Yongyong Hui,Xiaoqiang Zhao ,3,*

1 College of Electrical and Information Engineering,Lanzhou University of Technology,Lanzhou 730050,China

2 Key Laboratory of Gansu Advanced Control for Industrial Processes,Lanzhou University of Technology,Lanzhou 730050,China

3 National Experimental Teaching Center of Electrical and Control Engineering,Lanzhou University of Technology,Lanzhou 730050,China

Keywords:Batch process Monitoring Related and independent variables Global–local Support vector data description

A B S T R A C T In many batch processes,there are related or independence relationships among process variables.The traditional monitoring method usually carries out a single statistical model according to the related or independent method,and in the feature extraction there is not fully taken into account the characterization of fault information,it will make the process monitoring ineffective,so a fault monitoring method based on WGNPE(weighted global neighborhood preserving embedding)–GSVDD(greedy support vector data description)related and independent variables is proposed.First,mutual information method is used to separate the related variables and independent variables.Secondly,WGNPE method is used to extract the local and global structures of the related variables in batch process and highlight the fault information,GSVDD method is used to extract the process information of the independent variables quickly and effectively.Finally,the statistical monitoring model is established to achieve process monitoring based on WGNPE and GSVDD.The effectiveness of the proposed method was verified by the penicillin fermentation process.

1.Introduction

The batch process plays an important role in modern industry and is widely used in pharmaceutical,chemical production,food and semiconductor industries.Process monitoring is very important for ensuring product quality and production safety,and is widely concerned by many scholars,thus a variety of process monitoring methods are proposed.

With the development of computer control system,a large number of industrial process data are collected and stored,thus process monitoring is applied based on multivariate statistics.Since the batch process data are three-dimensional(batch×variable×sampling time),the multiway principal component analysis(MPCA)[1],multiway partial least squares(MPLS)[2]and multiway Independent component analysis(MICA)[3]are widely used in batch process monitoring.Afterwards,many scholars have made a series of improvements to the above algorithms,such as hierarchical PCA[4],dynamic PCA and dynamic PLS for on-line monitoring[5,6].These algorithms are for Gaussian or non-Gaussian statistical analysis,there are not considered the Gaussian and non-Gaussian mixed distribution.Huang et al.proposed PCA-ICA algorithm[7,8]to solve the mixed distribution problem of Gaussian and non-Gaussian.But the algorithm used global dimensionality reduction algorithm in process monitoring,which ignored the local structure of data.In recent years,many manifold learning algorithms have been widely used in pattern recognition,such as Locally Linear Embedding(LLE)[9],Isometric Mapping(ISOMAP)[10],Laplacian Eigenmaps(LE)[11],Locality Preserving Projections(LPP)[12]and Neighborhood Preserving Embedding(NPE)[13,14].The main idea of these manifold learning algorithms is to find the local neighborhood structure of data.

Both the global and local features of data should be well preserved for dimension reduction.Zhang et al.[15]proposed a global–local structural feature analysis based on global–local structure analysis(GLSA)algorithm,which combined the global structure and the local structure and constructed a dual-objective optimization function for dimension reduction.Yu[16]proposed a local global PCA algorithm that preserved local and global structures.Zhao et al.[17]proposed a fault diagnosis algorithm based on tensor global and local,which could extract the global and local structure better.However,a single model is not sensitive to the process of mixed distribution.

Support vector data description(SVDD)[18]algorithm has been widely used in recent years.Based on SVDD,a variety of multivariable process control charts are proposed,such as K chart[19],robust K chart[20]and D2chart[21].In addition,SVDD was used in non-Gaussian process monitoring[22–24].Liu et al.[25]combined SVDD and nonlinear PCA to improve the performance of process monitoring.Compared with conventional methods such as PCA/PLS algorithm,SVDD has better performance for non-Gaussian and nonlinear systems,and only a small portion of training data are required to detect a fault,which would make the fault detection more convenient.However,it is mainly on the non-Gaussian model,and ineffective for multiple distributions.

In actual industrial production,the relationship between variables are more complex.Some variables are related,and some are independent of each other.Mutual information(MI)[26]is a nonparametric,non-linear information measure method,which is used to explain the degree of a variable that contains an other variable.MI measures the amount of information shared by two variables,which are linear relationship or nonlinear relationship.MI has been widely used in data analysis,modeling and multivariate statistical process monitoring[27–29].

