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Multimode process monitoring based on data-driven method
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Multimode process monitoring based on data-driven method Wenyou Du, Yunpeng Fan, Yingwei Zhangn State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Liaoning, 110819, P.R. China Received 31 December 2015; received in revised form 27 October 2016; accepted 2 November 2016
Abstract In this paper, a new data-driven method and its application to process monitoring is proposed for handling the multimode process monitoring problem in the electro fused magnesia furnace (EFMF). Compared to conventional methods, the contributions are as follows: (1) New similarity between different mode is defined with weighted norm distance which can extract common and special feature of all modes respectively, and the similar degree is analyzed; (2) Multi-mode modeling method is then proposed based on the similarity defined above; (3) Fault caused by different section often performs abnormal in different subspace, so we applied the fault detecting indices with the multi-mode model. The experiment results show the effectiveness of the proposed method. & 2016 Published by Elsevier Ltd. on behalf of The Franklin Institute
1. Introduction In consideration of ensuring the safety of equipments and the quality of products, monitoring of process performance has become an indispensable issue. In order to enforce the rationality and effectiveness of monitoring, in the last few decades, multivariate statistical process monitoring (MSPM) has been intensively researched. Particularly, principal component analysis (PCA) and partial least squares (PLS) which are widely applied in the industrial process have been important approaches for monitoring of the process performance [1,4,5,18], and some improved methods, n Correspondence to: Northeastern University Shenyang City, Liaoning Province China, postcode 110819, P.R. China Tel. +86 139 4033 4407; fax +86 024 8368 8673. E-mail address: [email protected] (Y. Zhang).
http://dx.doi.org/10.1016/j.jfranklin.2016.11.002 0016-0032/& 2016 Published by Elsevier Ltd. on behalf of The Franklin Institute Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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such as kernel principal component analysis (KPCA) and kernel partial least squares (KPLS) have achieved great success in process monitoring and fault diagnosis [2,3,14,34]. Multi-way based method has also been proposed to deal with batch processes [6–8]. Recently, monitoring for multimode processes is an urgent mission for various reasons, such as in safety control, waste-stream reduction, consistency and quality improvement. In the industrial process, the same production line is often used to produce different products. Therefore, there are often different production modes in the same production line, which makes dynamical multimode batch process more complicated [9,10,13] and there exists data need to be analyzed. Recently, some research effort has been reported to solve the multimode process monitoring problem through some modified PCA/PLS methods. Zhao et al. [32] developed multiple PCA/PLS models for multimode process monitoring.In the preliminary step, however, a priori process knowledge is required to manually segment the historical operating data into multiple groups that correspond to different operating modes which is unconvinent. Moreover, a similarity threshold has to be predefined by user to incorporate the similar data groups. Those conditions are not desirable for automatic process monitoring in industrial practice. For example, the most intuitive remedy is to develop separate model corresponding to specific operating mode. However, since these operating modes are usually not well-defined whilst difficult to identify, it is critical to assess the similarities or equivalently, the differences for any two models. Otherwise, more and more models will unavoidably be constructed as the process proceeds [33]. Considering that the mode multiplicity is an inherent nature of multimode process, various strategies have been reported and can be applied into process monitoring [19]. One is to build the variable correlation model within each mode under the influence of other modes by multi-block partial least squares (MBPLS). Compared with MPLS, the advantage of MBPLS is mainly the easier modeling, and it considers both the roles of each smaller meaningful block and the integrated contribution of all blocks [16]. Further, Reinikainen and Hoskuldsson reported a priority PLS regression analysis method and its successful application to a multi-step industrial process, which gave multimode descending priority following operation time sequence. The idea was to compute the model information only on the basis of first mode. When no more significant latent vectors could be found from the first mode, the next model information was extracted from the data of the second mode so that the left process information that was not modeled by the previous mode would be explained by the following modes. Yu and Chiang presented a multimodes statistical analysis method to reduce the complexity of fast process analysis [20,21]. It revealed that which part variation within each mode was responsible for process variations and which part was dominated [12,17,22]. For process monitoring, the subPLS modeling algorithm was developed where the regression model was suitable for all measurements within the same mode. It was based on such a presupposition that despite the time-varying mode operation trajectory, the correlations between process and process variables should remain similar within the same mode. Further researches have also been developed [23–25]. Ge and Song [26] proposed a multimode process monitoring approach based on the Bayesian method. By transferring the traditional monitoring statistic to fault probability in each operation mode, monitoring results in different operation modes can be easily combined by the Bayesian inference. This approach requires modeling for all the modes. In our earlier work, the common score algorithm across data includes two versions, common score and common weight, which extract the similar scores and weights respectively. Actually, the algorithm is directly related with the common eigenvectors which will make the same contribution to processes [27–29]. However, the common information part cannot be obtained. In this work, a between-mode process modeling approach is proposed for transition analysis and Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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process monitoring in the electro fused magnesia furnace (EFMF) For multimode process, the reason why process dynamic effects on process monitoring performance while mode changes is an interesting issue. Generally speaking, we propose a new fault detecting method for multimode industrial process based on similarity analysis. The similarity analysis extracts features between modes by calculating the distance expressed by weighted matrix norm, which is the way calculating the similarity of load matrix. With this similarity, common and special features between modes are extracted respectively. Next, kernel principal component analysis is applied, by maximizing the covariance of common feature matrix; fault detection for dynamically changed modes can be carried out together in common part. Additionally, most methods for MSPM have an assumption that process variables follow the multivariable normal distribution as precondition. This is also a precondition for our algorithm. The rest of this paper is organized as follows. Multimode process modeling method based on similarity is proposed in Section 2. The multimode process fault detection method is proposed and analyzed in Section 3. The experiment results are given to show the effectiveness of the proposed method in Section 4. Finally, conclusions are summarized in Section 5.
