Optimization regression analysis

Another problem is to determine the optimal number of descriptors for the objects (patterns), such as for the structure of the molecule. A widespread observation is that one has to keep the number of descriptors as low as 20 % of the number of the objects in the dataset. However, this is correct only in case of ordinary Multilinear Regression Analysis. Some more advanced methods, such as Projection of Latent Structures (or. Partial Least Squares, PLS), use so-called latent variables to achieve both modeling and predictions. 

A non-linear regression analysis is employed using die Solver in Microsoft Excel spreadsheet to determine die values of and in die following examples. Example 1-5 (Chapter 1) involves the enzymatic reaction in the conversion of urea to ammonia and carbon dioxide and Example 11-1 deals with the interconversion of D-glyceraldehyde 3-Phosphate and dihydroxyacetone phosphate. The Solver (EXAMPLEll-l.xls and EXAMPLEll-3.xls) uses the Michaehs-Menten (MM) formula to compute v i- The residual sums of squares between Vg(,j, and v j is then calculated. Using guessed values of and the Solver uses a search optimization technique to determine MM parameters. The values of and in Example 11-1 are ... 

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. 

The separation of synthetic red pigments has been optimized for HPTLC separation. The structures of the pigments are listed in Table 3.1. Separations were carried out on silica HPTLC plates in presaturated chambers. Three initial mobile-phase systems were applied for the optimization A = n-butanol-formic acid (100+1) B = ethyl acetate C = THF-water (9+1). The optimal ratios of mobile phases were 5.0 A, 5.0 B and 9.0 for the prisma model and 5.0 A, 7.2 B and 10.3 C for the simplex model. The parameters of equations describing the linear and nonlinear dependence of the retention on the composition of the mobile phase are compiled in Table 3.2. It was concluded from the results that both the prisma model and the simplex method are suitable for the optimization of the separation of these red pigments. Multivariate regression analysis indicated that the components of the mobile phase interact with each other [79],... 

Fu et al. [16] analyzed a set of 57 compounds previously used by Lombardo and other workers also. Their molecular geometries were optimized using the semiempirical self-consistent field molecular orbital calculation AMI method. Polar molecular surface areas and molecular volumes were calculated by the Monte Carlo method. The stepwise multiple regression analysis was used to obtain the correlation equations between the log BB values of the training set compounds and their structural parameters. The following model was generated after removing one outlier (Eq. 50) ... 

In reference 88, response surfaces from optimization were used to obtain an initial idea about the method robustness and about the interval of the factors to be examined in a later robustness test. In the latter, regression analysis was applied and a full quadratic model was fitted to the data for each response. The method was considered robust concerning its quantitative aspect, since no statistically significant coefficients occurred. However, for qualitative responses, e.g., resolution, significant factors were found and the results were further used to calculate system suitability values. In reference 89, first a second-order polynomial model was fitted to the data and validated. Then response surfaces were drawn for... 

Repeating regression analysis with these newly established data, the equation with was optimal, although the statistical significance was worse, r = 0.77 for Equation 6 versus 0.96 for Equation 3. 

The total surface area of a FCC catalyst is the sum of the zeolite and matrix surface areas and is therefore not useful for optimizing of the catalysts. However, the ratio between the zeolite and the matrix surface areas (ZSA/MSA) is a valuable parameter, and has been used for optimization of vacuum gas oil catalysts [4] as well as catalysts for North Sea long residue feeds [9,13]. Additional information about the catalyst is also gained by studying the yields as a function of the zeolite surface area and as a function of the matrix surface area [9]. The regression analysis in this paper is performed at a constant conversion of 75 wt%. 

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. 

Multiple regression analysis is a useful statistical tool for the prediction of the effect of pH, suspension percentage, and composition of soluble and insoluble fractions of oilseed vegetable protein products on foam properties. Similar studies were completed with emulsion properties of cottonseed and peanut seed protein products (23, 24, 29, 30, 31). As observed with the emulsion statistical studies, these regression equations are not optimal, and predicted values outside the range of the experimental data should be used only with caution. Extension of these studies to include nonlinear (curvilinear) multiple regression equations have proven useful in studies on the functionality of peanut seed products (33). 