In this paper,we propose a batch process monitoring based on WGNPE–GSVDD related and independent variables.The variables are divided into related variables and independent variables.Traditional PCA and NPE algorithms assume that the variables satisfy the linear or nonlinear relationship,but in actual industrial process,some variables are independent and some are related,so the related and independent variables should be discussed separately.We use the mutual information method to divide the variables.If the MI value is close to zero,the two variables are irrelevant.If the MI value of all the variables is close to 0,the variable is independent.The variables are divided into related and independent variables based on the MI values.And then the related and independent variables are handled by corresponding algorithm separately.GNPE algorithm is used to extract the global and local structures and obtain feature matrix.Then the weighted matrix is obtained by using the kernel density estimation method for the obtained feature matrix to extract the useful information and suppress the noise.WGNPE algorithm is used to fault detection of related variables.The independent variables are first extracted by Greedy algorithm to reduce the computational complexity and computation time,and then use SVDD algorithm to detect faults.

2.Preliminaries

2.1.NPE algorithm

Neighborhood preserving embedding(NPE)is a local manifold algorithm.First,k nearest neighbors are chosen to seek the nearest neighbors of each sample.And then the weight coefficient matrix W is calculated,if the node i to j has an edge connection,then the edge weight is wij;if no connection,the weight value is 0.The weight coefficient matrix can be obtained by solving the optimal solution of Eq.(1).

In NPE algorithm,if wijcan reconstruct data point xiin the space Rm,it can also reconstruct the corresponding points yiin space Rm.Therefore,the mapping transformation matrix A can be obtained by solving the Eq.(2).

where,M=(I-W)Τ(I-W),the constraints are:yΤy=aΤXXΤa=1.This transforms the problem of the transformation matrix into the generalized eigenvalue of Eq.(3):

The smallest d eigenvalues(λ1≤ λ2,…,≤ λd)corresponding to the eigenvectors in Eq.(3)form a mapping transformation matrix A=(a1,a2,…,ad).Therefore,Y=AΤX,Y is the reduced dimension data matrix,and X is the original data.

2.2.SVDD algorithm

For the dataset X={xi,i=1,…,N},SVDD algorithm makes an original spatial data project to the feature space{?(xi),i=1,…,N}through the nonlinear conversion ? :X → F,and finds the smallest volume of the hypersphere which contains almost all data.Assumed that a is the sphere center and R is the radius of the hypersphere.Because of the influence of outliers,the relaxation factor ξ is introduced,and C is the penalty parameter[18].The problem can be described as:

The optimization problem of Eq.(4)is derived:

αiis the Lagrangian factor.The inner product〈?(xi)·?(xj)〉is replaced with the kernel function K(xi·xj)to convert the nonlinear problem of the low-dimensional space to the linear high-dimensional space:

Based on the optimization problems of Eqs.(5)and(6),we can get the center a and the radius R as follows:

where,xkis the support vector of the SVDD model.

For new samples xnew,the distance can be expressed as:

3.Batch Process Monitoring Based on WGNPE–GSVDD Related and Independent Variables

3.1.Unfolding of 3D data in batch process

The main difference between batch process and continuous process is that batch process adds a“batch”dimension.Therefore,batch process data form a three-dimensional data matrix X(I× J× K),where I represents the batch,J represents the number of variables,K represents the sample time.In this paper,the three-dimensional matrix X(I× J× K)is first unfolded into X(I× KJ)based on batches.X(I× KJ)is normalized in batch direction and then rearranged as X(KI×J).This hybrid approach not only allows for variations between batches,but also takes into account its dynamic behavior over time[30].The unfold method is shown as Fig.1.

Fig.1.Hybrid unfolding of 3D data in batch process.

3.2.The division of variables

The relationship between variables in practical industrial processes is more complex.Traditional methods usually assume that variables are related,but in industrial processes,some variables are linearly related,some are non-linearly related,and others are independent or weakly related.We can ignore the weak related between variables.Thus,independent variables and related variables should be separated respectively[29].

For variable matrix X=[x1,x2,…,xm]∈ Rn×m,the relationship between any two variables can be obtained by MI value.The magnitude of the MI value between two variables indicates whether the two variables are independent or related.When the MI value is close to 0,then the two variables are independent of each other.On the contrary,the two variables are related.

