Elsevier

Control Engineering Practice

Volume 22, January 2014, Pages 205-216
Control Engineering Practice

Online monitoring of nonlinear multivariate industrial processes using filtering KICA–PCA

https://doi.org/10.1016/j.conengprac.2013.06.017Get rights and content

Highlights

  • A novel filtering KICA–PCA (FKICA–PCA) is proposed.

  • Gaussian and non-Gaussian features are made to be comparable by using FKICA-PCA.

  • A novel contribution analysis scheme is developed for FKICA-PCA to diagnose faults.

  • The feasibility and effectiveness of FKICA–PCA have been validated on the TE process.

Abstract

In this paper, a novel approach for processes monitoring, termed as filtering kernel independent component analysis–principal component analysis (FKICA–PCA), is developed. In FKICA–PCA, first, a method to calculate the variance of independent component is proposed, which is significant to make Gaussian features and non-Gaussian features comparable and to select dominant components legitimately; second, Genetic Algorithm is used to determine the kernel parameter through minimizing false alarm rate and maximizing detection rate; furthermore, exponentially weighted moving average (EWMA) scheme is used to filter the monitoring indices of KICA–PCA to improve monitoring performance. In addition, a novel contribution analysis scheme is developed for FKICA–PCA to diagnosis faults. The feasibility and effectiveness of the proposed method are validated on the Tennessee Eastman (TE) process.

Introduction

Multivariate statistical process control and fault diagnosis (MSPC&FD), which can extract a lot of useful and important information from large numbers of process data, have achieved considerable developments in multivariate industrial processes (Li et al., 2011, MacGregor and Kourti, 1995, Muradore and Fiorini, 2012, Nomikos and MacGregor, 1995, Qin, 2003, Yu and Qin, 2009; Zhao et al., 2007, Zhou et al., 2011). For MSPC&FD, the data under normal process condition are first utilized to establish a statistical model; second, the model is used to monitor the process and detect fault online; finally, once a fault is detected, fault isolation and identification should be implemented to estimate the fault. Originally, a few MSPC charts, such as the multivariate Shewhart chart, the multivariate cumulative sum (MCUSUM) chart, and multivariate exponentially weighted moving average (MEWMA) chart, are used to monitor processes and improve product quality (Nomikos & MacGregor, 1995). Multivariate latent variable methods, such as principal component analysis (PCA) and partial least squares (PLS), are widely used in process monitoring and fault diagnosis (MacGregor & Kourti, 1995). PCA is probably the best known multivariate monitoring method, of which the main idea is to replace a large number of interrelated variables by a few uncorrelated variables (Wold, Esbensen, & Geladi, 1987). PCA has been successfully used in continuous processes and batch processes (Ku et al., 1995, Nomikos and MacGregor, 1994, Yao and Gao, 2007).

Nevertheless, PCA extracts only uncorrelated components because it is a second-order method. In order to compute the control limits of PCA, it is assumed that all process variables follow multivariate Gaussian distribution. Hence, PCA was challenged by non-Gaussian processes which could not be described completely by the first two moments. Independent component analysis (ICA) was proposed to cope with non-Gaussian characteristics. ICA depends on higher order statistics and extracts independent components (ICs), and while PCA gives just uncorrelated components. It should be pointed out that uncorrelated variables are only partly independent (Comon, 1994, Hyvärinen and Oja, 2000). In order to obtain ICs, it is required to maximize non-Gaussianity, minimize mutual information, or maximum likelihood estimation for ICA (Hyvärinen & Oja, 2000). More suitable than PCA, ICA was successfully applied in non-Gaussian processes (Lee et al., 2004, Lee et al., 2006, Wang et al., 2012). However, because all ICs calculated by the ICA algorithm have unit variances, they cannot be sorted according to variance and further to reduce the number of ICs. Some researchers proposed to sort ICs according to non-Gaussianity, 2-norm of columns/rows of mixing/de-mixing matrix, or variance of PCA score, but those methods have no clear physical meaning. On the contrary, in PCA, PCs can be sorted according to variance and further reduced by many criterions. Those criterions include AIC, MDL, imbedded error function (IEF), cumulative percent variance (CPV), Scree test on residual percent variance (RPV), average eigenvalue (AE), parallel analysis (PA), autocorrelation, cross validation on R Ratio or PRESS (CV), and variance of the reconstruction error criterion (VRE). In Valle, Li, and Qin (1999), the above methods were summarized and compared, and the results demonstrate that CPV, AE, PA, PRESS, and VRE are the most reliable methods. Recently, there are few studies on this problem. When ICs have different variances, criterions such as CPV, AE, PA, PRESS and VRE could be used to reduce the number of ICs.

