# Fault Detection for Multimodal Process Using Quality-Relevant Kernel Neighborhood Preserving Embedding

Authors: Yunpeng Fan; Wenyou Du; Yingwei Zhang; Xiaogang Wang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A new method named quality-relevant kernel neighborhood preserving embedding (QKNPE) has been proposed. Quality variables have been considered for the first time in kernel neighborhood preserving embedding (KNPE) method for monitoring multimodal process. In summary, the whole algorithm is a two-step process: first, to improve manifold structure and to deal with multimodal nonlinearity problem, the neighborhood preserving embedding technique is introduced; and second to monitoring the complete production process, the product quality variables are added in the objective function. Compared with the conventional monitoring method, the proposed method has the following advantages: (1) the hidden manifold which related to the character of industrial process has been embedded to a low dimensional space and the identifying information of the different mode of the monitored system has been extracted; (2) the product quality as an important factor has been considered for the first time in manifold method. In the experiment section, we applied this method to electrofused magnesia furnace (EFMF) process, which is a representative case study. The experimental results show the effectiveness of the proposed method.


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## Body

## 1. Introduction

Product quality monitoring has attracted a lot of attention like process monitoring in nearly two decades. Scholars and enterprises have made great efforts and obtained many worthy achievements [1]. Data-based process monitoring is an important means in solving some such issue [2–4]. Typical data-based process monitoring methods are used to deal with the input variable space, such as principal component analysis (KPCA) [5–7] or independent component analysis (KICA) [8–10]. Partial least squares (PLS) [11–14] or kernel partial least squares (KPLS) [15–18] can build the model of input/process variables and output/quality variables; however, the effect is not ideal for multimodal process [19, 20]. Because root causes of potential quality problems are in process variables [21], synthetically considering the whole variables is a subject worthy of study.Large-scale data can be obtained and stored from industrial process, which contain process variables and quality variables [22–24]. Storing and processing these received data are a difficult problem; we need the mathematical transformation and dimensionality reduction. Moreover, because process variable is usually measured directly, it does not cause big time delay generally. However, quality variable is measured by many complex means, such as chemical test; it will cause big time delay. Sample frequencies of process variables and quality variables are usually not synchronized [21]. This problem may increase the false alarm rate and missed alarm rate in the monitoring of multimodal process using the above traditional method. Manifold-based method, such as locally linear embedding [25–28], is a kind of method which keeps relative relationship among the data points from a high dimension space to a low dimension space. As a manifold learning framework, it calculates the Euclidean distance between every data point and its neighbors and then keeps this relationship to a lower dimensional manifold in the process of dimension reduction. Time serials of sample will not affect the effectiveness of the previous mentioned method.However, for strong-nonlinearity problems, it may not be used directly, because some data points may lay on nonmanifold surfaces. In this case, kernel technique was introduced [20], which projects the raw data into a higher dimensional linear space, that is, the feature space. The manifold structure of data is improved by this transformation. In addition, since change of feed, adjustment of production plan, and switching set value by human, industrial production processes have multimodal characteristics [29, 30]. The traditional monitoring methods are not available for multimodal processes monitoring as previously stated. The quality variables of each mode and the product quality variables were unnoticed in the existing manifold-based monitoring methods.This paper proposed a new manifold-based monitoring method used to detect fault in multimodal process, which considered the quality variables for the first time. We can decompose the process into two steps as follows: first, using KNPE technique to improve manifold structure and to deal with multimodal nonlinearity problem; second, adding preprocessed quality variables in the objective function to monitoring the complete production process. The proposed method has the following advantages: Firstly, the hidden manifold has been embedded in a low dimensional space and the identifying information of the different mode has been extracted. Secondly, the product quality has been considered in manifold method for the first time. To verify the effectiveness of the proposed method, EFMF is introduced in this paper. EFMF process will be described in the experiment section, which is a typical industrial object with characteristic of multimode and nonlinearity [31, 32]. Through experimental examples, detection capability of the algorithm proposed will be verified.The main parts of this paper are organized as follows. In Section2, KNPE algorithm and statistic are illustrated. In Section 3, we show how QKNPE works in process monitoring. An important example of EFMF process is studied to verify the effectiveness of the proposed method.