2. Multimode process modeling method based on similarity 2.1. K-means clustering method With the deep development of data mining, as an effective data mining method, more and more people concern clustering algorithm [15]. K-means clustering algorithm is one of the widely used partition clustering algorithms, which is suitable for processing large sample data. X A RNM is the data set collected from multi-mode production process, where N is the number of samples, M is the number of variables. Because in the actual industrial process, the operating data of all modes are mixed, the data operating in which mode are unknowable. In order to distinguish data, here the k-means clustering method (K - means) is used for clustering data set. The principle of K - means clustering algorithm is: K centers are selected randomly from an initial sample set as the initial clustering centers. In order to assign data points to the nearest class, the distances from each sample to cluster centers are calculated. The mean of each cluster is calculated to update the clustering center, then distances from data points to cluster centers are calculated, and they are assigned to the nearest class. Repeat this process, until each class no longer changes or clustering criterion function converges. The main steps are as follows: Step 1: Select any K objects from the N data as the initial clustering centers. Step 2: Compare the distances from sample points to center points, assign each point to its nearest class. Step 3: Recalculate the mean of each clustering, update the centers of all classes. Step 4: Repeat steps 2 and 3, until each clustering center no longer changes. After for Tclustering X, historical data are divided into K groups, which can be expressed as T T T X ¼P , where K is the number of modes. Total number of samples for each mode k X1 ; X2 ; …Xk is N i ¼ N. i¼1
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Clustered data are labeled, according to their modes. The same kinds of samples are the same labels. Here a label matrix is defined as 2 3T 6 7 Index ¼ 41; 1; …1 ; 2; 2; …2 ; …; c; c; …c 5 |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} N1
N2
Nc
2.2. Multimode modeling method in common feature information part Unlike other multimode modeling methods, the proposed method is based on the principles of similarity, which compares the similarities among load vectors in each mode and load vectors of all modes, and extracts common feature information of all modes. Then, the rest parts as the specific feature information are modeled respectively; this content will be introduced in Section 2.3. It is supposed that there is a multimode production process. X1 ; X2 ; …; Xk A RNM are training data of K modes. X ¼ ½XT1 ; XT2 ; …; XTd …; XTk T A RkNM is mixed data of K modes, where Xd ¼ ½xd1 ; xd2 ; …; xdM ; d ¼ 1; 2; …; k. Firstly, X is mapped to F space : T X-ΦðXÞ; ΦðXÞ ¼ ΦðX1 ÞT ; ΦðX2 ÞT ; …ΦðXk ÞT and ΦðXd Þ ¼ ½Φðxd1 Þ; Φðxd2 Þ; …; ΦðxdM Þ; d ¼ 1; 2; …; k: Covariance matrices of each mode and mixed mode data are as follows: 8 N > 1X > > ΦðXdi ÞΦðXdi ÞT ; d ¼ 1; 2; …; k > Cd ¼ > < Ni ¼ 1 kN > 1X > > C¼ ΦðXi ÞΦðXi ÞT > > : kN i ¼ 1
ð1Þ
Looking for principal components in high dimensional space, it needs to calculate eigenvectors of covariance matrix: 8 N > 1X > > C p ¼ λ p ¼ ΦðXdi ÞU pd ΦðXdi ÞT ; d ¼ 1; 2; …; k > d d d d > < Ni ¼ 1 ð2Þ kN > 1X > T > Cp ¼ λp ¼ ðΦðXi Þ UpÞΦðXi Þ > > : kN i ¼ 1 where λ is the eigenvalue, p is eigenvector of covariance matrix C. When λa 0, eigenvector is defined as: 8 N X > > > p ¼ vdj ΦðXj Þ; d ¼ 1; 2; …; k > d > < j¼1 ð3Þ kN X > > > vj ΦðXj Þ p¼ > > : j¼1 Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Kernel function Ki;j ¼ oΦðxi Þ; Φðxj Þ4 is introduced to Eq. (2). Then Eq. (3) is simplified as: Kd vd ¼ Nλd vd ; d ¼ 1; 2; …; k ð4Þ Kv ¼ KNλv The eigenvalues λ1 Z λ2 Z … Z λM are obtained from Eq. (4). The corresponding eigenvectors are v1 ; v2 ; …vM . When Singular values are decomposed, the kernel function will be centralized: ( Kd ¼ Kd IN Kd Kd IN þ IN Kd IN ; d ¼ 1; 2; …; k ð5Þ K ¼ K IkN K KIkN þ IkN KIkN where each element of IN A RNN is equal to 1=N, each element of IkN A RkNkN is equal to 1=kN. From the weighted norm, the following formula is obtained: m m X X ErosðXd ; X; wd Þ ¼ wdi jhvdi ; vi ij ¼ wdi j cos θi j; d ¼ 1; 2; …; k ð6Þ i¼1
i¼1
where w is weighted vector of the eigenvalue of model data Xc and the mixed mode data X, P m i ¼ 1 wdi ¼ 1, the angle between coefficient factors vdi and vi is θ i . The value of Eros is between 0 1. When Eros is 1, the two groups of data are the most similar. In order to calculate conveniently, weighted norm distance is used to calculate the similarity: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v u X m m m X X u DEros ðXd ; X; wd Þ ¼ 2 2 wdi jhvdi ; vi ij ¼ t2 2 wdi vdij vji ; d ¼ 1; 2; …; k ð7Þ i¼1 i¼1 j¼1 Like weighted norm Eros, weighted norm distance DEros reflects the similarity of two sets of data. Contrary to Eros, the smaller DEros is, the greater similarity Xc and X have. Namely, when ErosðXd ; X; wd Þ4ErosðXdþa ; X; wd Þ, then DEros ðXd ; X; wd Þ r DEros ðXdþa ; X; wd Þ. a is a random integer, which is less than k d. The maximum and the minimum weighted norm distances between two data sets can be calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m X X vdij vji ; d ¼ 1; 2; …; k Dmin ðXd ; X; wd Þ ¼ 2 2 wdi i¼1
j¼1
i¼1
j¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m X X vdij vji Dmax ðXd ; X; wd Þ ¼ 2 2 wdi
ð8Þ
When the similarity based on weighted norm distance between the dth model and the mixed data is in the range of Eq. (9), the two sets of data are considered to be similar Fault Diagnosis of Multimode Processes Based on Similarities. Dmin ðXd ; X; wd Þ rDðXd ; X; wd Þr ðDmax ðXd ; X; wd Þ þ Dmin ðXd ; X; wd ÞÞ=2
ð9Þ
According to the scope of similarity, similar part between the dth model and the mixed mode data is obtained. A new coefficient factor is vcd A v, where c represents the common section. Known from Eq. (7), as long as we get the weight vector of weighted norm distance wdi ; i ¼ 1; 2; …; m, the new coefficient factor vcd can be obtained. General steps of calculation of the weight vector are as follows: Step 1: Each row ofPthe eigenvalue matrix is summed up column is λj ¼ λj = M i ¼ 1 λij .
PM
i ¼ 1 λij ,
and the new value of each
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Step 2: Calculate the variance of the new value of each column, as the weight vector wi ¼ f ðλj Þ. P Step 3: Standardize weight vectors wi ¼ wi = M i ¼ 1 wi . Then we get the new coefficient factors vc1 ; vc2 ; …; vcd ; …; vck . The new coefficient factors consider the correlation between modes, but we need to consider the common characteristics with larger effect, so we need to extract the new coefficient factors further. That is to say, common column vectors of new coefficient factors are extracted to construct the public coefficient factors vc ¼ vc1 \ vc2 \ … P \ vcd \ … \ vck between different modes. Then c c common characteristic load matrix p ¼ kN j ¼ 1 vj ΦðXj Þ is obtained. The score matrix of the common part is as follows: tc ¼ ΦðXÞpc ¼ ΦðXÞΦðXÞT vc ¼ Kvc
ð10Þ
At this point, common information part ΦðX Þ of the modes can be obtained as: C
ΦðXC Þ ¼ tc pcT ¼ Kvc vcT ΦðXÞ
ð11Þ
2.3. Multimode modeling method in specific feature information part Mixed mode data are the sum of common information part and specific information part of the data, the mixed mode data can be represented as: ΦðXÞ ¼ ΦðXC Þ þ ΦðXS Þ
ð12Þ
According to kernel principal component analysis, the specific part of the score matrix can be expressed as: ts ¼ ΦðXÞps ¼ ΦðXÞΦðXÞT vs ¼ Kvs
ð13Þ
where ps is load matrix of specific feature information part. vs is coefficient factor of specific feature information part. The data of specific feature information part can be expressed as: ΦðXs Þ ¼ ts psT ¼ Kvs vsT ΦðXÞ
ð14Þ
Because the specific part ΦðXs Þ is a part of data set X, the specific part ΦðXs Þ can be divided into k modes completely according to the K-means cluster label Index. Corresponding score matrices ts of each mode data ΦðXsi Þði ¼ 1; 2; …; kÞ are divided into k groups, ts1 ; ts2 ; …; tsk . ΦðXsk Þ ¼ tsk psT ¼ tsk vsT ΦðXÞ
