Multiscale wavelet-based representation of data has been shown to be a powerful tool in feature extraction from practical process data. In this paper, this characteristic of multiscale representation is utilized to improve the prediction accuracy of some of the latent variable regression models, such as Principal Component Regression (PCR) and Partial Least Squares (PLS), by developing a multiscale latent variable regression (MSLVR) modeling algorithm. The idea is to decompose the input-output data at multiple scales using wavelet and scaling functions, construct multiple latent variable regression models at multiple scales using the scaled signal approximations of the data and then using cross-validation, and select among all MSLVR models the model which best describes the process. The main advantage of the MSLVR modeling algorithm is that it inherently accounts for the presence of measurement noise in the data by the application of the low-pass filters used in multiscale decomposition, which in turn improves the model robustness to measurement noise and enhances its prediction accuracy. The advantages of the developed MSLVR modeling algorithm are demonstrated using a simulated inferential model which predicts the distillate composition from measurements of some of the trays' temperatures. 1. Introduction Process models are an essential part of many process operations, such as model-based control [1, 2]. However, constructing empirical models using measurements of the process variables is associated with many difficulties, which include dealing with collinearity or redundancy in the variables and accounting for the presence of measurement noise in the data. Collinearity is common in models which involve large number of variables, such as Finite Impulse Response (FIR) models [3, 4] and inferential models. Collinearity increases the variance of the estimated model parameters, which degrades their accuracy of estimation. Many modeling techniques have been developed to deal with collinearity, which include Ridge Regression (RR) [5–7] and latent variable regression [3–5]. RR reduces the variations in model parameters by imposing a penalty on the norm of their estimated values. The latent variable regression models, on the other hand, use singular value decomposition to reduce the dimension of the input variables to provide a more conditioned set of inputs. Some of the popular latent variable regression model estimation techniques include the well-known Principal Component Regression (PCR) and Partial Least Squares (PLS) modeling methods [3–5]. Also, the
References
[1]
S. Qin and T. Badgwell, “An overview of industrial model predictive control technology,” in Proceedings of the 5th International Conference on Chemical Process Control (CPC '97), C. Garcia and J. Kantor, Eds., vol. 93 of AICHE Symposium Series 316, pp. 232–256, 1997.
[2]
R. Braatz and G. Mijares, “Control relevant identification and estimation,” in AICHE Annual Meeting, p. 183a, Miami Beach, Fla, USA, 1995.
[3]
B. M. Wise and N. L. Ricker, “Identification of finite impulse response models by principal components regression: frequency-response properties,” Process Control and Quality, vol. 4, no. 1, pp. 77–86, 1992.
[4]
N. L. Ricker, “The use of biased least-squares estimators for parameters in discrete-time pulse-response models,” Industrial and Engineering Chemistry Research, vol. 27, no. 2, pp. 343–350, 1988.
[5]
I. Frank and J. Friedman, “A statistical view of some chemometric regression tools,” Technometrics, vol. 35, no. 2, pp. 109–148, 1993.
[6]
A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics, vol. 42, no. 1, pp. 80–86, 2000.
[7]
J. McGregor, T. Kourti, and J. Kresta, “Multivariate identification: a study of several methods,” in IFAC ADCHEM Conference Proceedings, pp. 369–375, Toulouse, France, 1991.
[8]
B. R. Bakshi, “Multiscale analysis and modeling using wavelets,” Journal of Chemometrics, vol. 13, no. 3-4, pp. 415–434, 1999.
[9]
B. R. Bakshi and G. Stephanopoulos, “Representation of process trends-IV. Induction of real-time patterns from operating data for diagnosis and supervisory control,” Computers and Chemical Engineering, vol. 18, no. 4, pp. 303–332, 1994.
[10]
S. Palavajjhala, R. L. Motard, and B. Joseph, “Process identification using discrete wavelet transforms: design of prefilters,” AIChE Journal, vol. 42, no. 3, pp. 777–790, 1996.
[11]
B. R. Bakshi, “Multiscale PCA with application to multivariate statistical process monitoring,” AIChE Journal, vol. 44, no. 7, pp. 1596–1610, 1998.
[12]
A. N. Robertson, K. C. Park, and K. F. Alvin, “Extraction of impulse response data via wavelet transform for structural system identification,” Journal of Vibration and Acoustics, vol. 120, no. 1, pp. 252–260, 1998.
[13]
M. Nikolaou and P. Vuthandam, “FIR model identification: parsimony through Kernel compression with wavelets,” AIChE Journal, vol. 44, no. 1, pp. 141–150, 1998.
[14]
M. N. Nounou, “Dealing with collinearity in FIR models using multiscale estimation,” in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (CDC-ECC '05), pp. 8162–8167, Seville, Spain, December 2005.
[15]
M. N. Nounou, “Multiscale ARX process modeling,” in Proceedings of the 45th IEEE Conference on Decision and Control (CDC '06), pp. 823–828, San Diego, Calif, USA, December 2006.
[16]
M. N. Nounou and H. N. Nounou, “Enhanced prediction accuracy of fuzzy models using multiscale estimation,” in Proceedings of the 43rd IEEE Conference on Decision and Control (CDC '04), vol. 5, pp. 5170–5175, Atlantis, Bahamas, December 2004.
[17]
J. F. Carrier and G. Stephanopoulos, “Wavelet-based modulation in control-relevant process identification,” AIChE Journal, vol. 44, no. 2, pp. 341–360, 1998.
[18]
W. Massy, “Principal components regression in exploratory statistical research,” Journal of the American Statistical Association, vol. 60, pp. 234–246, 1965.
[19]
S. Wold, “Soft modeling. The basic design and some extensions,” in Systems under Indirect Observations, K. Joreskog and H. Wold, Eds., Elsevier, Amsterdam, The Netherlands, 1982.
[20]
E. C. Malthouse, A. C. Tamhane, and R. S. H. Mah, “Nonlinear partial least squares,” Computers and Chemical Engineering, vol. 21, no. 8, pp. 875–890, 1997.
[21]
I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 909–996, 1988.
[22]
G. Strang, “Wavelets and dilation equations: a brief introduction,” SIAM Review, vol. 31, no. 4, pp. 614–627, 1989.
[23]
S. G. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, no. 7, pp. 764–693, 1989.
[24]
M. N. Nounou and B. R. Bakshi, “On-line multiscale filtering of random and gross errors without process models,” AIChE Journal, vol. 45, no. 5, pp. 1041–1058, 1999.
[25]
G. Nason, “Wavelet shrinkage using cross-validation,” Journal of the Royal Statistical Society. Series B, vol. 58, no. 2, pp. 463–479, 1996.
[26]
M. Kano, K. Miyazaki, S. Hasebe, and I. Hashimoto, “Inferential control system of distillation compositions using dynamic partial least squares regression,” Journal of Process Control, vol. 10, no. 2, pp. 157–166, 2000.
