Concurrent Canonical Correlation Analysis Modeling for Quality-Relevant Monitoring

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1 Preprint, 11th IFAC Symposium on Dynamics and Control of Process Systems, including Biosystems June 6-8, 16. NTNU, Trondheim, Norway Concurrent Canonical Correlation Analysis Modeling for Quality-Relevant Monitoring Qinqin Zhu Qiang Liu, S. Joe Qin Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, CA 989, USA ( State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, Liaoning 11819, PR China ( School of Science and Engineering, the Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong 51817, China ( Abstract: Canonical correlation analysis (CCA) is a well-known data analysis technique that extracts multidimensional correlation structure between two groups of variables. Due to the advantages of CCA on quality prediction, CCA-based modeling and monitoring are discussed in this paper. To overcome the shortcoming of CCA that focuses on correlation but ignores variance information, a new concurrent CCA (CCCA) modeling method is proposed to completely decompose the input and output spaces into five subspaces, to retain the CCA efficiency in predicting the output while exploiting the variance structure for process monitoring using subsequent principal component decomposition in the input and output spaces, respectively. The corresponding monitoring statistics and control limits are then developed in these subspaces. The Tennessee Eastman process is used to demonstrate the effectiveness of CCCA-based monitoring methods. Keywords: Concurrent Canonical Correlation Analysis (CCCA), Quality-Relevant Monitoring 1. INTRODUCTION Multivariate statistical process monitoring based on process variables and quality variables has been widely used in industrial processes, including chemicals, polymers, microelectronics manufacturing and pharmaceutical processes (Nomikos and MacGregor (1995); Wise and Gallagher (1996); Qin (3); Chiang et al. ()). Among them, principal component analysis (PCA), partial least squares (PLS) and canonical correlation analysis (CCA) are three basic multivariate statistical methods. When the quality measurements are expensive or difficult to obtain, PCA is widely used for process monitoring on account of its ability to handle high-dimensional, noisy, and highly correlated data by projecting the data onto a lower-dimensional subspace that contains most of the variance of the original data (Wise and Gallagher (1996)). However, PCA-based monitoring methods are only effective for monitoring variations in process variables (X), and no information from the quality variables (Y) is extracted. In practical industrial processes, the variations of process variables may be compensated by feedback controllers, which will have no influence on the quality variables. Thus, monitoring only This work was supported in part by the Natural Science Foundation of China under Grant , Grant , Grant 61573, the China Postdoctoral Science Foundation Grant 13M5414, the International Postdoctoral Exchange Fellowship Program under Grant 13, and the Fundamental Research Funds for the Central Universities (N13181). on the process variables will increase the false alarm rates, making the models unreliable. In order to include the information of the quality variables, PLS-based and CCA-based modeling methods are employed to find the covariances or correlations between process variables and quality variables (MacGregor et al. (1994); Khan et al. (8)). PLS is a data decomposition method for maximizing covariances between X and Y, and it has been studied intensively. Li et al. (1) studied the effect of quality variables on the X-space decomposition and the geometric properties of the PLS structure. In standard PLS methods, the original space is decomposed into two subspaces, which are monitored by T and Q statistics (Qin (3)), respectively. Although it works well in some cases, there are several problems involved in this monitoring scheme. Firstly, PLS components that form T may contain variations orthogonal to Y, which are useless to predict Y. Moreover, PLS extracts principal components based on a maximum covariance criteria, and the residuals of X are not necessarily small. Thus, it is inappropriate to monitor the residual space with the Q statistic. In order to improve the monitoring performance, Zhou et al. (1) proposed a total PLS (T-PLS), which performs further decomposition and divides X-space into four parts. The T-PLS-based monitoring, however, decomposes the X-space unnecessarily into four parts, which can be concisely divided into input-relevant and output-relevant Copyright 16 IFAC 144