## 2. Kernel Neighborhood Preserving Embedding Algorithm and Statistic

### 2.1. The Theory of Kernel Neighborhood Preserving Embedding Algorithm

Nonlinear characteristics of complex industrial process limit the application of many linear processes monitoring method. The basic principle of kernel function method is as follows. We assume process data after standardization asXm=x1,x2,…,xn∈Rm×n, n is sample size obtained by sampling, and m is dimension of measure variable. We use a nonlinear mapping Φ· to map input data into a high dimensional feature space and then process the data in high dimensional feature space. We assume that the data points in the neighborhood fit locally linearly in the feature space. As shown in Figure 1, it can convert the computation of nonlinear high dimensional feature space after inner product operation into kernel function operation of the original data space by kernel function KXi,Xj=ΦTXiΦXj. The kernel function used commonly in process monitoring areas is radial basis kernel function KXi,Xj=exp⁡-Xi-Xj2/c, where c=rMσ2, r is parameter, M is the dimensions of the input data, and σ2 is variance [33]. ΦX=[Φx1,Φx2,…,Φxn]∈Rv is data of high dimensional feature space. v is the dimensions of the high dimensional feature space. The Euclidean distance between two points in high dimensional feature space can be calculated according to the following formula:(1)Φxi-Φxj=Φxi-ΦxjTΦxi-Φxj=ΦTxiΦxi+ΦTxjΦxj-2ΦTxiΦxj=Kxi,xi+Kxj,xj-2Kxi,xj.Figure 1
Data nonlinear relationship of EFMF.And then, the following weight matrix of KNPEW can be calculated according to the following formula [34]:(2)mineW=∑i=1nΦxi-∑j=1nWijΦxj2=∑i=1kKxi,xi-2∑j=1kWijKxi,xj+∑j=1k∑j=1kWijWijKxj,xjs.t.∑j=1nWij=1Wij=0,j∉Ji,i=1,…,n.Solving (2),(3)Wij=∑i=1nK-1xi,xjKx,xi+1-∑i=1n∑j=1nK-1xi,xjKx,xi∑i=1n∑j=1nK-1xi,xj.The global public informationΦXg=Φxg,1,…,Φxg,n∈Rd can be extracted by keeping local structure of the input data. The cost error of global public subspace is defined as (4)mineΦXg=∑i=1nΦxg,i-∑j=1nWijΦxg,j2=traceΦTXg-WΦTXgTΦTXg-WΦTXg=traceΦXgMΦTXgs.t.ΦTXgΦXg=In,where M=MT=I-WTI-W and In is n order unit matrix.Using the Lagrange multiplier method to deduce the global public spaceΦXg, the Lagrange function is(5)L=traceΦXgMΦTXg-λΦTXgΦXg-In.Setting Lagrange function asΦXg and partial derivative of λ as 0, we can obtain (6)∂L∂ΦXg=ΦXgM+ΦXgMT-2λΦXg=2ΦXgM-2λΦXg=0d×n∂L∂λ=ΦXgTΦXg-In=0n.The minimal solutionΦXg satisfies this form:(7)MΦXg=λΦXg.So the global public spaceΦXg is the first d minimum eigenvalue feature vector of M.