For many significantly nonlinear processes, neural-network-related PCA (NN-PCA) methods were further developed (Chen and Liao, 2002, Dong and McAvoy, 1996). However, NN-PCA methods are generally time-consuming in training the network and the number of PCs has to be determined before training. Therefore, kernel method was proposed to overcome these disadvantages of NN-PCA. The main idea of the kernel method is to transform the original nonlinear space into a kernel linear space of higher dimension through implementing a nonlinear kernel map. Then, linear methods such as PCA and ICA can be applied to the higher-dimensional linear space. KPCA (Alcala and Qin, 2010, Ge et al., 2009, Lee et al., 2004) and KICA(Lee et al., 2007, Zhang, 2008) have been applied to nonlinear process successfully. However, in KICA, because the mixing/de-mixing matrix is unknown, ICs can only be sorted according to non-Gaussianity or variance of KPCA score. Hence, reducing ICs for KICA is more difficult compared with ICA.

In many processes, Gaussian features and non-Gaussian features exist simultaneously. Obviously, a two-step method is an intuitive option, which uses ICA-related method to extract ICs first and then using PCA-related method to extract PCs from the residual error of the ICA model. Ge and Song (Ge & Song, 2007) applied ICA–PCA to continuous processes and demonstrated this two-step strategy is more effective than any one-step strategy (PCA or ICA). Zhao, Gao, and Wang (2009) applied KICA–PCA on nonlinear batch processes. However, a basic assumption for using ICA–PCA or KICA–PCA is that the process contains both Gaussian features and non-Gaussian features, where these two kinds of features are not comparable. In the existing two-step strategy, dominant ICs and dominant PCs are selected separately and it is difficult to judge whether one dominant IC is more important than one dominant PC or not. Hence, if either the Gaussian features or the non-Gaussian features are all noises, the existing ICA–PCA and KICA–PCA schemes are useless. This issue will be solved if a standard could be developed to compare ICs and PCs fairly and then sort them all together.

A common drawback of KPCA, KICA, and KICA–PCA is that selecting appropriate kernel function and determining its parameters are difficult. Some researchers suggest using experience or trial and error, which are quite subjective and may not give best choices. Jia, Xu, Liu, and Wang (2012) proposed to use Genetic Algorithm to optimize kernel function and its parameters for KPCA. In that study, the fitness function is based on the correct monitoring rate (CMR), the number of PCs, and the control limit of SPE, but the CMR is only calculated based on T2. In KICA–PCA, the best fitness function should be related to the monitoring abilities of I2, T2, and SPE, where these three indices should have appropriate weighting factors.

After a fault is detected, fault diagnosis is needed. For linear processes, variables' contribution plot for PCA or ICA is usually used to diagnose faults. Contribution plot does not need priori fault knowledge and it can be implemented online conveniently. Alcala and Qin (2011) made some analysis and generalization of contribution plot and pointed out that a well-defined contribution analysis scheme should have the following desirable properties: first, when no fault occurs, all variable contributions should have statistically the same mean, which can establish a level ground to compare the contributions when there is a fault; second, when a fault is mainly attributed to one variable, the contribution of that variable should be the largest. What's more, Alcala and Qin proposed a new form of relative contributions. However, for nonlinear processes, it is difficult to calculate variables' contribution plots for KPCA, KICA, and KICA–PCA, due to the kernel mapping. Cho, Lee, Choi, Lee, and Lee (2005) developed a nonlinear contribution analysis scheme for KPCA; however, there was no level ground to compare these contributions and the calculation is time consuming.