2 IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondheim, Norway parts instead. Qin and Zheng (13) put forward a concurrent PLS (CPLS) to overcome the drawbacks of T-PLS and monitor input-relevant and output-relevant faults separately. CPLS reduces the false alarm rates and imporves the monitoring performance. PLS related methods are variants of affine transformation of the process and quality variables, and are efficient for quality prediction since they simultaneously exploit the input structure while predicting the output. CCA, in contrast, focuses only on extracting the multidimensional correlation structure between X and Y, which enables it to build a more efficient prediction. CCA is widely used in signal processing, computer vision and behavioral studies (Hardoon et al. (4); Sherry and Henson (5)). CCAbased monitoring, however, has not been studied due to its lack of attention to variance structure in the data. In this paper, we discuss the advantages of CCA over PLS on quality-relevant prediction and the advantages of PLS over CCA for variable structure modeling and monitoring. We then propose a concurrent canonical correlation analysis (CCCA) algorithm for quality-relevant fault detection. CCCA decomposes the original data space into five subspaces: 1) correlation subspace (CRS), which is formed by the canonical variates that are directly relevant to the predictable variations of the output; ) output-principal subspace (OPS), the principal subspace of the unpredictable part of the output; 3) output-residual subspace (ORS), the residual subspace of the unpredictable part of the output; 4) input-principal subspace (IPS), the principal subspace of the input that is irrelevant to the prediction of the output; and 5) input-residual subspace (IRS), the residual subspace of the input which is useless to predict the output. The corresponding fault indices and control limits for the five subspaces are also developed. Moreover, some significant properties of CCCA are analyzed. The CCCA modeling and monitoring approach is inspired by the CPLS approach, while the efficiency in input-output prediction of CCA is incorporated in CCCA modeling. The remaining work of this paper is organized as follows. Fault detection based on CCA model is first presented in Section. A comparison between CCA and PLS will be demonstrated as well. In Section 3, the concurrent CCA modeling algorithm and the corresponding fault monitoring scheme are developed. The Tennessee Eastman process is employed to illustrate the effectiveness of CCCA against PLS, CCA and CPLS in Section 4. Finally, conclusion is presented in the last section.. CCA FOR QUALITY-RELEVANT MONITORING.1 CCA Modeling Hotelling (1936) studied the extension of PCA to extract the multidimensional correlation structure between two sets of variables, and developed CCA. Given input matrix X R n m consisting of n samples with m process variables, and output matrix Y R n p with p quality variables, CCA projects X and Y to a lower-dimensional space defined by a small number of latent variables (t 1,..., t A ), where A is the number of CCA factors. The mean-centered X and Y are decomposed as: { X = TP + E Y = TQ + F where T = [t 1,..., t A ] are the sets of canonical variates for X, P = [p 1,..., p A ] and Q = [q 1,..., q A ] are the loading vectors for X and Y, respectively. A brief algorithm for CCA is given in Appendix A. In this algorithm, the canonical variates T and U are calculated through weighting matrices R and C, respectively. { T = XR () U = YC (1) where U = [u 1,..., u A ] are the sets of canonical variates for Y, R = [r 1,..., r A ] and C = [c 1,..., c A ].. Comparison of CCA and PLS on Process Modeling Both CCA and PLS are used to extract the relation between process and quality variables. However, CCA maximizes the correlation between a linear combination of process and quality variables; while PLS aims to extract the covariance in the two sets of variables. PLS has been widely studied and used in multivariate statistical quality control. The objective function of PLS is max J = t t PLS,u PLS PLSu PLS (3) where t PLS and u PLS are score vectors for X and Y in PLS model. From the above equation, we can see