Regression Analysis and Parameter Estimation

Regression Analysis and Parameter Estimation By expanding Equation 7.7 and defining... 

Therefore, it is good engineering practice to use in reactor models so-called effective parameters by lumping several basic mechanisms or reaction steps into an overall phenomenon described by one single term in the respective reactor or reaction model. In this way, the parameters are always bound to the model equations by which they have been defined. The parameter values determined by regression analysis cannot be used independently in other model equations like physical properties, flow velocity etc., measurable independently without any model definition. This fundamental dilemma has some consequences in kinetic data analysis and parameter estimation that should always be kept in mind ... 

The authors are indebted to Messrs. L.T. Hillegers and A.G. Swenker for assistance in the non-linear regression analysis and the parameter estimation, to Mr. J.H.M. Palmen for experimental help, to Mr. G. Schuler for drawing the figures and to Mrs. 

On a practical level, the heuristic approach includes first collecting all the possible data during the experiments as a function of the parameters which are deemed to be important, i.e. concentrations, temperature, pressures, pH, catalyst concentration, volume, etc. Then the rate constants are estimated by regression analysis and the adequacy of the model is judged based on some criteria (like residual sums and parameter significance, which will be discussed further). If a researcher is not satisfied, then additional experiments are performed, followed by parameter estimation and sometimes simulations outside the studied parameter domain. The latter procedure provides the possibility to test the predictive power of a kinetic model. The kinetic model is then gradually improved and the experimental plan is modified, if needed. This process continues until the researcher is satisfied with the kinetic model. 

Nonlinear regression analysis of the H2-TPD was implemented in MATLAB 6 and parameter estimation was carried out to determine Arefi, AHj and the adsorption capacity Vmi for the required number of states. (Temperature mean-centring for Ki was applied in the estimation)... 

To indicate a relationship between the average acidity on a yearly basis and the changes occurring in the total interior emissions of SO2 and NOx, by means of a regression analysis the best estimates have been determined for the parameters in the following model ... 

Although the tested catalyst shows a good intrinsic selectivity for butadiene hydrogenation, the results evidenced the presence of severe diffusion limitations in spite of the thin active shell (230 im). The experimental data were modeled by Langmuir-Hinshelwood kinetic expressions derived from an elementary mechanism. Nine kinetic parameters were reliably estimated by means of a regression analysis and it is concluded that the proposed kinetic model provides a good fitting of the experimental observations. 

The problem of estimating the parameters 0 in (1) is a part of regression analysis, and its solution are estimates by the least squares method ... 

The first two examples show that the interaction of the model parameters and database parameters can lead to inaccurate estimates of the model parameters. Any use of the model outside the operating conditions (temperature, pressures, compositions, etc.) upon which the estimates are based will lead to errors in the extrapolation. These model parameters are effec tively no more than adjustable parameters such as those obtained in linear regression analysis. More comphcated models mav have more subtle interactions. Despite the parameter ties to theoiy, tliey embody not only the uncertainties in the plant data but also the uncertainties in the database. 

The following expressions can be used to estimate the temperature and enthalpy of steam. The expressions are based upon multiple regression analysis. The equation for temperature is accurate to within 1.5 % at 1,000 psia. The expression for latent heat is accurate to within + 3 % at 1,000 psia. Input data required to use these equations is the steam pressure in psia. The parameters in the equations are defined as t for temperature in F, for latent heat in Btu/lb, and P for pressure in psia. 

Appendix B. Parameter estimation and statistical analysis of regression... 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

The models with insignificant overall model regression as indicated by the F -value and with meaningless parameter estimates (with confidence limits) as indicated by r-values should be rejected. If rejection of the parameter does not lead to a physically nonsensical model stmcture, repeat parameter estimation and statistical analysis. 

Two models of practical interest using quantum chemical parameters were developed by Clark et al. [26, 27]. Both studies were based on 1085 molecules and 36 descriptors calculated with the AMI method following structure optimization and electron density calculation. An initial set of descriptors was selected with a multiple linear regression model and further optimized by trial-and-error variation. The second study calculated a standard error of 0.56 for 1085 compounds and it also estimated the reliability of neural network prediction by analysis of the standard deviation error for an ensemble of 11 networks trained on different randomly selected subsets of the initial training set [27]. 

The unknown model parameters will be obtained by minimizing a suitable objective function. The objective function is a measure of the discrepancy or the departure of the data from the model i.e., the lack of fit (Bard, 1974 Seinfeld and Lapidus, 1974). Thus, our problem can also be viewed as an optimization problem and one can in principle employ a variety of solution methods available for such problems (Edgar and Himmelblau, 1988 Gill et al. 1981 Reklaitis, 1983 Scales, 1985). Finally it should be noted that engineers use the term parameter estimation whereas statisticians use such terms as nonlinear or linear regression analysis to describe the subject presented in this book. 

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