cooling water valve StickingIDV(15) Condenser cooling water valve StickingIDV(16) Unknown UnknownIDV(17) Unknown UnknownIDV(18) Unknown UnknownIDV(19) Unknown UnknownIDV(20) Unknown Unknown

1

5

2

3

4

A

D

E

C

FI

FI

FI

FI

FIFI

FI

FI

FI

FI

XA

XB

XC

XD

XE

XF

XG

XH

XD

XE

XF

XG

XH

XA

XB

XC

XD

XE

XF

Compressor

Ana

lyze

rA

naly

zer

Ana

lyze

r

6

7

8

9

10

11

12

SC

PI

PI

PI

JI

TI

Condenser

CWS

CWS

CWR

Reactor

CWR

Stripper

TI

TI

TI

TILI

LI

LI

Purge

Product

Stm

Cond

Vapliqseparator

Figure 2 Control scheme for the TE process

8 Mathematical Problems in Engineering

Sample number0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

11

12

13

14

15

16

17

18

19

2

Labe

l

Figure 3 Different modes of the training data

Table 3 Missed detection rates () of 12 faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 075 025 1 01252 35 85 55 18754 0 0 0 05 0375 0 1125 06 0 0 0 07 0 0 0 08 3375 3875 425 250010 82375 16 89875 787511 225 85 45 112512 1375 1625 15 075013 16125 265 19625 1162514 0 35 0125 0

that the results of SPCS-BIP are better than the results ofMPPCA and PCA-GMM

Figure 5 shows the monitoring performances of fault 10It is easy to see that the FDI of MPPCA-1198792 and the BIP ofPCA-GMM cannot detect the fault effectively in Figures 4(a)and 4(c) In the figure more than half of the fault sampleswere regarded as the normal samples while compared tothe performances of MPPCA-1198792 and PCA-GMM the FDIof MPPCA-SPE shows some improvements However themonitoring performance of MPPCA-SPE does not match theperformance of SPCS-BIP We can find this point both inFigure 4 and Table 3

42 CSTR This study simulated the CSTR process describedby Yoon and MacGregor [49] The diagram of the process ispresented in Figure 6 Due to the fact that the CSTR processconsisted of only one operating unit the number of subblockswas selected as 1

Table 4 Missed detection rates () of two faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 90 100 988 6562 288 996 318 28

In the modeling stage 1000 samples which include 500mode 1 samples and 500mode 2 samples were collected as thetraining data set In the testing stage 1000 samples of mode 2were tested and two faults were introduced to the process asfollows

Case 1 A step of 1 K was added in the cooling watertemperature 119879

119862from the 500th sample

Case 2 A 2 kmol(m3 sdotmin) step was added in the inlet soluteconcentration 119862

119860119860from the 500th sample

In the MMPCA algorithm when the variance contri-bution was selected as 85 the dimension of feature spacein MPPCA is 10 So in order to compare the monitoringperformances of these algorithms in the same situation thenumber of PCs in PCA-GMM and the selected sparse PCs inSPCA-BIPwere both selected as 10The 99 control limit wasassigned to all three algorithms

The same as TEP the FR of these algorithms are 04 012 and 1 respectively In an industry process FR lowerthan 005 is acceptable [28]

The data sets of two faults in mode 2 were tested and theMR were listed in Table 4 In the table the smallest misseddetection rates are shown in bold

As shown in Table 4 the SPCS-BIP algorithm has shownthe best performance for these two faults compared withother algorithms listed in the table It is obvious that neitherMPPCA nor PCA-GMM algorithms can detect the faultbecause their missed detection rates were high In those fouralgorithms only the SPCS-BIP was based on the selectionPCs so the improvements in the proposed sparse principalcomponents selection can be demonstrated through thebetter monitoring performance of the SPCS-BIP algorithm

Fault 1 is a bias in cooling water temperature 119879119862 Due to

the control loop in the CSTR process these would be a biasin outlet temperature 119879 and then the cooling water flow rate119865119862would increase In Figure 7 both the MMPCA and PCA-

GMMalgorithms could not detect fault 1 effectively accordingto the performances of those shown in Figures 7(a) 7(b)and 7(c) In Figure 7(d) it is obvious that the monitoringperformance of SPCS-BIP is much better than the othersThe reason is that the correct classification for each subgroupby using E-M algorithm and the PCs selected by SPCS arediscriminative and could construct the subspace that containsthe important fault information for abnormal data

Fault 2 is a bias in inlet solute concentration 119862119860119860

Thendue to the control loop in the CSTR process there wouldbe biases in outlet concentration 119862 and outlet temperature119879 According to the performances shown in Figure 8(b) theFDI of MPPCA-SPE could not detect fault 2 Comparedto the MPPCA-SPE the FDI of MPPCA-1198792 showed some

Mathematical Problems in Engineering 9

0 500 1000 1500 20000

001

002

003

004

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

0005

001

0015

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 4 Monitoring performance of the normal process

0 500 1000 1500 20000

002

004

006

008

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

05

1

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 5 Monitoring performances of fault 10 in TEP

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

22 Construction of Finite Gaussian Mixture Model Basedon EM For the process running at multiple operating con-dition owing to the mean shifts or covariance changes theassumption of multivariate Gaussian distribution becomesinvalid [21 22] In this situation the local Gaussian dis-tribution is still appropriate to characterize each subsetof measurement data from the same operating conditionsTherefore the finite Gaussian mixture model is prime suitedto represent the data sources driven by different operatingmodes [13 24 25]

To construct a FGMM given a set of training samplesX isin

R119873times119898 the log-likelihood function can be expressed as

log 119871 (X Θ) =119873

sum

119895=1

log(119871

sum

119894=1

120596119894119892 (119909119895| 120579119894)) (3)

and the parameter estimation problem is formulated as

Θ = argmaxΘ

(log 119871 (X Θ)) (4)

where Θ = 120596119894 120583119894 Σ1 120596

119871 120583119871 Σ119871 120596119894(1 le 119894 le 119871)

is the prior probabilities and 119871 is the number of Gaussiancomponents included in FGMM 120583

119894is the mean vector and

Σ119894is the covariance matrixThere are a lot of learning algorithms such as maximum

likelihood estimation (MLE) EM and the F-J algorithm thathave been put forward formixturemodel estimation [39 40]As amore tractable numerical strategy the EM algorithm hasbeen well used in practice to estimate the maximum like-lihood distribution parameter [39] EM algorithm is imple-mented iteratively by means of repeating the expectation step(E-step) andmaximization step (M-step) to calculate the pos-terior probabilities and then the corresponding distributionparameters until a convergence criterion of the log-likelihoodfunction is satisfied Given the training data X and an initialestimate Θ(0) = 120596

(0)

1 120583(0)

1 Σ(0)

1 120596

(0)

119871 120583(0)

119871 Σ(0)

119871 the

iterative E-step and M-step are expressed as follows(i) E-step

119875(119904)(119862119897| x119895) =

120596(119904)

119897119892 (x119895| 120583(119904)

119897 Σ(119904)

119897)

sum119871

119894=1120596(119904)

119894119892 (x119895| 120583(119904)

119894 Σ(119904)

119894)

(5)

where119875(119904)(119862119897| x119895) denotes the posterior probability of the 119895th

training sample within the 119897th Gaussian component at the 119904thiteration

(ii) M-step

120583(119904+1)

119897=

sum119873

119895=1119875(119904)(119862119897| x119895) x119895

sum119873

119895=1119875(119904) (119862

119897| x119895)

Σ(119904+1)

119897

=

sum119873

119895=1119875(119904)(119862119897| x119895) (x119895minus 120583(119904+1)

119897) (x119895minus 120583(119904+1)

119897)119879

sum119873

119895=1119875(119904) (119862

119897| x119895)

120596(119904+1)

119897=

sum119873

119895=1119875(119904)(119862119897| x119895)

119873

(6)

where 120583(119904+1)119897

Σ(119904+1)119897

and 120596(119904+1)119897

are the mean covariance andprior probability of the 119897thGaussian component at the (119904+1)thiteration respectively

3 Fault Detection with Sparse PrincipalComponent Selection andBayesian Inference-Based Probability

In this section the idea of SPCS-BIP algorithm formultimodeprocess monitoring is demonstrated in detail We first intro-duce the Bayesian inference-based probability which canderive the confidence boundary around the normal operatingregions for process monitoring and fault detection Thenthe sparse principal component selection was introduced forselecting the key Pcs related with fault information Finallythe steps of this algorithm were given

31 Bayesian Inference-Based Probability In the previoussection the FGMM has been constructed and it is essentialto further derive the confidence boundary around the normaloperating regions for process monitoring and fault detectionDue to the multimodality of mixture distribution it is reallydifficult to capture the analytical boundary of the densityfunction 119901(119909 | Θ) in a certain confidence level

In the proposed monitoring approach given an arbitrarymonitored sample 119909

119905belonging to eachGaussian component

Bayesian inference strategy is used to calculate the posteriorprobability as follows

119875 (119909119905isin 119862119897) = 119875 (119862

119897| 119909119905) =

119875 (119862119897) sdot 119901 (119909

119905| 119862119897)

119901 (119909119905)

=119875 (119862119897) sdot 119901 (119909

119905| 119862119897)

sum119871

119894=1119875 (119862119894) sdot 119901 (119909

119905| 119862119894)

(7)

which can also be formulated as

119875 (119909119905isin 119862119897) =

120596119897119892 (119909119905| 120583119897 Σ119897)

sum119871

119894=1120596119894119892 (119909119905| 120583119894 Σ119894)

(119897 = 1 2 119871) (8)

Given that each component 119862119897follows a unimodal Gaus-

sian distribution the squared Mahalanobis distance of 119909119905

from the center of 119862119897follows 1205942 distribution provided that

119909119894belongs to 119862

119897

119863((119909119905 119862119897) | 119909119905isin 119862119897) = (119909

119905minus 120583119897)119879

Σminus1

119897(119909119905minus 120583119897) 1205942

119898 (9)

Under the assumption that 119909119905isin 119862119897and 1205942

119898has 119898 degree

of freedom 119863((119909119905 119862119897) | 119909

119905isin 119862119897) denotes the squared

Mahalanobis distance between 119909119905and the mean center of

119862119897 Owing to colinearity Σ

119897is usually ill-conditioned and

the following regularized Mahalanobis distance is utilizedinstead to avoid too wide confidence regions

119863119903((119909119905 119862119897) | 119909119905isin 119862119897)

= (119909119905minus 120583119897)119879

(Σ119897+ 120576119868)minus1

(119909119905minus 120583119897)

(10)

4 Mathematical Problems in Engineering

where the function of 120576 is to remove the ill condition ofcovariance matrix Σ

119897by adding a positive number to all the

diagonal entriesFor the monitored sample 119909

119905 a local Mahalanobis

distance-based probability index relative to each Gaussiancomponent 119862

119897can be defined as

119875(119897)

119871(119909119905)

= Pr 119863 ((119909 119862119897) | 119909 isin 119862

119896) le 119863 ((119909

119894 119862119897) | 119909119905isin 119862119897)

(11)

or

119875(119897)

119871(119909119905) = Pr 119863

119903((119909 119862

119897) | 119909 isin 119862

119897)

le 119863119903((119909119894 119862119897) | 119909119905isin 119862119897)

(12)

Given the appropriate degree of freedom 119875(119897)119871(119909119905) can be

computed by integrating the 1205942 probability density functionUnder a given confidence level this index has the functionof indicating whether the monitored sample is normal orabnormal provided that it belongs to the correspondingGaussian component A global BIP index is proposed tocombine the local probability metrics across all the Gaussianclusters because the random characteristic of eachmonitoredsample may come from multiple Gaussian components withthe corresponding posterior probabilitiesThe formulation ofBIP index for the monitored sample 119909