that the scaling of the variables will affect the solutions of PLS, since it is based on a maximum covariance criteria. For the principal components, apart from extracting the ones that reflect the relation between input and output variables, PLS also considers the effects of the input structures. Thus, PLS exploits the variable structure well. In practical industrial processes, the input and output data are sampled from a set of different sensors, and the variance of the values from a given sensor may be affected by noise and is unrelated to the importance of the received information. There may be one pair of directions in the two spaces that has a high covariance due to high variable magnitudes but has a high noise level, while another pair of directions has an almost perfect correlation but a small variable magnitude and therefore low covariance. Therefore, PLS is not appropriate for quality prediction and monitoring in these cases. In contrast, CCA tries to maximize the correlation between X and Y, t CCA u CCA max J = (4) t CCA,u CCA t CCA u CCA where t CCA and u CCA are score vectors for X and Y in CCA model. Correlation between process and quality variables is invariant to the variable magnitude (Borga et al. (1997)). Therefore, the magnitudes of the variances of the values from sensors have no impact on the result of CCA, and the components extracted by CCA can build a more efficient prediction..3 CCA-based Monitoring To perform process monitoring on a new data sample x and y, CCA projects them on input data space as x = ˆx+ 145

3 IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondheim, Norway x, where ˆx = PR x Span{P} x = ( I PR ) x Span{R} Similar to PCA and standard PLS (Qin (3)), T and Q statistics are used to monitor the variations in principal and residual subspaces for CCA. Under the assumption that latent vectors are normally distributed with zero means, the monitoring statistics and control limits of T and Q are defined as: T = t Λ 1 t A(n 1) n(n 1) F A,n A,α (5) Q = x gχ h (6) where t = R x, Λ = 1 n 1 diag{λ 1, λ,..., λ A }. F A,n A,α is F-distribution with A and n A degrees of freedom, and the confidence level is defined by α by (1 α) 1%. gχ h is the χ -distribution with scaling factor g and h degrees of freedom. The calculations of g and h can be found in MacGregor et al. (1994). Then if T exceeds the control limit, a quality-relevant fault is detected with (1 α) 1% confidence. If Q exceeds its control limit, a fault in the residual space will be detected with (1 α) 1% confidence. This monitoring scheme works well in some cases. However, it suffers from the same problems of PLS, which are the orthogonal components in the principal component subspace and the potential large variances in the residual subspace. 3. CCCA FOR PROCESS MONITORING 3.1 Concurrent Canonical Correlation Analysis Inspired by the improvement from PLS to CPLS on fault monitoring in the work of Qin and Zheng (13), CCCAbased monitoring method is proposed. To realize a complete monitoring scheme of the quality variables and exclude unrelated variables, CCCA decomposes the data into five spaces: CRS, responsible for the predictable output, IPS, monitoring input relevant faults, IRS, monitoring potentially output-relevant faults, OPS and ORS, monitoring abnormal variations that are output-relevant and unpredictable from input. The CCCA algorithm for data with multiple input and multiple output is given in Table 1. After performing CCCA, the matrices X and Y can be decomposed as: X = T c R c + T x P x + X (7) Y = T c Q c + T y P y + Ỹ (8) where the loadings R c R m Ac, P x R m Ax, Q c R p Ac and P y R p Ay characterize the CCCA model, and the scores T c R n Ac, T x R n Ax and T y R n Ay represent the correlations in X that are related to predictable part of Y, variations in X that are useless to predict Y, and variations in Y that are unpredictable from X, respectively. Given a new data sample x new and y new, CCCA projects them as follows: Table 1. The Concurrent CCA Algorithm 1. Scale X and Y to zero-mean. Perform CCA on input matrix X and output matrix Y to give R, T, Q, and P (Appendix A).. Perform singular value decomposition (SVD) on the predictable output Ŷ = TQ, Ŷ = U c D c Vc = T c Q c where T c = U c is comprised by left singular vectors, and Q c = V c D c contains nonzero singular values in descending order and the corresponding right singular vectors. The number of components is