- Email: [email protected]
Fault Diagnosis based on DPCA and CA
I.A. Karimi and Rajagopalan Srinivasan (Editors), Proceedings of the 11th International Symposium on Process Systems Engineering, 15-19 July 2012, Singapore c 2012 Elsevier B.V. All rights reserved.
Fault Diagnosis based on DPCA and CA Celina Rea, Ruben Morales-Menendez ∗, Juan C. Tudón Martínez, Ricardo A. Ramírez Mendoza, Luis E. Garza Castañon Tecnológico de Monterrey, Campus Monterrey, Av. Eugenio Garza Sada 2501, Col. Tecnológico, 64,849 Monterrey NL, México
Abstract A comparison of two fault detection methods based in process history data is presented. The selected methods are Dynamic Principal Component Analysis (DPCA) and Correspondence Analysis (CA). The study is validated with experimental databases taken from an industrial process. The performance of methods is compared using the Receiver Operating Characteristics (ROC) graph with respect to several tuning parameters. The diagnosis step for both methods was implemented through Contribution Plots. The effects of each parameter are discussed and some guidelines for using these methods are proposed.
1. Motivation Industrial process have grown in integration and complexity. Monitoring only by humans is risky and sometimes impossible. Faults are always present, early Fault Detection and Isolation (FDI) systems can help operators to avoid abnormal event progression. DPCA and CA are two techniques based on statistical models coming from experimental data that can be used for fault diagnosis, Detroja et al. (2006b). These approaches are well known in some domains; but, there are several questions in the fault diagnosis. A ROC graph is a technique for visualizing, organizing and selecting classifiers based on their performance. ROC graph has been extended for use in diagnostic systems. In the published research works, the number of data for model’s learning the model, sampling rate, number of principal components/axes, thresholds have not been studied under same experimental databases.
2. Fundamentals A brief comparative review of both DPCA and CA approaches is presented focus in modelling, detection and diagnosis. Modeling for both DPCA and CA. Both methods need a statistical model of the process under normal operating conditions. The data set need be scaled to zero mean and unit variance. CA requires nt observations for p variables having a form of X (t) = [X1 (t) . . . X p (t) ](nt ×p) ; while DPCA additionally includes some past observations (i.e. wtime delay) X (t) = [X1 (t) . . . X1 (t − w) . . . X p (t) . . . X p (t − w)](nt ×(p ·[w+1])) . Based on X (t) matrix, two subspaces are built. The Principal Subspace captures major faults in the process, and the Residual Subspace considers minor faults and correlation rupture. A SCREE test, determines the number of principal and residual components (axes). By plotting the eigenvalues (singular values) of X (t) for DPCA (CA), the principal components (axes) are the first k components (axes) before the inflection point is located, while ∗ [email protected]
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Fault Diagnosis based on DPCA and CA
Process
...
...
Process
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...
the remainders are the residuals components (axes). Even DPCA and CA models cannot be compared. The representation given by CA would appear to be better able to capture inter-relationships between variables and samples, Detroja et al. (2006a). Fig. 1 (left) summarizes the model’s learning for both methods.
Monitoring the process variables under normal operating conditions
Monitoring the process variables under normal operating conditions
Normalize the experimental data Compute T Hotelling statistic Q and are off-line computed
No Yes Compute Q statistic No
A fault is detected
Figure 1. Model’s learning algorithm (left) and online detection (right) for DPCA & CA.