### 2.2. Calculations ofT2 and SPE Statistic for Kernel Neighborhood Preserving Embedding

The method based on KNPE process monitoring can reducet, the false alarm rate and missed alarm rate, and improve the accuracy by separating the data space into global public space and the local special subspace.We project the matrix from the high dimension space to low dimension space asΦXg=GTΦX. According to (4), we can obtain (8)mineG=traceΦTXG-WΦTXGTΦTXG-WΦTXG=traceGTΦXMΦTXGs.t.GGT=Iv,where G is mapping projection matrix from high dimension space to low dimension space. The constraint condition for the projection mapping matrix is GTG=Id. Lagrange multiplier can be used to solve it as follows:(9)L=traceGTΦXMΦTXG-λGTG-Id∂L∂G=ΦXMΦTXG+ΦXMΦTXTG-2λG=2ΦXMΦTXG-2λG=0v×d∂L∂λ=GTG-Id=0d×d.SettingG=ΦXS, coefficient matrix is S=s1,…,sd∈Rn. Equation (9) can be transformed to(10)ΦXMΦTXΦXS=λΦXSΦTXΦXMΦTXΦXS=λΦTXΦXSKMKS=λKS.Perform eigenvalue decomposition of matrixMK. And then the projection mapping matrix S from high dimension space to low dimension space can be gotten by reconstructing eigenvector referring to the d smallest eigenvalue.Based on the subspace KNPE separation modeling process is as follows:(11)ΦTX=ΦTXgGT+ΦXl=ΦTXgGT+ΦXl,1,…,ΦXl,C,where ΦXg is global public subspace, ΦXl is each local special subspace, ΦXl,i∈Rv, i=1,…,C, and C is the number of modes in the multiple mode process.For the new sample dataΦXnew, the global public subspace Hotelling T2 statistics and SPE statistics of each local special subspace are as follows:(12)Lnew=GTΦXnew=STΦTXΦXnew=STKTXnew,XTg2=LnewTΛ-1LnewSPE=Φ~Xnew2=ΦTXnew-LnewTGT2=ΦTXnew-ΦTXnewΦXSSTΦTX2=ΦTXnewI-ΦXSSTΦTX2=I-ΦXSSTΦTXΦXnew2=I-ΦXSSTΦTXΦXnewTI-ΦXSSTΦTXΦXnew=ΦTXnewI-ΦXSSTΦTXI-ΦXSSTΦTXΦXnew=ΦTXnewI-2ΦXSSTΦTX+ΦXSSTΦTXΦXSSTΦTXΦXnew=ΦTXnewΦXnew-2ΦTXnewΦXSSTΦTXΦXnew+ΦTXnewΦXSSTΦTXΦXSSTΦTXΦXnew=KXnew,Xnew-2KXnew,XSSTKTXnew,X+KXnew,XSSTKSSTKTXnew,X,where Λ=ΦXgΦTXg/n-1 is covariance matrix of global public subspace on the training set.


## 2.1. The Theory of Kernel Neighborhood Preserving Embedding Algorithm

Nonlinear characteristics of complex industrial process limit the application of many linear processes monitoring method. The basic principle of kernel function method is as follows. We assume process data after standardization asXm=x1,x2,…,xn∈Rm×n, n is sample size obtained by sampling, and m is dimension of measure variable. We use a nonlinear mapping Φ· to map input data into a high dimensional feature space and then process the data in high dimensional feature space. We assume that the data points in the neighborhood fit locally linearly in the feature space. As shown in Figure 1, it can convert the computation of nonlinear high dimensional feature space after inner product operation into kernel function operation of the original data space by kernel function KXi,Xj=ΦTXiΦXj. The kernel function used commonly in process monitoring areas is radial basis kernel function KXi,Xj=exp⁡-Xi-Xj2/c, where c=rMσ2, r is parameter, M is the dimensions of the input data, and σ2 is variance [33]. ΦX=[Φx1,Φx2,…,Φxn]∈Rv is data of high dimensional feature space. v is the dimensions of the high dimensional feature space. The Euclidean distance between two points in high dimensional feature space can be calculated according to the following formula:(1)Φxi-Φxj=Φxi-ΦxjTΦxi-Φxj=ΦTxiΦxi+ΦTxjΦxj-2ΦTxiΦxj=Kxi,xi+Kxj,xj-2Kxi,xj.Figure 1
Data nonlinear relationship of EFMF.And then, the following weight matrix of KNPEW can be calculated according to the following formula [34]:(2)mineW=∑i=1nΦxi-∑j=1nWijΦxj2=∑i=1kKxi,xi-2∑j=1kWijKxi,xj+∑j=1k∑j=1kWijWijKxj,xjs.t.∑j=1nWij=1Wij=0,j∉Ji,i=1,…,n.Solving (2),(3)Wij=∑i=1nK-1xi,xjKx,xi+1-∑i=1n∑j=1nK-1xi,xjKx,xi∑i=1n∑j=1nK-1xi,xj.The global public informationΦXg=Φxg,1,…,Φxg,n∈Rd can be extracted by keeping local structure of the input data. The cost error of global public subspace is defined as (4)mineΦXg=∑i=1nΦxg,i-∑j=1nWijΦxg,j2=traceΦTXg-WΦTXgTΦTXg-WΦTXg=traceΦXgMΦTXgs.t.ΦTXgΦXg=In,where M=MT=I-WTI-W and In is n order unit matrix.Using the Lagrange multiplier method to deduce the global public spaceΦXg, the Lagrange function is(5)L=traceΦXgMΦTXg-λΦTXgΦXg-In.Setting Lagrange function asΦXg and partial derivative of λ as 0, we can obtain (6)∂L∂ΦXg=ΦXgM+ΦXgMT-2λΦXg=2ΦXgM-2λΦXg=0d×n∂L∂λ=ΦXgTΦXg-In=0n.The minimal solutionΦXg satisfies this form:(7)MΦXg=λΦXg.So the global public spaceΦXg is the first d minimum eigenvalue feature vector of M.