119905is given by

BIP =119871

sum

119897=1

119875 (119862119897| 119909119905) 119875(119897)

119871(119909119905) (13)

where the posterior probability 119875(119862119897| 119909119905) is used to incor-

porate the contribution of each local Gaussian component tothe overall probabilistic index As 0 le 119875(119897)

119871(119909119905) le 1 we have

0 le BIP le119871

sum

119897=1

119875 (119862119897| 119909119905) = 1 (14)

Under the preset confidence level (1minus120572) 100 the processis determined within normal operation if

BIP le 1 minus 120572 (15)

Otherwise the process operation is treated out of control

32 Sparse Principal Component Selection Sparse represen-tation has proven to be an extremely powerful tool foracquiring representing and compressing high-dimensionaldata [41ndash43] This success is mainly because of the factthat the important reconstruction information of data suchas process data and time series data has naturally sparserepresentations with respect to fixed bases or concatenationsof such bases Qiao et al [44] proposed that the graphsconstructed by the 119871

1-norm have the advantage of greater

robustness to data noise automatic sparsity and adaptiveneighborhood for individual datum What is more anotherimportant advantage is that sparse representation has thepotential discriminative ability since most nonzero elements

are located on the samples in the same class as the representedsample

Given the training sample L = [1198971 1198972 119897119873] isin R119873times119898

a test sample H isin R119898 the solution to the sparse represen-tation problem can be obtained by solving the following ℓ1-minimization problem

(ℓ1) 120573 = argmin 1003817100381710038171003817120573

1003817100381710038171003817

subject to L120573 = H

Card (120573) le 119896

(16)

where 120573 = [1205731 1205732 120573

119873] are the sparse representation

coefficients and Card(120573) denotes the number of nonzeroelements of 120573 From the perspective of statistics formula(16) can be named the Lasso criterion Lasso is a penalizedleast squares algorithm which was originally by quadraticprogramming imposing a constraint on the 119871

1norm of

the regression coefficients Thus the Lasso estimates 120573 areobtained by minimizing the Lasso criterion

120573 = argmin120573

1003817100381710038171003817100381710038171003817100381710038171003817

H minussum

119894=1

L119894120573119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

+ 120582

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

subject to Card (120573) le 119896

(17)

where 120582 is nonnegative However only using the 1198711-norm

penalty in Lasso has its limitation Zou et al [45] proposedthat if there is a group of variables among which the pairwisecorrelations are very high lasso tends to select any variablefrom the group and does not consider which one is selectedFortunately elastic net was put forward by Zou et al whichovercomes the limitation of only using the 119871

1-norm penalty

It is known that combining the 1198711-norm and 119871

2-norm

penalty can result in grouping effectiveness in regression andthus enhance the prediction accuracy For any nonnegative1205821and 120582

2 the elastic net estimates 120573en are given by

120573en = argmin120573

1003817100381710038171003817100381710038171003817100381710038171003817

H minus

119873

sum

119894=1

L119894120573119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

+ 1205821

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

+ 1205822

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

2

subject to Card (120573) le 119896

(18)

In brief it is expected that the elastic net is used to groupa set of sparse coefficients to construct the sparse alignmentmatrices in which the sparse representation informationor the potential discriminative information is encoded toenhance the discriminative ability in an unsupervised man-ner

33 Fault Detection with SPCS and BIP The key problemfor monitoring the multimode process is to select a suitablemodel and choose the subspace spanned by key PCs In theIntroduction we had put forward the fact that the subspace

Mathematical Problems in Engineering 5

Offline modeling

Use EM algorithm to learn the GMM andestimate the model parameters

For each submodel normalize the trainingdata

Obtain the basic PCs using SVDdecomposition

Obtain the sparse PCs of normal data byconstructing elastic net regression

End

Online modeling

Current data normalization

Obtain the sparse PCs of each testingsample by constructing elastic net

regression

Calculate BIP index value of each testingsample

Exceed limit

Next

No

There is a fault in process

YesSpecify a confidence level and constantcontrol limit

Figure 1 The steps of SPCS-BIP algorithm for process monitoring

spanned by the first several PCs with largest explainedvariance does not always have fault information

In the following part a novel multimode process mon-itoring approach based on SPCS and BIP is proposed Thisapproach is in a just-in-time form For each sample an elasticnet regression between all PCs and the sample is constructedand solved The PCs which have nonzero regression coeffi-cients are retained while other PCs are rejected That meansfor each sample we can pick out the most discriminativebases and the others are set to zero Its concrete calculatingsteps are summarized in Figure 1

Offline Modeling

(1) Collect a set of historical training data under allpossible operating conditions

(2) Use the EM algorithm to learn the Gaussian mixturemodel and estimate the model parameter set Θ =

1205831 Σ1 1205961 120583

119896 Σ119896 120596119896 based on the iterative steps

(3) For each submodel get a normal operational obser-vation set X =isin R119873times119898 where 119873 is the number ofsamples and 119898 is the number of variables This set isdenoted as the training set for threshold determiningA testing set Y isin R119878times119898 with both normal andabnormal observations is given for testing

(4) Normalize the training data through the mean valueand variance of each variable

(5) Obtain all principal components using SVD decom-position The training data X is reconstructed by X =

sum119898

119894=1t119894p119879119894 where t

119894is the score vector and p

119894is the

loading vector(6) For training sample x

119895(119895 = 1 2 119873) construct an

elastic net regression between each observation valueof training data and loading vector Pmade of PCs instep (5) according to 120573

119895= argmin

120573x119895minussum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

(7) Corresponding to the nonzero representation coef-ficients 120573

1198951

1205731198952

120573119895119896

construct a new loadingvector P

119895= [p1198951

p1198952

p119895119896

](8) Specify a confidence (1minus120572) 100 and constant control

limit 1 minus 120572

Online Monitoring

(1) Normalize the current time point data by using meanvalues and variance of the training data

(2) Obtain the loading vector P from offline modeling(3) When a test sample y

119895isin R119898 (119895 = 1 2 119878) is

coming construct an elastic net regression betweenthe sample and loading vector Pmade of PCs in step(2) according to 120573 = argmin

120573y119895minus sum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

6 Mathematical Problems in Engineering

(4) Corresponding to the nonzero representation coef-ficients 120573

1198971

1205731198972

120573119897119896

construct a new loadingvector P

119895= [p1198971

p1198972

p119897119896

](5) Generate theBIP control chartwith the calculatedBIP

index values for all the monitored samples If the BIPindex of a test sample is lower than the control limitwhichmeans the sample is normal go to step (1) Elsethere is a fault in the process

4 Case Studies on the TE and CSTR Process

In this case study the TE benchmark and CSTR processare introduced to verify the effectiveness of the SPCS-BIPalgorithm PCA-GMM is the classic algorithm formultimodeprocessing monitoring And the fault detection index (FDI)is similar to Bayesian inference probability (BIP) So here acomparison was made between SPCS-BIP and PCA-GMMIn addition to verify the improvements of SPCS algorithmwhich can select sparse PCs a comparison was performedbetween the SPCS-BIP algorithm and theMPPCA algorithm

41 Tennessee Eastman Process As a well-known benchmarkprocess the Tennessee Eastman process which was pre-sented by Downs and Vogel has been widely applied toevaluate and compare the efficiency of process monitoringtechniques [46 47] The schematic diagram of the processis illustrated in Figure 2 This process consists of five majorunit operations a reactor a product condenser a vapor-liquidseparator a recycle compressor and a product stripper Inaddition there are six modes of process operation as listedin Table 1 The variables can be divided into three categoriescomposition variables continuous process variables andmanipulated variables In our study only modes 1 and 3were simulated through the Simulink programs developedon the basis of the decentralized control strategy designed byRicker [48]The Simulink programs can be downloaded fromhttpdeptswashingtoneducontrolLARRYTEdownloadhtml The 31 selected monitoring variables contained 9manipulated variables and 22 continuous process variablesThus these variables were divided into five subblocks accord-ing to five units However given that only two variables wereallocated to each the compressor unit and the condenser unitthere were four variables assigned to the other three relatedsubblocks As a result the total of 31 variables was dividedinto three subblocks

There are 20 faults in the multimode TE process whichare listed in Table 2 Among these faults the root causes of thefaults 16ndash20 are unknown [46 47] What is more to simplifyinterpretation the amplitudes of faults 3 9 and 15 are sosmall It is difficult to detect so only the remaining 12 faultswere considered in this study In the modeling stage 2000normal samples which include 1000mode 1 samples and 1000mode 3 samples were collected as the training data set Inthe testing stage 1000 samples of mode 1 were tested firstand then the process switches to mode 3 As a result the testdata set consists of 1000 samples of mode 1 and 1000 samplesof mode 3 And faults occurred from the 1200th sample Aset of 20 faults in multimode TE process which are listed

Table 1 Six process operation modes of TE process

Mode GH mass ratio Production rate1 5050 7038 kghG and 7038 kghH2 1090 1048 kghG and 12669 kghH3 9010 10000 kghG and 1111 kghH4 5050 Maximum5 1090 Maximum6 9010 Maximum

in Table 2 are simulated and the corresponding process dataare collected for testingThe following simulations are run inMATLAB 830 (2014a) environment Here two indicatorswhich are FR (FR) and MR (MR) are often introduced tomeasure the result of process monitoring FR is the rate ofnormal data classified as fault dataMR is the rate of fault dataclassified as normal rate

In the MPPCA algorithm and PCA-GMM algorithmwhen the variance contribution was selected as 85 thedimension of feature space inMPPCAand the number of PCsin PCA-GMM were each selected as 18 In order to comparethemonitoring performances of these algorithms in the samesituation the selected sparse PCs of each mode in SPCS-BIPwere selected as 18 The 99 control limit was assigned to allthree algorithms

First Figure 3 shows that the different submodes can besuccessfully divided by the EM algorithm used in this paperAnd by using other algorithms themodes also can be dividedcorrectly In other words how to correctly divide the trainingdata into multiple subset is not a problem by many relatedalgorithms

The normal process was tested by different algorithm andthe results are shown in Figure 4 In this figure it is hard tofigure out which algorithmrsquos FR is lower In the figure mostsamples of each algorithm are lower than the control limitAnd by calculation the FR of these algorithms are 0333008 025 and 108 respectively corresponding to Fig-ures 4(a) 4(b) 4(c) and 4(d) The monitoring performancesof these three algorithms suggest that the FR are acceptableNext the data sets of 12 faults in mode 3 were tested and theMR of these three algorithms are listed in Table 3 with thesmallest MR shown in bold

From Table 3 we observe that the monitoring perfor-mance of SPCS-BIP is the best compared to the MPPCAand PCA-GMM algorithms for all 12 faults Here we takethe further analysis In comparison with MPPCA and PCA-GMMalgorithms the SPCS-BIP algorithm can exactly dividethe process data into subgroups corresponding to differentmodes by using the E-M algorithm and in each submodeSPCS-BIP can select the most important PCs that have mostrelation with the fault Due to the fact that the subspacespanned by the PCs was monitored by BIP most of the PCsare related to themain process of chemical industrial processand only little PCs are related to the fault process SPCSare discriminative by constructing an elastic net regressionbetween all PCs and each sample So in Table 3 we observe

Mathematical Problems in Engineering 7

Table 2 Process faults for the multimode TE process

Faults number Disturbance state TypeIDV(1) AC feed ratio B composition constant (Stream 4) StepIDV(2) B composition AC ratio constant (Stream 4) StepIDV(3) D feed temperature (Stream 2) StepIDV(4) Reactor cooling water inlet temperature StepIDV(5) Condenser cooling water inlet temperature StepIDV(6) A feed loss (Stream 1) StepIDV(7) C header pressure loss reduced availability (Stream 4) StepIDV(8) A B and C feed composition (Stream 4) Random variationIDV(9) D feed temperature (Stream 2) Random variationIDV(10) C feed temperature (Stream 4) Random variationIDV(11) Reactor cooling water inlet temperature Random variationIDV(12) Condenser cooling water inlet temperature Random variationIDV(13) Reaction kinetics Slow driftIDV(14) Reactor cooling water valve StickingIDV(15) Condenser cooling water valve StickingIDV(16) Unknown UnknownIDV(17) Unknown UnknownIDV(18) Unknown UnknownIDV(19) Unknown UnknownIDV(20) Unknown Unknown