Detection for DPCA. The variables must be normalized, XS . Based on the correlation matrix R, the eigenvalues λi and eigenvectors Vi are obtained. The eigenvalues must be organized in decreasing order. The eigenvectors form two matrices: V[1,k] which is the principal component transformation and V[k+1,p·[w+1]] , known as residual transformation matrix. The projections to the subspaces are based on a linear transformation with no correlation between them. Mapping from multivariate approach to a scalar demands two statistics: T 2 Hotelling and Q statistics. The T 2 measures the deviation of variables in a data set from their mean values. Hotelling T 2 chart is based on the concept of statistical distance and it monitors the change in the mean vector T 2 statistic is obtained by TX2 S ; a i
normal operating point can be established as Tα2 = (nt − 1)Fα (k, nt − k)/(knt − k) Fα distribution with k and nt − k degrees of freedom, α = 0.95. If T 2 > Tα2 a fault is detected. Q statistic will detect changes on the residual directions. The Q statistic is obtained as QXS = XSi GXSTi . The threshold for normal operating conditions is given by i 1 √ Qα = θ1 hoCα 2θ2 /θ1 + 1 + θ2 h0 (h0 − 1)θ12 h0 If QXS >Qα a fault is detected. i
−1/2
−1/2
Detection for CA. CA is an optimization problem with P = Dr [(1/g)X −rcT ]Dc T and A Dμ B = SV D(P). For choosing the principal and residual axes, a SCREE plot can be made with the singular values obtained in Dμ . The Greenacre’s criteria was used for selection of k principal axis. The coordinates of the row and column profile points for the new principal axis can be computed by projecting on A and B, with only the first k −1/2 −1/2 columns retained, with F = Dr A Dμ and G = Dc BDμT . Matrix F gives the new row coordinates for the row cloud. Using a new measurement vector X = [X1 . . . Xm ]T , p the row sum is given by r = ∑i=1 Xi and the new row score is f = [r−1 xT G Dμ−1 ]T . The 2 T statistic for CA is defined as Ti2 = f T Dμ−2 f , where Dμ contains the first k-largest singular values. The threshold is computed through Tα2 . For residual axes, Q statistic
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allows to detect any significant deviation. Considering Qi = ResT Res, and the control limit as Qα = μQ ±Cα ∗ σQ where Cα is a confidence limit of 95% according to its normal deviation N (μQ , σQ ). Fig. 1 (right) shows the scheme for online detection. Diagnosis for both DPCA and CA. The T 2 statistic gives a variable which is calculated by using all variables; it shows the changes that occurred in all variables. The T 2 chart does not give information on which variable or variables is faulty. Miller et al. (1998) introduced a method for determining the contribution of each of the p variables in the T 2 computation. Scores of the PC are used for monitoring in T 2 chart; upon a faulty signal, first the contribution of each variable in the normalized scores is computed. Then total contributions of each variable are determined and plotted. The plot that shows the contribution of each variable in T 2 chart at time k is called the variable contribution plot, Kosebalaban and Cinar (2001). When T 2 value at time k is above the upper control limit, the variable contribution are calculated and plotted to diagnose the variable(s) that caused the fault alarm in multivariate T 2 chart. The variables with higher contribution, are isolated t as the most probable fault: Conti = Zi2 / ∑nj=1 Z 2j where Z = RVr is a generated residual for DPCA. CA method cannot diagnose the fault neither; for this purpose, a contribution plot base on the residuals of the Q statistic are used, Z = Res. The contribution plots can be plotted over time when a faulty signal alarms, the change in contribution of process variables gives more information about the root cause of the faulty condition at that time period.
3. Experimental Setup An industrial Heat Exchanger (HE) was the test-bed for this research. The HE uses steam vapor for heating water. The operating point is 70% of steam vapor flow (FT2 ), 38% of water flow (FT1 ), and the input water temperature (T T1 ) was at 23o C, which give an outlet water temperature (T T2 ) of 34o C. More than 20 experimental tests were done. For each sensor an abrupt fault as a bias of { ±5, ±6, ±8 } σ of the signal were implemented Prakash et al. (2002). Also, the number of data were 200, 500,..., 5000, and the sampling rate: 1,2,...,10.
4. Analysis of the Tuning Parameters. Number of Principal Components (PC)/axes. Data compression is an important aspect of multivariate statistical tools. SCREE plot indicates the percentage of accumulated information variation versus the components/axes. The components/axes, when the slope of the plot does not change, defines the number of PC/axes. Figure 2 (A) shows the percentage accumulated variation information versus the number of components for DPCA, and versus number of axes for CA is shown in Fig. 2 (B). Each plot includes databases with 100, 200, 500 and 1,000 data vectors. For DPCA, 5 PC describe 92-94 % of the information; 6 or more components do not contribute significantly. The number of data vector of process variables do not impact in the curve behavior. For CA the number of data affects the curve: 100-200 data have lower percentage accumulated variation than 500-1000 data vectors for 1 or 2 axes. After 3 axes there is not difference. According to Greenacre criteria, Greenacre and Blasius (1995), three axes are recommended. Thresholds. A False Negative (FN) is when there is a fault, but it is not detected; while a False Positive (FP) is when there is no fault, but one is detected. True Positive (TP) is when there is a fault, and it is detected; while, True Negative (TN) is when there is no
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Fault Diagnosis based on DPCA and CA
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Figure 2. Average variation accumulated versus components for DPCA (A), and versus number of axes for CA (B). ROC curves for outlet water temperature (T T2 ) transmitter using DPCA (C), and for outlet water temperature (T T2 ) transmitter CA (D). Frequency of errors for different sampling time (E) and for different number of training data (F).