## 2.2. Calculations ofT2 and SPE Statistic for Kernel Neighborhood Preserving Embedding

The method based on KNPE process monitoring can reducet, the false alarm rate and missed alarm rate, and improve the accuracy by separating the data space into global public space and the local special subspace.We project the matrix from the high dimension space to low dimension space asΦXg=GTΦX. According to (4), we can obtain (8)mineG=traceΦTXG-WΦTXGTΦTXG-WΦTXG=traceGTΦXMΦTXGs.t.GGT=Iv,where G is mapping projection matrix from high dimension space to low dimension space. The constraint condition for the projection mapping matrix is GTG=Id. Lagrange multiplier can be used to solve it as follows:(9)L=traceGTΦXMΦTXG-λGTG-Id∂L∂G=ΦXMΦTXG+ΦXMΦTXTG-2λG=2ΦXMΦTXG-2λG=0v×d∂L∂λ=GTG-Id=0d×d.SettingG=ΦXS, coefficient matrix is S=s1,…,sd∈Rn. Equation (9) can be transformed to(10)ΦXMΦTXΦXS=λΦXSΦTXΦXMΦTXΦXS=λΦTXΦXSKMKS=λKS.Perform eigenvalue decomposition of matrixMK. And then the projection mapping matrix S from high dimension space to low dimension space can be gotten by reconstructing eigenvector referring to the d smallest eigenvalue.Based on the subspace KNPE separation modeling process is as follows:(11)ΦTX=ΦTXgGT+ΦXl=ΦTXgGT+ΦXl,1,…,ΦXl,C,where ΦXg is global public subspace, ΦXl is each local special subspace, ΦXl,i∈Rv, i=1,…,C, and C is the number of modes in the multiple mode process.For the new sample dataΦXnew, the global public subspace Hotelling T2 statistics and SPE statistics of each local special subspace are as follows:(12)Lnew=GTΦXnew=STΦTXΦXnew=STKTXnew,XTg2=LnewTΛ-1LnewSPE=Φ~Xnew2=ΦTXnew-LnewTGT2=ΦTXnew-ΦTXnewΦXSSTΦTX2=ΦTXnewI-ΦXSSTΦTX2=I-ΦXSSTΦTXΦXnew2=I-ΦXSSTΦTXΦXnewTI-ΦXSSTΦTXΦXnew=ΦTXnewI-ΦXSSTΦTXI-ΦXSSTΦTXΦXnew=ΦTXnewI-2ΦXSSTΦTX+ΦXSSTΦTXΦXSSTΦTXΦXnew=ΦTXnewΦXnew-2ΦTXnewΦXSSTΦTXΦXnew+ΦTXnewΦXSSTΦTXΦXSSTΦTXΦXnew=KXnew,Xnew-2KXnew,XSSTKTXnew,X+KXnew,XSSTKSSTKTXnew,X,where Λ=ΦXgΦTXg/n-1 is covariance matrix of global public subspace on the training set.


