Errors-in-variables method

If the assumption of normality is grossly violated, ML estimates of the parameters can only be obtained using the error-in-variables" method where besides the parameters, we also estimate the true (error-free) value of the measured variables. In particular, assuming that Ey i is known, the parameters are obtained by minimizing the following objective function... 

Over the years two ML estimation approaches have evolved (a) parameter estimation based an implicit formulation of the objective function and (b) parameter and state estimation or "error in variables" method based on an explicit formulation of the objective function. In the first approach only the parameters are estimated whereas in the second the true values of the state variables as well as the values of the parameters are estimated. In this section, we are concerned with the latter approach. 

This is the so-called error in variables method. The formulation of the above optimality criterion was based on the following assumptions ... 

The error in variables method can be simplified to weighted least squares estimation if the independent variables are assumed to be known precisely or if they have a negligible error variance compared to those of the dependent variables. In practice however, the VLE behavior of the binary system dictates the choice of the pairs (T,x) or (T,P) as independent variables. In systems with a... 

Parameter estimation is also an important activity in process design, evaluation, and control. Because data taken from chemical processes do not satisfy process constraints, error-in-variable methods provide both parameter estimates and reconciled data estimates that are consistent with respect to the model. These problems represent a special class of optimization problem because the structure of least squares can be exploited in the development of optimization methods. A review of this subject can be found in the work of Biegler et al. (1986). 

Chapter 9 deals with the general problem of joint parameter estimation data reconciliation. Starting from the typical parameter estimation problem, the more general formulation in terms of the error-in-variable methods is described, where measurement errors in all variables are considered. Some solution techniques are also described here. 

In the error-in-variable method, measurement errors in all variables are treated in the calculation of the parameters. Thus, EVM provides both parameter estimates and reconciled data estimates that are consistent with respect to the model. 

In this chapter, the general problem of joint parameter estimation and data reconciliation was discussed. First, the typical parameter estimation problem was analyzed, in which the independent variables are error-free, and aspects related to the sequential processing of the information were considered. Later, the more general formulation in terms of the error-in-variable method (EVM), where measurement errors in all variables are considered in the parameter estimation problem, was stated. Alternative solution techniques were briefly discussed. Finally, joint parameter-state estimation in dynamic processes was considered and two different approaches, based on filtering techniques and nonlinear programming techniques, were discussed. 

Example 3.B Radiographic calibration by error-in-variables method... 

The application of the Error-in-Variables method [72], which considers the variance in all measured parameters to obtain more precise estimates has also led to a better agreement with experimental values [73]. 

Numerous reports are available [19,229-248] on the development and analysis of the different procedures of estimating the reactivity ratio from the experimental data obtained over a wide range of conversions. These procedures employ different modifications of the integrated form of the copolymerization equation. For example, intersection [19,229,231,235], (KT) [236,240], (YBR) [235], and other [242] linear least-squares procedures have been developed for the treatment of initial polymer composition data. Naturally, the application of the non-linear procedures allows one to obtain more accurate estimates of the reactivity ratios. However, majority of the calculation procedures suffers from the fact that the measurement errors of the independent variable (the monomer feed composition) are not considered. This simplification can lead in certain cases to significant errors in the estimated kinetic parameters [239]. Special methods [238, 239, 241, 247] were developed to avoid these difficulties. One of them called error-in-variables method (EVM) [239, 241, 247] seems to be the best. EVM implies a statistical approach to the general problem of estimating parameters in mathematical models when the errors in all measured variables are taken into account. Though this method requires more information than do ordinary non-linear least-squares procedures, it provides more reliable estimates of rt and r2 as well as their confidence limits. 

Kim, I.-W., T. F. Edgar, and N. H. Bell, Parameter estimation for a laboratory water-gas-shift reactor using a nonlinear error-in-variables method, Comput. Chem. Eng., 15, 361-367 (1991). 

Multiresponse analysis can provide fuller information, as illustrated in Investigation 11. There is a growing literature on error-in-variables methods. 

Errors in variables methods are particularly suited for parameter estimation of copolymerization models not only because they provide a better estimation in general but also, because it is relatively easy to incorporate error structures due to the different techniques used in measuring copolymer properties (i.e. spectroscopy, chromatography, calorimetry etc.). The error structure for a variety of characterization techniques has already been identified and used in conjunction with EVM for the estimation of the reactivity ratios for styrene acrylonitrile copolymers (12). 

The free-radical copolymerization of acrylamide with three common cationic comonomers diallyldimethylammonium chloride, dimethyl-aminoethyl methacrylate, and dimethylaminoethyl acrylate, has been investigated. Polymerizations were carried out in solution and inverse microsuspension with azocyanovaleric acid, potassium persulfate, and azobisisobutyronitrile over the temperature range 45 to 60 C. The copolymer reactivity ratios were determined with the error-in-variables method by using residual monomer concentrations measured by high-performance liquid chromatography. This combination of estimation procedure and analytical technique has been found to be superior to any methods previously used for the estimation of reactivity ratios for cationic acrylamide copolymers. A preliminary kinetic investigation of inverse microsuspension copolymerization at high monomer concentrations is also discussed. 

The error-in-variables method was used to estimate the reactivity ratios. This method was developed by Reilly et al. (57, 58), and it was first applied for the determination of reactivity ratios by O Driscoll, Reilly, and co-workers (59, 60). In this work, a modified version by MacGregor and Sutton (61) adapted by Gloor (62) for a continuous stirred tank reactor was used. The error-in-variables method shows two important advantages compared to the other common methods for the determination of copolymer reactivity ratios, which are statistically incorrect, as for example, Fineman-Ross (63) or Kelen-Tiidos (64). First, it accounts for the errors in both dependent and independent variables the other estimation methods assume the measured values of monomer concentration and copolymer composition have no variance. Second, it computes the joint confidence region for the reactivity ratios, the area of which is proportional to the total estimation error. 

Gloor, P. Estimation of Reactivity Ratios Using the Error-in-Variables Method and Data Collected from a Continuous Stirred Tank Reactor, MIPPT-Report, McMaster University, Hamilton, Ontario, Canada, 1987. 

With some classical polymerization reactions, e.g., styrene with methyl metacrylate, deviations were observed, e.g., by Fukuda et and O Driscoll and Reilly, whereby a penultimate unit effect was discussed and corrections introduced. This is expected on the basis of the Markov chain model because in this model only the last added unit controls the next step. As the correction method of O Driscoll also took errors in the variables into account, it is called the error in variables method (EVM). ... 

High, M.S., and Danner, R.P., Treatment of gas-solid adsorption data by the error-in-variables method, AlChE J., 32(7), 1138-1145 (1986). 

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