fault and no fault is detected. The sum of TP and FN are the total faulty cases (TFC) and the sum of TN and FP are total healthy cases (THC). The probability of detection P Pd = TTFC while the probability of false alarms Pf a = TFP HC . For every condition, when [(Pd > 0.9) and (Pf a < 0.1)], an optimum is achieved by having a good compromise between right detection and minimum false alarms. ROC curves show a relation between opportune detection probability Pd versus false alarms probability Pf a . Table 1 summarizes the performance of [(Pd > 0.9) and (Pf a < 0.1)] criteria for faults sensors. DPCA shows successful results; while, CA exhibits some troubles mainly with Q statistics. Fig. 2 (C and D) shows ROC curves, where the thresholds were modified in different percentage. Fig. 2 (C) shows that DPCA is successful for both statistics. However, Fig. 2 (D) shows that for CA only T 2 statistics works well. Q statistics for CA, is only based on the residuals axes, which does not capture the variation of the residual per axes, Table 1. Sampling rate. The process variables were sampled at 1, 2, 5, 8 or 10 s. Fig. 2 (E) shows the number of times that the [(Pd > 0.9) and (Pf a < 0.1)] criteria was violated for each statistics for different sampling rate. There is a good performance for both methods when sampling rate is 1 or 2 s; however, the probabilities for missing detections or increasing false alarms grows up for sampling times greater than 5 s. The Q statistic for CA was avoided because its low performance. Number of Training Data. Fig. 2 (F) shows the number of times that the [(Pd > 0.9) and (Pf a < 0.1)] criteria was violated for each statistics for the number of training data. The number of data is a key issue for learning a statistical model. It can be seen that after 1,500 training data (25 min, 1 s sampling rate) both methods improve their performance.
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Table 1. Average performance for different sensors based on ROC curves. Sensor Transmitter T T2 FT1 FT2 T T1
Number Tests 11 15 12 11
DPCA-T 2 100 100 91.67 81.82
DPCA-Q 100 86.67 100 90.91
CA-T 2 100 40 91.67 81.82
CA-Q 9.09 6.67 91.67 0
Diagnosis. There is a good performance with DPCA for the 4 faulty sensors, while CA shows 90.9% performance when a fault occurs in (T T2 ) and 46.6% if the fault is in (FT1 ). Contributions of 4 variables were compared to the same statistic and choose the variables corresponding to the relatively large contributions as the possible causes for faults. Instead of comparing the absolute contribution and the corresponding control limit, the use of the relative contribution is more convenient way to identify faulty variables Choi and Lee (2005).
5. Conclusions Given experimental data a multivariate statistical model can be learned. SCREE plot and the Greenacre criteria guide the complexity of the model based on the minimum number of components/axes. The statistical model for CA is more sensible to the number of data than the model for DPCA; but, defined the minimum number of components/axes, the number of data does not have influence. The ROC graphs are a useful tool for visualizing and evaluating fault detection algorithms. A detection probability (Pd > 0.9) and false alarm probability (Pf a < 0.1.) is a good criteria for choosing the sampling rate, number of data and thresholds. Sampling rate was the most important parameter. Based on ROC graphs, DPCA outperforms CA in fault detection, because the Q statistics does not work well. Diagnosis could be implemented in both methods through contribution plots.
References Choi, S., Lee, I., 2005. Multiblock PLS-based Localized Process Diagnosis . J of Process Control 15 (3), 295–306. Detroja, K., Gudi, R., Patwardhan, S., 2006a. Fault Diagnosis using Correspondence Analysis: Implementation Issues and Analysis. In: IEEE Int Conf on Ind Tech. pp. 1374 – 1379. Detroja, K., Gudi, R., Patwardhan, S., Roy, K., 2006b. Fault Detection and Isolation Using Correspondence Analysis . Ind Eng Chem Res 45, 223 – 235. Greenacre, M., Blasius, J., 1995. Correspondence Analysis in the Social Sciences. A Press. Kosebalaban, F., Cinar, A., 2001. Integration of Multivariate SPM and FDD by Parity Space Technique for a Food Pasteurization Process. Computers and Chemical Eng 25, 473–491. Miller, P., Swanson, R., Heckler, C., 1998. Contribution Plots: A Missing Link in Multivariate Quality Control . Appl. Math. and Comp. 8, 775–792. Prakash, J., Patwardhan, S., Narasimhan, S., 2002. A Supervisory Approach to Fault-Tolerant Control of Linear Multivariable Systems. Ind. Eng. Chem. Res. 41, 2270–2281.