## 3. Quality-Relevant Kernel Neighborhood Preserving Embedding Method

### 3.1. Theory of Quality-Relevant Kernel Neighborhood Preserving Embedding

KNPE method can deal with the nonlinear problem better, but it did not consider the change of process quality variables which industrial process is most concerned with. That is one of the reasons why false alarm and missed alarm happen frequently. The covariance information represents relationship of global public subspace and quality variables. In multimodal subspace separation, we make global public spaceΦXg=Φxg,1,…,Φxg,d∈Rn to keep the local structural information of high dimension space of input data and at the same time make the covariance information of score matrix of output quality variable Y=y1,…,yn∈Rm after standardization largest. T=t1,…,td∈Rn is score matrix of input data variable ΦTX of each mode, where T=ΦTXR and R=r1,…,rd∈Rv is the load matrix of ΦTX. U=u1,…,ud∈Rn is score matrix of each model output variable. Namely, U=YTC. C=c1,…,cd∈Rm is load matrix of YT. n is the total length of various models. The relationship of global public subspace and quality variable information is fully extracted. The extraction of the global public space will meet requirements of complex industrial process and improve the accuracy of the multimode process monitoring. The model using QKNPE method to extract the global public subspace is as follows:(13)minJ=trΦxg,iMΦTxg,i-Φxg,iΦTXri-Φxg,iYTcis.t.ΦTxg,iΦxg,i=InriTri=1ciTci=1,where min⁡trΦxg,iMΦTxg,i corresponds to (4) and min⁡-Φxg,iΦTXri-Φxg,iYTci means maximizing the projection of global public data along the score directions of ΦTX and YT. Thus, we can make the point Φxg,i in the global public subspace establish the relations with quality variables. Φxg,i is obtained by using the Lagrange multiplier method. The Lagrange function is as follows:(14)L=trΦxg,iMΦTxg,i-Φxg,iΦTXri-Φxg,iYTci-λ1ΦTxg,iΦxg,i-In-λ2riTri-1-λ3ciTci-1.Setting the partial derivative of Lagrange function forΦxg,i, ri, ci, and λi as 0, (15)∂L∂Φxg,i=Φxg,iM+Φxg,iMT-riTΦX-ciTY-2λ1Φxg,i=2Φxg,iM-riTΦX-ciTY-2λ1Φxg,i=01×n∂L∂ri=-ΦXΦTxg,i-2λ2ri=0v×1∂L∂ci=-YΦTxg,i-2λ3ci=0m×1∂L∂λ1=Φxg,iTΦxg,i-In=0n∂L∂λ2=riTri-1=0∂L∂λ3=ciTci-1=0.It is available by deriving(16)2MΦTxg,i-ΦTXri-YTci=2λ1ΦTxg,iΦTXri=-12λ2ΦTXΦXΦTxg,iYTci=-12λ3YTYΦTxg,i.The only least solutionΦxg,i will meet the following form:(17)2M+ΦTXΦX2λ2+YjTYj2λ3ΦTxg,i=2λ1ΦTxg,i.So the feature decomposition of matrix 2M+K/2λ2+YjTYj/2λ3 is made. The eigenvectors which the d smallest eigenvalue corresponds to are the best refactoring point matrix ΦXg. The load vector of each model input data and output quality variables is(18)ri=-ΦXΦTxg,i2λ2ci=-YΦTxg,i2λ3.


### 3.2. Process Monitoring Based on Quality-Relevant Kernel Neighborhood Preserving Embedding

According to (12), we set Φxg,i=GiTΦX and(19)mineGi=traceGiTΦXMΦTXGi+GiTΦXΦTXΦXΦTXGi2λ2+GiTΦXYTYΦTXGi2λ3s.t.GiTGi=1,where G=G1,G2,…,Gd∈Rv is mapping matrix from high dimension space to low dimension space. The constraint condition is GiTGi=1. And then(20)L=traceGiTΦXMΦTXGi+GiTΦXΦTXΦXΦTXGi2λ2+GiTΦXYTYΦTXGi2λ3-λGiTGi-1∂L∂Gi=ΦXMΦTXGi+ΦXMΦTXTGi+ΦXΦTXΦXΦTXGi2λ2+ΦXΦTXΦXΦTXTGi2λ2+ΦXYTYΦTXGi2λ3+ΦXYTYΦTXGiT2λ3-2λGi=2ΦXMΦTXGi+ΦXKΦTXGiλ2+ΦXYTYΦTXGiλ3-2λGi=0v×1∂L∂λ=GiTGi-1=0.LetGi=ΦXSi and let S=S1,…,Sd∈Rn; (20) can be transformed as(21)2ΦXMΦTXΦXSi+ΦXKΦTXΦXSiλ2+ΦXYTYΦTXΦXSiλ3=2λΦXSi2ΦTXΦXMΦTXΦXSi+ΦTXΦXKΦTXΦXSiλ2+ΦTXΦXYTYΦTXΦXSiλ3=2λΦTXΦXSi2KMKSi+KKKSiλ2+KYTYKSiλ3=2λKSi.The minimal solution ofSi meets the following form:(22)2MK+KKλ2+YTYKλ3Si=2λSi.So the feature decomposition for matrix2MK+KK/λ2+YTYK/λ3 is made. The eigenvector corresponding to the d smallest eigenvalue is projection mapping matrix Si from high dimension space to low dimension space.Thus, the following equation can be gotten:(23)ΦTX=ΦTXgGT+ΦXl=ΦTXgGT+ΦXl,1,…,ΦXl,C,where ΦXl are each local special subspace and ΦXl,i∈Rv, i=1,…,C.The flow chart of the whole multimode process monitoring method is shown in Figure2.Figure 2
The monitoring flow of multimode processes based on QKNPE.