1

5

2

3

4

A

D

E

C

FI

FI

FI

FI

FIFI

FI

FI

FI

FI

XA

XB

XC

XD

XE

XF

XG

XH

XD

XE

XF

XG

XH

XA

XB

XC

XD

XE

XF

Compressor

Ana

lyze

rA

naly

zer

Ana

lyze

r

6

7

8

9

10

11

12

SC

PI

PI

PI

JI

TI

Condenser

CWS

CWS

CWR

Reactor

CWR

Stripper

TI

TI

TI

TILI

LI

LI

Purge

Product

Stm

Cond

Vapliqseparator

Figure 2 Control scheme for the TE process

8 Mathematical Problems in Engineering

Sample number0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

11

12

13

14

15

16

17

18

19

2

Labe

l

Figure 3 Different modes of the training data

Table 3 Missed detection rates () of 12 faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 075 025 1 01252 35 85 55 18754 0 0 0 05 0375 0 1125 06 0 0 0 07 0 0 0 08 3375 3875 425 250010 82375 16 89875 787511 225 85 45 112512 1375 1625 15 075013 16125 265 19625 1162514 0 35 0125 0

that the results of SPCS-BIP are better than the results ofMPPCA and PCA-GMM

Figure 5 shows the monitoring performances of fault 10It is easy to see that the FDI of MPPCA-1198792 and the BIP ofPCA-GMM cannot detect the fault effectively in Figures 4(a)and 4(c) In the figure more than half of the fault sampleswere regarded as the normal samples while compared tothe performances of MPPCA-1198792 and PCA-GMM the FDIof MPPCA-SPE shows some improvements However themonitoring performance of MPPCA-SPE does not match theperformance of SPCS-BIP We can find this point both inFigure 4 and Table 3

42 CSTR This study simulated the CSTR process describedby Yoon and MacGregor [49] The diagram of the process ispresented in Figure 6 Due to the fact that the CSTR processconsisted of only one operating unit the number of subblockswas selected as 1

Table 4 Missed detection rates () of two faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 90 100 988 6562 288 996 318 28

In the modeling stage 1000 samples which include 500mode 1 samples and 500mode 2 samples were collected as thetraining data set In the testing stage 1000 samples of mode 2were tested and two faults were introduced to the process asfollows

Case 1 A step of 1 K was added in the cooling watertemperature 119879

119862from the 500th sample

Case 2 A 2 kmol(m3 sdotmin) step was added in the inlet soluteconcentration 119862

119860119860from the 500th sample

In the MMPCA algorithm when the variance contri-bution was selected as 85 the dimension of feature spacein MPPCA is 10 So in order to compare the monitoringperformances of these algorithms in the same situation thenumber of PCs in PCA-GMM and the selected sparse PCs inSPCA-BIPwere both selected as 10The 99 control limit wasassigned to all three algorithms

The same as TEP the FR of these algorithms are 04 012 and 1 respectively In an industry process FR lowerthan 005 is acceptable [28]

The data sets of two faults in mode 2 were tested and theMR were listed in Table 4 In the table the smallest misseddetection rates are shown in bold

As shown in Table 4 the SPCS-BIP algorithm has shownthe best performance for these two faults compared withother algorithms listed in the table It is obvious that neitherMPPCA nor PCA-GMM algorithms can detect the faultbecause their missed detection rates were high In those fouralgorithms only the SPCS-BIP was based on the selectionPCs so the improvements in the proposed sparse principalcomponents selection can be demonstrated through thebetter monitoring performance of the SPCS-BIP algorithm

Fault 1 is a bias in cooling water temperature 119879119862 Due to

the control loop in the CSTR process these would be a biasin outlet temperature 119879 and then the cooling water flow rate119865119862would increase In Figure 7 both the MMPCA and PCA-

GMMalgorithms could not detect fault 1 effectively accordingto the performances of those shown in Figures 7(a) 7(b)and 7(c) In Figure 7(d) it is obvious that the monitoringperformance of SPCS-BIP is much better than the othersThe reason is that the correct classification for each subgroupby using E-M algorithm and the PCs selected by SPCS arediscriminative and could construct the subspace that containsthe important fault information for abnormal data

Fault 2 is a bias in inlet solute concentration 119862119860119860

Thendue to the control loop in the CSTR process there wouldbe biases in outlet concentration 119862 and outlet temperature119879 According to the performances shown in Figure 8(b) theFDI of MPPCA-SPE could not detect fault 2 Comparedto the MPPCA-SPE the FDI of MPPCA-1198792 showed some

Mathematical Problems in Engineering 9

0 500 1000 1500 20000

001

002

003

004

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

0005

001

0015

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 4 Monitoring performance of the normal process

0 500 1000 1500 20000

002

004

006

008

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

05

1

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 5 Monitoring performances of fault 10 in TEP

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where the function of 120576 is to remove the ill condition ofcovariance matrix Σ

119897by adding a positive number to all the

diagonal entriesFor the monitored sample 119909

119905 a local Mahalanobis

distance-based probability index relative to each Gaussiancomponent 119862

119897can be defined as

119875(119897)

119871(119909119905)

= Pr 119863 ((119909 119862119897) | 119909 isin 119862

119896) le 119863 ((119909

119894 119862119897) | 119909119905isin 119862119897)

(11)

or

119875(119897)

119871(119909119905) = Pr 119863

119903((119909 119862

119897) | 119909 isin 119862

119897)

le 119863119903((119909119894 119862119897) | 119909119905isin 119862119897)

(12)

Given the appropriate degree of freedom 119875(119897)119871(119909119905) can be

computed by integrating the 1205942 probability density functionUnder a given confidence level this index has the functionof indicating whether the monitored sample is normal orabnormal provided that it belongs to the correspondingGaussian component A global BIP index is proposed tocombine the local probability metrics across all the Gaussianclusters because the random characteristic of eachmonitoredsample may come from multiple Gaussian components withthe corresponding posterior probabilitiesThe formulation ofBIP index for the monitored sample 119909

119905is given by

BIP =119871

sum

119897=1

119875 (119862119897| 119909119905) 119875(119897)

119871(119909119905) (13)

where the posterior probability 119875(119862119897| 119909119905) is used to incor-

porate the contribution of each local Gaussian component tothe overall probabilistic index As 0 le 119875(119897)

119871(119909119905) le 1 we have

0 le BIP le119871

sum

119897=1

119875 (119862119897| 119909119905) = 1 (14)

Under the preset confidence level (1minus120572) 100 the processis determined within normal operation if

BIP le 1 minus 120572 (15)

Otherwise the process operation is treated out of control

32 Sparse Principal Component Selection Sparse represen-tation has proven to be an extremely powerful tool foracquiring representing and compressing high-dimensionaldata [41ndash43] This success is mainly because of the factthat the important reconstruction information of data suchas process data and time series data has naturally sparserepresentations with respect to fixed bases or concatenationsof such bases Qiao et al [44] proposed that the graphsconstructed by the 119871

1-norm have the advantage of greater

robustness to data noise automatic sparsity and adaptiveneighborhood for individual datum What is more anotherimportant advantage is that sparse representation has thepotential discriminative ability since most nonzero elements

are located on the samples in the same class as the representedsample

Given the training sample L = [1198971 1198972 119897119873] isin R119873times119898

a test sample H isin R119898 the solution to the sparse represen-tation problem can be obtained by solving the following ℓ1-minimization problem

(ℓ1) 120573 = argmin 1003817100381710038171003817120573

1003817100381710038171003817

subject to L120573 = H

Card (120573) le 119896

(16)

where 120573 = [1205731 1205732 120573

119873] are the sparse representation

coefficients and Card(120573) denotes the number of nonzeroelements of 120573 From the perspective of statistics formula(16) can be named the Lasso criterion Lasso is a penalizedleast squares algorithm which was originally by quadraticprogramming imposing a constraint on the 119871

1norm of

the regression coefficients Thus the Lasso estimates 120573 areobtained by minimizing the Lasso criterion

120573 = argmin120573

1003817100381710038171003817100381710038171003817100381710038171003817

H minussum

119894=1

L119894120573119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

+ 120582

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

subject to Card (120573) le 119896

(17)

where 120582 is nonnegative However only using the 1198711-norm

penalty in Lasso has its limitation Zou et al [45] proposedthat if there is a group of variables among which the pairwisecorrelations are very high lasso tends to select any variablefrom the group and does not consider which one is selectedFortunately elastic net was put forward by Zou et al whichovercomes the limitation of only using the 119871

1-norm penalty

It is known that combining the 1198711-norm and 119871

2-norm

penalty can result in grouping effectiveness in regression andthus enhance the prediction accuracy For any nonnegative1205821and 120582

2 the elastic net estimates 120573en are given by

120573en = argmin120573

1003817100381710038171003817100381710038171003817100381710038171003817

H minus

119873

sum

119894=1

L119894120573119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

+ 1205821

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

+ 1205822

119873

sum

119894=1

10038161003816100381610038161205731198941003816100381610038161003816

2

subject to Card (120573) le 119896

(18)

In brief it is expected that the elastic net is used to groupa set of sparse coefficients to construct the sparse alignmentmatrices in which the sparse representation informationor the potential discriminative information is encoded toenhance the discriminative ability in an unsupervised man-ner

33 Fault Detection with SPCS and BIP The key problemfor monitoring the multimode process is to select a suitablemodel and choose the subspace spanned by key PCs In theIntroduction we had put forward the fact that the subspace

Mathematical Problems in Engineering 5

Offline modeling

Use EM algorithm to learn the GMM andestimate the model parameters

For each submodel normalize the trainingdata

Obtain the basic PCs using SVDdecomposition

Obtain the sparse PCs of normal data byconstructing elastic net regression

End

Online modeling

Current data normalization

Obtain the sparse PCs of each testingsample by constructing elastic net

regression

Calculate BIP index value of each testingsample

Exceed limit

Next

No

There is a fault in process

YesSpecify a confidence level and constantcontrol limit

Figure 1 The steps of SPCS-BIP algorithm for process monitoring

spanned by the first several PCs with largest explainedvariance does not always have fault information

In the following part a novel multimode process mon-itoring approach based on SPCS and BIP is proposed Thisapproach is in a just-in-time form For each sample an elasticnet regression between all PCs and the sample is constructedand solved The PCs which have nonzero regression coeffi-cients are retained while other PCs are rejected That meansfor each sample we can pick out the most discriminativebases and the others are set to zero Its concrete calculatingsteps are summarized in Figure 1

Offline Modeling

(1) Collect a set of historical training data under allpossible operating conditions

(2) Use the EM algorithm to learn the Gaussian mixturemodel and estimate the model parameter set Θ =

1205831 Σ1 1205961 120583

119896 Σ119896 120596119896 based on the iterative steps

(3) For each submodel get a normal operational obser-vation set X =isin R119873times119898 where 119873 is the number ofsamples and 119898 is the number of variables This set isdenoted as the training set for threshold determiningA testing set Y isin R119878times119898 with both normal andabnormal observations is given for testing

(4) Normalize the training data through the mean valueand variance of each variable

(5) Obtain all principal components using SVD decom-position The training data X is reconstructed by X =

sum119898

119894=1t119894p119879119894 where t

119894is the score vector and p

119894is the

loading vector(6) For training sample x

119895(119895 = 1 2 119873) construct an

elastic net regression between each observation valueof training data and loading vector Pmade of PCs instep (5) according to 120573

