Linear Inferential Modeling: Theoretical Perspectives, Extensions, and Comparative Analysis

Affiliation(s)

Chemical Engineering Program, Texas A & M University at Qatar, Doha, Qatar.

Electrical and Computer Engineering Program, Texas A & M University at Qatar, Doha, Qatar.

Chemical Engineering Program, Texas A & M University at Qatar, Doha, Qatar.

Electrical and Computer Engineering Program, Texas A & M University at Qatar, Doha, Qatar.

ABSTRACT

Inferential models are widely used in the chemical industry to infer key process variables, which are challenging or expensive to measure, from other more easily measured variables. The aim of this paper is three-fold: to present a theoretical review of some of the well known linear inferential modeling techniques, to enhance the predictive ability of the regularized canonical correlation analysis (RCCA) method, and finally to compare the performances of these techniques and highlight some of the practical issues that can affect their predictive abilities. The inferential modeling techniques considered in this study include full rank modeling techniques, such as ordinary least square (OLS) regression and ridge regression (RR), and latent variable regression (LVR) techniques, such as principal component regression (PCR), partial least squares (PLS) regression, and regularized canonical correlation analysis (RCCA). The theoretical analysis shows that the loading vectors used in LVR modeling can be computed by solving eigenvalue problems. Also, for the RCCA method, we show that by optimizing the regularization parameter, an improvement in prediction accuracy can be achieved over other modeling techniques. To illustrate the performances of all inferential modeling techniques, a comparative analysis was performed through two simulated examples, one using synthetic data and the other using simulated distillation column data. All techniques are optimized and compared by computing the cross validation mean square error using unseen testing data. The results of this comparative analysis show that scaling the data helps improve the performances of all modeling techniques, and that the LVR techniques outperform the full rank ones. One reason for this advantage is that the LVR techniques improve the conditioning of the model by discarding the latent variables (or principal components) with small eigenvalues, which also reduce the effect of the noise on the model prediction. The results also show that PCR and PLS have comparable performances, and that RCCA can provide an advantage by optimizing its regularization parameter.

Inferential models are widely used in the chemical industry to infer key process variables, which are challenging or expensive to measure, from other more easily measured variables. The aim of this paper is three-fold: to present a theoretical review of some of the well known linear inferential modeling techniques, to enhance the predictive ability of the regularized canonical correlation analysis (RCCA) method, and finally to compare the performances of these techniques and highlight some of the practical issues that can affect their predictive abilities. The inferential modeling techniques considered in this study include full rank modeling techniques, such as ordinary least square (OLS) regression and ridge regression (RR), and latent variable regression (LVR) techniques, such as principal component regression (PCR), partial least squares (PLS) regression, and regularized canonical correlation analysis (RCCA). The theoretical analysis shows that the loading vectors used in LVR modeling can be computed by solving eigenvalue problems. Also, for the RCCA method, we show that by optimizing the regularization parameter, an improvement in prediction accuracy can be achieved over other modeling techniques. To illustrate the performances of all inferential modeling techniques, a comparative analysis was performed through two simulated examples, one using synthetic data and the other using simulated distillation column data. All techniques are optimized and compared by computing the cross validation mean square error using unseen testing data. The results of this comparative analysis show that scaling the data helps improve the performances of all modeling techniques, and that the LVR techniques outperform the full rank ones. One reason for this advantage is that the LVR techniques improve the conditioning of the model by discarding the latent variables (or principal components) with small eigenvalues, which also reduce the effect of the noise on the model prediction. The results also show that PCR and PLS have comparable performances, and that RCCA can provide an advantage by optimizing its regularization parameter.

Cite this paper

M. Madakyaru, M. Nounou and H. Nounou, "Linear Inferential Modeling: Theoretical Perspectives, Extensions, and Comparative Analysis,"*Intelligent Control and Automation*, Vol. 3 No. 4, 2012, pp. 376-389. doi: 10.4236/ica.2012.34042.