## 3.1. Theory of Quality-Relevant Kernel Neighborhood Preserving Embedding

KNPE method can deal with the nonlinear problem better, but it did not consider the change of process quality variables which industrial process is most concerned with. That is one of the reasons why false alarm and missed alarm happen frequently. The covariance information represents relationship of global public subspace and quality variables. In multimodal subspace separation, we make global public spaceΦXg=Φxg,1,…,Φxg,d∈Rn to keep the local structural information of high dimension space of input data and at the same time make the covariance information of score matrix of output quality variable Y=y1,…,yn∈Rm after standardization largest. T=t1,…,td∈Rn is score matrix of input data variable ΦTX of each mode, where T=ΦTXR and R=r1,…,rd∈Rv is the load matrix of ΦTX. U=u1,…,ud∈Rn is score matrix of each model output variable. Namely, U=YTC. C=c1,…,cd∈Rm is load matrix of YT. n is the total length of various models. The relationship of global public subspace and quality variable information is fully extracted. The extraction of the global public space will meet requirements of complex industrial process and improve the accuracy of the multimode process monitoring. The model using QKNPE method to extract the global public subspace is as follows:(13)minJ=trΦxg,iMΦTxg,i-Φxg,iΦTXri-Φxg,iYTcis.t.ΦTxg,iΦxg,i=InriTri=1ciTci=1,where min⁡trΦxg,iMΦTxg,i corresponds to (4) and min⁡-Φxg,iΦTXri-Φxg,iYTci means maximizing the projection of global public data along the score directions of ΦTX and YT. Thus, we can make the point Φxg,i in the global public subspace establish the relations with quality variables. Φxg,i is obtained by using the Lagrange multiplier method. The Lagrange function is as follows:(14)L=trΦxg,iMΦTxg,i-Φxg,iΦTXri-Φxg,iYTci-λ1ΦTxg,iΦxg,i-In-λ2riTri-1-λ3ciTci-1.Setting the partial derivative of Lagrange function forΦxg,i, ri, ci, and λi as 0, (15)∂L∂Φxg,i=Φxg,iM+Φxg,iMT-riTΦX-ciTY-2λ1Φxg,i=2Φxg,iM-riTΦX-ciTY-2λ1Φxg,i=01×n∂L∂ri=-ΦXΦTxg,i-2λ2ri=0v×1∂L∂ci=-YΦTxg,i-2λ3ci=0m×1∂L∂λ1=Φxg,iTΦxg,i-In=0n∂L∂λ2=riTri-1=0∂L∂λ3=ciTci-1=0.It is available by deriving(16)2MΦTxg,i-ΦTXri-YTci=2λ1ΦTxg,iΦTXri=-12λ2ΦTXΦXΦTxg,iYTci=-12λ3YTYΦTxg,i.The only least solutionΦxg,i will meet the following form:(17)2M+ΦTXΦX2λ2+YjTYj2λ3ΦTxg,i=2λ1ΦTxg,i.So the feature decomposition of matrix 2M+K/2λ2+YjTYj/2λ3 is made. The eigenvectors which the d smallest eigenvalue corresponds to are the best refactoring point matrix ΦXg. The load vector of each model input data and output quality variables is(18)ri=-ΦXΦTxg,i2λ2ci=-YΦTxg,i2λ3.