Shannon entropy of a variable can be calculated by the following:

where,the probability density of the variable x is p(x),and the joint probability density for the two variables x1and x2is p(x1,x2),so the MI value can be obtained by:

The Eq.(11)can be written as the form of entropy as follows:

where,H(x1)and H(x2)are the entropy of x1and x2respectively,H(x1,x2)is a joint entropy of x1and x2,which can be obtained by:

The mutual information vector is created for the variable xi(i=1,2,…,m),and the elements in its vector are the mutual information values xiwith the other m-1 variables.The random matrix R′=[r1′,r2′,…,rm′-1] ∈ Rn×m-1is established,which satisfies the Gaussian distribution of 0 mean and the unit variance.The MI values of variable xiand m-1 variables in matrix R′are calculated respectively and the vector qi∈Rm-1is obtained.Variable xiis not related to any of the variables in R′.Therefore,the value of qiwould be relatively small.If variable xiin the data matrix X is independent of the other variables,the corresponding MI value would be close to the 0 vector.Similarly,the larger piindicates that the variable xiis related to other variables.The norm of piis calculated and expressed as Di= ‖pi‖.Small Diindicates that xiis independent of other variables.The control limit can be obtained by calculating the norm of qi,which is expressed as Ni= ‖qi‖[29].In order to overcome the influence of random factors,the random factor c is introduced in this paper.If we ignore the weak correlation of variables,we can choose a larger factor c,otherwise,select a smaller correlation factor.So the control limit is chosen as cNi.When Diis larger than cNi,xiis considered to be related to other variables,and if xiis less than cNi,xiis considered to be independent of the other variables.

By dividing variable set X,the original data can be divided into two parts,that is,the related variables and the independent variables(the matrix of the related variables is XR,the matrix of the independent variables is XI),so the original data can be expressed as:

3.3.Modeling of WGNPE–GSVDD related and independent variables

NPE can extract the local feature structure of the data by reconstructing the neighborhood points.Although the detailed feature information can be extracted,the data points outside the neighborhood points are discarded,and the global structural features of the data are ignored.After the variables are divided,the global and local structures are extracted by WGNPE algorithm for the related variables.SVDD algorithm can be used to establish statistical models for the independent variables,but the computational complexity is N3(N is the number of training samples).Since the large-scale data would cause the“dimension disaster”in the calculation of the kernel function matrix,it is necessary to extract the feature samples in the modeling data to reduce the computational complexity before modeling.Therefore,the combination method of Greedy and SVDD is used to extract the independent variables.

3.3.1.WGNPE in related variables

In this paper,NPE algorithm is used to extract the local features of the data.The objective function is shown in Eq.(15)

where M=(I-W)Τ(I-W),the constraint condition is aTXXTa=1.

In the global structure preservation,the neighbor features are also embedded to better reflect the density information of the distribution of data points,and the local mean of the samples is used to replace the whole mean.The goal of the global structure is to find projection vector s that maximizes the following objective function:

where,Xi=[x1，x2，…，xnTis the local mean matrix,G= （X-Xi）T（X-Xi）is the global structure matrix.

GNPE algorithm considers both objective Eqs.(15)and(16)together,and the global and local feature extraction objective function is as follows:

Eq.(17)can be solved by the Lagrangian multiplier method and eventually be converted to the generalized eigenvalue problem,that is:

Supposed p1,p2,…,pdare the eigenvectors corresponding to the largest d eigenvalues in Eq.(18),the process of GNPE algorithm which projects the original data X into the low-dimensional space Y can be expressed as:

where,Y is the low-dimensional structure of original data X.

Because GNPE algorithm does not take into account the different contributions in low-dimensional space.So,a weighting method is proposed to highlight the fault information and suppress the noise through a weight matrix[31],which can be expressed as:

where,W=diag（w1，w2，…，wd）∈R is a diagonal matrix.This paper uses the probability density estimation to enhance the useful information and suppress the noise.First,the sampling point is projected into the feature space,and then the kernel density estimate is used to estimate the density.A single variable kernel density function is as follows:

Fig.2.WGNPE–GSVDD modeling flowchart for the related and independent variables.

where,y is the sampling data,yiis the observed value in the data set,h is the width of the window,and n is the number of observations.There,we select the Gaussian kernel function,so the kernel density is estimated as:

In general,the width of the window has an important influence on the estimation of the kernel density.The optimal window width is related to the sampling points,the distribution of the data and the choice of the kernel function.In this paper,the best window width method,namely MISE(mean integrated squared error)method[32]is used.