119895= argmin

120573x119895minussum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

(7) Corresponding to the nonzero representation coef-ficients 120573

1198951

1205731198952

120573119895119896

construct a new loadingvector P

119895= [p1198951

p1198952

p119895119896

](8) Specify a confidence (1minus120572) 100 and constant control

limit 1 minus 120572

Online Monitoring

(1) Normalize the current time point data by using meanvalues and variance of the training data

(2) Obtain the loading vector P from offline modeling(3) When a test sample y

119895isin R119898 (119895 = 1 2 119878) is

coming construct an elastic net regression betweenthe sample and loading vector Pmade of PCs in step(2) according to 120573 = argmin

120573y119895minus sum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

6 Mathematical Problems in Engineering

(4) Corresponding to the nonzero representation coef-ficients 120573

1198971

1205731198972

120573119897119896

construct a new loadingvector P

119895= [p1198971

p1198972

p119897119896

](5) Generate theBIP control chartwith the calculatedBIP

index values for all the monitored samples If the BIPindex of a test sample is lower than the control limitwhichmeans the sample is normal go to step (1) Elsethere is a fault in the process

4 Case Studies on the TE and CSTR Process

In this case study the TE benchmark and CSTR processare introduced to verify the effectiveness of the SPCS-BIPalgorithm PCA-GMM is the classic algorithm formultimodeprocessing monitoring And the fault detection index (FDI)is similar to Bayesian inference probability (BIP) So here acomparison was made between SPCS-BIP and PCA-GMMIn addition to verify the improvements of SPCS algorithmwhich can select sparse PCs a comparison was performedbetween the SPCS-BIP algorithm and theMPPCA algorithm

41 Tennessee Eastman Process As a well-known benchmarkprocess the Tennessee Eastman process which was pre-sented by Downs and Vogel has been widely applied toevaluate and compare the efficiency of process monitoringtechniques [46 47] The schematic diagram of the processis illustrated in Figure 2 This process consists of five majorunit operations a reactor a product condenser a vapor-liquidseparator a recycle compressor and a product stripper Inaddition there are six modes of process operation as listedin Table 1 The variables can be divided into three categoriescomposition variables continuous process variables andmanipulated variables In our study only modes 1 and 3were simulated through the Simulink programs developedon the basis of the decentralized control strategy designed byRicker [48]The Simulink programs can be downloaded fromhttpdeptswashingtoneducontrolLARRYTEdownloadhtml The 31 selected monitoring variables contained 9manipulated variables and 22 continuous process variablesThus these variables were divided into five subblocks accord-ing to five units However given that only two variables wereallocated to each the compressor unit and the condenser unitthere were four variables assigned to the other three relatedsubblocks As a result the total of 31 variables was dividedinto three subblocks

There are 20 faults in the multimode TE process whichare listed in Table 2 Among these faults the root causes of thefaults 16ndash20 are unknown [46 47] What is more to simplifyinterpretation the amplitudes of faults 3 9 and 15 are sosmall It is difficult to detect so only the remaining 12 faultswere considered in this study In the modeling stage 2000normal samples which include 1000mode 1 samples and 1000mode 3 samples were collected as the training data set Inthe testing stage 1000 samples of mode 1 were tested firstand then the process switches to mode 3 As a result the testdata set consists of 1000 samples of mode 1 and 1000 samplesof mode 3 And faults occurred from the 1200th sample Aset of 20 faults in multimode TE process which are listed

Table 1 Six process operation modes of TE process

Mode GH mass ratio Production rate1 5050 7038 kghG and 7038 kghH2 1090 1048 kghG and 12669 kghH3 9010 10000 kghG and 1111 kghH4 5050 Maximum5 1090 Maximum6 9010 Maximum

in Table 2 are simulated and the corresponding process dataare collected for testingThe following simulations are run inMATLAB 830 (2014a) environment Here two indicatorswhich are FR (FR) and MR (MR) are often introduced tomeasure the result of process monitoring FR is the rate ofnormal data classified as fault dataMR is the rate of fault dataclassified as normal rate

In the MPPCA algorithm and PCA-GMM algorithmwhen the variance contribution was selected as 85 thedimension of feature space inMPPCAand the number of PCsin PCA-GMM were each selected as 18 In order to comparethemonitoring performances of these algorithms in the samesituation the selected sparse PCs of each mode in SPCS-BIPwere selected as 18 The 99 control limit was assigned to allthree algorithms

First Figure 3 shows that the different submodes can besuccessfully divided by the EM algorithm used in this paperAnd by using other algorithms themodes also can be dividedcorrectly In other words how to correctly divide the trainingdata into multiple subset is not a problem by many relatedalgorithms

The normal process was tested by different algorithm andthe results are shown in Figure 4 In this figure it is hard tofigure out which algorithmrsquos FR is lower In the figure mostsamples of each algorithm are lower than the control limitAnd by calculation the FR of these algorithms are 0333008 025 and 108 respectively corresponding to Fig-ures 4(a) 4(b) 4(c) and 4(d) The monitoring performancesof these three algorithms suggest that the FR are acceptableNext the data sets of 12 faults in mode 3 were tested and theMR of these three algorithms are listed in Table 3 with thesmallest MR shown in bold

From Table 3 we observe that the monitoring perfor-mance of SPCS-BIP is the best compared to the MPPCAand PCA-GMM algorithms for all 12 faults Here we takethe further analysis In comparison with MPPCA and PCA-GMMalgorithms the SPCS-BIP algorithm can exactly dividethe process data into subgroups corresponding to differentmodes by using the E-M algorithm and in each submodeSPCS-BIP can select the most important PCs that have mostrelation with the fault Due to the fact that the subspacespanned by the PCs was monitored by BIP most of the PCsare related to themain process of chemical industrial processand only little PCs are related to the fault process SPCSare discriminative by constructing an elastic net regressionbetween all PCs and each sample So in Table 3 we observe

Mathematical Problems in Engineering 7

Table 2 Process faults for the multimode TE process

Faults number Disturbance state TypeIDV(1) AC feed ratio B composition constant (Stream 4) StepIDV(2) B composition AC ratio constant (Stream 4) StepIDV(3) D feed temperature (Stream 2) StepIDV(4) Reactor cooling water inlet temperature StepIDV(5) Condenser cooling water inlet temperature StepIDV(6) A feed loss (Stream 1) StepIDV(7) C header pressure loss reduced availability (Stream 4) StepIDV(8) A B and C feed composition (Stream 4) Random variationIDV(9) D feed temperature (Stream 2) Random variationIDV(10) C feed temperature (Stream 4) Random variationIDV(11) Reactor cooling water inlet temperature Random variationIDV(12) Condenser cooling water inlet temperature Random variationIDV(13) Reaction kinetics Slow driftIDV(14) Reactor cooling water valve StickingIDV(15) Condenser cooling water valve StickingIDV(16) Unknown UnknownIDV(17) Unknown UnknownIDV(18) Unknown UnknownIDV(19) Unknown UnknownIDV(20) Unknown Unknown

1

5

2

3

4

A

D

E

C

FI

FI

FI

FI

FIFI

FI

FI

FI

FI

XA

XB

XC

XD

XE

XF

XG

XH

XD

XE

XF

XG

XH

XA

XB

XC

XD

XE

XF

Compressor

Ana

lyze

rA

naly

zer

Ana

lyze

r

6

7

8

9

10

11

12

SC

PI

PI

PI

JI

TI

Condenser

CWS

CWS

CWR

Reactor

CWR

Stripper

TI

TI

TI

TILI

LI

LI

Purge

Product

Stm

Cond

Vapliqseparator

Figure 2 Control scheme for the TE process

8 Mathematical Problems in Engineering

Sample number0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

11

12

13

14

15

16

17

18

19

2

Labe

l

Figure 3 Different modes of the training data

Table 3 Missed detection rates () of 12 faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 075 025 1 01252 35 85 55 18754 0 0 0 05 0375 0 1125 06 0 0 0 07 0 0 0 08 3375 3875 425 250010 82375 16 89875 787511 225 85 45 112512 1375 1625 15 075013 16125 265 19625 1162514 0 35 0125 0

that the results of SPCS-BIP are better than the results ofMPPCA and PCA-GMM

Figure 5 shows the monitoring performances of fault 10It is easy to see that the FDI of MPPCA-1198792 and the BIP ofPCA-GMM cannot detect the fault effectively in Figures 4(a)and 4(c) In the figure more than half of the fault sampleswere regarded as the normal samples while compared tothe performances of MPPCA-1198792 and PCA-GMM the FDIof MPPCA-SPE shows some improvements However themonitoring performance of MPPCA-SPE does not match theperformance of SPCS-BIP We can find this point both inFigure 4 and Table 3

42 CSTR This study simulated the CSTR process describedby Yoon and MacGregor [49] The diagram of the process ispresented in Figure 6 Due to the fact that the CSTR processconsisted of only one operating unit the number of subblockswas selected as 1

Table 4 Missed detection rates () of two faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 90 100 988 6562 288 996 318 28

In the modeling stage 1000 samples which include 500mode 1 samples and 500mode 2 samples were collected as thetraining data set In the testing stage 1000 samples of mode 2were tested and two faults were introduced to the process asfollows

Case 1 A step of 1 K was added in the cooling watertemperature 119879

119862from the 500th sample

Case 2 A 2 kmol(m3 sdotmin) step was added in the inlet soluteconcentration 119862

119860119860from the 500th sample

In the MMPCA algorithm when the variance contri-bution was selected as 85 the dimension of feature spacein MPPCA is 10 So in order to compare the monitoringperformances of these algorithms in the same situation thenumber of PCs in PCA-GMM and the selected sparse PCs inSPCA-BIPwere both selected as 10The 99 control limit wasassigned to all three algorithms

The same as TEP the FR of these algorithms are 04 012 and 1 respectively In an industry process FR lowerthan 005 is acceptable [28]

The data sets of two faults in mode 2 were tested and theMR were listed in Table 4 In the table the smallest misseddetection rates are shown in bold

As shown in Table 4 the SPCS-BIP algorithm has shownthe best performance for these two faults compared withother algorithms listed in the table It is obvious that neitherMPPCA nor PCA-GMM algorithms can detect the faultbecause their missed detection rates were high In those fouralgorithms only the SPCS-BIP was based on the selectionPCs so the improvements in the proposed sparse principalcomponents selection can be demonstrated through thebetter monitoring performance of the SPCS-BIP algorithm

Fault 1 is a bias in cooling water temperature 119879119862 Due to

the control loop in the CSTR process these would be a biasin outlet temperature 119879 and then the cooling water flow rate119865119862would increase In Figure 7 both the MMPCA and PCA-

GMMalgorithms could not detect fault 1 effectively accordingto the performances of those shown in Figures 7(a) 7(b)and 7(c) In Figure 7(d) it is obvious that the monitoringperformance of SPCS-BIP is much better than the othersThe reason is that the correct classification for each subgroupby using E-M algorithm and the PCs selected by SPCS arediscriminative and could construct the subspace that containsthe important fault information for abnormal data

Fault 2 is a bias in inlet solute concentration 119862119860119860

Thendue to the control loop in the CSTR process there wouldbe biases in outlet concentration 119862 and outlet temperature119879 According to the performances shown in Figure 8(b) theFDI of MPPCA-SPE could not detect fault 2 Comparedto the MPPCA-SPE the FDI of MPPCA-1198792 showed some

Mathematical Problems in Engineering 9

0 500 1000 1500 20000

001

002

003

004

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

0005

001

0015

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 4 Monitoring performance of the normal process

0 500 1000 1500 20000

002

004

006

008

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

05

1

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 5 Monitoring performances of fault 10 in TEP

