Regression Techniques

Partial least-squares path modeling with latent variables (PLS), a newer, general method of handling regression problems, is finding wide apphcation in chemometrics. This method allows the relations between many blocks of data ie, data matrices, to be characterized (32—36). Linear and multiple regression techniques can be considered special cases of the PLS method. 

Some formulas, such as equation 98 or the van der Waals equation, are not readily linearized. In these cases a nonlinear regression technique, usually computational in nature, must be appHed. For such nonlinear equations it is necessary to use an iterative or trial-and-error computational procedure to obtain roots to the set of resultant equations (96). Most of these techniques are well developed and include methods such as successive substitution (97,98), variations of Newton s rule (99—101), and continuation methods (96,102). 

Implementation Issues A critical factor in the successful application of any model-based technique is the availability of a suitaole dynamic model. In typical MPC applications, an empirical model is identified from data acquired during extensive plant tests. The experiments generally consist of a series of bump tests in the manipulated variables. Typically, the manipulated variables are adjusted one at a time and the plant tests require a period of one to three weeks. The step or impulse response coefficients are then calculated using linear-regression techniques such as least-sqiiares methods. However, details concerning the procedures utihzed in the plant tests and subsequent model identification are considered to be proprietary information. The scaling and conditioning of plant data for use in model identification and control calculations can be key factors in the success of the apphcation. 

Multiple regression techniques have been applied by investigators to determine the coefficients in a plume rise equation containing both of the above terms ... 

Versions of Volume I exist for C, Basic, and Pascal. Matlab enthusiasts will find some coverage of optimization (and nonlinear regression) techniques in... 

Whenever a linear relationship between dependent and independent variables (ordinate-resp. abscissa-values) is obtained, the straightforward linear regression technique is used the equations make for a simple implementation, even on programmable calculators. 

Ripley, B. D., and Tompson, M., Regression Techniques for the Detection of Analytical Bias, Analyst 112, April 1987, 377-383. 

Garden, J. S., Mitchell, D. G., and Mills, W. N., Nonconstant Variance Regression Techniques for Calibration-Curve-Based Analysis, Anal. Chem. 52, 1980, 2310-2315. 

Very often empirical equations can be developed from plant data using multiple regression techniques. The main advantage of this approach is that the correlations are often linear, can be easily coupled to optimization algorithms, do not cause convergence problems and are easily transferred from one computer to another. However, there are disadvantages, namely,... 

In the subsequent paragraphs and sections of this chapter we will see that these two issues impose requirements that transgress the abilities of simple smoothing filters and conventional regression techniques. 

Kinetic analysis of the data obtained in differential reactors is straightforward. One may assume that rates arc directly measured for average concentrations between the inlet and the outlet composition. Kinetic analysis of the data produced in integral reactors is more difficult, as balance equations can rarely be solved analytically. The kinetic analysis requires numerical integration of balance equations in combination with non-linear regression techniques and thus it requires the use of computers. 

It must be emphasized that Procrustes analysis is not a regression technique. It only involves the allowed operations of translation, rotation and reflection which preserve distances between objects. Regression allows any linear transformation there is no normality or orthogonality restriction to the columns of the matrix B transforming X. Because such restrictions are released in a regression setting Y = XB will fit Y more closely than the Procrustes match Y = XR (see Section 35.3). 

Note that the lipophilicity parameter log P is defined as a decimal logarithm. The parabolic equation is only non-linear in the variable log P, but is linear in the coefficients. Hence, it can be solved by multiple linear regression (see Section 10.8). The bilinear equation, however, is non-linear in both the variable P and the coefficients, and can only be solved by means of non-linear regression techniques (see Chapter 11). It is approximately linear with a positive slope (/ ,) for small values of log P, while it is also approximately linear with a negative slope b + b for large values of log P. The term bilinear is used in this context to indicate that the QSAR model can be resolved into two linear relations for small and for large values of P, respectively. This definition differs from the one which has been introduced in the context of principal components analysis in Chapter 17. 

B. Waiczack and D.L. Massart, The radial basis functions — partial least squares approach as a flexible non-linear regression technique. Anal. Chim. Acta, 331 (1996) 177-185. 

Instrument calibration is done during the analysis of samples by interspersing standards among the samples. Following completion of the samples and standards, a linear calibration curve is estimated from the response of the standards using standard linear regression techniques. The calibration constants obtained from each run are used only for the samples quantitated in that run. Drastic changes or lack of linearity may indicate a problem with the detector. 

One of the most widely used experimental plans is that of the factorial design, or some variation of it (two of the techniques in the following section utilize it). By multiple regression techniques, the relationships... 

Draper NR, Smith H (1981) Applied regression techniques, 2nd edn. Wiley, New York Edgeworth FY (1887) On observations relating to several quantities. Hermathena 6 279... 

Garden JS, Mitchell DG, Mills WH (1980) Nonconstant variances regression techniques for calibration-curve-based analysis. Anal Chem 52 2310... 

Regression Algorithms. The fitting of structural models to X-ray scattering data requires utilization of nonlinear regression techniques. The respective methods and their application are exhausted by Draper and Smith [270], Moreover, the treatment of nonlinear regression in the Numerical Recipes [154] is recommended. 

Calculating the Solution for Regression Techniques Part 1 - Multivariate Regression Made Simple... 

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