The weight matrix W highlights the useful information and suppress the noise in fault detection,therefore,the relatively smaller densities indicate that there are deviations between the normal data and the test data,and they should be larger weighted.Similarly,a larger density value indicates less deviation,the weighted value doesn't need to change.The weighted value can be expressed as follows:

where,δ is the density threshold.β is the embedded weight.According to the empirical,the range of δ is 0.01–0.1,the range of β is 3–50.In practical applications,the value δ and β can be obtained by the normal batch,a major principle is that the selection of δ and β should be ensure that process monitoring of the normal batch data is not affected.^p（y^i（k-1））is the kernel density estimate of the i-th embedded value at(k-1)sample point.

After the weighted matrix W is obtained,the projection yipof the low-dimensional space is calculated by

3.3.2.GSVDD in independent variables

In the independent variable space,the modeling samples are extracted by Greedy algorithm[33]to reduce the computational complexity and computation time.The feature extraction of Greedy algorithm is to find a feature subset Et=[et1,et2,…,ets]T(s ≤ N)in the modeling residual data set E=[e1,e2,…,eN]T∈ RN×m.

where,αijis the undetermined coefficient.Greedy algorithm can be described as the following optimization problem:

where,?(Et)=[φ(et1),φ(et2),…,φ(ets)]T,Δi=[αi1,αi2,…,αis]T.

When Etis given,Δican be obtained by minimizing e(Et,Δ):

where,Kt=φ(Xt)·φ(Xt)T,kt(ei)=[K(ei,et1),K(ei,et2),…,K(ei,ets)],bring the Eq.(26)into Eq.(25):

After extracting the features of the samples by Greedy algorithm,the SVDD model is established.The independent statistical model is used for process monitoring.The flow chart of WGNPE–GSVDD modeling based on the related and independent variables is shown in Fig.2.

Fig.3.Penicillin fermentation process.

4.The WGNPE–GSVDD Batch Process Monitoring Steps

4.1.Offline modeling steps

Step 1 The sample data of the batch process are scaled and then MI method is used to divide into independent variables and related variables.

Step 2 GNPE algorithm is used to extract the global and local structure of the related variables;

Step 3 The kernel density estimation method is used to estimate the kernel density of the obtained projection matrix,and WGNPE algorithm is obtained.

Step 4 The obtained WGNPE algorithm is used for dimensional reduction and statistical analysis of the related variables;

Step 5 For the obtained independent variable,Greedy algorithm is used for feature extraction;

Step 6 SVDD algorithm is used to establish a statistical model for the extracted manifold structures.

4.2.Online monitoring steps

Step 1 Get the online sample Xnewand standardize it;

Step 2 The online process variables Xnewis divided into two parts according to Step 1 in offline modeling;

Step 3 Use WGNPE algorithm to obtain the related statistical modeland use GSVDD algorithm to obtain the independent statistical model;

Step 4 If the statistic exceeds the control limit,then a fault detected.

5.Algorithm Simulation Verification

Penicillin fermentation process is a typical batch process with nonlinear,dynamic,time-varying,multi-stage and other characteristics.The process is comprised by three stages:the cell growth stage,the penicillin synthesis stage and the cell autolysis stage.In all these processes,many factors can affect the efficiency of the penicillin fermentation process,such as temperature,pH,substrate concentration and dissolved oxygen concentration.

In order to facilitate the study of batch process,this paper uses Pensim2.0 penicillin fermentation process which proposed by Birol et al.[34].This platform has been widely used in batch process monitoring[35–37].The basic flow chart of the fermentation process is shown in Fig.3.The whole duration of each batch is 400 h,the sampling time is set to 1 h.The obtained 30 normal batches have different initial conditions but in the normal range.All variables are added to the noise.The selected 10 process variables are as the monitoring variables(see Table 1),we can get three-dimensional matrix X(30×10×400).

Table 1Process variables

Pensim2.0 simulation platform provides three types of fault,namely,Aeration rate fault,agitator power fault and substrate feed rate fault.We introduce 10 fault batches which are shown in Table 2.

Table 2Fault batches in the penicillin fermentation process

The division operation of independent and related variables is implemented.First,a random matrix R′with 10 variables is generated.MI vectors niand miare obtained,and Diof each variable is calculated.The bar graph of each Diillustrates in Fig.4.An extremely low bar implies that the corresponding variable is independent from others.We can see that variables 1,3,9,10 are independent from others and variables 2,4,5,6,7,8 are related each other.The division results of independent variables with different constants c are listed in Table 3.The division results of independent variables of constants 1.2 and 1.3 are the same.For the constant 1.4,the variable 2 are classified into related variable space.The bar graph of variable 2 in Fig.4 is visually low,this means that variable 2 should be put in the independent variable space.For each variable,norm Niis very small because each element of niis the MI value of two unrelated variables.

