Error In Variables Method Research Articles

Acrylonitrile and Glycidyl Methacrylate Copolymers: Nuclear Magnetic Resonance Characterization.

Copolymers containing acrylonitrile (A) and glycidyl methacrylate (G) units of different compositions were synthesized by free radical solution polymerization. The reactivity ratios were estimated by Kelen Tudos and nonlinear error in variable methods. The triad sequence distributions in terms of A- and G-centered triads have been obtained from 13C{1H} NMR spectroscopy. The complete spectral assignments in terms of compositional and configurational sequences of the overlapping carbon and proton spectra of these copolymers were done with the help of distortionless enhancement by polarization transfer (DEPT), two-dimensional proton-detected heteronuclear correlation (inverse-HETCOR), and total correlated spectroscopy (TOCSY) experiments. The Monte Carlo simulation was used to study the effect of the fractional conversion on the triad fractions.

Estimation of activation parameters for the propagation rate constant of styrene

For a temperature range of −11.8–92.6°C, the propagation rate constant kp of styrene has been determined with the use of pulsed-laser polymerization (PLP). The temperature dependency of the obtained kp data was evaluated using the Arrhenius equation. The NLLS error-in-variables method (EVM) is recommended for this fit. The resulting activation energy is 32.6 kJ mol−1 and the pre-exponential factor is 107.66 dm3 mol−1 s−1. A joint confidence interval for these parameters is given. © 1996 John Wiley & Sons, Inc.

Compositional Assignments and Sequence Distribution of Vinylidene Chloride−Methyl Acrylate Copolymers Using One- and Two-Dimensional NMR Spectroscopy

The microstructure of vinylidene chloride−methyl acrylate copolymers prepared by a photopolymerization process using uranyl ion as photosensitizer is analyzed by one- and two-dimensional NMR spectroscopy. Sequence distribution was calculated from 13C{1H} NMR spectra of the copolymers. A 13C distortionless enhancement by polarization transfer spectrum was used to differentiate between the resonance signals of methoxy and methylene units in the copolymer. Comonomer reactivity ratios were determined using Kelen−Tudos and the nonlinear error in variables methods. The sequence distribution of vinylidene chloride- and methyl acrylate-centered triads determined from 13C{1H} NMR spectra of copolymers is in good agreement with triad concentrations calculated from statistical model and Monte Carlo simulation methods. 2D heteronuclear single quantum correlation and correlated spectroscopy spectrum was used to analyze the complex 1H NMR spectrum.

Copolymerization behavior of 2-vinyl-5-methyl furan

The title monomer was copolymerized with methyl methacrylate, isobutyl methacrylate n-butyl acrylate. The results of the copolymerization experiments were analyzed using a nonlinear least squares error-invariables method in order to obtain reactivity ratios for each monomer pair. In all cases the 2-vinyl-5-methyl furan proved to be the more reactive monomer, although this monomer is not as reactive as some other vinyl heterocycles have proved to be in copolymerizations with these same or similar comonomers.

Evaluation of parameter estimation methods for ozone disinfection kinetics

Inactivation data with ozone in a CSTR system were analyzed using error-in-variables method (EVM), forced linearization and classical nonlinear regression. Their relative efficiency was compared statistically. The Monte Carlo simulations demonstrated that the EVM produces parameter estimates with the least bias and minimum variance of residuals from the regression. Although used traditionally, the forced linearization would give the biased parameter estimates of kinetics and lead to the potential of misinterpreting the type of inactivation. Requiring the ad hoc assumption of first order decay rate, the classical nonlinear regression is of little use in practice. Fitting the kinetic model to experimental data shows that the behavior of the bacteria and virus inactivation can be well described by predictions from EVM estimates. The resultant coefficient of dilution ( n), defined as equation (1) was found to be much higher than one, indicating that the ozone concentration is a dominant factor over the contact time in determining the efficiency of ozone disinfection processes.

Reactivity ratios and microstructure determination of vinyl acetate–alkyl acrylate copolymers by NMR spectroscopy

Abstract(Vinyl acetate)/(ethyl acrylate) (V/E) and (vinyl acetate)/(butyl acrylate) (V/B) copolymers were prepared by free radical solution polymerization. 1H‐NMR spectra of copolymers were used for calculation of copolymer composition. The copolymer composition data were used for determining reactivity ratios for the copolymerization of vinyl acetate with ethyl acrylate and butyl acrylate by Kelen‐Tudos (KT) and nonlinear Error in Variables methods (EVM). The reactivity ratios obtained are rv = 0.03 ± 0.03, rE = 4.68 ± 1.70 (KT method); rv = 0.03 ± 0.01, rE = 4.60 ± 0.65 (EV method) for (V/E) copolymers and rv − 0.03 ± 0.01, rB − 6.67 ± 2.17 (KT method); rv = 0.03 ± 0.01, rB = 7.43 ± 0.71 (EV method) for (V/B) copolymers. Microstructure was obtained in terms of the distribution of V‐ and E‐centered triads and V‐ and B‐centered triads for (V/E) and (V/B) copolymers respectively. Homonuclear 1H 2D‐COSY NMR spectra were also recorded to ascertain the existence of coupling between protons in (V/E) as well as (V/B) copolymers. © 1995 John Wiley & Sons, Inc.