ð15Þ
Specific part of each model has its particularity and independence, which requires to be monitored respectively. Csk ¼
1 1 s sT ΦðXsk ÞT ΦðXsk Þ ¼ ΦðXÞT vs tsT k tk v ΦðXÞ Nk Nk
ð16Þ
Covariance maximization problem is converted to the problem of solving the eigenvalue and eigenvector: Csk rk ¼ βrk
ð17Þ
where β is the eigenvalue, rk is the corresponding eigenvector. Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Because eigenvector is defined as rk ¼ ΦðXÞT jk and jk is coefficient factor of specific part, Eq. (17) can be simplified as follows: 1 s sT s sT v t t v Kjk ¼ βjk Nk k k
ð18Þ
The coefficient factor jk is obtained by the singular value decomposition to Eq. (18). The corresponding score vector tsk;c can be expressed as: tsk;c ¼ ΦðXsk Þrk;c ¼ tsk vsT Kjk;c
ð19Þ
where rk;c is the 85% contribution part of load vector rk ,jk;c is coefficient factor of load vector, [11,35] K is the centralized kernel matrix, then ΦðXc S Þ is expressed as: ^ sk Þ ¼ ΦðXsk Þrk;c rTk;c ¼ tsk vsT Kjk;c jTk;c ΦðXÞ ΦðX
ð20Þ
The residuals of the specific part is expressed as: Ek ¼ ΦðXsk Þrk;s rTk;s ¼ tsk vsT Kjk;s jTk;s ΦðXÞ
ð21Þ
Then model of specific information part for each mode is expressed as: ^ sk Þ þ Ek ΦðXsk Þ ¼ ΦðX
ð22Þ
To sum up, with multi-mode fault detection method based on the similarity the modeling of common feature information part and specific feature information part have been completed. 3. Multimode process fault detection method based on similarity Common part model of all modes and specific part model of each mode are used for fault detection. A new data xnew A R1M is obtained in the multimode process. Steps of multimode process fault detection method based on similarity are as follows: Step 1: Sample point xnew is standardized, new kernel function ½Knew i;j ¼ oΦðxnew;i Þ; Φðxj Þ4 is constructed,Knew is centralized to Knew . Step 2: Common feature information part is obtained, T 2 statistic of common feature information part is calculated, here T 2 statistic is denoted by CT new 2 : 8 N X > > < tC ¼ opc ; Φðxnew Þ4 ¼ vc Kðxnew ; xi Þ new ð23Þ i¼1 > > : CT 2 ¼ tc Λ 1 tc T new
new
new
Under the condition that the process is normal and the data follow a multivariate normal distribution, the T 2 statistic is related to an F distribution considering that the population mean and covariance are estimated from data [30] NðN lÞ CT 2 F l;N l lðN 2 1Þ
ð24Þ
where F l;N l is an F distribution with l and N l degrees of freedom where l equals to the number of eigenvalues obtained before and N equals to the number of variables. For a given Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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significance level α the process is considered normal if CT 2new r CT α 2 ¼
lðN 2 1Þ F l;N l;α NðN lÞ
ð25Þ
If the calculated statistic overruns the control limit, the common feature information part is considered to be abnormal. Otherwise, monitoring process is considered under the normal condition. Step 3: Specific feature information part is obtained, T 2 and SPE statistics of specific feature information part are calculated, here T 2 and SPE statistics are denoted by ST 2new and SSPE new : 8 2 ST ¼ tsnew;k β 1 tsnew;k T > > < new;k N R X X ð26Þ > ts2 ts2 SSPE new;k ¼ ‖enew;k ‖2 ¼ > new;i new;i : i¼1
i¼1
The process is considered normal if SSPEnew;k r δ2α
ð27Þ
where δ2α denotes the upper control limit for SPE with a significance level α. It is calculated as follows: 0 qffiffiffiffiffiffiffiffiffiffiffiffi 11=h0 cα 2θ2 h20 θ h ðh 1Þ 2 0 0 A δ2α ¼ θ1 @ þ1þ θ1 θ21
ð28Þ
Fig. 1. Schematic diagram of the EFMF with multimode manufacture process. Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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where θi ¼
M X
λij ; i ¼ 1; 2; 3
ð29Þ
j ¼ lþ1
h0 ¼ 1
2θ1 θ3 3θ22
ð30Þ
l is the number of retained principal components and cα is the normal deviate corresponding to the upper 1 α percentile. Note that this result is derived under the conditions that sample vector follows a multivariate normal distribution [31]. If one of the calculated statistics overruns the control limit, the specific feature information part is considered to be abnormal. Otherwise, monitoring process is considered under the normal condition. 4. Experiment results The electro-fused magnesia furnace (EFMF) is one of the main equipments used to produce electro-fused magnesia, which belongs to a kind of mine hot electric arc furnaces. With the development of technology of melting, EFMF has already got extensively application in the industry. EFMF refining technology can enhance the quality and increase the production variety. In the melting process, leakage of furnace is a kind of power quality problems, which occurs in the power system when the gas aggregates quickly in the magnesia. The Schematic diagram is shown in Fig. 1. There are mainly two different modes for EFMF. The differences between two modes include materials used and products manufactured. In mode A, the materials are powdery magnesium, and the product is fused magnesia crystals, in this mode, the temperature is higher and need higher current correspondingly. In mode B, the materials are magnesium blocks, and the product is heavy-burned magnesia which need lower current and generate lower temperature. Since different material has different resistance, pressure in the EFMF for two modes is also different, values of variables generated from the two modes has not only similarity but also specific characteristics respectively. So, Electro-fused magnesia furnace production process is used for fault detection to verify the effectiveness of the proposed method based on similarity. Temperature is a very important variable in the process of EFMF. It is determined by the current value of EFMF. In order to verify the effectiveness of multimode fault detection method based on similarity, 9 variable values are selected at runtime of EFMF, including the measured values of three electrode currents and six temperatures of EFMF. In our experiment, there are totally 200,000 sample data collected from the factory during one week. Through the K-means clustering analysis method, the sample data are divided into two modes, including 113,212 sample data in mode A and 86,788 sample data in mode B. The whole data set are labeled by experts as normal or abnormal, then 300 normal data for each mode are selected randomly as training set, and 400 data including abnormal data for each mode are selected as testing set. Fault 1 is the abnormal of the first electrode current. The advantage of the proposed similarity decomposition based statistical monitoring method is that, similar component of different mode is extracted forming the common part. When operating mode varies dynamically yet cannot be known online immediately, monitoring on common part is more stable which means there will be less False Positives or False Negatives comparing to Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Fig. 2. T 2 statistic of normal test data from mode A with model developed on the common part.
Fig. 3. T 2 statistic of normal test data from mode A with model developed on the specific part.
monitoring data with certain model develop on one mode. So in this part, we design two experiments using data without fault and data with fault respectively. Model on common part, mode A, and mode B are established for monitoring respectively as a comparison. First of all, the 400 test samples of mode A which are all normal data are used to test the effectiveness of multimode fault detection method based on similarity decomposition. Models on the common part, specific part, and Mode B are established respectively and the corresponding control limits are also developed. Here, according to the criterion mentioned forward, the p-value which is the number of principal components is set 3. Fig. 2 is the T 2 statistic of test data in the
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Fig. 4. T 2 and SPE statistics of normal test data from mode A with model developed in mode B.