Fig.4.The division of the related and independent variables.

Table 3The division results of different c

MPCA,MNPE,MGNPE,MPCA-SVDD algorithms and WGNPE–GSVDD algorithm proposed in this paper are applied in penicillin fermentation process.The fault detection rate and false alarm rate are shown in Tables 4 and 5.We can see that the proposed algorithm is superior to other algorithms except the 5th and 8th faults.Especially for some minor faults,other conventional algorithms cannot effectively detect,but the proposed algorithm can detect better,in the fault 3,4,7,MPCA,MNPE,MGNPE,and MPCA-SVDD algorithms cannot detecteffectively,The proposed WGNPE–GSVDD algorithm has high detection rate and low false alarm rate,the detection effect is better.In all 10 faults,two faults with different fault variables and different fault types are selected for specific analysis and description.There,we choose the fault 2,4 for analysis.

Table 4Fault detect rate(FDR)

Table 5Fault alarm rate(FAR)

Fig.5.is the comparison charts of average fault detection rate and average false alarm rate under the all 10 faults,we can see that the average FDR of proposed algorithm is higher than MPCA,MNPE,MGNPE and MPCA-SVDD algorithms,and the average FAR of proposed algorithm is lower than other algorithms,this indicates that the monitoring performance of the proposed algorithm is superior to MPCA,MNPE,MGNPE and MPCA-SVDD algorithms.

Fig.5.Comparison charts of:(a)average fault detection rates,(b)average false alarm rates,under the 10 faults.

Fig.6.The monitoring charts for fault 2.(a)SPE monitoring chart of MPCA,(b)T2 monitoring chart of MPCA,(c)SPE monitoring chart of MNPE,(d)T2 monitoring chart of MNPE,(e)SPE monitoring chart of MGNPE,(f)T2 monitoring chart of MGNPE,(g)SPE monitoring chart of MPCA-SVDD related space,(h)T2 monitoring chart of MPCA-SVDD related space,(i)R2 monitoring chart of the MPCA-SVDD independent space.(j)SPE monitoring chart of the WGNPE–GSVDD related space,(k)T2 monitoring chart of the WGNPE–GSVDD related space,(l)R2 monitoring chart of the WGNPE–GSVDD independent space.

Fig.6(continued).

Fig.7.The monitoring charts for fault 4.(a)SPE monitoring chart of MPCA,(b)T2 monitoring chart of MPCA,(c)SPE monitoring chart of MNPE,(d)T2 monitoring chart of MNPE,(e)SPE monitoring chart of MGNPE,(f)T2 monitoring chart of MGNPE,(g)SPE monitoring chart of MPCA-SVDD related space,(h)T2 monitoring chart of MPCA-SVDD related space,(i)R2 monitoring chart of the MPCA-SVDD independent space.(j)SPE monitoring chart of the WGNPE–GSVDD related space,(k)T2 monitoring chart of the WGNPE–GSVDD related space,(l)R2 monitoring chart of the WGNPE–GSVDD independent space.

Fig.7(continued).

Fault 2 is the aeration rate adds-8%of the step fault between 200–400 h.Fig.6 shows the monitoring charts of MPCA,MNPE,MGNPE,MPCA-SVDD and WGNPE–GSVDD algorithms under fault 2,respectively.Fig.6a–b are the SPE and T2monitoring charts of MPCA algorithm under fault 2,which can be detected fault quickly.However,in Fig.6a,it is exceeded control limit at the 40th sampling point,false alarm occurs.In Fig.6b there are many false alarms between 0 to 200 points;Fig.6c–d are the SPE and T2monitoring charts of MNPE algorithm under fault 2,which can be detected quickly.However,there are multiple false alarms at the sampling points of 0 to 200.Fig.6e–fare the SPE and T2monitoring graphs of MGNPE algorithm under fault 2,which can be detected quickly.There also are multiple false alarms.Fig.6g–i are the SPE,T2and R2monitoring charts of MPCA-SVDD algorithm,R2can quickly detect the fault,there are false alarms between 0 and 100 points,SPE and T2cannot detect the fault because the fault variables are in independent variables.Fig.6j–l are the SPE,T2and R2monitoring charts of WGNPE–GSVDD algorithm,we can see that R2can detect the fault immediately when the fault occurs,and has small false alarm rate in the normal state.Compared with other algorithms,this algorithm is excellent,because WGNPE–GSVDD algorithm eliminates the interference of irrelevant variables while maintaining the advantages of WGNPE and GSVDD algorithms.After the division of the related variables and the independent variables,it can judge that the fault variables of fault 2 exists in the independent variables.