Synthesis and characterization of methacrylic acid–ethyl acrylate copolymers

AbstractCopolymerization of methacrylic acid (MAA) and ethyl acrylate (EA) was performed in aqueous medium in the presence of an emulsifier with MAA as the major component (53.7–82.3 mol %). Emulsion stability was studied as a function of concentration and the nature of the emulsifier (ionic or nonionic). Copolymers were characterized using IR and 1H‐NMR spectroscopy. 13C‐NMR spectra of MAA–EA copolymers have been discussed in terms of their triad monomer sequence distributions. Reactivity ratios of the MAA–EA pair were determined using a nonlinear least‐square errors in variables method. Partition of the monomers in aqueous and organic phases was studied using a gas chromatographic technique. A two‐loci polymerization mechanism was proposed on the basis of this partition behavior and the calculated composition of the oligomeric radicals formed in the initial stages of polymerization. © 1994 John Wiley & Sons, Inc.

Reduction of ion‐exchange equilibria data using an error in variables approach

AbstractAn error in variables method (EVM) was used to reduce binary ion‐exchange equilibria data. Ion‐exchange equilibria data were collected via batch equilibrium experiments. Three binary ion‐exchange systems were studied: the Na+1‐Cd+2, Na+1‐Cu+2, and Cu+2‐Cd+2 systems with a strong acid synthetic ion‐exchange resin. The Wilson model and a three‐suffix, two‐parameter Margules equation were used to model the dependency of the resin phase activity coefficients with resin‐phase composition. The EVM was used to determine the best‐fit estimates of the Wilson and Margules equation binary resin‐phase activity coefficient parameters. Statistical analysis of the data fits indicates that although both equations were able to adequately model the resin phase nonidealities, the Wilson model provided the superior data fits based on the minimized objective functions.

Reactivity ratios and microstructure of vinyl acetate–butyl methacrylate copolymers: NMR study

AbstractVinyl acetate/butyl methacrylate (V/B) copolymers were prepared by free‐radical solution polymerization in benzene. Compositions of the copolymers have been obtained from 1H‐NMR spectroscopy. Kelen–Tüdös (KT), the nonlinear error in variables method (EVM), and 13C{1H}‐NMR spectra of the copolymers were utilized to calculate the comonomer reactivity ratios. The reactivity from different methods are rV = 0.06 ± 0.02, rB = 26.34 ± 8.49 (KT); rV = 0.06 ± 0.01, rB = 26.88 ± 3.59 (EVM); and rV = 0.06, rB = 18.20 (13C‐NMR spectroscopy). The microstructure in terms of V‐ and B‐centered triad fractions was obtained using 13C{1H}‐NMR spectra of copolymers. The copolymerization behavior of V/B copolymers as a function of conversion was also studied. © 1994 John Wiley & Sons, Inc.

Microstructure determination of styrene/ethyl acrylate copolymers by NMR spectroscopy and its correlation with glass transition temperature

AbstractStyrene/ethyl acrylate (S/E) copolymers of different compositions were prepared by bulk polymerization using benzoyl peroxide as an initiator. Copolymer compositions were determined from NMR spectra and reactivity ratios r1 and r2 were calculated using a non‐linear least‐squares errors‐in‐variables method (EVM) which gives more reliable values of r1 and r2. The copolymerization parameters such as terminal, penultimate reactivity ratios, diad concentrations, conditional probabilities, number‐average sequence lengths and run numbers in the copolymers were calculated by evaluating 13C{1H} NMR spectra. The observed sequence concentrations of triads as determined from 13C{1H} NMR spectra were in good agreement with those calculated from the reactivity ratios. The coisotacticity parameter (σ) was calculated from NMR spectroscopy data using the oxymethylene resonance signal in 1H NMR spectra. The glass transition temperatures (Tg) of various copolymers were obtained from Barton's equation (using diad relation) and are in agreement with those determined by means of differential scanning calorimetry.

Microstructure Determination of Styrene-Butyl Acrylate Copolymers by NMR Spectroscopy

Styrene-butyl acrylate (S/B) copolymers of different composition were prepared by bulk polymerization using a free radical initiator. Copolymer composition was determined from NMR spectra and monomer reactivity ratios were calculated using the nonlinear least square Error-in-Variable method (EVM). Copolymerization parameters such as triad concentrations, conditional probabilities and number-average sequence lengths in the copolymers were calculated from 13C{1H} NMR spectra. The observed triad sequence concentrations as determined from 13C{1H} NMR spectra were in good agreement with those calculated from the reactivity ratios. Coisotacticity parameter (σ) has been calculated from NMR spectroscopy using oxymethylene resonance signal in 1H NMR spectra.