Fig. 5. T 2 statistic of test data including fault 1 from mode B with model developed on the common part.
common part. It can be seen from the chart that statistics are not beyond the control limits. Fig. 3 (a) and (b) are the statistics of test data in specific part. It can be seen from the chart that T 2 and SPE statistics do not overrun the control limits. Fig. 4(a) and (b) are the statistics with model developed in mode B. It can be seen from the chart that T 2 and SPE statistics overrun the control limits seriously. It is considered that the test data does not belong to mode B, but belongs to the normal data of model A. Actually the EFMF indeed working in mode A. In order to further verify fault detection performance of the proposed multimode fault detection method based on similarity, a set of 400 samples in mode B containing fault data is tested, fault 1 caused by actuator failure starts from the 251th sample, resulting in abnormal current value. Similar with the following experiment, models on the common part, specific part, Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Fig. 6. Statistic of test data including fault 1 from mode B with model developed on the specific part.
and Mode B are established respectively and the corresponding control limits are also developed. With the same way, the p-value is also calculated and set 3. Fig. 5 is the simulation results of the proposed multimode fault detection method based on similarity. It can be seen from the charts that fault starts from the 251th sample. The fault is detected not only in the common part but also with the model developed in mode B as shown in Fig. 7(a) and (b). To the contrast Fig. 6(a) and (b) are the statistics on specific part. It can be seen obviously that fault 1 cannot be detected correctly. Generally speaking, two experiment results demonstrate that, monitoring on the common part obtained with the proposed similarity based decomposition method is more stable. When mode changes dynamically, fault can be detected with the same model effectively. As a comparison, another novel multimode monitoring approach based on Between-Mode Transition which has the same industrial background with our paper was cited in this experiment. In this cited article, the manifold is learned to extract the common part of between-mode transition [36]. Normal data from mode A and data with fault 1 from mode B are both detected with our proposed method and the cited method. The effectiveness of different approaches is evaluated via using FAR and MAR defined as follows: False Positive (FP) donates those normal samples alarmed as fault samples. True Negative (TN) donates those normal samples not alarmed as fault samples. True Positive (TP) donates those abnormal samples alarmed as fault samples. False Negative (FN) donates those abnormal samples not alarmed as fault samples.
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Fig. 7. T 2 and SPE statistics of test data including fault 1 from mode B with model developed in mode B.
Table 1 MAR and FAR for the two approaches. Data type
Approach
FAR(%)
MAR(%)
Normal
Between-mode Transition based(referenced) similarity based(ours)
1.34 1.54
– –
Fault 1
Between-mode Transition based(referenced) similarity based(ours)
2.21 2.50
6.38 0.92
Then, we define False Alarmed Rate FAR ¼ FP/ (FPþTN); Missing Alarm Rate MAR ¼ FN/ (TPþFN). Knowing from the above definition, approaches with lower MAR are more sensitive in fault detection, correspondingly, those approaches with lower FAR are more stable in fault detection. Average value of FAR and MAR are calculated with ten times of repeated experiments, each time we choose 300 normal samples for training randomly. Experiment results with MAR and FAR are shown in Table 1. As shown in Table 1, when using the normal data from mode A, FARs for both methods are relatively low, this means the two methods are stable and few normal data are false alarmed as fault. However, when using the data with fault 1from mode B, although FARs of the two methods are still very close, MAR of our proposed method is quite low, which means hardly any Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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faults are missed. This result demonstrates monitoring on common part obtained by similarity decomposition can extract the common feature between different modes more effectively, and are more sensitive when detecting fault 1. 5. Conclusions In this paper, a multimode process fault detection method based on the similarity is proposed. This method can effectively extract common information part and specific information part of the multimode data for fault detection. K-means method is used for data clustering, mode classification of history data, and calculation of load matrices of each mode data and load matrices of mixed modes data. According to the comparative value, common information part is determined, and the rest is the specific information part. This method not only studies the correlation among modes and obtains the common information part, but also analyzes the local data of each mode, maintain the local characteristics, and avoid direct analysis of mixed data which causes the local information loss. In addition, this method can also be applied to the nonlinear fault detection problem. In order to prove the effectiveness and accuracy of the method, the feasibility and the superiority of the proposed method is proved in the simulation of EFMF production process. Acknowledgement This work is supported by China's National 973 program (2009CB320600 and 2009CB320604) and NSF in China (61325015 and 61273163). References [1] C. Alcala, S.J. Qin, Reconstruction-based contribution for process monitoring, Automatica 45 (7) (2009) 1593–1600. [2] S. Lane, E.B. Martin, R.K. Kooijmans, Performance monitoring of a multi-product semi-batch process, J. Process Control 11 (1) (2001) 1–11. [3] S.J. Qin, Statistical process monitoring: basics and beyond, J. Chemom. 17 (8-9) (2003) 480–502. [4] C. Shahabi, D. Yan, Real-time pattern isolation and recognition over immersive sensor data streams, in: Proceedings of the 9th International Conference on Multi-Media Modeling, pp. 93–113, 2003. [5] S.J. Qin, S. Valle, M.J. Piovoso, On unifying multiblock analysis with application to decentralized process monitoring, J. Chemom. 15 (9) (2001) 715–742. [6] C.K. Yoo, J.M. Lee, P.A. Vanrolleghem, I.B. Lee, On-line monitoring of batch processes using multiway independent component analysis, Chemom. Intell. Lab. Syst. 71 (2) (2004) 151–163. [7] P. Nomikos, J.F. MacGregor, Monitoring batch processes using multiway principal component analysis, AIChE J. 40 (8) (1994) 1361–1375. [8] D. Zhou, G. Li, S.J. Qin, Total projection to latent structures for process monitoring, AIChE J. 56 (1) (2010) 168–178. [9] U. Thissen, H. Swierenga, P.D. Weijer, Multivariate statistical process control using mixture modeling, J. Chemom. 19 (2) (2005) 23–31. [10] S.J. Qin, Y. Zheng, Quality-relevant and process-relevant fault monitoring with concurrent projection to latent structures, AIChE J. 59 (2) (2013) 496–504. [11] T. Chen, Y. Sun, Probabilistic contribution analysis for statistical process monitoring: a missing variable approach, Control Eng. Pract. 17 (4) (2009) 469–477. [12] Y. Zhang, J. An, C. Ma, Fault detection of non-gaussian process based on model migration, IEEE Trans. Control Syst. Technol. 21 (5) (2013) 1517–1526. [13] G. Birol, C. Undey, A. Cinar, A modular simulation package for feed-batch fermentation: penicillin production, Comput. Chem. Eng. 26 (11) (2002) 1553–1565. Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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[14] Y. Zhang, Z. Hu, Multivariate process monitoring and analysis based on multi-scale KPLS, Chem. Eng. Res. Des. 89 (12) (2011) 2667–2678. [15] H. Xia, J. Zhuang, D.H. Yu, Novel soft subspace clustering with multi-objective evolutionary approach for highdimensional data, Pattern Recognit. 46 (9) (2013) 562–2575. [16] S.W. Choi, I.B. Lee, Multiblock PLS–based localized process diagnosis, J. Process Control 15 (3) (2005) 295–306. [17] Y. Zhang, S. Li, Z. Hu, Improved multi-scale kernel principal component analysis and its application for fault detection, Chem. Eng. Res. Des. 90 (9) (2012) 1271–1280. [18] W. Li, H.H. Yue, C. Valle, S.J. Qin, Recursive PCA for adaptive process monitoring, J. Process Control 10 (2) (2000) 471–486. [19] Y. Zhang, Y. Teng, Process data modeling using modified kernel partial least squares, Chem. Eng. Sci. 65 (24) (2010) 6353–6361. [20] J. Yu, A nonlinear kernel Gaussian mixture model based inferential monitoring approach for fault detection and diagnosis of chemical process, Chem. Eng. Sci. 68 (2) (2012) 506–519. [21] L.H. Chiang, E.L. Russell, R.D. Braatz, Fault diagnosis in chemical process using Fisher discriminant analysis, discriminant partial least squares, and principal component analysis, Chemom. Intell. Lab. Syst. 50 (2) (2000) 243–252. [22] Y. Zhang, H. Zhou, T. Chai, S.J. Qin, Decentralized fault diagnosis of large-scale processes using multiblock kernel partial least squares, IEEE Trans. Ind. Inform. 6 (1) (2010) 3–10. [23] D. Dong, T.J. McAvoy, Nonlinear principal component analysis: based on principal curves and neural networks, Com. Chem. Eng. 20 (2) (1996) 65–78. [24] M.A. Kramer, Non-linear principal component analysis using auto associative neural networks, AIChE J. 37 (5) (1991) (233-24). [25] L. Xu, A.L. Yuille, Robust principal component analysis by self-organizing rules based on statistical physics approach, IEEE Trans., Neural Netw. 6 (1) (1995) 131–143. [26] Z.Q. Ge, Z.H. Song, Multimode process monitoring based on Bayesian method, J. Chemom. 23 (11-12) (2009) 636–650. [27] Y.S. Qi, P. Wang, X.J. Gao, Fault detection and diagnosis of multiphase batch process based on kernel principal component analysis-principal component analysis, Control Theory Appl. 29 (6) (2012) 754–764. [28] P.M. Frank, Fault diagnosis in the dynamic system using analytical acknowledge-based fault redundancy-a survey and some new results, Automatic 26 (6) (1990) 459–474. [29] R. Patton, P.M. Frank, R. Clark, Fault Diagn. Dyn. Syst. (1989) 166–189. [30] N.D. Tracy, J.C. Young, R.L. Mason, Multivariate control charts for individual observations, J. Qual. Technol. 24 (1992) 88–95. [31] J.E. Jackson, G. Mudholkar, Control procedures for residuals associated with principal component analysis, Technometrics, 21 (1979) 341–349. [32] S.J. Zhao, J. Zhang, Y.M. Xu, Monitoring of processes with multiple operating modes through multiple principal component analysis models, Ind. Eng. Chem. Res 43 (2004) 7025–7035. [33] S.J. Zhao, J. Zhang, Y.M. Xu, Performance monitoring of processes with multiple operating modes through multiple PLS models, J. Process Control 16 (2006) 763–772. [34] J. Chen, J. Liu, Mixture principal component analysis models for process monitoring, Ind. Eng. Chem. Res 38 (1999) 1478–1488. [35] Sergio Valle, Weihua Li, S. Joe Qin, Selection of the Number of Principal Components: the Variance of the Reconstruction Error Criterion with a Comparison to Other Methods Ind, Eng. Chem. Res. 38 (1999) 4389–4401. [36] Y.W. Zhang, S. Li, Modeling and monitoring between-mode transition of multimodes processes, IEEE Trans. Ind. Inform. 9 (4) (2013) 2248–2255.