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Offline modeling

Use EM algorithm to learn the GMM andestimate the model parameters

For each submodel normalize the trainingdata

Obtain the basic PCs using SVDdecomposition

Obtain the sparse PCs of normal data byconstructing elastic net regression

End

Online modeling

Current data normalization

Obtain the sparse PCs of each testingsample by constructing elastic net

regression

Calculate BIP index value of each testingsample

Exceed limit

Next

No

There is a fault in process

YesSpecify a confidence level and constantcontrol limit

Figure 1 The steps of SPCS-BIP algorithm for process monitoring

spanned by the first several PCs with largest explainedvariance does not always have fault information

In the following part a novel multimode process mon-itoring approach based on SPCS and BIP is proposed Thisapproach is in a just-in-time form For each sample an elasticnet regression between all PCs and the sample is constructedand solved The PCs which have nonzero regression coeffi-cients are retained while other PCs are rejected That meansfor each sample we can pick out the most discriminativebases and the others are set to zero Its concrete calculatingsteps are summarized in Figure 1

Offline Modeling

(1) Collect a set of historical training data under allpossible operating conditions

(2) Use the EM algorithm to learn the Gaussian mixturemodel and estimate the model parameter set Θ =

1205831 Σ1 1205961 120583

119896 Σ119896 120596119896 based on the iterative steps

(3) For each submodel get a normal operational obser-vation set X =isin R119873times119898 where 119873 is the number ofsamples and 119898 is the number of variables This set isdenoted as the training set for threshold determiningA testing set Y isin R119878times119898 with both normal andabnormal observations is given for testing

(4) Normalize the training data through the mean valueand variance of each variable

(5) Obtain all principal components using SVD decom-position The training data X is reconstructed by X =

sum119898

119894=1t119894p119879119894 where t

119894is the score vector and p

119894is the

loading vector(6) For training sample x

119895(119895 = 1 2 119873) construct an

elastic net regression between each observation valueof training data and loading vector Pmade of PCs instep (5) according to 120573

119895= argmin

120573x119895minussum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

(7) Corresponding to the nonzero representation coef-ficients 120573

1198951

1205731198952

120573119895119896

construct a new loadingvector P

119895= [p1198951

p1198952

p119895119896

](8) Specify a confidence (1minus120572) 100 and constant control

limit 1 minus 120572

Online Monitoring

(1) Normalize the current time point data by using meanvalues and variance of the training data

(2) Obtain the loading vector P from offline modeling(3) When a test sample y

119895isin R119898 (119895 = 1 2 119878) is

coming construct an elastic net regression betweenthe sample and loading vector Pmade of PCs in step(2) according to 120573 = argmin

120573y119895minus sum119898

119894=1p1198941205731198942+

1205821sum119898

119894=1|120573119894| + 1205822sum119898

119894=11205732 subject to Card(120573) le 119896

6 Mathematical Problems in Engineering

(4) Corresponding to the nonzero representation coef-ficients 120573

1198971

1205731198972

120573119897119896

construct a new loadingvector P

119895= [p1198971

p1198972

p119897119896

](5) Generate theBIP control chartwith the calculatedBIP

index values for all the monitored samples If the BIPindex of a test sample is lower than the control limitwhichmeans the sample is normal go to step (1) Elsethere is a fault in the process

4 Case Studies on the TE and CSTR Process

In this case study the TE benchmark and CSTR processare introduced to verify the effectiveness of the SPCS-BIPalgorithm PCA-GMM is the classic algorithm formultimodeprocessing monitoring And the fault detection index (FDI)is similar to Bayesian inference probability (BIP) So here acomparison was made between SPCS-BIP and PCA-GMMIn addition to verify the improvements of SPCS algorithmwhich can select sparse PCs a comparison was performedbetween the SPCS-BIP algorithm and theMPPCA algorithm

41 Tennessee Eastman Process As a well-known benchmarkprocess the Tennessee Eastman process which was pre-sented by Downs and Vogel has been widely applied toevaluate and compare the efficiency of process monitoringtechniques [46 47] The schematic diagram of the processis illustrated in Figure 2 This process consists of five majorunit operations a reactor a product condenser a vapor-liquidseparator a recycle compressor and a product stripper Inaddition there are six modes of process operation as listedin Table 1 The variables can be divided into three categoriescomposition variables continuous process variables andmanipulated variables In our study only modes 1 and 3were simulated through the Simulink programs developedon the basis of the decentralized control strategy designed byRicker [48]The Simulink programs can be downloaded fromhttpdeptswashingtoneducontrolLARRYTEdownloadhtml The 31 selected monitoring variables contained 9manipulated variables and 22 continuous process variablesThus these variables were divided into five subblocks accord-ing to five units However given that only two variables wereallocated to each the compressor unit and the condenser unitthere were four variables assigned to the other three relatedsubblocks As a result the total of 31 variables was dividedinto three subblocks

There are 20 faults in the multimode TE process whichare listed in Table 2 Among these faults the root causes of thefaults 16ndash20 are unknown [46 47] What is more to simplifyinterpretation the amplitudes of faults 3 9 and 15 are sosmall It is difficult to detect so only the remaining 12 faultswere considered in this study In the modeling stage 2000normal samples which include 1000mode 1 samples and 1000mode 3 samples were collected as the training data set Inthe testing stage 1000 samples of mode 1 were tested firstand then the process switches to mode 3 As a result the testdata set consists of 1000 samples of mode 1 and 1000 samplesof mode 3 And faults occurred from the 1200th sample Aset of 20 faults in multimode TE process which are listed

Table 1 Six process operation modes of TE process

Mode GH mass ratio Production rate1 5050 7038 kghG and 7038 kghH2 1090 1048 kghG and 12669 kghH3 9010 10000 kghG and 1111 kghH4 5050 Maximum5 1090 Maximum6 9010 Maximum

in Table 2 are simulated and the corresponding process dataare collected for testingThe following simulations are run inMATLAB 830 (2014a) environment Here two indicatorswhich are FR (FR) and MR (MR) are often introduced tomeasure the result of process monitoring FR is the rate ofnormal data classified as fault dataMR is the rate of fault dataclassified as normal rate

In the MPPCA algorithm and PCA-GMM algorithmwhen the variance contribution was selected as 85 thedimension of feature space inMPPCAand the number of PCsin PCA-GMM were each selected as 18 In order to comparethemonitoring performances of these algorithms in the samesituation the selected sparse PCs of each mode in SPCS-BIPwere selected as 18 The 99 control limit was assigned to allthree algorithms

First Figure 3 shows that the different submodes can besuccessfully divided by the EM algorithm used in this paperAnd by using other algorithms themodes also can be dividedcorrectly In other words how to correctly divide the trainingdata into multiple subset is not a problem by many relatedalgorithms

The normal process was tested by different algorithm andthe results are shown in Figure 4 In this figure it is hard tofigure out which algorithmrsquos FR is lower In the figure mostsamples of each algorithm are lower than the control limitAnd by calculation the FR of these algorithms are 0333008 025 and 108 respectively corresponding to Fig-ures 4(a) 4(b) 4(c) and 4(d) The monitoring performancesof these three algorithms suggest that the FR are acceptableNext the data sets of 12 faults in mode 3 were tested and theMR of these three algorithms are listed in Table 3 with thesmallest MR shown in bold

From Table 3 we observe that the monitoring perfor-mance of SPCS-BIP is the best compared to the MPPCAand PCA-GMM algorithms for all 12 faults Here we takethe further analysis In comparison with MPPCA and PCA-GMMalgorithms the SPCS-BIP algorithm can exactly dividethe process data into subgroups corresponding to differentmodes by using the E-M algorithm and in each submodeSPCS-BIP can select the most important PCs that have mostrelation with the fault Due to the fact that the subspacespanned by the PCs was monitored by BIP most of the PCsare related to themain process of chemical industrial processand only little PCs are related to the fault process SPCSare discriminative by constructing an elastic net regressionbetween all PCs and each sample So in Table 3 we observe

Mathematical Problems in Engineering 7

Table 2 Process faults for the multimode TE process

Faults number Disturbance state TypeIDV(1) AC feed ratio B composition constant (Stream 4) StepIDV(2) B composition AC ratio constant (Stream 4) StepIDV(3) D feed temperature (Stream 2) StepIDV(4) Reactor cooling water inlet temperature StepIDV(5) Condenser cooling water inlet temperature StepIDV(6) A feed loss (Stream 1) StepIDV(7) C header pressure loss reduced availability (Stream 4) StepIDV(8) A B and C feed composition (Stream 4) Random variationIDV(9) D feed temperature (Stream 2) Random variationIDV(10) C feed temperature (Stream 4) Random variationIDV(11) Reactor cooling water inlet temperature Random variationIDV(12) Condenser cooling water inlet temperature Random variationIDV(13) Reaction kinetics Slow driftIDV(14) Reactor cooling water valve StickingIDV(15) Condenser cooling water valve StickingIDV(16) Unknown UnknownIDV(17) Unknown UnknownIDV(18) Unknown UnknownIDV(19) Unknown UnknownIDV(20) Unknown Unknown

1

5

2

3

4

A

D

E

C

FI

FI

FI

FI

FIFI

FI

FI

FI

FI

XA

XB

XC

XD

XE

XF

XG

XH

XD

XE

XF

XG

XH

XA

XB

XC

XD

XE

XF

Compressor

Ana

lyze

rA

naly

zer

Ana

lyze

r

6

7

8

9

10

11

12

SC

PI

PI

PI

JI

TI

Condenser

CWS

CWS

CWR

Reactor

CWR

Stripper

TI

TI

TI

TILI

LI

LI

Purge

Product

Stm

Cond

Vapliqseparator

Figure 2 Control scheme for the TE process

8 Mathematical Problems in Engineering

Sample number0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

11

12

13

14

15

16

17

18

19

2

Labe

l

Figure 3 Different modes of the training data

Table 3 Missed detection rates () of 12 faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 075 025 1 01252 35 85 55 18754 0 0 0 05 0375 0 1125 06 0 0 0 07 0 0 0 08 3375 3875 425 250010 82375 16 89875 787511 225 85 45 112512 1375 1625 15 075013 16125 265 19625 1162514 0 35 0125 0

that the results of SPCS-BIP are better than the results ofMPPCA and PCA-GMM

Figure 5 shows the monitoring performances of fault 10It is easy to see that the FDI of MPPCA-1198792 and the BIP ofPCA-GMM cannot detect the fault effectively in Figures 4(a)and 4(c) In the figure more than half of the fault sampleswere regarded as the normal samples while compared tothe performances of MPPCA-1198792 and PCA-GMM the FDIof MPPCA-SPE shows some improvements However themonitoring performance of MPPCA-SPE does not match theperformance of SPCS-BIP We can find this point both inFigure 4 and Table 3

42 CSTR This study simulated the CSTR process describedby Yoon and MacGregor [49] The diagram of the process ispresented in Figure 6 Due to the fact that the CSTR processconsisted of only one operating unit the number of subblockswas selected as 1

Table 4 Missed detection rates () of two faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 90 100 988 6562 288 996 318 28

In the modeling stage 1000 samples which include 500mode 1 samples and 500mode 2 samples were collected as thetraining data set In the testing stage 1000 samples of mode 2were tested and two faults were introduced to the process asfollows

Case 1 A step of 1 K was added in the cooling watertemperature 119879

119862from the 500th sample

Case 2 A 2 kmol(m3 sdotmin) step was added in the inlet soluteconcentration 119862

119860119860from the 500th sample

In the MMPCA algorithm when the variance contri-bution was selected as 85 the dimension of feature spacein MPPCA is 10 So in order to compare the monitoringperformances of these algorithms in the same situation thenumber of PCs in PCA-GMM and the selected sparse PCs inSPCA-BIPwere both selected as 10The 99 control limit wasassigned to all three algorithms

The same as TEP the FR of these algorithms are 04 012 and 1 respectively In an industry process FR lowerthan 005 is acceptable [28]