Fault Diagnosis based on DPCA and CA Celina Rea, Ruben Morales-Menendez ∗, Juan C. Tudón Martínez, Ricardo A. Ramírez Mendoza, Luis E. Garza Castañon Tecnológico de Monterrey, Campus Monterrey, Av. Eugenio Garza Sada 2501, Col. Tecnológico, 64,849 Monterrey NL, México
Abstract A comparison of two fault detection methods based in process history data is presented. The selected methods are Dynamic Principal Component Analysis (DPCA) and Correspondence Analysis (CA). The study is validated with experimental databases taken from an industrial process. The performance of methods is compared using the Receiver Operating Characteristics (ROC) graph with respect to several tuning parameters. The diagnosis step for both methods was implemented through Contribution Plots. The effects of each parameter are discussed and some guidelines for using these methods are proposed.
1. Motivation Industrial process have grown in integration and complexity. Monitoring only by humans is risky and sometimes impossible. Faults are always present, early Fault Detection and Isolation (FDI) systems can help operators to avoid abnormal event progression. DPCA and CA are two techniques based on statistical models coming from experimental data that can be used for fault diagnosis, Detroja et al. (2006b). These approaches are well known in some domains; but, there are several questions in the fault diagnosis. A ROC graph is a technique for visualizing, organizing and selecting classifiers based on their performance. ROC graph has been extended for use in diagnostic systems. In the published research works, the number of data for model’s learning the model, sampling rate, number of principal components/axes, thresholds have not been studied under same experimental databases.
2. Fundamentals A brief comparative review of both DPCA and CA approaches is presented focus in modelling, detection and diagnosis. Modeling for both DPCA and CA. Both methods need a statistical model of the process under normal operating conditions. The data set need be scaled to zero mean and unit variance. CA requires nt observations for p variables having a form of X (t) = [X1 (t) . . . X p (t) ](nt ×p) ; while DPCA additionally includes some past observations (i.e. wtime delay) X (t) = [X1 (t) . . . X1 (t − w) . . . X p (t) . . . X p (t − w)](nt ×(p ·[w+1])) . Based on X (t) matrix, two subspaces are built. The Principal Subspace captures major faults in the process, and the Residual Subspace considers minor faults and correlation rupture. A SCREE test, determines the number of principal and residual components (axes). By plotting the eigenvalues (singular values) of X (t) for DPCA (CA), the principal components (axes) are the first k components (axes) before the inflection point is located, while ∗ [email protected]
281
Fault Diagnosis based on DPCA and CA
Process
...
...
Process
...
...
the remainders are the residuals components (axes). Even DPCA and CA models cannot be compared. The representation given by CA would appear to be better able to capture inter-relationships between variables and samples, Detroja et al. (2006a). Fig. 1 (left) summarizes the model’s learning for both methods.
Monitoring the process variables under normal operating conditions
Monitoring the process variables under normal operating conditions
Normalize the experimental data Compute T Hotelling statistic Q and are off-line computed
No Yes Compute Q statistic No
A fault is detected
Figure 1. Model’s learning algorithm (left) and online detection (right) for DPCA & CA.
Detection for DPCA. The variables must be normalized, XS . Based on the correlation matrix R, the eigenvalues λi and eigenvectors Vi are obtained. The eigenvalues must be organized in decreasing order. The eigenvectors form two matrices: V[1,k] which is the principal component transformation and V[k+1,p·[w+1]] , known as residual transformation matrix. The projections to the subspaces are based on a linear transformation with no correlation between them. Mapping from multivariate approach to a scalar demands two statistics: T 2 Hotelling and Q statistics. The T 2 measures the deviation of variables in a data set from their mean values. Hotelling T 2 chart is based on the concept of statistical distance and it monitors the change in the mean vector T 2 statistic is obtained by TX2 S ; a i
normal operating point can be established as Tα2 = (nt − 1)Fα (k, nt − k)/(knt − k) Fα distribution with k and nt − k degrees of freedom, α = 0.95. If T 2 > Tα2 a fault is detected. Q statistic will detect changes on the residual directions. The Q statistic is obtained as QXS = XSi GXSTi . The threshold for normal operating conditions is given by i 1 √ Qα = θ1 hoCα 2θ2 /θ1 + 1 + θ2 h0 (h0 − 1)θ12 h0 If QXS >Qα a fault is detected. i
−1/2
−1/2
Detection for CA. CA is an optimization problem with P = Dr [(1/g)X −rcT ]Dc T and A Dμ B = SV D(P). For choosing the principal and residual axes, a SCREE plot can be made with the singular values obtained in Dμ . The Greenacre’s criteria was used for selection of k principal axis. The coordinates of the row and column profile points for the new principal axis can be computed by projecting on A and B, with only the first k −1/2 −1/2 columns retained, with F = Dr A Dμ and G = Dc BDμT . Matrix F gives the new row coordinates for the row cloud. Using a new measurement vector X = [X1 . . . Xm ]T , p the row sum is given by r = ∑i=1 Xi and the new row score is f = [r−1 xT G Dμ−1 ]T . The 2 T statistic for CA is defined as Ti2 = f T Dμ−2 f , where Dμ contains the first k-largest singular values. The threshold is computed through Tα2 . For residual axes, Q statistic
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Celina Rea et. al.