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Multimode process monitoring based on data-driven method Wenyou Du, Yunpeng Fan, Yingwei Zhangn State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Liaoning, 110819, P.R. China Received 31 December 2015; received in revised form 27 October 2016; accepted 2 November 2016
Abstract In this paper, a new data-driven method and its application to process monitoring is proposed for handling the multimode process monitoring problem in the electro fused magnesia furnace (EFMF). Compared to conventional methods, the contributions are as follows: (1) New similarity between different mode is defined with weighted norm distance which can extract common and special feature of all modes respectively, and the similar degree is analyzed; (2) Multi-mode modeling method is then proposed based on the similarity defined above; (3) Fault caused by different section often performs abnormal in different subspace, so we applied the fault detecting indices with the multi-mode model. The experiment results show the effectiveness of the proposed method. & 2016 Published by Elsevier Ltd. on behalf of The Franklin Institute
1. Introduction In consideration of ensuring the safety of equipments and the quality of products, monitoring of process performance has become an indispensable issue. In order to enforce the rationality and effectiveness of monitoring, in the last few decades, multivariate statistical process monitoring (MSPM) has been intensively researched. Particularly, principal component analysis (PCA) and partial least squares (PLS) which are widely applied in the industrial process have been important approaches for monitoring of the process performance [1,4,5,18], and some improved methods, n Correspondence to: Northeastern University Shenyang City, Liaoning Province China, postcode 110819, P.R. China Tel. +86 139 4033 4407; fax +86 024 8368 8673. E-mail address: [email protected] (Y. Zhang).
http://dx.doi.org/10.1016/j.jfranklin.2016.11.002 0016-0032/& 2016 Published by Elsevier Ltd. on behalf of The Franklin Institute Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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such as kernel principal component analysis (KPCA) and kernel partial least squares (KPLS) have achieved great success in process monitoring and fault diagnosis [2,3,14,34]. Multi-way based method has also been proposed to deal with batch processes [6–8]. Recently, monitoring for multimode processes is an urgent mission for various reasons, such as in safety control, waste-stream reduction, consistency and quality improvement. In the industrial process, the same production line is often used to produce different products. Therefore, there are often different production modes in the same production line, which makes dynamical multimode batch process more complicated [9,10,13] and there exists data need to be analyzed. Recently, some research effort has been reported to solve the multimode process monitoring problem through some modified PCA/PLS methods. Zhao et al. [32] developed multiple PCA/PLS models for multimode process monitoring.In the preliminary step, however, a priori process knowledge is required to manually segment the historical operating data into multiple groups that correspond to different operating modes which is unconvinent. Moreover, a similarity threshold has to be predefined by user to incorporate the similar data groups. Those conditions are not desirable for automatic process monitoring in industrial practice. For example, the most intuitive remedy is to develop separate model corresponding to specific operating mode. However, since these operating modes are usually not well-defined whilst difficult to identify, it is critical to assess the similarities or equivalently, the differences for any two models. Otherwise, more and more models will unavoidably be constructed as the process proceeds [33]. Considering that the mode multiplicity is an inherent nature of multimode process, various strategies have been reported and can be applied into process monitoring [19]. One is to build the variable correlation model within each mode under the influence of other modes by multi-block partial least squares (MBPLS). Compared with MPLS, the advantage of MBPLS is mainly the easier modeling, and it considers both the roles of each smaller meaningful block and the integrated contribution of all blocks [16]. Further, Reinikainen and Hoskuldsson reported a priority PLS regression analysis method and its successful application to a multi-step industrial process, which gave multimode descending priority following operation time sequence. The idea was to compute the model information only on the basis of first mode. When no more significant latent vectors could be found from the first mode, the next model information was extracted from the data of the second mode so that the left process information that was not modeled by the previous mode would be explained by the following modes. Yu and Chiang presented a multimodes statistical analysis method to reduce the complexity of fast process analysis [20,21]. It revealed that which part variation within each mode was responsible for process variations and which part was dominated [12,17,22]. For process monitoring, the subPLS modeling algorithm was developed where the regression model was suitable for all measurements within the same mode. It was based on such a presupposition that despite the time-varying mode operation trajectory, the correlations between process and process variables should remain similar within the same mode. Further researches have also been developed [23–25]. Ge and Song [26] proposed a multimode process monitoring approach based on the Bayesian method. By transferring the traditional monitoring statistic to fault probability in each operation mode, monitoring results in different operation modes can be easily combined by the Bayesian inference. This approach requires modeling for all the modes. In our earlier work, the common score algorithm across data includes two versions, common score and common weight, which extract the similar scores and weights respectively. Actually, the algorithm is directly related with the common eigenvectors which will make the same contribution to processes [27–29]. However, the common information part cannot be obtained. In this work, a between-mode process modeling approach is proposed for transition analysis and Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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process monitoring in the electro fused magnesia furnace (EFMF) For multimode process, the reason why process dynamic effects on process monitoring performance while mode changes is an interesting issue. Generally speaking, we propose a new fault detecting method for multimode industrial process based on similarity analysis. The similarity analysis extracts features between modes by calculating the distance expressed by weighted matrix norm, which is the way calculating the similarity of load matrix. With this similarity, common and special features between modes are extracted respectively. Next, kernel principal component analysis is applied, by maximizing the covariance of common feature matrix; fault detection for dynamically changed modes can be carried out together in common part. Additionally, most methods for MSPM have an assumption that process variables follow the multivariable normal distribution as precondition. This is also a precondition for our algorithm. The rest of this paper is organized as follows. Multimode process modeling method based on similarity is proposed in Section 2. The multimode process fault detection method is proposed and analyzed in Section 3. The experiment results are given to show the effectiveness of the proposed method in Section 4. Finally, conclusions are summarized in Section 5.