Fault 4 is the Agitator power which adds-1.25 ramp disturbance fault between 150–400 h.Due to the presence of the control system and the magnitude of the fault,the fault feature in the measured data could not be expressed immediately,this causes some troubles in fault detection.Fig.7 shows the monitoring charts of MPCA,MNPE,MGNPE,MPCA-SVDD and WGNPE–GSVDD algorithms under fault batch 4,respectively.Fig.7a–b are the SPE and T2monitoring charts of MPCA algorithm under fault 4.We can see that the fault is detected at about the 300th point,there is a greater delay and higher false alarm rate.Fig.7c–d are the SPE and T2monitoring charts of MNPE algorithm under fault4,it cannot detect the fault when adds the fault at 150th point,until the 300th point the fault is detected,and there are more false alarms.Fig.7e–f are the SPE and T2monitoring charts of MGNPE algorithm under fault 4.In Fig.7e,the fault is detected until the 270th point,and there are many false alarms.In Fig.7f,the fault is detected until the 250th point,and there are also many false alarms.Fig.7g–i are the SPE,T2and R2monitoring charts of MPCA-SVDD algorithm under fault 4,the fault is detected at the 280th point in Fig.7g,there are greater delay,also false alarms are at first 30 points.In Fig.7h,the fault is detected until the 260th point,there is delay,but less false alarms.In Fig.7i,the fault cannot be detected,because the fault variables are related variables.Fig.7j–l are the SPE,T2and R2monitoring charts of WGNPE–GSVDD algorithm under fault 4.We can see that the SPE monitoring chart can detect the fault at the 200th point,which can detect the fault ahead of nearly 100 points compared to other algorithms,fault detection is fast and the false alarm rate is lower.The T2monitoring chart can detect fault before the 200th point,the false alarm rate is also lower.The R2monitoring chart cannot detect the fault,it means the fault variable exists in the related variables.Compared with other algorithms,the proposed algorithm in this paper has high fault detection rate,low time delay and low false alarm rate.And by dividing related variables and independent variables,we can judge that the fault 4 exists in the related variables.

Once the fault is detected,the next step is to find which variable causes this fault.The variable contribution plot method identifies the fault variable by computing the contribution index.Process variable with the highest contribution may cause the fault most possibly.For WGNPE–GSVDD,the variables can be divided into related and independent variables,then,the fault can be detected.Because the WGNPE–GSVDD has better fault detection ability,the fault variable contribution will be highest than other variables,and by dividing related variables and independent variables,we can quickly determine the fault variable,this would be accurate for fault diagnosis.

6.Conclusions

In this paper,a WGNPE–GSVDD algorithm based on related and independent variables is proposed.The related variables and independent variables are separated by mutual information and then statistically analyzed by using the corresponding statistical algorithm.This algorithm combines with the advantages of WGNPE to extract the characteristics of the related variables and GSVDD to extract the characteristics of independent variables.Through confirmation by penicillin simulation,the proposed algorithm is superior to MPCA,MNPE,MGNPE,MPCASVDD algorithms,and has better monitoring effect on batch process monitoring.

Nomenclature

A transformation matrix of NPE

C penalty parameter of SVDD

E residual data set of GSVDD

Etfeature subset of GSVDD

H(x) Shannon entropy of a variable x

H(x1,x2)a joint entropy of x1and x2

h the width of the window in KDE

JGNPE(p) objective function of GNPE

P loading matrix of GNPE

^p（z） kernel density function of variable z

R the radius of SVDD

R′ random matrix

R2monitoring index in support vector

S loading matrix of PCA

SPE monitoring index in residual space

T2monitoring index in feature space

W weight coefficient matrix of NPE

W weighted matrix of WGNPE

X normal batch process data

XIindependent variables

Xnewthe test data

Xilocal mean matrix of PCA

XRrelated variables

Y score matrix of LGSPA

αiLagrangian factor of SVDD

β embedded weight of WGNPE

δ density threshold of WGNPE

λ eigenvalue of GNPE

ξ relaxation factor of SVDD

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China(No.61763029).