denoted as A 1 c. 3. Obtain the unpredictable output Ỹc = Y T c Q c, and perform PCA with A y principle components, Ỹ c = T y P y + Ỹ This gives the output-principal scores T y and outputresiduals Ỹ. 4. Obtain the output-irrelevant input by projecting on the orthogonal complement of Span R c, Xc = X T c R c, where R c = RQ V c D 1 c, and R c = ( ) R 1 c R c R c, and perform PCA with A x principal components, X c = T x P x + X to yield the input-principal scores T x and inputresiduals X. 1 Different from CPLS in Qin and Zheng (13), A c in CCCA does not equal to the number of all non-zero singular values. Instead, A c is determined by the first A c singular values, whose sum exceeds the 95% of sum of all the singular values for the first time. x new = ˆx c + x c = R c t c + P x t x + x (9) y new = ŷ c + ỹ c = Q c t c + P y t y + ỹ (1) where ˆx c = R c t c, x c = x new R c t c = P x t x + x (11) ŷ c = Q c t c, ỹ c = y new Q c t c = P y t y + ỹ (1) and t c = R c x new (13) t x = P x x c (14) t y = P y ỹc (15) For the residual part x and ỹ, we have x = x c P x t x = ( I P x P ) x xc (16) ỹ = ỹ c P y t y = ( I P y P ) y ỹc (17) 3. Properties of CCCA Several important properties are hold for the CCCA algorithm, and they are convenient to calculate the monitoring statistics in Table, which will be developed in the next subsection. Lemma 1. X c R c =, P x R c = and XR c =. Proof: From the step and 4 of CCCA algorithm in Table 1, we have T c = U c = ŶcV c D 1 c X c = X T c R c = XR c 146

4 IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondheim, Norway Thus, X c R c = (X T c R c)r c =. Also, due to X c = T x P x + X and X = (I P x P x ) X c, we have X c R c = T x P x R c + XR c = T x P x R c + (I P x P x ) X c R c = T x P x R c = which shows that XR c =. Finally, T x T x = (n 1)Λ x = diag{λ x1,..., λ xax } and λ xi is the variance of principal component, then T x T x P x R c = (n 1)Λ x P x R c = Therefore, P x R c =. Lemma. t x = P x x new, and x = (I P x P x )x new. Proof: From Lemma 1, we have t x = P x x c = P x (x new R c t c ) = P x x new Similar for x, = P x x new P x R c ( R c R c ) 1 tc x = (I P x P x ) x c = (I P x P x )(x new R c t c ) = (I P x P x )x new (I P x P x )R c ( R c R c ) 1 tc = (I P x P x )x new The last equivalence is derived from XR c =. 3.3 CCCA-based Process Monitoring CCCA-based monitoring scheme works in a similar way as CCA. First, we build a CCCA model from normal data sets X and Y. Then for a new sample, all scores and residuals are calculated through Eq Eq. 17. Finally, several control plots are constructed with corresponding control limits, which are used for fault detection. In multivariate statistical process monitoring, T and Q statistics are widely used for monitoring systematic part and residual part of the process variations (Zhou et al. (1)), respectively. In CCCA, the variations in the CRS, IPS and OPS subspaces contain systematic part, and it is suitable to use T statistic. The residual part in IRS and ORS subspaces represents the residual variations, which should be monitored with Q index instead (Jackson and Mudholkar (1979)). For a sample x and y, the T statistic for the predictable part ŷ can be calculated by Tc = t c Λ 1 c t c = x R c Λ 1 c R c x (18) where Λ c = diag{λ c1, λ c,..., λ cac } and λ ci denotes the variances of the principal components. The input-relevant scores in Eq. 14 and residuals in Eq. 16 can be monitored like PCA by T and Q as follow, Tx = t x Λ 1 x t x = x c P x Λ 1 x P x x c (19) Q x = x x = x ( c I Px P ) x xc () where ( I P x P x ) = ( I Px P x ), and Λx = 1 n 1 T x T x. Table. Monitoring Statistics and Control Limits Statistics Calculation Control Limit Tc t c Λ 1 c t c A c(n 1) n(n A F c) A c,n A c,α Tx t x Λ 1 x t x A x(n 1) n(n A F x) A x,n A x,α Ty t y Λ 1 y t y A y(n 1) n(n A F y) A y,n A y,α Q x x x g x χ h x,α Q y ỹ ỹ g y χ h y,α n, number of training samples; A c, number of principal components in the CRS subspace; A x, number of principal components in IPS space; A y, number of principal components in OPS subspace; the calculation of g x, h x, g y and h y can be found in Qin (3). Similarly, the output-relevant scores in Eq. 15 and residuals in Eq. 17 are monitored by Ty = t y Λ 1 y t y = ỹc P y Λ 1 y P y ỹc (1) Q y = ỹ ỹ = ỹc ( I Py P ) y ỹc () where ( ) I P y P ( ) y = I Py P y, and Λy = 1 n 1 T y T y. Assume that the data are sampled from a multivariate normal distribution. Then the control limits of T c, T x