allows to detect any significant deviation. Considering Qi = ResT Res, and the control limit as Qα = μQ ±Cα ∗ σQ where Cα is a confidence limit of 95% according to its normal deviation N (μQ , σQ ). Fig. 1 (right) shows the scheme for online detection. Diagnosis for both DPCA and CA. The T 2 statistic gives a variable which is calculated by using all variables; it shows the changes that occurred in all variables. The T 2 chart does not give information on which variable or variables is faulty. Miller et al. (1998) introduced a method for determining the contribution of each of the p variables in the T 2 computation. Scores of the PC are used for monitoring in T 2 chart; upon a faulty signal, first the contribution of each variable in the normalized scores is computed. Then total contributions of each variable are determined and plotted. The plot that shows the contribution of each variable in T 2 chart at time k is called the variable contribution plot, Kosebalaban and Cinar (2001). When T 2 value at time k is above the upper control limit, the variable contribution are calculated and plotted to diagnose the variable(s) that caused the fault alarm in multivariate T 2 chart. The variables with higher contribution, are isolated t as the most probable fault: Conti = Zi2 / ∑nj=1 Z 2j where Z = RVr is a generated residual for DPCA. CA method cannot diagnose the fault neither; for this purpose, a contribution plot base on the residuals of the Q statistic are used, Z = Res. The contribution plots can be plotted over time when a faulty signal alarms, the change in contribution of process variables gives more information about the root cause of the faulty condition at that time period.
3. Experimental Setup An industrial Heat Exchanger (HE) was the test-bed for this research. The HE uses steam vapor for heating water. The operating point is 70% of steam vapor flow (FT2 ), 38% of water flow (FT1 ), and the input water temperature (T T1 ) was at 23o C, which give an outlet water temperature (T T2 ) of 34o C. More than 20 experimental tests were done. For each sensor an abrupt fault as a bias of { ±5, ±6, ±8 } σ of the signal were implemented Prakash et al. (2002). Also, the number of data were 200, 500,..., 5000, and the sampling rate: 1,2,...,10.
4. Analysis of the Tuning Parameters. Number of Principal Components (PC)/axes. Data compression is an important aspect of multivariate statistical tools. SCREE plot indicates the percentage of accumulated information variation versus the components/axes. The components/axes, when the slope of the plot does not change, defines the number of PC/axes. Figure 2 (A) shows the percentage accumulated variation information versus the number of components for DPCA, and versus number of axes for CA is shown in Fig. 2 (B). Each plot includes databases with 100, 200, 500 and 1,000 data vectors. For DPCA, 5 PC describe 92-94 % of the information; 6 or more components do not contribute significantly. The number of data vector of process variables do not impact in the curve behavior. For CA the number of data affects the curve: 100-200 data have lower percentage accumulated variation than 500-1000 data vectors for 1 or 2 axes. After 3 axes there is not difference. According to Greenacre criteria, Greenacre and Blasius (1995), three axes are recommended. Thresholds. A False Negative (FN) is when there is a fault, but it is not detected; while a False Positive (FP) is when there is no fault, but one is detected. True Positive (TP) is when there is a fault, and it is detected; while, True Negative (TN) is when there is no
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Fault Diagnosis based on DPCA and CA
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Figure 2. Average variation accumulated versus components for DPCA (A), and versus number of axes for CA (B). ROC curves for outlet water temperature (T T2 ) transmitter using DPCA (C), and for outlet water temperature (T T2 ) transmitter CA (D). Frequency of errors for different sampling time (E) and for different number of training data (F).