2. Multimode process modeling method based on similarity 2.1. K-means clustering method With the deep development of data mining, as an effective data mining method, more and more people concern clustering algorithm [15]. K-means clustering algorithm is one of the widely used partition clustering algorithms, which is suitable for processing large sample data. X A RNM is the data set collected from multi-mode production process, where N is the number of samples, M is the number of variables. Because in the actual industrial process, the operating data of all modes are mixed, the data operating in which mode are unknowable. In order to distinguish data, here the k-means clustering method (K - means) is used for clustering data set. The principle of K - means clustering algorithm is: K centers are selected randomly from an initial sample set as the initial clustering centers. In order to assign data points to the nearest class, the distances from each sample to cluster centers are calculated. The mean of each cluster is calculated to update the clustering center, then distances from data points to cluster centers are calculated, and they are assigned to the nearest class. Repeat this process, until each class no longer changes or clustering criterion function converges. The main steps are as follows: Step 1: Select any K objects from the N data as the initial clustering centers. Step 2: Compare the distances from sample points to center points, assign each point to its nearest class. Step 3: Recalculate the mean of each clustering, update the centers of all classes. Step 4: Repeat steps 2 and 3, until each clustering center no longer changes. After for Tclustering X, historical data are divided into K groups, which can be expressed as T T T X ¼P , where K is the number of modes. Total number of samples for each mode k X1 ; X2 ; …Xk is N i ¼ N. i¼1
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Clustered data are labeled, according to their modes. The same kinds of samples are the same labels. Here a label matrix is defined as 2 3T 6 7 Index ¼ 41; 1; …1 ; 2; 2; …2 ; …; c; c; …c 5 |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} N1
N2
Nc
2.2. Multimode modeling method in common feature information part Unlike other multimode modeling methods, the proposed method is based on the principles of similarity, which compares the similarities among load vectors in each mode and load vectors of all modes, and extracts common feature information of all modes. Then, the rest parts as the specific feature information are modeled respectively; this content will be introduced in Section 2.3. It is supposed that there is a multimode production process. X1 ; X2 ; …; Xk A RNM are training data of K modes. X ¼ ½XT1 ; XT2 ; …; XTd …; XTk T A RkNM is mixed data of K modes, where Xd ¼ ½xd1 ; xd2 ; …; xdM ; d ¼ 1; 2; …; k. Firstly, X is mapped to F space : T X-ΦðXÞ; ΦðXÞ ¼ ΦðX1 ÞT ; ΦðX2 ÞT ; …ΦðXk ÞT and ΦðXd Þ ¼ ½Φðxd1 Þ; Φðxd2 Þ; …; ΦðxdM Þ; d ¼ 1; 2; …; k: Covariance matrices of each mode and mixed mode data are as follows: 8 N > 1X > > ΦðXdi ÞΦðXdi ÞT ; d ¼ 1; 2; …; k > Cd ¼ > < Ni ¼ 1 kN > 1X > > C¼ ΦðXi ÞΦðXi ÞT > > : kN i ¼ 1
ð1Þ
Looking for principal components in high dimensional space, it needs to calculate eigenvectors of covariance matrix: 8 N > 1X > > C p ¼ λ p ¼ ΦðXdi ÞU pd ΦðXdi ÞT ; d ¼ 1; 2; …; k > d d d d > < Ni ¼ 1 ð2Þ kN > 1X > T > Cp ¼ λp ¼ ðΦðXi Þ UpÞΦðXi Þ > > : kN i ¼ 1 where λ is the eigenvalue, p is eigenvector of covariance matrix C. When λa 0, eigenvector is defined as: 8 N X > > > p ¼ vdj ΦðXj Þ; d ¼ 1; 2; …; k > d > < j¼1 ð3Þ kN X > > > vj ΦðXj Þ p¼ > > : j¼1 Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Kernel function Ki;j ¼ oΦðxi Þ; Φðxj Þ4 is introduced to Eq. (2). Then Eq. (3) is simplified as: Kd vd ¼ Nλd vd ; d ¼ 1; 2; …; k ð4Þ Kv ¼ KNλv The eigenvalues λ1 Z λ2 Z … Z λM are obtained from Eq. (4). The corresponding eigenvectors are v1 ; v2 ; …vM . When Singular values are decomposed, the kernel function will be centralized: ( Kd ¼ Kd IN Kd Kd IN þ IN Kd IN ; d ¼ 1; 2; …; k ð5Þ K ¼ K IkN K KIkN þ IkN KIkN where each element of IN A RNN is equal to 1=N, each element of IkN A RkNkN is equal to 1=kN. From the weighted norm, the following formula is obtained: m m X X ErosðXd ; X; wd Þ ¼ wdi jhvdi ; vi ij ¼ wdi j cos θi j; d ¼ 1; 2; …; k ð6Þ i¼1
i¼1
where w is weighted vector of the eigenvalue of model data Xc and the mixed mode data X, P m i ¼ 1 wdi ¼ 1, the angle between coefficient factors vdi and vi is θ i . The value of Eros is between 0 1. When Eros is 1, the two groups of data are the most similar. In order to calculate conveniently, weighted norm distance is used to calculate the similarity: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v u X m m m X X u DEros ðXd ; X; wd Þ ¼ 2 2 wdi jhvdi ; vi ij ¼ t2 2 wdi vdij vji ; d ¼ 1; 2; …; k ð7Þ i¼1 i¼1 j¼1 Like weighted norm Eros, weighted norm distance DEros reflects the similarity of two sets of data. Contrary to Eros, the smaller DEros is, the greater similarity Xc and X have. Namely, when ErosðXd ; X; wd Þ4ErosðXdþa ; X; wd Þ, then DEros ðXd ; X; wd Þ r DEros ðXdþa ; X; wd Þ. a is a random integer, which is less than k d. The maximum and the minimum weighted norm distances between two data sets can be calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m X X vdij vji ; d ¼ 1; 2; …; k Dmin ðXd ; X; wd Þ ¼ 2 2 wdi i¼1
j¼1
i¼1
j¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m X X vdij vji Dmax ðXd ; X; wd Þ ¼ 2 2 wdi
ð8Þ
When the similarity based on weighted norm distance between the dth model and the mixed data is in the range of Eq. (9), the two sets of data are considered to be similar Fault Diagnosis of Multimode Processes Based on Similarities. Dmin ðXd ; X; wd Þ rDðXd ; X; wd Þr ðDmax ðXd ; X; wd Þ þ Dmin ðXd ; X; wd ÞÞ=2
ð9Þ
According to the scope of similarity, similar part between the dth model and the mixed mode data is obtained. A new coefficient factor is vcd A v, where c represents the common section. Known from Eq. (7), as long as we get the weight vector of weighted norm distance wdi ; i ¼ 1; 2; …; m, the new coefficient factor vcd can be obtained. General steps of calculation of the weight vector are as follows: Step 1: Each row ofPthe eigenvalue matrix is summed up column is λj ¼ λj = M i ¼ 1 λij .