and T y can be obtained by an F -distribution, and the control limits for Q x and Q y are gained by a χ -distribution (Nomikos and MacGregor (1995)). The control limits of these indices are listed in Table. According to the monitoring statistics and control limits in Table, we can monitor the input-relevant and outputrelevant faults as follows: (1) If T c exceeds its control limit, a quality-relevant fault will be detected with (1 α) 1% confidence. () If T x exceeds its control limit, the fault will be identified as quality irrelevant but process relevant with (1 α) 1% confidence. These faults can be paid less attention if only quality variables are considered significant. (3) If Q x exceeds its control limit, a potentially qualityrelevant fault will be detected with (1 α) 1% confidence, since it may contain the variations that are not excited in the training dataset. (4) If T y or Q y exceeds their control limits, a qualityrelevant fault will be detected, which is unpredictable from the input. 4. TENNESSEE EASTMAN PROCESS CASE STUDIES Tennessee Eastman process (TEP) (Downs and Vogel (1993)) is widely used as a benchmark process for evaluating the process monitoring methods such as PCA and PLS. The process produces two products (G and H) from four reactants (A, C, D and E). The reactions are: A(g) + C(g) + D(g) G(l) A(g) + C(g) + E(g) H(l) A(g) + E(g) X(l) 3D(g) X(l) where X(l) is byproduct. There are 41 measurements and 1 manipulated variables in this process, which are listed in 147

5 IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondheim, Norway Table 3. Fault Detection Rate for TEP Using PLS, CCA, CPLS and CCCA (%) Disturbances PLS CCA CPLS CCCA IDV(1) IDV() IDV(5) IDV(6) IDV(8) IDV(1) IDV(1) IDV(13) Table 4. False Alarm Rate for TEP Using PLS, CCA, CPLS and CCCA (%) Disturbances PLS CCA CPLS CCCA IDV() IDV(3) IDV(4) IDV(9) IDV(11) IDV(15) Downs and Vogel (1993). The details of the whole process can be found in Chiang et al. (1). In this case study, PLS, CCA, CPLS and CCCA are performed on TEP. For all these monitoring schemes, XMEAS(1-36) and XMV(1-11) are regarded as the process variables, where XMEAS(1-36) are the process measurements and XMV(1-11) are the manipulated variables. XMEAS(37-41) are selected as the quality variables, which are quality measurements. We use 5 normal samples to build PLS, CCA, CPLS and CCCA models. The samples are all centered to zero mean and scaled to unit variance. The number of factors for CCA and PLS are 4 and 34, determined by cross-validation. For CPLS, A c = 5, A y = 5 and A x = 8. And for CCCA, A c = 5, A y = 5 and A x = 3. In this study, we use 13 faulty sample sets and one normal sample set for fault detection (Downs and Vogel (1993)). Each sample set consists of 96 samples. In order to compare the effectiveness of the four models, it is necessary to identify whether a fault is related to the output Y or not. Here, the strategy in Zhou et al. (1) is employed to divide the 14 sample sets into two groups, quality-relevant group and quality-irrelevant group, and their monitoring results are listed in Table 3 and Table 4, respectively. For PLS and CCA, we use T index to monitor the qualityrelevant faults, while for CPLS and CCCA, T c and T y are employed to monitor the process. From Table 3 and Table 4, we can see that CCCA has higher fault detection rates and lower false alarm rates than PLS, CCA and CPLS in most cases. This is because: (1) Compared with PLS and CCA, CCCA also monitors the variations in the OPS subspace, which will affect the quality variables but are not predictable from the process variables; () Compared with CPLS, CCCA maximizes the correlation between X and Y, and the extracted T c T x 6 4 T y Q x Fig. 1. CCCA-based Monitoring Result for IDV() T 6 4 Q Fig.. CCA-based Monitoring Result for IDV() principal components are more relevant than those of CPLS. Moreover, the fact that CCA has lower false alarm rates than PLS in Table 4 also shows the second reason. Apart from the higher fault detection rates and lower false alarm rates, CCCA outperforms PLS and CCA since it can decompose the spaces completely and monitor the quality changes efficiently. For example, the monitoring results for IDV() of CCCA and CCA in Downs and Vogel (1993) are presented in Fig. 1 and, and this is a step disturbance in B composition. Although both CCA and CCCA detect this disturbance in all subspaces, Tc of CCCA-based model tends to return