Here, n, v, and p represent a specific growth rate, a specific substrate consumption rate, and a specific product formation rate, respectively. and are the mean values of data used for regression analysis and a, bp and C are the coefficients in the regression models that are determined based on selected operating data in a database. This model was linked with the dynamic programming method and successfully applied to the simulation and onhne optimization of glutamic acid production and Baker s yeast production. 

Optimization techniques may be classified as parametric statistical methods and nonparametric search methods. Parametric statistical methods, usually employed for optimization, are full factorial designs, half factorial designs, simplex designs, and Lagrangian multiple regression analysis [21]. Parametric methods are best suited for formula optimization in the early stages of product development. Constraint analysis, described previously, is used to simplify the testing protocol and the analysis of experimental results. 

By executing the steps of the analytical process, we can take advantage of most of the basic methods of chemometrics, e.g., statistics including analysis of variance, experimental design and optimization, regression modeling, and methods of time series analysis. 

A further reduction of experimental effort may be achieved by the selection of special designs developed, for example, by HARTLEY [1959], BOX and BEHNKEN [1960], WESTLAKE [1965], and others. In these designs the ratio of experiments to the number of coefficients necessary is reduced almost to unity. (This situation is somewhat different from regression analysis or random selection of experiments where, in principle, k experiments or measurements are sufficient to estimate k parameters of a model. In experimental design the optimized number of experiments is derived from statistical consideration to encompass as much variation of the factors as possible.)... 

However, a simple linear relationship does not usually exist. A clear example is the optimization of the pH in RPLC. The window diagram approach was applied to this problem by Deming et al. [550,551,552]. They measured the retention of each solute at a series of pH values (9 in ref. [550], 4 in refs. [551,552]) and fitted the experiments to eqn.(3.70). This is a three-parameter equation and hence a minimum of three experiments is required for it to be applied as a description of the retention surface. If more data points are available, the equation can be fitted to the data by regression analysis. 

The kinetic and deactivation models were fitted by non-linear regression analysis against the experimental data using the Modest software, especially designed for the various tasks -simulations, parameter estimation, sensitivity analysis, optimal design of experiments, performance optimization - encountered in mathematical modelling [6], The main interest was to describe the epoxide conversion. The kinetic model could explain the data as can be seen in Fig. 1 and 2, which represent the data sets obtained at 70 °C and 75°C, respectively. The model could also explain the data for hydrogenated alkyltetrahydroanthraquinone. 

We next sought to improve upon the results of Equations 1 and 2 by determining optimal values of n for Equation 3. This was carried out by regression analysis with Equation 3b, and the results appear in Table III. With one exception, the optimal ns all fall between 0.5 and 1.0. The overall mean n for both properties in Table IV was 0.68 it= 0.18, with the optimal n for reflectance closer to 0.5, and the optimal n for folding endurance closer to 1.0. 

Randic, M. and Basak, S.C. (1999b). Multiple Regression Analysis with Optimal Molecular Descriptors. SAR QSAR Environ.Res., in press. 

The coupled thermodynamic analysis, i.e. the calculation of the coefficients Gj in Eq. (3.204) is performed using the multiple linear regression analysis omitting the statistically non-important terms according to the Student test on the chosen confidence level. As the optimizing criterion for the best fit between the experimental and calculated temperatures of primary crystallization, the following condition was used for all the p measured points... 

The above analysis is, of course, based on the assumption of simple order reactions under Tafel operation and on the availability of sufficiently accurate data ( 5-10%). With complex reaction kinetics, for example, those involving surface adsorption terms (Eq. 16), a nonlinear regression analysis would yield the best estimate of a, Uj, and for a possible kinetic model. In all cases, use of these parameters for predicting the performance of an electrochemical reactor or the selectivity of a reaction scheme should be restricted within the potential, concentration, and temperature range that they were determined. We should stress here that kinetic information is presently scanty for complex, multiple electrochemical reactions, yet it is essential for the design, optimization, and control of electrochemical processes. 

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