The data sets of two faults in mode 2 were tested and theMR were listed in Table 4 In the table the smallest misseddetection rates are shown in bold

As shown in Table 4 the SPCS-BIP algorithm has shownthe best performance for these two faults compared withother algorithms listed in the table It is obvious that neitherMPPCA nor PCA-GMM algorithms can detect the faultbecause their missed detection rates were high In those fouralgorithms only the SPCS-BIP was based on the selectionPCs so the improvements in the proposed sparse principalcomponents selection can be demonstrated through thebetter monitoring performance of the SPCS-BIP algorithm

Fault 1 is a bias in cooling water temperature 119879119862 Due to

the control loop in the CSTR process these would be a biasin outlet temperature 119879 and then the cooling water flow rate119865119862would increase In Figure 7 both the MMPCA and PCA-

GMMalgorithms could not detect fault 1 effectively accordingto the performances of those shown in Figures 7(a) 7(b)and 7(c) In Figure 7(d) it is obvious that the monitoringperformance of SPCS-BIP is much better than the othersThe reason is that the correct classification for each subgroupby using E-M algorithm and the PCs selected by SPCS arediscriminative and could construct the subspace that containsthe important fault information for abnormal data

Fault 2 is a bias in inlet solute concentration 119862119860119860

Thendue to the control loop in the CSTR process there wouldbe biases in outlet concentration 119862 and outlet temperature119879 According to the performances shown in Figure 8(b) theFDI of MPPCA-SPE could not detect fault 2 Comparedto the MPPCA-SPE the FDI of MPPCA-1198792 showed some

Mathematical Problems in Engineering 9

0 500 1000 1500 20000

001

002

003

004

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

0005

001

0015

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 4 Monitoring performance of the normal process

0 500 1000 1500 20000

002

004

006

008

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

05

1

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 5 Monitoring performances of fault 10 in TEP

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

(4) Corresponding to the nonzero representation coef-ficients 120573

1198971

1205731198972

120573119897119896

construct a new loadingvector P

119895= [p1198971

p1198972

p119897119896

](5) Generate theBIP control chartwith the calculatedBIP

index values for all the monitored samples If the BIPindex of a test sample is lower than the control limitwhichmeans the sample is normal go to step (1) Elsethere is a fault in the process

4 Case Studies on the TE and CSTR Process

In this case study the TE benchmark and CSTR processare introduced to verify the effectiveness of the SPCS-BIPalgorithm PCA-GMM is the classic algorithm formultimodeprocessing monitoring And the fault detection index (FDI)is similar to Bayesian inference probability (BIP) So here acomparison was made between SPCS-BIP and PCA-GMMIn addition to verify the improvements of SPCS algorithmwhich can select sparse PCs a comparison was performedbetween the SPCS-BIP algorithm and theMPPCA algorithm

41 Tennessee Eastman Process As a well-known benchmarkprocess the Tennessee Eastman process which was pre-sented by Downs and Vogel has been widely applied toevaluate and compare the efficiency of process monitoringtechniques [46 47] The schematic diagram of the processis illustrated in Figure 2 This process consists of five majorunit operations a reactor a product condenser a vapor-liquidseparator a recycle compressor and a product stripper Inaddition there are six modes of process operation as listedin Table 1 The variables can be divided into three categoriescomposition variables continuous process variables andmanipulated variables In our study only modes 1 and 3were simulated through the Simulink programs developedon the basis of the decentralized control strategy designed byRicker [48]The Simulink programs can be downloaded fromhttpdeptswashingtoneducontrolLARRYTEdownloadhtml The 31 selected monitoring variables contained 9manipulated variables and 22 continuous process variablesThus these variables were divided into five subblocks accord-ing to five units However given that only two variables wereallocated to each the compressor unit and the condenser unitthere were four variables assigned to the other three relatedsubblocks As a result the total of 31 variables was dividedinto three subblocks

There are 20 faults in the multimode TE process whichare listed in Table 2 Among these faults the root causes of thefaults 16ndash20 are unknown [46 47] What is more to simplifyinterpretation the amplitudes of faults 3 9 and 15 are sosmall It is difficult to detect so only the remaining 12 faultswere considered in this study In the modeling stage 2000normal samples which include 1000mode 1 samples and 1000mode 3 samples were collected as the training data set Inthe testing stage 1000 samples of mode 1 were tested firstand then the process switches to mode 3 As a result the testdata set consists of 1000 samples of mode 1 and 1000 samplesof mode 3 And faults occurred from the 1200th sample Aset of 20 faults in multimode TE process which are listed

Table 1 Six process operation modes of TE process

Mode GH mass ratio Production rate1 5050 7038 kghG and 7038 kghH2 1090 1048 kghG and 12669 kghH3 9010 10000 kghG and 1111 kghH4 5050 Maximum5 1090 Maximum6 9010 Maximum

in Table 2 are simulated and the corresponding process dataare collected for testingThe following simulations are run inMATLAB 830 (2014a) environment Here two indicatorswhich are FR (FR) and MR (MR) are often introduced tomeasure the result of process monitoring FR is the rate ofnormal data classified as fault dataMR is the rate of fault dataclassified as normal rate

In the MPPCA algorithm and PCA-GMM algorithmwhen the variance contribution was selected as 85 thedimension of feature space inMPPCAand the number of PCsin PCA-GMM were each selected as 18 In order to comparethemonitoring performances of these algorithms in the samesituation the selected sparse PCs of each mode in SPCS-BIPwere selected as 18 The 99 control limit was assigned to allthree algorithms

First Figure 3 shows that the different submodes can besuccessfully divided by the EM algorithm used in this paperAnd by using other algorithms themodes also can be dividedcorrectly In other words how to correctly divide the trainingdata into multiple subset is not a problem by many relatedalgorithms

The normal process was tested by different algorithm andthe results are shown in Figure 4 In this figure it is hard tofigure out which algorithmrsquos FR is lower In the figure mostsamples of each algorithm are lower than the control limitAnd by calculation the FR of these algorithms are 0333008 025 and 108 respectively corresponding to Fig-ures 4(a) 4(b) 4(c) and 4(d) The monitoring performancesof these three algorithms suggest that the FR are acceptableNext the data sets of 12 faults in mode 3 were tested and theMR of these three algorithms are listed in Table 3 with thesmallest MR shown in bold

From Table 3 we observe that the monitoring perfor-mance of SPCS-BIP is the best compared to the MPPCAand PCA-GMM algorithms for all 12 faults Here we takethe further analysis In comparison with MPPCA and PCA-GMMalgorithms the SPCS-BIP algorithm can exactly dividethe process data into subgroups corresponding to differentmodes by using the E-M algorithm and in each submodeSPCS-BIP can select the most important PCs that have mostrelation with the fault Due to the fact that the subspacespanned by the PCs was monitored by BIP most of the PCsare related to themain process of chemical industrial processand only little PCs are related to the fault process SPCSare discriminative by constructing an elastic net regressionbetween all PCs and each sample So in Table 3 we observe

Mathematical Problems in Engineering 7

Table 2 Process faults for the multimode TE process

Faults number Disturbance state TypeIDV(1) AC feed ratio B composition constant (Stream 4) StepIDV(2) B composition AC ratio constant (Stream 4) StepIDV(3) D feed temperature (Stream 2) StepIDV(4) Reactor cooling water inlet temperature StepIDV(5) Condenser cooling water inlet temperature StepIDV(6) A feed loss (Stream 1) StepIDV(7) C header pressure loss reduced availability (Stream 4) StepIDV(8) A B and C feed composition (Stream 4) Random variationIDV(9) D feed temperature (Stream 2) Random variationIDV(10) C feed temperature (Stream 4) Random variationIDV(11) Reactor cooling water inlet temperature Random variationIDV(12) Condenser cooling water inlet temperature Random variationIDV(13) Reaction kinetics Slow driftIDV(14) Reactor cooling water valve StickingIDV(15) Condenser cooling water valve StickingIDV(16) Unknown UnknownIDV(17) Unknown UnknownIDV(18) Unknown UnknownIDV(19) Unknown UnknownIDV(20) Unknown Unknown

1

5

2

3

4

A

D

E

C

FI

FI

FI

FI

FIFI

FI

FI

FI

FI

XA

XB

XC

XD

XE

XF

XG

XH

XD

XE

XF

XG

XH

XA

XB

XC

XD

XE

XF

Compressor

Ana

lyze

rA

naly

zer

Ana

lyze

r

6

7

8

9

10

11

12

SC

PI

PI

PI

JI

TI

Condenser

CWS

CWS

CWR

Reactor

CWR

Stripper

TI

TI

TI

TILI

LI

LI

Purge

Product

Stm

Cond

Vapliqseparator

Figure 2 Control scheme for the TE process

8 Mathematical Problems in Engineering

Sample number0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

11

12

13

14

15

16

17

18

19

2

Labe

l

Figure 3 Different modes of the training data

Table 3 Missed detection rates () of 12 faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 075 025 1 01252 35 85 55 18754 0 0 0 05 0375 0 1125 06 0 0 0 07 0 0 0 08 3375 3875 425 250010 82375 16 89875 787511 225 85 45 112512 1375 1625 15 075013 16125 265 19625 1162514 0 35 0125 0

that the results of SPCS-BIP are better than the results ofMPPCA and PCA-GMM

Figure 5 shows the monitoring performances of fault 10It is easy to see that the FDI of MPPCA-1198792 and the BIP ofPCA-GMM cannot detect the fault effectively in Figures 4(a)and 4(c) In the figure more than half of the fault sampleswere regarded as the normal samples while compared tothe performances of MPPCA-1198792 and PCA-GMM the FDIof MPPCA-SPE shows some improvements However themonitoring performance of MPPCA-SPE does not match theperformance of SPCS-BIP We can find this point both inFigure 4 and Table 3

42 CSTR This study simulated the CSTR process describedby Yoon and MacGregor [49] The diagram of the process ispresented in Figure 6 Due to the fact that the CSTR processconsisted of only one operating unit the number of subblockswas selected as 1

Table 4 Missed detection rates () of two faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 90 100 988 6562 288 996 318 28

In the modeling stage 1000 samples which include 500mode 1 samples and 500mode 2 samples were collected as thetraining data set In the testing stage 1000 samples of mode 2were tested and two faults were introduced to the process asfollows

Case 1 A step of 1 K was added in the cooling watertemperature 119879

119862from the 500th sample

Case 2 A 2 kmol(m3 sdotmin) step was added in the inlet soluteconcentration 119862

119860119860from the 500th sample

In the MMPCA algorithm when the variance contri-bution was selected as 85 the dimension of feature spacein MPPCA is 10 So in order to compare the monitoringperformances of these algorithms in the same situation thenumber of PCs in PCA-GMM and the selected sparse PCs inSPCA-BIPwere both selected as 10The 99 control limit wasassigned to all three algorithms

The same as TEP the FR of these algorithms are 04 012 and 1 respectively In an industry process FR lowerthan 005 is acceptable [28]

The data sets of two faults in mode 2 were tested and theMR were listed in Table 4 In the table the smallest misseddetection rates are shown in bold

As shown in Table 4 the SPCS-BIP algorithm has shownthe best performance for these two faults compared withother algorithms listed in the table It is obvious that neitherMPPCA nor PCA-GMM algorithms can detect the faultbecause their missed detection rates were high In those fouralgorithms only the SPCS-BIP was based on the selectionPCs so the improvements in the proposed sparse principalcomponents selection can be demonstrated through thebetter monitoring performance of the SPCS-BIP algorithm

Fault 1 is a bias in cooling water temperature 119879119862 Due to

the control loop in the CSTR process these would be a biasin outlet temperature 119879 and then the cooling water flow rate119865119862would increase In Figure 7 both the MMPCA and PCA-