M. Madakyaru, M. Nounou and H. Nounou, "Linear Inferential Modeling: Theoretical Perspectives, Extensions, and Comparative Analysis,"

References

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[2] R. Weber and C. B. Brosilow, “The Use of Secondary Measurement to Improve Control,” AIChE Journal, Vol. 18, No. 3, 1972, pp. 614-623. doi:10.1002/aic.690180323

[3] B. Joseph and C. B. Brosilow, “Inferential Control Processes,” AIChE Journal, Vol. 24, No. 3, 1978, pp. 485-509. doi:10.1002/aic.690240313

[4] M. Morari and G. Stephanopoulos, “Optimal Selection of Secondary Measurements within the Framework of State Estimationin the Presence of Persistent Unknown Disturbances,” AIChE Journal, Vol. 26, No. 2, 1980, pp. 247-259. doi:10.1002/aic.690260207

[5] I. Frank and J. Friedman, “A Statistical View of Some Chemometric Regression Tools,” Technometrics, Vol. 35, No. 2, 1993, pp. 109-148. doi:10.1080/00401706.1993.10485033

[6] A. Hoerl and R. Kennard, “Ridge Regression Based Estimation for Nonorthogonal Problems,” Technometrics, Vol. 8, 1970, pp. 27-52.

[7] J. McGregor, T. Kourti and J. Kresta, “Multivariate Identification: A Study of Several Methods,” IFAC ADCHEM Proceedings, Toulouse, Vol. 4, 1991, pp. 145-156.

[8] M. N. Nounou, “Dealing with Collinearity in Fir Models Using Bayesian Shrinkage,” Industrial and Engineering Chemistry Research, Vol. 45, 2006, pp. 292-298. doi:10.1021/ie048897m

[9] B. R. Kowalski and M. B. Seasholtz, “Recent Developments in Multivariate Calibration,” Journal of Chemometrics, Vol. 5, 1991, pp. 129-145. doi:10.1002/cem.1180050303

[10] M. Stone and R. J. Brooks, “Continuum Regression: Cross-Validated Sequentially Constructed Prediction Embracing Ordinaryleast Squares, Partial Least Squares and Principal Components Regression,” Journal of the Royal Statistical Society B, Vol. 52, No. 2, 1990, pp. 237-269.

[11] S. Wold, “Soft Modeling: The Basic Design and Some Extensions, Systems under Indirect Observations,” Elsevier, Amsterdam, 1982.

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[13] H. Hotelling, “Relations between Two Sets of Variables,” Biometrika, Vol. 28, 1936, pp. 321-377.

[14] F. R. Bach and M. I. Jordan, “Kernel Independent Component Analysis,” Journal of Machine Learning Research, Vol. 3, No. 1, 2002, pp. 1-48.

[15] S. S. D. R. Hardoon and J. Shawetaylor, “Canonical Correlation Analysis: An Overview with Application to Learning Methods,” Neural Computation, Vol. 16, No. 12, 2004, pp. 2639-2664. doi:10.1162/0899766042321814

[16] M. Borga, T. Landelius and H. Knutsson, “A Unified Approach to PCA, PLS, MLR and CCA, Technical Report,” Technical Report, Linkoping University, 1997.

[17] T. Mejdell and S. Skogestad, “Estimation of Distillation Compositions from Multiple Temperature Measurements Using Partial Least Squares Regression,” Industrial & Engineering Chemistry Research, Vol. 30, 1991, pp. 2543-2555. doi:10.1021/ie00060a007

[18] M. kano, K. Miyazaki, S. Hasebe and I. Hashimoto, “Inferential Control System of distillation Compositions Using Dynamicpartial Least Squares Regression,” Journal of Process Control, Vol. 10, No. 2, 2000, pp. 157-166. doi:10.1016/S0959-1524(99)00027-X

[19] T. Mejdell and S. Skogestad, “Composition Estimator in a Pilot-Plant Distillation Column,” Industrial & Engineering Chemistry Research, Vol. 30, 1991, pp. 2555-2564. doi:10.1021/ie00060a008

[20] Y. Hiroyuki, Y. B. Hideki, F. C. E. O. Hiromu and F. Hideki, “Canonical Correlation Analysis for Multivariate Regression and Its Application to Metabolic Fingerprinting,” Biochemical Engineering Journal, Vol. 40, No. 2, 2008, pp. 199-204.

