Regression Routine Example

Referring to Table 11.3, there are a = 3 treatments, making a — 1 = 2 indicator variables, 7. 

As earlier, the regression when both /j and I2 equal zero is for treatment 3. The new codes are presented in Table 11.9. The model is 

Note that Fc for treatment 27.91 is the same as determined from the covariance analysis. 

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Nonlinearity is a subject the specifics of which are not prolifically or extensively discussed as a specific topic in the multivariate calibration literature, to say the least. Textbooks routinely cover the issues of multiple linear regression and nonlinearity, but do not cover the issue with full-spectrum methods such as PCR and PLS. Some discussion does exist relative to multiple linear regression, for example in Chemometrics A Textbook by D.L. Massart et al. [6], see Section 2.1, Linear Regression (pp. 167-175) and Section 2.2, Non-linear Regression, (pp. 175-181). The authors state,... 

Principal component regression (PCR) is an extension of PCA with the purpose of creating a predictive model of the Y-data using the X or measurement data. For example, if X is composed of temperatures and pressures, Y may be the set of compositions that results from thermodynamic considerations. Piovoso and Kosanovich (1994) used PCR and a priori process knowledge to correlate routine pressure and temperature measurements with laboratory composition measurements to develop a predictive model of the volatile bottoms composition on a vacuum tower. 

Robust, multimethod regression codes are required to optimize the rate parameters, also in view of possible strong correlations. For example, the BURENL routine, specifically developed for regression analysis of kinetic schemes (Donati and Buzzi-Ferraris, 1974 Villa et al., 1985) has been used in the case of SCR modeling activities. The adaptive simplex optimization method Amoeba was used for minimization of the objective function Eq. (35) when evaluating kinetic parameters for NSRC and DOC. 

The two models in Sections VI. 1 and VI.2 have been solved by a numerical method based on a finite difference routine BAND (j).718,20 To solve a non-linear model, iteration with trail values is required. Furthermore, double iterations are needed in cases, for example, when it is required to optimize the thickness of the PBER, or to regress the key parameters from experimental data. These complex situations make the convergences of the solution difficult. 

For fitting the binary interaction parameters nonlinear regression methods are applied, which allow adjusting the parameters in such a way that a minimum deviation of an arbitrary chosen objective function F is obtained. For this job, for example, the Simplex-Nelder-Mead method (21j can be applied successfully. The Simplex-Nelder-Mead method in contrast to many other methods [22] is a simple search routine, which does not need the first and the second derivate of the objective function with respect to the different variables. This has the great advantage that computational problems, such as "underflow or overflow with the arbitrarily chosen initial parameters can be avoided. 

There is no necessity in practice for the manual calculation of all these results, which would clearly be too tedious for routine use. The application of a spreadsheet program to some regression data is demonstrated in Section 5.9. Every advantage should also be taken of the extra facilities provided by programs such as Minitab, for example plots of residuals against x or y values, normal probability plots for the residuals, etc. (see also Section 5.15). 

They fall into two groups - those derived from regression analysis of historical process data and first-principle types which rely on engineering calculations. First-principle techniques still require some historical data to calibrate the model and to check its accuracy. While the vendors of first-principle techniques might argue that the volume of data required is less, the key to the success of both techniques is the quality of the data. The use of routinely collected data, for example from a plant history database, can often cause inaccuracies in the end result. 

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