GMMalgorithms could not detect fault 1 effectively accordingto the performances of those shown in Figures 7(a) 7(b)and 7(c) In Figure 7(d) it is obvious that the monitoringperformance of SPCS-BIP is much better than the othersThe reason is that the correct classification for each subgroupby using E-M algorithm and the PCs selected by SPCS arediscriminative and could construct the subspace that containsthe important fault information for abnormal data

Fault 2 is a bias in inlet solute concentration 119862119860119860

Thendue to the control loop in the CSTR process there wouldbe biases in outlet concentration 119862 and outlet temperature119879 According to the performances shown in Figure 8(b) theFDI of MPPCA-SPE could not detect fault 2 Comparedto the MPPCA-SPE the FDI of MPPCA-1198792 showed some

Mathematical Problems in Engineering 9

0 500 1000 1500 20000

001

002

003

004

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

0005

001

0015

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 4 Monitoring performance of the normal process

0 500 1000 1500 20000

002

004

006

008

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

05

1

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 5 Monitoring performances of fault 10 in TEP

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 2 Process faults for the multimode TE process

Faults number Disturbance state TypeIDV(1) AC feed ratio B composition constant (Stream 4) StepIDV(2) B composition AC ratio constant (Stream 4) StepIDV(3) D feed temperature (Stream 2) StepIDV(4) Reactor cooling water inlet temperature StepIDV(5) Condenser cooling water inlet temperature StepIDV(6) A feed loss (Stream 1) StepIDV(7) C header pressure loss reduced availability (Stream 4) StepIDV(8) A B and C feed composition (Stream 4) Random variationIDV(9) D feed temperature (Stream 2) Random variationIDV(10) C feed temperature (Stream 4) Random variationIDV(11) Reactor cooling water inlet temperature Random variationIDV(12) Condenser cooling water inlet temperature Random variationIDV(13) Reaction kinetics Slow driftIDV(14) Reactor cooling water valve StickingIDV(15) Condenser cooling water valve StickingIDV(16) Unknown UnknownIDV(17) Unknown UnknownIDV(18) Unknown UnknownIDV(19) Unknown UnknownIDV(20) Unknown Unknown

1

5

2

3

4

A

D

E

C

FI

FI

FI

FI

FIFI

FI

FI

FI

FI

XA

XB

XC

XD

XE

XF

XG

XH

XD

XE

XF

XG

XH

XA

XB

XC

XD

XE

XF

Compressor

Ana

lyze

rA

naly

zer

Ana

lyze

r

6

7

8

9

10

11

12

SC

PI

PI

PI

JI

TI

Condenser

CWS

CWS

CWR

Reactor

CWR

Stripper

TI

TI

TI

TILI

LI

LI

Purge

Product

Stm

Cond

Vapliqseparator

Figure 2 Control scheme for the TE process

8 Mathematical Problems in Engineering

Sample number0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

11

12

13

14

15

16

17

18

19

2

Labe

l

Figure 3 Different modes of the training data

Table 3 Missed detection rates () of 12 faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 075 025 1 01252 35 85 55 18754 0 0 0 05 0375 0 1125 06 0 0 0 07 0 0 0 08 3375 3875 425 250010 82375 16 89875 787511 225 85 45 112512 1375 1625 15 075013 16125 265 19625 1162514 0 35 0125 0

that the results of SPCS-BIP are better than the results ofMPPCA and PCA-GMM

Figure 5 shows the monitoring performances of fault 10It is easy to see that the FDI of MPPCA-1198792 and the BIP ofPCA-GMM cannot detect the fault effectively in Figures 4(a)and 4(c) In the figure more than half of the fault sampleswere regarded as the normal samples while compared tothe performances of MPPCA-1198792 and PCA-GMM the FDIof MPPCA-SPE shows some improvements However themonitoring performance of MPPCA-SPE does not match theperformance of SPCS-BIP We can find this point both inFigure 4 and Table 3

42 CSTR This study simulated the CSTR process describedby Yoon and MacGregor [49] The diagram of the process ispresented in Figure 6 Due to the fact that the CSTR processconsisted of only one operating unit the number of subblockswas selected as 1

Table 4 Missed detection rates () of two faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 90 100 988 6562 288 996 318 28

In the modeling stage 1000 samples which include 500mode 1 samples and 500mode 2 samples were collected as thetraining data set In the testing stage 1000 samples of mode 2were tested and two faults were introduced to the process asfollows

Case 1 A step of 1 K was added in the cooling watertemperature 119879

119862from the 500th sample

Case 2 A 2 kmol(m3 sdotmin) step was added in the inlet soluteconcentration 119862

119860119860from the 500th sample

In the MMPCA algorithm when the variance contri-bution was selected as 85 the dimension of feature spacein MPPCA is 10 So in order to compare the monitoringperformances of these algorithms in the same situation thenumber of PCs in PCA-GMM and the selected sparse PCs inSPCA-BIPwere both selected as 10The 99 control limit wasassigned to all three algorithms

The same as TEP the FR of these algorithms are 04 012 and 1 respectively In an industry process FR lowerthan 005 is acceptable [28]

The data sets of two faults in mode 2 were tested and theMR were listed in Table 4 In the table the smallest misseddetection rates are shown in bold

As shown in Table 4 the SPCS-BIP algorithm has shownthe best performance for these two faults compared withother algorithms listed in the table It is obvious that neitherMPPCA nor PCA-GMM algorithms can detect the faultbecause their missed detection rates were high In those fouralgorithms only the SPCS-BIP was based on the selectionPCs so the improvements in the proposed sparse principalcomponents selection can be demonstrated through thebetter monitoring performance of the SPCS-BIP algorithm

Fault 1 is a bias in cooling water temperature 119879119862 Due to

the control loop in the CSTR process these would be a biasin outlet temperature 119879 and then the cooling water flow rate119865119862would increase In Figure 7 both the MMPCA and PCA-

GMMalgorithms could not detect fault 1 effectively accordingto the performances of those shown in Figures 7(a) 7(b)and 7(c) In Figure 7(d) it is obvious that the monitoringperformance of SPCS-BIP is much better than the othersThe reason is that the correct classification for each subgroupby using E-M algorithm and the PCs selected by SPCS arediscriminative and could construct the subspace that containsthe important fault information for abnormal data

Fault 2 is a bias in inlet solute concentration 119862119860119860

Thendue to the control loop in the CSTR process there wouldbe biases in outlet concentration 119862 and outlet temperature119879 According to the performances shown in Figure 8(b) theFDI of MPPCA-SPE could not detect fault 2 Comparedto the MPPCA-SPE the FDI of MPPCA-1198792 showed some

Mathematical Problems in Engineering 9

0 500 1000 1500 20000

001

002

003

004

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

0005

001

0015

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 4 Monitoring performance of the normal process

0 500 1000 1500 20000

002

004

006

008

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

05

1

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 5 Monitoring performances of fault 10 in TEP

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Sample number0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

11

12

13

14

15

16

17

18

19

2

Labe

l

Figure 3 Different modes of the training data

Table 3 Missed detection rates () of 12 faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 075 025 1 01252 35 85 55 18754 0 0 0 05 0375 0 1125 06 0 0 0 07 0 0 0 08 3375 3875 425 250010 82375 16 89875 787511 225 85 45 112512 1375 1625 15 075013 16125 265 19625 1162514 0 35 0125 0

that the results of SPCS-BIP are better than the results ofMPPCA and PCA-GMM

Figure 5 shows the monitoring performances of fault 10It is easy to see that the FDI of MPPCA-1198792 and the BIP ofPCA-GMM cannot detect the fault effectively in Figures 4(a)and 4(c) In the figure more than half of the fault sampleswere regarded as the normal samples while compared tothe performances of MPPCA-1198792 and PCA-GMM the FDIof MPPCA-SPE shows some improvements However themonitoring performance of MPPCA-SPE does not match theperformance of SPCS-BIP We can find this point both inFigure 4 and Table 3

42 CSTR This study simulated the CSTR process describedby Yoon and MacGregor [49] The diagram of the process ispresented in Figure 6 Due to the fact that the CSTR processconsisted of only one operating unit the number of subblockswas selected as 1

Table 4 Missed detection rates () of two faults

Faults number MPPCA-1198792 MPPCA-SPE PCA-GMM SPCS-BIP1 90 100 988 6562 288 996 318 28

In the modeling stage 1000 samples which include 500mode 1 samples and 500mode 2 samples were collected as thetraining data set In the testing stage 1000 samples of mode 2were tested and two faults were introduced to the process asfollows

Case 1 A step of 1 K was added in the cooling watertemperature 119879

119862from the 500th sample

Case 2 A 2 kmol(m3 sdotmin) step was added in the inlet soluteconcentration 119862

119860119860from the 500th sample

In the MMPCA algorithm when the variance contri-bution was selected as 85 the dimension of feature spacein MPPCA is 10 So in order to compare the monitoringperformances of these algorithms in the same situation thenumber of PCs in PCA-GMM and the selected sparse PCs inSPCA-BIPwere both selected as 10The 99 control limit wasassigned to all three algorithms

The same as TEP the FR of these algorithms are 04 012 and 1 respectively In an industry process FR lowerthan 005 is acceptable [28]

The data sets of two faults in mode 2 were tested and theMR were listed in Table 4 In the table the smallest misseddetection rates are shown in bold

As shown in Table 4 the SPCS-BIP algorithm has shownthe best performance for these two faults compared withother algorithms listed in the table It is obvious that neitherMPPCA nor PCA-GMM algorithms can detect the faultbecause their missed detection rates were high In those fouralgorithms only the SPCS-BIP was based on the selectionPCs so the improvements in the proposed sparse principalcomponents selection can be demonstrated through thebetter monitoring performance of the SPCS-BIP algorithm

Fault 1 is a bias in cooling water temperature 119879119862 Due to

the control loop in the CSTR process these would be a biasin outlet temperature 119879 and then the cooling water flow rate119865119862would increase In Figure 7 both the MMPCA and PCA-

GMMalgorithms could not detect fault 1 effectively accordingto the performances of those shown in Figures 7(a) 7(b)and 7(c) In Figure 7(d) it is obvious that the monitoringperformance of SPCS-BIP is much better than the othersThe reason is that the correct classification for each subgroupby using E-M algorithm and the PCs selected by SPCS arediscriminative and could construct the subspace that containsthe important fault information for abnormal data

Fault 2 is a bias in inlet solute concentration 119862119860119860

Thendue to the control loop in the CSTR process there wouldbe biases in outlet concentration 119862 and outlet temperature119879 According to the performances shown in Figure 8(b) theFDI of MPPCA-SPE could not detect fault 2 Comparedto the MPPCA-SPE the FDI of MPPCA-1198792 showed some

Mathematical Problems in Engineering 9

0 500 1000 1500 20000

001

002

003

004

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

0005

001

0015

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 4 Monitoring performance of the normal process

0 500 1000 1500 20000

002

004

006

008

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

05

1

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 5 Monitoring performances of fault 10 in TEP

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 500 1000 1500 20000

001

002

003

004

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

0005

001

0015

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 4 Monitoring performance of the normal process

0 500 1000 1500 20000

002

004

006

008

Sample

FDI

(a) MPPCA-1198792

0 500 1000 1500 20000

05

1

Sample

FDI

(b) MPPCA-SPE

0 500 1000 1500 20000

05

1

Sample

BIP

(c) PCA-GMM

0 500 1000 1500 20000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 5 Monitoring performances of fault 10 in TEP

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Solventflow

Pure Asolute flow

Coolingwater flow

SP SP

T C

1 2

M

11 10 7

CAS T0

TCFC

Fs

3 4

FA9

CAA

8

Figure 6 Diagram of the CSTR process

0 200 400 600 800 10000

001

002

003

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

0005

001

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

Sample

BIP

(d) SPCS-BIP

Figure 7 Monitoring performance of fault 1 in CSTR

improvements Both PCA-GMM and the proposed SPCS areall using BIP In Figures 8(c) and 8(d) we could hardly seewhich algorithm is better However in Table 4 we couldobviously find that the SPCS-BIP is better Even comparedto MPPCA-1198792 the proposed algorithm has a little advantagethan MPPCA-1198792