[21] R. Rosipal and N. Kramer, “Overview and Recent Advances in Partial Least Squares. Subspace, Latent Structure and Feature Selection Techniques,” Lecture Notes in Computer Science, Vol. 3940, 2006, pp. 34-51. doi:10.1007/11752790_2

[22] P. Geladi and B. R. Kowalski, “Partial Least Square Regression: A Tutorial,” Analytica Chimica Acta, Vol. 185, No. 1, 1986, pp. 1-17. doi:10.1016/0003-2670(86)80028-9

[23] S. Wold, “Cross-Validatory Estimation of the Number of Components in Factor and Principal Components Models,” Technometrics, Vol. 20, No. 4, 1978, p. 397. doi:10.1080/00401706.1978.10489693

[24] O. Yeniay and A. Goktas, “A Comparison of Partial Least Squares Regression with Other Prediction Methods,” Hacettepe Journal of Mathematics and Statistics, Vol. 31, 2002, pp. 99-111.

[25] P. D. Wentzell and L. V. Montoto, “Comparison of Principal Components Regression and Partial Least Square Regression through Generic Simulations of Complex Mixtures,” Chemometrics and Intelligent Laboratory Systems, Vol. 65, 2003, pp. 257-279. doi:10.1016/S0169-7439(02)00138-7

[1] J. V. Kresta, T. E. Marlin and J. F. McGregor, “Development of Inferential Process Models Using PLS,” Computers & Chemical Engineering, Vol. 18, No. 7, 1994, pp. 597-611. doi:10.1016/0098-1354(93)E0006-U

[2] R. Weber and C. B. Brosilow, “The Use of Secondary Measurement to Improve Control,” AIChE Journal, Vol. 18, No. 3, 1972, pp. 614-623. doi:10.1002/aic.690180323

[3] B. Joseph and C. B. Brosilow, “Inferential Control Processes,” AIChE Journal, Vol. 24, No. 3, 1978, pp. 485-509. doi:10.1002/aic.690240313

[4] M. Morari and G. Stephanopoulos, “Optimal Selection of Secondary Measurements within the Framework of State Estimationin the Presence of Persistent Unknown Disturbances,” AIChE Journal, Vol. 26, No. 2, 1980, pp. 247-259. doi:10.1002/aic.690260207

[5] I. Frank and J. Friedman, “A Statistical View of Some Chemometric Regression Tools,” Technometrics, Vol. 35, No. 2, 1993, pp. 109-148. doi:10.1080/00401706.1993.10485033

[6] A. Hoerl and R. Kennard, “Ridge Regression Based Estimation for Nonorthogonal Problems,” Technometrics, Vol. 8, 1970, pp. 27-52.

[7] J. McGregor, T. Kourti and J. Kresta, “Multivariate Identification: A Study of Several Methods,” IFAC ADCHEM Proceedings, Toulouse, Vol. 4, 1991, pp. 145-156.

[8] M. N. Nounou, “Dealing with Collinearity in Fir Models Using Bayesian Shrinkage,” Industrial and Engineering Chemistry Research, Vol. 45, 2006, pp. 292-298. doi:10.1021/ie048897m

[9] B. R. Kowalski and M. B. Seasholtz, “Recent Developments in Multivariate Calibration,” Journal of Chemometrics, Vol. 5, 1991, pp. 129-145. doi:10.1002/cem.1180050303

[10] M. Stone and R. J. Brooks, “Continuum Regression: Cross-Validated Sequentially Constructed Prediction Embracing Ordinaryleast Squares, Partial Least Squares and Principal Components Regression,” Journal of the Royal Statistical Society B, Vol. 52, No. 2, 1990, pp. 237-269.