fault and no fault is detected. The sum of TP and FN are the total faulty cases (TFC) and the sum of TN and FP are total healthy cases (THC). The probability of detection P Pd = TTFC while the probability of false alarms Pf a = TFP HC . For every condition, when [(Pd > 0.9) and (Pf a < 0.1)], an optimum is achieved by having a good compromise between right detection and minimum false alarms. ROC curves show a relation between opportune detection probability Pd versus false alarms probability Pf a . Table 1 summarizes the performance of [(Pd > 0.9) and (Pf a < 0.1)] criteria for faults sensors. DPCA shows successful results; while, CA exhibits some troubles mainly with Q statistics. Fig. 2 (C and D) shows ROC curves, where the thresholds were modified in different percentage. Fig. 2 (C) shows that DPCA is successful for both statistics. However, Fig. 2 (D) shows that for CA only T 2 statistics works well. Q statistics for CA, is only based on the residuals axes, which does not capture the variation of the residual per axes, Table 1. Sampling rate. The process variables were sampled at 1, 2, 5, 8 or 10 s. Fig. 2 (E) shows the number of times that the [(Pd > 0.9) and (Pf a < 0.1)] criteria was violated for each statistics for different sampling rate. There is a good performance for both methods when sampling rate is 1 or 2 s; however, the probabilities for missing detections or increasing false alarms grows up for sampling times greater than 5 s. The Q statistic for CA was avoided because its low performance. Number of Training Data. Fig. 2 (F) shows the number of times that the [(Pd > 0.9) and (Pf a < 0.1)] criteria was violated for each statistics for the number of training data. The number of data is a key issue for learning a statistical model. It can be seen that after 1,500 training data (25 min, 1 s sampling rate) both methods improve their performance.
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Table 1. Average performance for different sensors based on ROC curves. Sensor Transmitter T T2 FT1 FT2 T T1
Number Tests 11 15 12 11
DPCA-T 2 100 100 91.67 81.82
DPCA-Q 100 86.67 100 90.91
CA-T 2 100 40 91.67 81.82
CA-Q 9.09 6.67 91.67 0
Diagnosis. There is a good performance with DPCA for the 4 faulty sensors, while CA shows 90.9% performance when a fault occurs in (T T2 ) and 46.6% if the fault is in (FT1 ). Contributions of 4 variables were compared to the same statistic and choose the variables corresponding to the relatively large contributions as the possible causes for faults. Instead of comparing the absolute contribution and the corresponding control limit, the use of the relative contribution is more convenient way to identify faulty variables Choi and Lee (2005).
5. Conclusions Given experimental data a multivariate statistical model can be learned. SCREE plot and the Greenacre criteria guide the complexity of the model based on the minimum number of components/axes. The statistical model for CA is more sensible to the number of data than the model for DPCA; but, defined the minimum number of components/axes, the number of data does not have influence. The ROC graphs are a useful tool for visualizing and evaluating fault detection algorithms. A detection probability (Pd > 0.9) and false alarm probability (Pf a < 0.1.) is a good criteria for choosing the sampling rate, number of data and thresholds. Sampling rate was the most important parameter. Based on ROC graphs, DPCA outperforms CA in fault detection, because the Q statistics does not work well. Diagnosis could be implemented in both methods through contribution plots.
References Choi, S., Lee, I., 2005. Multiblock PLS-based Localized Process Diagnosis . J of Process Control 15 (3), 295–306. Detroja, K., Gudi, R., Patwardhan, S., 2006a. Fault Diagnosis using Correspondence Analysis: Implementation Issues and Analysis. In: IEEE Int Conf on Ind Tech. pp. 1374 – 1379. Detroja, K., Gudi, R., Patwardhan, S., Roy, K., 2006b. Fault Detection and Isolation Using Correspondence Analysis . Ind Eng Chem Res 45, 223 – 235. Greenacre, M., Blasius, J., 1995. Correspondence Analysis in the Social Sciences. A Press. Kosebalaban, F., Cinar, A., 2001. Integration of Multivariate SPM and FDD by Parity Space Technique for a Food Pasteurization Process. Computers and Chemical Eng 25, 473–491. Miller, P., Swanson, R., Heckler, C., 1998. Contribution Plots: A Missing Link in Multivariate Quality Control . Appl. Math. and Comp. 8, 775–792. Prakash, J., Patwardhan, S., Narasimhan, S., 2002. A Supervisory Approach to Fault-Tolerant Control of Linear Multivariable Systems. Ind. Eng. Chem. Res. 41, 2270–2281.