Another drawback of KICA–PCA is that the monitoring is only based on the information at current moment and the historical information is ignored. What's more, the monitoring indices always fluctuate up and down around the control limits after a fault occurs, i.e., the monitoring results are not consistent, especially for small faults.

Based on the above analysis, in this paper, a novel approach, termed as filtering kernel independent component analysis–principal component analysis (FKICA–PCA), is developed. This study has the following main contributions. First, a method to calculate the variance of independent component (VIC) is proposed, which is significant to make Gaussian features and non-Gaussian features comparable and select dominant components legitimately. Second, Genetic Algorithm is used to determine the kernel parameter through minimizing false alarm rate and maximizing detection rate of the three indices weighted by their corresponding energy. Third, exponentially weighted moving average (EWMA) is used to filter the monitoring indices of KICA–PCA to further improve monitoring performance. Finally, a novel contribution analysis scheme is developed for FKICA–PCA to diagnosis faults. The feasibility and effectiveness of the proposed approach are validated on the Tennessee Eastman (TE) process.

The rest of this paper is organized as follows. Section 2 is some introductions and discussions about KICA–PCA. Section 3 is the detailed illustration of the proposed approach, which includes VIC method, optimizing of kernel parameter, filtered KICA–PCA, and the nonlinear variable contribution analysis. Section 4 is the case study on the TE process. Section 5 presents some conclusions.

Section snippets

Data whitening using kernel mapping

Given that a data matrix X(m×n) is obtained from a nonlinear process, where m is the number of variables, n is the number of observations. In order to implement linear methods such as ICA and PCA, linear relationships can be obtained in a space of higher dimensionality by using a nonlinear mapping Φ(). First, X should be normalized to have zero mean value and unit variance; then,Φ=Φ(X),Φ():Rm×nF(featurespaceofhigherdimensionality)The covariance matrix in F is given bySF=1nΦΦT

Actually, Φ()

Calculate the variance of independent component

As discussed in Introduction section and Section 2.2, all ICs given by ICA algorithm have unit variances, and hence, ICs cannot be sorted by variance and further to reduce the dimension. Variance indicates the energy of a feature or signal. In process monitoring, sorting all features by variance and remove the features with smaller variance is significant. For example, in PCA, all criterions for feature selection such as CPV, AE, PRESS and VRE, are based on the sorting with PC's variance. In

A numerical example of nonlinear mixture of signals

In order to test the VIC method developed in Section 3.1, four independent signals are mixed thorough a nonlinear form. To be specific, t=0:1:499, s1=square(0.2×t), s2=sin(0.2×t), s2 are 500 samples from a uniform distribution in the interval [1,1], and s3 are 500 samples from a standard normal distribution. The variances of these four signals are redefined by s1=1×s1/std(s1), s2=0.8×s2/std(s2), s3=0.6×s3/std(s3), and s4=0.4×s4/std(s4). Then, their variance are 1, 0.64, 0.36, and 0.16

Conclusions

In this study, a novel monitoring approach, FKICA–PCA is developed. In FKICA–PCA, first, a method to calculate the variance of independent component (VIC) is developed, which makes Gaussian feature and non-Gaussian feature comparable. This VIC method can also be used for ICA, ICA–PCA, and KICA. Second, genetic algorithm is used to optimizing the kernel parameter by minimizing false alarm rate and maximizing detection rate. Third, EWMA is used to filter the three monitoring indices to get a

Acknowledgment

This work was supported by National Natural Science Foundation of China (61074081), the Fundamental Research Funds for the Central Universities (RC1101), Doctoral Fund of Ministry of Education of China (20100010120011), Beijing Nova Program (2011025), and the Fok Ying-Tong Education Foundation (131060).

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