Error-in-variables models

Duever, T.A., S. E. Keeler, P.M. Reilly, J.H. Vera, and P.A. Williams, "An Application of the Error-In-Variables Model-Parameter Estimation from van Ness-type Vapour-Liqud Equilibrium Experiments", Chem. Eng Sci., 42, 403-412 (1987). 

Reilly, P.M. and H. Patino-Leal, "A Bayessian Study of the Error in Variables Model", Technometrics, 23, 221-231 (1981). 

Valko, P. and S. Vajda, "An Extended Marquardt -type Procedure for Fitting Error-In-Variables Models", Computers Chem. Eng., 11, 37-43 (1987). 

Valko, P., and Vadja, S. (1987). An extended Marquardt-type procedure for fitting error-in-variable models. Comput. Chem. Eng. 11, 37-43. 

M52 Fitting an error-in-variables model of the form F(Z,P)=0 modified Patino-Leal - Reilly method 5200 5460... 

Computational the most demanding task is locating the minimum of the function (3.89) at step (ii). Since the Gauss-Newtan-Marquardt algorithm is a robust and efficient way of solving the nonlinear least squares problem discussed in Section 3.3, we would like to extend it to error—in-variables models. First we show, however, that this extension is not obvious, and the apparently simplest approach does not work. 

P.M. Reilly and H. Patino-Leal, A Bayesian study of the error-in-variables model. Technometrics, 23 (1981) 221-227. 

The next step in the protocol answers the question about what is the best method to estimate the reactivity ratios. Historically, because of its simplicity, linearization techniques such as the Fineman-Ross, Kelen-Tudos, and extended Kelen-Tudos methods have been used. Easily performed on a simple calculator, these techniques suffer from inaccuracies due to the linearization of the inherently nonlinear Mayo-Lewis model. Such techniques violate basic assumptions of linear regression and have been repeatedly shown to be invalid [117, 119, 126]. Nonlinear least squares (NLLS) techniques and other more advanced nonlinear techniques such as the error-in-variables-model (EVM) method have been readily available for several decades [119, 120, 126, 127]. 

Brar and Sunita [58] described a method for the analysis of acrylonitrile-butyl acrylate (A/B) copolymers of different monomer compositions. Copolymer compositions were determined by elemental analyses and comonomers reactivity ratios were determined using a non-linear least squares errors-in-variables model. Terminal and penultimate reactivity ratios were calculated using the observed distribution determined from C( H)-NMR spectra. The triad sequence distribution was used to calculate diad concentrations, conditional probability parameters, number-average sequence lengths and block character of the copolymers. The observed triad sequence concentrations determined from C( H)-NMR spectra agreed well with those calculated from reactivity ratios. 

Schoukens, J., Pintelon, R. Rolain, Y. (1997). Recent Advances in Total Least Squares Techniques and Errors-In-Variables Modeling, SIAM, chapter Maximum Likelihood Estimation of Errors-In-Variables Models Using a Sample Covariance Matrix Obtained from Small Data Sets, pp. 59-68. 

Brar and Sunita [184] have reported the reactivity ratios (r) for the acrylonitrile]A)-methyl acrylate (M) monomer pair using the errors in variables model (EVM) [185, 186] with the use of a computer program written by O Driscoll and co-workers [187]. The primary structure factors monomer composition, diad/triad sequence distribution, conditional probabilities, and number-average sequence lengths of acrylonitrile-methyl acrylate copolymers were determined on the basis of C H -NMR analyses and compared with those calculated from reactivity ratios as determined from EVM program. The diad sequence calculated from C H)-NMR (proton decoupled C-NMR) spectra was correlated with the Tg of A/M copolymers. 

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