Experimental regression coefficient, application

This example refers to the verification of lack of fit of the second-order regression model, that was obtained by application of CCOD in Example 2.51. Experimental outcomes are shown in Table 2.165. The calculated regression coefficients form this regression model ... 

Regression coefficients b2, b5 and b7 are statistically insignificant so that associated factors when applying the method of steepest ascent are fixed at corresponding levels. Other significant regression coefficients are symmetrical, which has been proved by successful application of the method of steepest ascent. Due to the fact that the optimum is in the vicinity of the experimental region, it is possible to switch to a second-order model. 

The simplest form of regression is multiple linear regression (MLR), Y = XB + E. Here, X contains the descriptors, [di... dn] B contains the regression coefficients, Y contains the figures of merit and E contains the residuals. One well known example of MLR is the relationship shown in Equation (6.9). This model requires a few well-characterized parameters d. .. dn, which are usually derived from experimental measurements or from QM calculations. There are several applications of MLR in catalysis, eg., the quantitative analysis of ligand effects (QALE) model developed by Fernandez et al. [90]. 

Computer simulation of separations for binary mobile phases containing a single strong solvent are usually based on one of the functions Rm = a log (Xs) + b or Rp = a (Xs) + b (Xs) + c where a, b and c are regression constants and Xs is the mole fraction of strong solvent [167]. A certain amount of experimental data is required to determine the regression coefficients after which additional Rp values can be estimated by interpolation. This approach can only estimate results for compounds included in the initial experiments. Results for simulation of two-dimensional separations based on sequential application of the above equations for one-dimensional separations varied from poor to reasonable without obvious reasons for the variation [61,168,169]. 

The following table provides the partition coefficients (or distribution coefficients), K = Cs/Cv (solid/vapor), at various temperatures, for application in gas chromatographic headspace analysis.1,2 The values marked with an asterisk were determined from a linear regression of experimental data. 

The nonlinear least squares regression program PEST [79] was used to fit the proposed correlation relating the time invariant Sherwood number to overall Peclet numbers for circular pools given by Eq. (91) to the seven experimentally determined Sh values presented in Fig. 12b, in order to estimate the empirical coefficients fi, y2> and y3. The experimental, overall Sherwood number correlation applicable to circular TCE pool dissolution in water saturated, homogeneous porous media can be expressed by the following relationship ... 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

In the same way as for the % x 2 design, we can select the data for these 12 experiments from the experimental results of table 3.8 (2 effervescent tablet factor study) and estimate the coefficients by multi-linear regression. (The linear combinations method is not applicable here.) The results, given in table 3.36, are almost identical to those found with the full design and reported in table 3.9. 

Regression analysis is one of the main tools in generating mathematical models by fitting a model equation to experimental data. In general, the regression analysis is based on the application of the least squares method for the estimation of unknown coefficients in the model equation. This method minimizes the sum of squares of the differences between the experimental values of the dependent variable, and those estimated by the model, y . Polynomials of various degrees are often used to describe complex non-linear relationships between the dependent and independent variables because the model equation is linear with respect to the unknown coefficients, and therefore the procedure for the calculation of the coefficients reduces to the solution of a system of linear simultaneous equations. 

Successful prediction of the system, ignoring possible ion association, could be made by regressing ternary experimental data for the activity coefficients and/or equilibrium constants. However, these regression results would be applicable only to that ternary system due to the compensating effects. Regression results are most "portable" when based on binary solution data. 

The application of linear and nonlinear regression analysis to fit mathematical models to experimental data and to evaluate the unknown parameters of these models (see Chap. 7) requires the repetitive solution of sets of linear algebraic equations. In addition, the ellipse formed by the correlation coefficient matrix in the parameter hyperspace of these systems must be searched in the direction of the major and minor axes. The directions of these axes are defined by the eigenvectors of the correlation coefficient matrix, and the relative lengths of the axes are measured by the eigenvalues of the correlation coefficient matrix. 

---