## 3.2. Process Monitoring Based on Quality-Relevant Kernel Neighborhood Preserving Embedding

According to (12), we set Φxg,i=GiTΦX and(19)mineGi=traceGiTΦXMΦTXGi+GiTΦXΦTXΦXΦTXGi2λ2+GiTΦXYTYΦTXGi2λ3s.t.GiTGi=1,where G=G1,G2,…,Gd∈Rv is mapping matrix from high dimension space to low dimension space. The constraint condition is GiTGi=1. And then(20)L=traceGiTΦXMΦTXGi+GiTΦXΦTXΦXΦTXGi2λ2+GiTΦXYTYΦTXGi2λ3-λGiTGi-1∂L∂Gi=ΦXMΦTXGi+ΦXMΦTXTGi+ΦXΦTXΦXΦTXGi2λ2+ΦXΦTXΦXΦTXTGi2λ2+ΦXYTYΦTXGi2λ3+ΦXYTYΦTXGiT2λ3-2λGi=2ΦXMΦTXGi+ΦXKΦTXGiλ2+ΦXYTYΦTXGiλ3-2λGi=0v×1∂L∂λ=GiTGi-1=0.LetGi=ΦXSi and let S=S1,…,Sd∈Rn; (20) can be transformed as(21)2ΦXMΦTXΦXSi+ΦXKΦTXΦXSiλ2+ΦXYTYΦTXΦXSiλ3=2λΦXSi2ΦTXΦXMΦTXΦXSi+ΦTXΦXKΦTXΦXSiλ2+ΦTXΦXYTYΦTXΦXSiλ3=2λΦTXΦXSi2KMKSi+KKKSiλ2+KYTYKSiλ3=2λKSi.The minimal solution ofSi meets the following form:(22)2MK+KKλ2+YTYKλ3Si=2λSi.So the feature decomposition for matrix2MK+KK/λ2+YTYK/λ3 is made. The eigenvector corresponding to the d smallest eigenvalue is projection mapping matrix Si from high dimension space to low dimension space.Thus, the following equation can be gotten:(23)ΦTX=ΦTXgGT+ΦXl=ΦTXgGT+ΦXl,1,…,ΦXl,C,where ΦXl are each local special subspace and ΦXl,i∈Rv, i=1,…,C.The flow chart of the whole multimode process monitoring method is shown in Figure2.Figure 2
The monitoring flow of multimode processes based on QKNPE.