5 Conclusions

An algorithm using sparse principal component selectionand Bayesian inference-based probability (SPCS-BIP) wasproposed in this study Given that the modern industrialprocesses typically have multiple operating modes BIPis utilized to compute the posterior probabilities of eachmonitored sample belonging to the multiple components

and derive an integrated global probabilistic index forfault detection of multimode processes In each submodewe use the sparse principal component selection to selectthe key PCs that have the best relation with fault Thisalgorithm constructs an elastic net regression between allPCs and each sample and then selects PCs according tothe nonzero regression coefficients which indicate the dis-criminative expression of the sample Finally the TE andCSTR processes were employed to verify the superiority ofthe SPCS-BIP algorithm The monitoring performances ofMPPCA PCA-GMM and SPCS-BIP methods are discussedcompared to those of the MPPCA and PCA-GMM algo-rithms and the monitoring performances of the SPCS-BIPalgorithm were found to be the best ones among the threealgorithms

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 200 400 600 800 10000

05

1

Sample

FDI

(a) MPPCA-1198792

0 200 400 600 800 10000

002

004

006

008

Sample

FDI

(b) MPPCA-SPE

0 200 400 600 800 10000

05

1

Sample

BIP

(c) PCA-GMM

0 200 400 600 800 10000

05

1

SampleBI

P

(d) SPCS-BIP

Figure 8 Monitoring performance of fault 2 in CSTR

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 61375007) and ShanghaiScience and Research Projects (Grant nos 15JC140060015JC1401700)

References

[1] J-M Lee C Yoo S W Choi P A Vanrolleghem and I-B Lee ldquoNonlinear process monitoring using kernel principalcomponent analysisrdquo Chemical Engineering Science vol 59 no1 pp 223ndash234 2004

[2] Z Ge and Z Song ldquoMixture Bayesian regularization method ofPPCA for multimode process monitoringrdquo AIChE Journal vol56 no 11 pp 2838ndash2849 2010

[3] D Kim and I-B Lee ldquoProcess monitoring based on probabilis-tic PCArdquo Chemometrics and Intelligent Laboratory Systems vol67 no 2 pp 109ndash123 2003

[4] H D Jin Y-H Lee G Lee and C Han ldquoRobust recursive prin-cipal component analysis modeling for adaptive monitoringrdquoIndustrial and Engineering Chemistry Research vol 45 no 2 pp696ndash703 2006

[5] C Zhao and F Gao ldquoFault-relevant Principal Component Anal-ysis (FPCA) method for multivariate statistical modeling andprocess monitoringrdquo Chemometrics and Intelligent LaboratorySystems vol 133 pp 1ndash16 2014

[6] C Tong A Palazoglu and X Yan ldquoAn adaptive multimodeprocess monitoring strategy based on mode clustering andmode unfoldingrdquo Journal of Process Control vol 23 no 10 pp1497ndash1507 2013

[7] J Liu and D-S Chen ldquoOperational performance assessmentand fault isolation for multimode processesrdquo Industrial andEngineering Chemistry Research vol 49 no 8 pp 3700ndash37142010

[8] Z Ge Z Song and F Gao ldquoReview of recent research ondata-based process monitoringrdquo Industrial and EngineeringChemistry Research vol 52 no 10 pp 3543ndash3562 2013

[9] B R Bakshi ldquoMultiscale PCA with application to multivariatestatistical process monitoringrdquoAIChE Journal vol 44 no 7 pp1596ndash1610 1998

[10] X Wang U Kruger and B Lennox ldquoRecursive partial leastsquares algorithms for monitoring complex industrial pro-cessesrdquo Control Engineering Practice vol 11 no 6 pp 613ndash6322003

[11] Z Ge and Z Song ldquoProcess monitoring based on inde-pendent Component Analysis-Principal Component Analysis(ICA-PCA) and similarity factorsrdquo Industrial and EngineeringChemistry Research vol 46 no 7 pp 2054ndash2063 2007

[12] Y Hu H Ma and H Shi ldquoRobust online monitoring basedon spherical-kernel partial least squares for nonlinear processeswith contaminated modeling datardquo Industrial and EngineeringChemistry Research vol 52 no 26 pp 9155ndash9164 2013

[13] Y Ma and H Shi ldquoMultimode process monitoring basedon aligned mixture factor analysisrdquo Industrial amp EngineeringChemistry Research vol 53 no 2 pp 786ndash799 2014

[14] J-M Lee I-B Lee and C Yoo ldquoStatistical process monitoringwith independent component analysisrdquo Journal of Process Con-trol vol 14 no 5 pp 467ndash485 2004

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[15] Q P He S J Qin and J Wang ldquoA new fault diagnosis methodusing fault directions in Fisher discriminant analysisrdquo AIChEJournal vol 51 no 2 pp 555ndash571 2005

[16] C Lee S W Choi and I-B Lee ldquoVariable reconstructionand sensor fault identification using canonical variate analysisrdquoJournal of Process Control vol 16 no 7 pp 747ndash761 2006

[17] J-M Lee S J Qin and I-B Lee ldquoFault detection and diagnosisbased on modified independent component analysisrdquo AIChEJournal vol 52 no 10 pp 3501ndash3514 2006

[18] D-H Hwang and C Han ldquoReal-time monitoring for a processwith multiple operating modesrdquo Control Engineering Practicevol 7 no 7 pp 891ndash902 1999

[19] Z Ge and Z Song ldquoMultimode process monitoring based onBayesian methodrdquo Journal of Chemometrics vol 23 no 12 pp636ndash650 2009

[20] Q P He Q P He and J Wang ldquoFault detection usingthe k-nearest neighbor rule for semiconductor manufacturingprocessesrdquo IEEETransactions on SemiconductorManufacturingvol 20 no 4 pp 345ndash354 2007

[21] S Natarajan and R Srinivasan ldquoMulti-model based processcondition monitoring of offshore oil and gas production pro-cessrdquo Chemical Engineering Research and Design vol 88 no 5-6 pp 572ndash591 2010

[22] Y S Ng and R Srinivasan ldquoAn adjoined multi-model approachfor monitoring batch and transient operationsrdquo Computers andChemical Engineering vol 33 no 4 pp 887ndash902 2009

[23] J Liu and D-S Chen ldquoFault detection and identification usingmodified bayesian classification on PCA subspacerdquo Industrialand Engineering Chemistry Research vol 48 no 6 pp 3059ndash3077 2009

[24] S J Zhao J Zhang and Y M Xu ldquoMonitoring of processeswithmultiple operatingmodes throughmultiple principle com-ponent analysis modelsrdquo Industrial and Engineering ChemistryResearch vol 43 no 22 pp 7025ndash7035 2004

[25] J Yu and S J Qin ldquoMultimode process monitoring withbayesian inference-based finite gaussian mixture modelsrdquoAIChE Journal vol 54 no 7 pp 1811ndash1829 2008

[26] Z Ge and Z Song ldquoMaximum-likelihood mixture factoranalysis model and its application for process monitoringrdquoChemometrics and Intelligent Laboratory Systems vol 102 no1 pp 53ndash61 2010

[27] Z Ge F Gao and Z Song ldquoTwo-dimensional Bayesian mon-itoring method for nonlinear multimode processesrdquo ChemicalEngineering Science vol 66 no 21 pp 5173ndash5183 2011

[28] Q Jiang and X Yan ldquoChemical processes monitoring basedon weighted principal component analysis and its applicationrdquoChemometrics and Intelligent Laboratory Systems vol 119 pp 11ndash20 2012

[29] N Lu F Gao and F Wang ldquoSub-PCA modeling and on-linemonitoring strategy for batch processesrdquoAIChE Journal vol 50no 1 pp 255ndash259 2004

[30] GDiana andC Tommasi ldquoCross-validationmethods in princi-pal component analysis a comparisonrdquo Statistical Methods andApplications vol 11 no 1 pp 71ndash82 2002

[31] I T Jolliffe ldquoA note on the use of principal components inregressionrdquo Journal of the Royal Statistical Society Series CApplied Statistics vol 31 no 3 pp 300ndash303 1982

[32] T Togkalidou R D Braatz B K Johnson O Davidson andA Andrews ldquoExperimental design and inferential modeling inpharmaceutical crystallizationrdquoAIChE Journal vol 47 no 1 pp160ndash168 2001

[33] H C Peng F Long and C Ding ldquoFeature selection basedon mutual information criteria of max-dependency max-relevance and min-redundancyrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 27 no 8 pp 1226ndash12382005

[34] Q Jiang X Yan and W Zhao ldquoFault detection and diagnosisin chemical processes using sensitive principal componentanalysisrdquo Industrial and Engineering Chemistry Research vol 52no 4 pp 1635ndash1644 2013

[35] A Arbel I H Rinard and R Shinnar ldquoDynamics and controlof fluidized catalytic crackers 3 Designing the control systemchoice of manipulated and measured variables for partialcontrolrdquo Industrial and Engineering Chemistry Research vol 35no 7 pp 2215ndash2233 1996

[36] H Q Wang Z H Song and P Li ldquoFault detection behaviorand performance analysis of principal component analysisbased process monitoring methodsrdquo Industrial and EngineeringChemistry Research vol 41 no 10 pp 2455ndash2464 2002

[37] V H Nguyen and J-C Golinval ldquoFault detection based onkernel principal component analysisrdquo Engineering Structuresvol 32 no 11 pp 3683ndash3691 2010

[38] S W Choi and I-B Lee ldquoNonlinear dynamic process moni-toring based on dynamic kernel PCArdquo Chemical EngineeringScience vol 59 no 24 pp 5897ndash5908 2004

[39] R O Duda P E Hart and D G Stork Pattern ClassificationWiley New York NY USA 2001

[40] P Paalanen J-K Kamarainen J Ilonen and H KalviainenldquoFeature representation and discrimination based on Gaussianmixture model probability densities-practices and algorithmsrdquoPattern Recognition vol 39 no 7 pp 1346ndash1358 2006

[41] L K Ren and W M Lv ldquoFault detection via sparse repre-sentation for semiconductor manufacturing processesrdquo IEEETransactions on SemiconductorManufacturing vol 27 no 2 pp252ndash259 2014

[42] L Csato and M Opper ldquoSparse representation for gaussianprocess modelsrdquo Advances in Neural Information ProcessingSystems vol 49 no 1 pp 444ndash450 2001

[43] Z H Lai ldquoSparse local discriminant projections for discrim-inant knowledge extraction and classificationrdquo IET ComputerVision vol 6 no 6 pp 551ndash559 2012

[44] L Qiao S Chen and X Tan ldquoSparsity preserving projectionswith applications to face recognitionrdquo Pattern Recognition vol43 no 1 pp 331ndash341 2010

[45] H Zou T Hastie and R Tibshirani ldquoSparse principal compo-nent analysisrdquo Journal of Computational and Graphical Statis-tics vol 15 no 2 pp 265ndash286 2006

[46] J J Downs and E F Vogel ldquoPlant-wide industrial processcontrol problemrdquo Computers amp Chemical Engineering vol 17no 3 pp 245ndash255 1993

[47] P R Lyman and C Georgakis ldquoPlant-wide control of thetennessee Eastman problemrdquo Computers and Chemical Engi-neering vol 19 no 3 pp 321ndash331 1995

[48] N L Ricker ldquoDecentralized control of the tennessee eastmanchallenge processrdquo Journal of Process Control vol 6 no 4 pp205ndash221 1996

[49] S Yoon and J F MacGregor ldquoFault diagnosis with multivariatestatistical models part I using steady state fault signaturesrdquoJournal of Process Control vol 11 no 4 pp 387ndash400 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of