to normal, which reflects the effect of the feedback controllers in the process. Since Tc shows the variations predictable from the input and Tx presents the variations related to the input, the monitoring scheme of CCCA successfully separates them and improves the accuracy of fault detection. For the step disturbance IDV(4) that comes from reactor cooling water inlet temperature (Downs and Vogel (1993)), the monitoring results of CCCA and CCA are presented in Fig. 3 and 4. In this scenario, due to the cascade controller in the system, the variation of the reactor cooling water inlet temperature will not affect the output variables. For CCCA, the monitoring result shows that this fault is quality irrelevant, which should receive less attention during the process. CCA, however, detects the disturbance both in T and Q statistics, raising the alarm incorrectly. 148

6 IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondheim, Norway 3 1 T c 3 1 T x 3 1 T y 6 4 Q x Fig. 3. CCCA-based Monitoring Result for IDV(4) 3 1 T 6 4 Q Fig. 4. CCA-based Monitoring Result for IDV(4) 5. CONCLUSIONS In this article, concurrent CCA is proposed to monitor the process-relevant and quality-relevant faults separately. The input and output spaces are concurrently projected into five subspaces, and the corresponding monitoring statistics and control limits are also developed. The application results of TEP shows that CCCA gains higher fault detection rates and lower false alarm rates than CCA, PLS and CPLS. CCCA also outperforms other methods with its comprehensive decomposition. The comparison of CCA and PLS performed in this paper shows the advantages of CCA on quality prediction. Future work is to extend CCCA model to nonlinear cases. APPENDIX A: CCA ALGORITHM 1 Scale the data to zero mean. Perform eigenvalue decomposition to calculate the square root factors, [V x, D x ] = eig(x X) [V y, D y ] = eig(y Y) Σ 1 xx = V x D 1 x Σ 1 V x V y yy = V y D 1 y 3 Perform SVD to calculate weighting matrices R and C, [U z, S z, U z ] = svd(σ 1 xx X YΣ 1 yy ) R = Σ 1 xx U z C = Σ 1 yy V z 4 Obtain canonical variates T and U for X and Y, T = XR U = YC REFERENCES Borga, M., Landelius, T., and Knutsson, H. (1997). A unified approach to PCA, PLS, MLR and CCA. Chiang, L.H., Braatz, R.D., and Russell, E.L. (1). Fault detection and diagnosis in industrial systems. Springer Science & Business Media. Chiang, L.H., Russell, E.L., and Braatz, R.D. (). Fault diagnosis in chemical processes using fisher discriminant analysis, discriminant partial least squares, and principal component analysis. Chemometrics and Intelligent Laboratory Systems, 5(), Downs, J.J. and Vogel, E.F. (1993). A plant-wide industrial process control problem. Computers & Chemical Engineering, 17(3), Hardoon, D.R., Szedmak, S., and Shawe-Taylor, J. (4). Canonical correlation analysis: An overview with application to learning methods. Neural computation, 16(1), Hotelling, H. (1936). Relations between two sets of variates. Biometrika, Jackson, J.E. and Mudholkar, G.S. (1979). Control procedures for residuals associated with principal component analysis. Technometrics, 1(3), Khan, A.A., Moyne, J.R., and Tilbury, D.M. (8). Virtual metrology and feedback control for semiconductor manufacturing processes using recursive partial least squares. Journal of Process Control, 18(1), Li, G., Qin, S.J., and Zhou, D. (1). Geometric properties of partial least squares for process monitoring. Automatica, 46(1), 4 1. MacGregor, J.F., Jaeckle, C., Kiparissides, C., and Koutoudi, M. (1994). Process monitoring and diagnosis by multiblock PLS methods. AIChE Journal, 4(5), Nomikos, P. and MacGregor, J.F. (1995). Multivariate SPC charts for monitoring batch processes. Technometrics, 37(1), Qin, S.J. (3). Statistical process monitoring: basics and beyond. Journal of chemometrics, 17(8-9), Qin, S.J. and Zheng, Y. (13). Quality-relevant and process-relevant fault monitoring with concurrent projection to latent structures. AIChE Journal, 59(), Sherry, A. and Henson, R.K. (5). Conducting and interpreting canonical correlation analysis in personality research: A user-friendly primer. Journal of Personality Assessment, 84(1), Wise, B.M. and Gallagher, N.B. (1996). The process chemometrics approach to process monitoring and fault detection. Journal of Process Control, 6(6), Zhou, D., Li, G., and Qin, S.J. (1). Total projection to latent structures for process monitoring. AIChE Journal, 56(1),