PM
i ¼ 1 λij ,
and the new value of each
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Step 2: Calculate the variance of the new value of each column, as the weight vector wi ¼ f ðλj Þ. P Step 3: Standardize weight vectors wi ¼ wi = M i ¼ 1 wi . Then we get the new coefficient factors vc1 ; vc2 ; …; vcd ; …; vck . The new coefficient factors consider the correlation between modes, but we need to consider the common characteristics with larger effect, so we need to extract the new coefficient factors further. That is to say, common column vectors of new coefficient factors are extracted to construct the public coefficient factors vc ¼ vc1 \ vc2 \ … P \ vcd \ … \ vck between different modes. Then c c common characteristic load matrix p ¼ kN j ¼ 1 vj ΦðXj Þ is obtained. The score matrix of the common part is as follows: tc ¼ ΦðXÞpc ¼ ΦðXÞΦðXÞT vc ¼ Kvc
ð10Þ
At this point, common information part ΦðX Þ of the modes can be obtained as: C
ΦðXC Þ ¼ tc pcT ¼ Kvc vcT ΦðXÞ
ð11Þ
2.3. Multimode modeling method in specific feature information part Mixed mode data are the sum of common information part and specific information part of the data, the mixed mode data can be represented as: ΦðXÞ ¼ ΦðXC Þ þ ΦðXS Þ
ð12Þ
According to kernel principal component analysis, the specific part of the score matrix can be expressed as: ts ¼ ΦðXÞps ¼ ΦðXÞΦðXÞT vs ¼ Kvs
ð13Þ
where ps is load matrix of specific feature information part. vs is coefficient factor of specific feature information part. The data of specific feature information part can be expressed as: ΦðXs Þ ¼ ts psT ¼ Kvs vsT ΦðXÞ
ð14Þ
Because the specific part ΦðXs Þ is a part of data set X, the specific part ΦðXs Þ can be divided into k modes completely according to the K-means cluster label Index. Corresponding score matrices ts of each mode data ΦðXsi Þði ¼ 1; 2; …; kÞ are divided into k groups, ts1 ; ts2 ; …; tsk . ΦðXsk Þ ¼ tsk psT ¼ tsk vsT ΦðXÞ
ð15Þ
Specific part of each model has its particularity and independence, which requires to be monitored respectively. Csk ¼
1 1 s sT ΦðXsk ÞT ΦðXsk Þ ¼ ΦðXÞT vs tsT k tk v ΦðXÞ Nk Nk
ð16Þ
Covariance maximization problem is converted to the problem of solving the eigenvalue and eigenvector: Csk rk ¼ βrk
ð17Þ
where β is the eigenvalue, rk is the corresponding eigenvector. Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Because eigenvector is defined as rk ¼ ΦðXÞT jk and jk is coefficient factor of specific part, Eq. (17) can be simplified as follows: 1 s sT s sT v t t v Kjk ¼ βjk Nk k k
ð18Þ
The coefficient factor jk is obtained by the singular value decomposition to Eq. (18). The corresponding score vector tsk;c can be expressed as: tsk;c ¼ ΦðXsk Þrk;c ¼ tsk vsT Kjk;c
ð19Þ
where rk;c is the 85% contribution part of load vector rk ,jk;c is coefficient factor of load vector, [11,35] K is the centralized kernel matrix, then ΦðXc S Þ is expressed as: ^ sk Þ ¼ ΦðXsk Þrk;c rTk;c ¼ tsk vsT Kjk;c jTk;c ΦðXÞ ΦðX
ð20Þ
The residuals of the specific part is expressed as: Ek ¼ ΦðXsk Þrk;s rTk;s ¼ tsk vsT Kjk;s jTk;s ΦðXÞ
ð21Þ
Then model of specific information part for each mode is expressed as: ^ sk Þ þ Ek ΦðXsk Þ ¼ ΦðX
ð22Þ
To sum up, with multi-mode fault detection method based on the similarity the modeling of common feature information part and specific feature information part have been completed. 3. Multimode process fault detection method based on similarity Common part model of all modes and specific part model of each mode are used for fault detection. A new data xnew A R1M is obtained in the multimode process. Steps of multimode process fault detection method based on similarity are as follows: Step 1: Sample point xnew is standardized, new kernel function ½Knew i;j ¼ oΦðxnew;i Þ; Φðxj Þ4 is constructed,Knew is centralized to Knew . Step 2: Common feature information part is obtained, T 2 statistic of common feature information part is calculated, here T 2 statistic is denoted by CT new 2 : 8 N X > > < tC ¼ opc ; Φðxnew Þ4 ¼ vc Kðxnew ; xi Þ new ð23Þ i¼1 > > : CT 2 ¼ tc Λ 1 tc T new
new
new
Under the condition that the process is normal and the data follow a multivariate normal distribution, the T 2 statistic is related to an F distribution considering that the population mean and covariance are estimated from data [30] NðN lÞ CT 2 F l;N l lðN 2 1Þ
ð24Þ
where F l;N l is an F distribution with l and N l degrees of freedom where l equals to the number of eigenvalues obtained before and N equals to the number of variables. For a given Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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significance level α the process is considered normal if CT 2new r CT α 2 ¼
lðN 2 1Þ F l;N l;α NðN lÞ
ð25Þ
If the calculated statistic overruns the control limit, the common feature information part is considered to be abnormal. Otherwise, monitoring process is considered under the normal condition. Step 3: Specific feature information part is obtained, T 2 and SPE statistics of specific feature information part are calculated, here T 2 and SPE statistics are denoted by ST 2new and SSPE new : 8 2 ST ¼ tsnew;k β 1 tsnew;k T > > < new;k N R X X ð26Þ > ts2 ts2 SSPE new;k ¼ ‖enew;k ‖2 ¼ > new;i new;i : i¼1
i¼1
The process is considered normal if SSPEnew;k r δ2α
ð27Þ
where δ2α denotes the upper control limit for SPE with a significance level α. It is calculated as follows: 0 qffiffiffiffiffiffiffiffiffiffiffiffi 11=h0 cα 2θ2 h20 θ h ðh 1Þ 2 0 0 A δ2α ¼ θ1 @ þ1þ θ1 θ21
ð28Þ
Fig. 1. Schematic diagram of the EFMF with multimode manufacture process. Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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where θi ¼
M X
λij ; i ¼ 1; 2; 3
ð29Þ
j ¼ lþ1
h0 ¼ 1
2θ1 θ3 3θ22
ð30Þ
l is the number of retained principal components and cα is the normal deviate corresponding to the upper 1 α percentile. Note that this result is derived under the conditions that sample vector follows a multivariate normal distribution [31]. If one of the calculated statistics overruns the control limit, the specific feature information part is considered to be abnormal. Otherwise, monitoring process is considered under the normal condition. 4. Experiment results The electro-fused magnesia furnace (EFMF) is one of the main equipments used to produce electro-fused magnesia, which belongs to a kind of mine hot electric arc furnaces. With the development of technology of melting, EFMF has already got extensively application in the industry. EFMF refining technology can enhance the quality and increase the production variety. In the melting process, leakage of furnace is a kind of power quality problems, which occurs in the power system when the gas aggregates quickly in the magnesia. The Schematic diagram is shown in Fig. 1. There are mainly two different modes for EFMF. The differences between two modes include materials used and products manufactured. In mode A, the materials are powdery magnesium, and the product is fused magnesia crystals, in this mode, the temperature is higher and need higher current correspondingly. In mode B, the materials are magnesium blocks, and the product is heavy-burned magnesia which need lower current and generate lower temperature. Since different material has different resistance, pressure in the EFMF for two modes is also different, values of variables generated from the two modes has not only similarity but also specific characteristics respectively. So, Electro-fused magnesia furnace production process is used for fault detection to verify the effectiveness of the proposed method based on similarity. Temperature is a very important variable in the process of EFMF. It is determined by the current value of EFMF. In order to verify the effectiveness of multimode fault detection method based on similarity, 9 variable values are selected at runtime of EFMF, including the measured values of three electrode currents and six temperatures of EFMF. In our experiment, there are totally 200,000 sample data collected from the factory during one week. Through the K-means clustering analysis method, the sample data are divided into two modes, including 113,212 sample data in mode A and 86,788 sample data in mode B. The whole data set are labeled by experts as normal or abnormal, then 300 normal data for each mode are selected randomly as training set, and 400 data including abnormal data for each mode are selected as testing set. Fault 1 is the abnormal of the first electrode current. The advantage of the proposed similarity decomposition based statistical monitoring method is that, similar component of different mode is extracted forming the common part. When operating mode varies dynamically yet cannot be known online immediately, monitoring on common part is more stable which means there will be less False Positives or False Negatives comparing to Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Fig. 2. T 2 statistic of normal test data from mode A with model developed on the common part.