## 4. The Experimental Results

The EFMF is one type of the main equipment for the production of magnesia, and it belongs to mine heat electric arc furnace. Arc is heat source. It can well smelt magnesia by concentrating heat. Overall equipment of fused magnesia furnace generally includes transformer, circuit short net, electrode lifting gear, electrode, and furnace. Furnace shell is commonly rounded and slightly tapered. For the convenience of easing shell molten lead, the ring is welded in a furnace shell wall. The mobile car is equipped below furnace, so that clinker after complete melting can be moved to fixed station and cooled [20].The EFMF produces high temperature to complete melting process by introducing large current. The temperature of area near the electrode is high. The temperature of area far away from the electrode is low. Once the temperature of the area around electrode is too high, it is easy to cause security incident. However, once the temperature of area far away from the electrode is too low, lots of material will be wasted. It will seriously affect the product yield and quality. It is required to timely detect abnormal and failure in the process: therefore, the process monitoring for working process of fused magnesia furnace is very necessary and meaningful.The difference of magnesite stone raw materials and the difference of charging operation are corresponding to change of characteristics of the magnesium furnace smelting process and process data has strong nonlinear characteristic. This paper selects the process data obtained through the two conditions with and without charging smelting process of the massive magnesite, mixed massive magnesite, and powder magnesite of smelting process as the six different operation modes to model. For simplifying process dividing the model, this section assumes that each model is isometric. The definition of different modes in EFMF is shown in Table1. We use process monitoring method based on the proposed QKNPE subspace separation to monitor the work process. To show good effectiveness of the proposed method, LLE and KPLS are used for comparison. According to the comparing results, we can verify the efficiency and accuracy of the proposed QKNPE subspace separation method for monitoring the multimodal process.Table 1
The definition of different modes in EFMF.
Mode
Raw material
Whether to add raw material
1
Magnesite block
YES
2
Magnesite block
NO
3
Mixture
YES
4
Mixture
NO
5
Magnesite powder
YES
6
Magnesite powder
NOFirstly, we select sampling data from six modes of EFMF for offline modeling, and all of these sampling data are obtained under normal working condition. Each group of data contains three phase current values and three key variables. The modeling data set contains 1500 samples. For validating the monitoring performance of the proposed method for multimode process, we use 6 groups which contain 1500 samples data with fault. Faults 1, 2, and 3 happen from 200th, 400th, and 700th samples, respectively. The reason is excessive heating. Faults 4, 5, and 6, which are caused by actuator stuck fault, start from 900th, 1200th, and 1400th samples, respectively.Using multimodal process modeling and monitoring methods based on LLE, KPLS, and QKNPE subspace separation for fault 1 to fault 3,T2 and SPE statistics are shown in the diagrams in Figures 3–5. According to the results, it can be found that the local special subspace of LLE and KPLS has high nonresponse rates and low accuracy compared with QKNPE. Global public space monitoring of KPLS and QKNPE and local special subspace monitoring have high accuracy and low nonresponse rates, while QKNPE has lower rate of false positives. For faults 4–6, T2 and SPE are shown in Figures 6–8. We can find that the local special subspace of LLE monitoring for faults 4–6 has high nonresponse rates and low accuracy and the global public subspace monitoring, global public space monitoring of KPLS and QKNPE, and local special subspace monitoring have high accuracy and low nonresponse rates, while QKNPE has lower rate of false positives.Figure 3
Monitoring results for fault 1.
(a)
Monitoring based on LLE for fault 1
(b)
Monitoring based on KPLS for fault 1
(c)
Monitoring based on QKNPE for fault 1Figure 4
Monitoring results for fault 2.
(a)
Monitoring based on LLE for fault 2
(b)
Monitoring based on KPLS for fault 2
(c)
Monitoring based on QKNPE for fault 2Figure 5
Monitoring results for fault 3.
(a)
Monitoring based on LLE for fault 3
(b)
Monitoring based on KPLS for fault 3
(c)
Monitoring based on QKNPE for fault 3Figure 6
Monitoring results for fault 4.
(a)
Monitoring based on LLE for fault 4
(b)
Monitoring based on KPLS for fault 4
(c)
Monitoring based on QKNPE for fault 4Figure 7
Monitoring results for fault 5.
(a)
Monitoring based on LLE for fault 5
(b)
Monitoring based on KPLS for fault 5
(c)
Monitoring based on QKNPE for fault 5Figure 8
Monitoring results for fault 6.
(a)
Monitoring based on LLE for fault 6
(b)
Monitoring based on KPLS for fault 6
(c)
Monitoring based on QKNPE for fault 6In fused magnesia furnace multimodal process, the monitoring accuracy, the rate of false positives, and nonresponse rates of six faults based on LLE, KPLS, and QKNPE subspace separation method are shown in Table2. The experimental results show the process monitoring method based on QKNPE has very good monitoring effectiveness.Table 2
The monitoring accuracy rate, false alarm rate, missing alarm rate based on LLE, KPLS and QKNPE methods.
Failure
Method
Statistics
Accuracy
Rate of false alarm
Rate of missing report
1
LLE
T
2
73.07
1.51
30.82
SPE
99.60
3.02
0
KPLS
T
2
99.03
6.73
0
SPE
98.93
5.13
0
QKNPE
T
2
99.93
0.50
0
SPE
99.53
3.52
0
2
LLE
T
2
99.40
2.01
0.09
SPE
86.93
2.01
17.08
KPLS
T
2
97.40
6.22
0
SPE
58.07
3.56
55.86
QKNPE
T
2
99.80
0.75
0
SPE
99.07
3.26
0.09
3
LLE
T
2
98.87
2.43
0
SPE
61.60
1.29
70.79
KPLS
T
2
95.67
6.98
0
SPE
96.45
2.30
4.57
QKNPE
T
2
99.60
0.86
0
SPE
98.73
2.72
0
4
LLE
T
2
63.00
3.23
87.52
SPE
84.33
1.67
36.61
KPLS
T
2
91.07
8.42
8.65
SPE
92.05
1.88
14.91
QKNPE
T
2
96.27
0.89
7.99
SPE
95.07
2.34
8.82
5
LLE
T
2
76.27
5.59
96.01
SPE
86.87
1.50
59.47
KPLS
T
2
87.80
11.09
11.33
SPE
92.93
2.13
18.24
QKNPE
T
2
96.87
1.25
10.63
SPE
95.60
2.42
12.29
6
LLE
T
2
88.53
5.58
93.07
SPE
92.07
1.43
98.02
KPLS
T
2
87.42
11.19
16.41
SPE
94.15
2.41
31.58
QKNPE
T
2
97.60
1.43
25.84
SPE
96.20
2.64
19.80


## 5. Conclusion

In this paper, the method based on QKNPE is proposed and applied to multimodal process monitoring. This method can solve the problem of nonlinear multimodal process monitoring. KPLS method can solve the problems that manifold method cannot directly be applied to nonlinear multimodal process monitoring and KPLS changes did not explain ability of quality variables. In monitoring fused magnesia furnace smelting process, QKNPE method can use subspace separation method and considering the advantages of the relationship between input variable and output variable quality, it can improve the monitoring accuracy, can reduce false positives and omission, and can be effective to detect the fault during the process of multimode.



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*Source: 210125-2015-10-25.xml*