Reduced successive quadratic programming strategy for errors-in-variables estimation

Parameter estimation problems in process engineering represent a special class of nonlinear optimization problems, because the maximum likelihood structure of the objective function can be exploited. Within this class, the errors in variables method (EVM) is particularly interesting. Here we seek a weighted least-squares fit to the measurements with an underdetermined process model. Thus, both the number of variables and degrees of freedom available for optimization increase linearly with the number of data sets. Large optimization problems of this type can be particularly challenging and expensive to solve because, for general-purpose nonlinear programming (NLP) algorithms, the computational effort increases at least quadratically with problem size. In this study we develop a tailored NLP strategy for EVM problems. The method is based on a reduced Hessian approach to successive quadratic programming (SQP), but with the decomposition performed separately for each data set. This leads to the elimination of all variables but the model parameters, which are determined by a QP coordination step. In this way the computational effort remains linear in the number of data sets. Moreover, unlike previous approaches to the EVM problem, global and superlinear properties of the SQP algorithm apply naturally. Also, the method directly incorporates inequality constraints on the model parameters (although not on the fitted variables). This approach is demonstrated on five example problems with up to 102 degrees of freedom. Compared to general-purpose NLP algorithms, large improvements in computational performance are observed.

Estimation of a simple class of multivariate rational expectations models: A test of the new classical model at a sectoral level

This paper extends the results obtained by Pagan (1984) and Turkington (1985) for single equation rational expectations (RE) models to multivariate RE models and shows that the errorsin-variables method and the substitution method discussed in Wickens (1982) lead to exactly the same likelihood function. The paper also considers multivariate RE models with unanticipated variables and includes an empirical application to the problem of testing the natural rate-rational expectations (NR-RE) hypothesis at the disaggregate level using U.S. data over the period 1955–1985.

Parameter estimation for a laboratory water—gas-shift reactor using a nonlinear error-in-variables method

The weighted least squares (WLS) method and the error-in-variables method (EVM) have been compared for the estimation of parameters in a laboratory water—gas-shift reactor. In EVM, measurement errors in all variables are treated in the calculation of regression coefficients, whereas WLS only considers errors in the dependent variables. While EVM is computationally more intensive, the method provides both parameter estimates and reconciled data estimates, providing unbiased estimates and more accurate predictions. The prediction of the temperature profiles through the reactor bed by the EVM was more accurate than those obtained by LS. In addition, the true estimated values of the variables were used to check for systematic measurement errors through a t-test.

Robust error‐in‐variables estimation using nonlinear programming techniques

AbstractFor systems described by algebraic or differential equation models where all variables are subject to error, the error‐in‐variables method (EVM) for parameter estimation has been shown to be superior to standard least‐squares techniques. Previous EVM algorithms were developed assuming linear (or linearized) model equations. Unfortunately, many chemical engineering processes operate in strongly nonlinear regions where linear approximations may be inaccurate. In this paper, new algorithms using nonlinear programming techniques for the error‐in‐variables methods are proposed. In addition, a method for discerning when these methods are necessary is discussed. The proposed algorithms are compared to the least‐squares method and traditional error‐in‐variable approaches. Improved parameter estimates for several steady‐state nonlinear processes are demonstrated.

Error in Variables Parameter Estimation

The application of the errors in variables method (EVM) to environmental engineering practice using, as an example, the determination of kinetic parameters from a laboratory biodegradability study is presented. Model parameters must often be estimated from field or laboratory observations to describe biological treatability studies, adsorption, and water quality modeling. Such models consist of one or more linear or nonlinear equations. Usually, a function of the observed variables is designated as the “dependent.” The model parameter values minimizing the sum of squares of deviations between the observed and predicted “dependent” variable is accepted as “best.” Even for linear models, this method is biased when the independent variables are measured with error. Forced linearization of instrinsically nonlinear models may produce biased point estimates or interval estimates of parameter. Nonlinear parameter estimation methods when some or all of the variables are measured with error, designated as EVM, recently have been developed and utilized in chemical engineering for the reduction of thermodynamic data.

Rigorous approach to kinetic parameters: polymerization of styrene with benzoyl peroxide in carbon tetrachloride

AbstractBoth the rate constant of initiator dissociation, kd, and the chain transfer constant, Cx, are determined by monitoring the rates of monomer and transfer agent conversions in a chain polymerization. Statistical analysis that takes into account the errors of experimental variables has been used to treat the kinetic data for the polymerization of styrene initiated by benzoyl peroxide (Bz2O2) in carbon tetrachloride at 60°C. Values of kd and Cx produced by this error‐in‐variable method (EVM) are in accordance with those reported in literature. More important, the EVM algorithm always results in the smallest standard deviations of estimates, and hence, it is superior to the usual least‐squares calculations. Other distinct features of the method are outlined. The initiator efficiency for Bz2O2 is estimated to be 0.72 under the present conditions.

The Efficient Estimation of Econometric Models with Rational Expectations

The efficient estimation of econometric models with rational expectations by the substitution method usually entails the use of a non-linear simultaneous equations estimator and hence is not very attractive computationally. It is shown that for a wide class of problems efficient estimates can also be obtained with standard estimation methods if instead an errors in variables approach is used. For many other problems, although not providing fully efficient estimates because rationality is not imposed, the errors in variables method will still have a strong appeal because otherwise it uses all of the structural information in the model; unlike the substitution method, when an incomplete information set is used, it guarantees consistent estimates; and it is easy to compute.