Fig. 3. T 2 statistic of normal test data from mode A with model developed on the specific part.
monitoring data with certain model develop on one mode. So in this part, we design two experiments using data without fault and data with fault respectively. Model on common part, mode A, and mode B are established for monitoring respectively as a comparison. First of all, the 400 test samples of mode A which are all normal data are used to test the effectiveness of multimode fault detection method based on similarity decomposition. Models on the common part, specific part, and Mode B are established respectively and the corresponding control limits are also developed. Here, according to the criterion mentioned forward, the p-value which is the number of principal components is set 3. Fig. 2 is the T 2 statistic of test data in the
Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Fig. 4. T 2 and SPE statistics of normal test data from mode A with model developed in mode B.
Fig. 5. T 2 statistic of test data including fault 1 from mode B with model developed on the common part.
common part. It can be seen from the chart that statistics are not beyond the control limits. Fig. 3 (a) and (b) are the statistics of test data in specific part. It can be seen from the chart that T 2 and SPE statistics do not overrun the control limits. Fig. 4(a) and (b) are the statistics with model developed in mode B. It can be seen from the chart that T 2 and SPE statistics overrun the control limits seriously. It is considered that the test data does not belong to mode B, but belongs to the normal data of model A. Actually the EFMF indeed working in mode A. In order to further verify fault detection performance of the proposed multimode fault detection method based on similarity, a set of 400 samples in mode B containing fault data is tested, fault 1 caused by actuator failure starts from the 251th sample, resulting in abnormal current value. Similar with the following experiment, models on the common part, specific part, Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Fig. 6. Statistic of test data including fault 1 from mode B with model developed on the specific part.
and Mode B are established respectively and the corresponding control limits are also developed. With the same way, the p-value is also calculated and set 3. Fig. 5 is the simulation results of the proposed multimode fault detection method based on similarity. It can be seen from the charts that fault starts from the 251th sample. The fault is detected not only in the common part but also with the model developed in mode B as shown in Fig. 7(a) and (b). To the contrast Fig. 6(a) and (b) are the statistics on specific part. It can be seen obviously that fault 1 cannot be detected correctly. Generally speaking, two experiment results demonstrate that, monitoring on the common part obtained with the proposed similarity based decomposition method is more stable. When mode changes dynamically, fault can be detected with the same model effectively. As a comparison, another novel multimode monitoring approach based on Between-Mode Transition which has the same industrial background with our paper was cited in this experiment. In this cited article, the manifold is learned to extract the common part of between-mode transition [36]. Normal data from mode A and data with fault 1 from mode B are both detected with our proposed method and the cited method. The effectiveness of different approaches is evaluated via using FAR and MAR defined as follows: False Positive (FP) donates those normal samples alarmed as fault samples. True Negative (TN) donates those normal samples not alarmed as fault samples. True Positive (TP) donates those abnormal samples alarmed as fault samples. False Negative (FN) donates those abnormal samples not alarmed as fault samples.
Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Fig. 7. T 2 and SPE statistics of test data including fault 1 from mode B with model developed in mode B.
Table 1 MAR and FAR for the two approaches. Data type
Approach
FAR(%)
MAR(%)
Normal
Between-mode Transition based(referenced) similarity based(ours)
1.34 1.54
– –
Fault 1
Between-mode Transition based(referenced) similarity based(ours)
2.21 2.50
6.38 0.92
Then, we define False Alarmed Rate FAR ¼ FP/ (FPþTN); Missing Alarm Rate MAR ¼ FN/ (TPþFN). Knowing from the above definition, approaches with lower MAR are more sensitive in fault detection, correspondingly, those approaches with lower FAR are more stable in fault detection. Average value of FAR and MAR are calculated with ten times of repeated experiments, each time we choose 300 normal samples for training randomly. Experiment results with MAR and FAR are shown in Table 1. As shown in Table 1, when using the normal data from mode A, FARs for both methods are relatively low, this means the two methods are stable and few normal data are false alarmed as fault. However, when using the data with fault 1from mode B, although FARs of the two methods are still very close, MAR of our proposed method is quite low, which means hardly any Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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faults are missed. This result demonstrates monitoring on common part obtained by similarity decomposition can extract the common feature between different modes more effectively, and are more sensitive when detecting fault 1. 5. Conclusions In this paper, a multimode process fault detection method based on the similarity is proposed. This method can effectively extract common information part and specific information part of the multimode data for fault detection. K-means method is used for data clustering, mode classification of history data, and calculation of load matrices of each mode data and load matrices of mixed modes data. According to the comparative value, common information part is determined, and the rest is the specific information part. This method not only studies the correlation among modes and obtains the common information part, but also analyzes the local data of each mode, maintain the local characteristics, and avoid direct analysis of mixed data which causes the local information loss. In addition, this method can also be applied to the nonlinear fault detection problem. In order to prove the effectiveness and accuracy of the method, the feasibility and the superiority of the proposed method is proved in the simulation of EFMF production process. Acknowledgement This work is supported by China's National 973 program (2009CB320600 and 2009CB320604) and NSF in China (61325015 and 61273163). References [1] C. Alcala, S.J. Qin, Reconstruction-based contribution for process monitoring, Automatica 45 (7) (2009) 1593–1600. [2] S. Lane, E.B. Martin, R.K. Kooijmans, Performance monitoring of a multi-product semi-batch process, J. Process Control 11 (1) (2001) 1–11. [3] S.J. Qin, Statistical process monitoring: basics and beyond, J. Chemom. 17 (8-9) (2003) 480–502. [4] C. Shahabi, D. Yan, Real-time pattern isolation and recognition over immersive sensor data streams, in: Proceedings of the 9th International Conference on Multi-Media Modeling, pp. 93–113, 2003. [5] S.J. Qin, S. Valle, M.J. Piovoso, On unifying multiblock analysis with application to decentralized process monitoring, J. Chemom. 15 (9) (2001) 715–742. [6] C.K. Yoo, J.M. Lee, P.A. Vanrolleghem, I.B. Lee, On-line monitoring of batch processes using multiway independent component analysis, Chemom. Intell. Lab. Syst. 71 (2) (2004) 151–163. [7] P. Nomikos, J.F. MacGregor, Monitoring batch processes using multiway principal component analysis, AIChE J. 40 (8) (1994) 1361–1375. [8] D. Zhou, G. Li, S.J. Qin, Total projection to latent structures for process monitoring, AIChE J. 56 (1) (2010) 168–178. [9] U. Thissen, H. Swierenga, P.D. Weijer, Multivariate statistical process control using mixture modeling, J. Chemom. 19 (2) (2005) 23–31. [10] S.J. Qin, Y. Zheng, Quality-relevant and process-relevant fault monitoring with concurrent projection to latent structures, AIChE J. 59 (2) (2013) 496–504. [11] T. Chen, Y. Sun, Probabilistic contribution analysis for statistical process monitoring: a missing variable approach, Control Eng. Pract. 17 (4) (2009) 469–477. [12] Y. Zhang, J. An, C. Ma, Fault detection of non-gaussian process based on model migration, IEEE Trans. Control Syst. Technol. 21 (5) (2013) 1517–1526. [13] G. Birol, C. Undey, A. Cinar, A modular simulation package for feed-batch fermentation: penicillin production, Comput. Chem. Eng. 26 (11) (2002) 1553–1565. Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002
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Please cite this article as: W. Du, et al., Multimode process monitoring based on data-driven method, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j.jfranklin.2016.11.002