[11] S. Wold, “Soft Modeling: The Basic Design and Some Extensions, Systems under Indirect Observations,” Elsevier, Amsterdam, 1982.

[12] E. Malthouse, A. Tamhane and R. Mah, “Non-Linear Partial Least Squares,” Computers and Chemical Engineering, Vol. 21, 1997, pp. 875-890. doi:10.1016/S0098-1354(96)00311-0

[13] H. Hotelling, “Relations between Two Sets of Variables,” Biometrika, Vol. 28, 1936, pp. 321-377.

[14] F. R. Bach and M. I. Jordan, “Kernel Independent Component Analysis,” Journal of Machine Learning Research, Vol. 3, No. 1, 2002, pp. 1-48.

[15] S. S. D. R. Hardoon and J. Shawetaylor, “Canonical Correlation Analysis: An Overview with Application to Learning Methods,” Neural Computation, Vol. 16, No. 12, 2004, pp. 2639-2664. doi:10.1162/0899766042321814

[16] M. Borga, T. Landelius and H. Knutsson, “A Unified Approach to PCA, PLS, MLR and CCA, Technical Report,” Technical Report, Linkoping University, 1997.

[17] T. Mejdell and S. Skogestad, “Estimation of Distillation Compositions from Multiple Temperature Measurements Using Partial Least Squares Regression,” Industrial & Engineering Chemistry Research, Vol. 30, 1991, pp. 2543-2555. doi:10.1021/ie00060a007

[18] M. kano, K. Miyazaki, S. Hasebe and I. Hashimoto, “Inferential Control System of distillation Compositions Using Dynamicpartial Least Squares Regression,” Journal of Process Control, Vol. 10, No. 2, 2000, pp. 157-166. doi:10.1016/S0959-1524(99)00027-X

[19] T. Mejdell and S. Skogestad, “Composition Estimator in a Pilot-Plant Distillation Column,” Industrial & Engineering Chemistry Research, Vol. 30, 1991, pp. 2555-2564. doi:10.1021/ie00060a008

[20] Y. Hiroyuki, Y. B. Hideki, F. C. E. O. Hiromu and F. Hideki, “Canonical Correlation Analysis for Multivariate Regression and Its Application to Metabolic Fingerprinting,” Biochemical Engineering Journal, Vol. 40, No. 2, 2008, pp. 199-204.

[21] R. Rosipal and N. Kramer, “Overview and Recent Advances in Partial Least Squares. Subspace, Latent Structure and Feature Selection Techniques,” Lecture Notes in Computer Science, Vol. 3940, 2006, pp. 34-51. doi:10.1007/11752790_2

[22] P. Geladi and B. R. Kowalski, “Partial Least Square Regression: A Tutorial,” Analytica Chimica Acta, Vol. 185, No. 1, 1986, pp. 1-17. doi:10.1016/0003-2670(86)80028-9

[23] S. Wold, “Cross-Validatory Estimation of the Number of Components in Factor and Principal Components Models,” Technometrics, Vol. 20, No. 4, 1978, p. 397. doi:10.1080/00401706.1978.10489693

[24] O. Yeniay and A. Goktas, “A Comparison of Partial Least Squares Regression with Other Prediction Methods,” Hacettepe Journal of Mathematics and Statistics, Vol. 31, 2002, pp. 99-111.

[25] P. D. Wentzell and L. V. Montoto, “Comparison of Principal Components Regression and Partial Least Square Regression through Generic Simulations of Complex Mixtures,” Chemometrics and Intelligent Laboratory Systems, Vol. 65, 2003, pp. 257-279. doi:10.1016/S0169-7439(02)00138-7