Linear regression technique

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. 

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. 

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. 

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. 

As predicted, the points next to the ends of the profile give unreliable (negative) estimates because the fluxes at the end points themselves become a substantial fraction of the fluxes in the neighborhood. In addition, noise in sections with rather flat gradients becomes a problem. More elaborated non-linear regression techniques should be used to handle this specific problem. Nevertheless, the profile between IS and SO/rm, where steep Ce variations are observed, shows evidence for substantial changes of the Off which seems to hint at much faster diffusion when this element is only in trace amounts in the apatite lattice, o... 

When the Mg isotope data are cast in terms of the new 8 values, they can be subjected to well-established linear regression techniques to extract the best-fit exponent p. A handy equation that converts the original 8 values to the new, linear 8 values is... 

The fundamental principles of pK determination by CE rely upon measuring the ionic effective mobility of the solute as a function of pH. Consequently, the pK value is obtained by fitting an equilibrium equation to the effective mobility and pH data with a non-linear regression technique (Figure 1). Depending on the chemical character of the compound... 

It was found that the requirements were satisfied for application of the linear regression technique to species mass concentrations in a multicomponent aerosol. The results of 254 particle size distributions measured at China Lake in 1979 indicate that the normalized fine aerosol volume distribution remained approximately constant. The agreement between the calculated and measrued fine particle scattering coefficients was excellent. The measured aerosol sulfur mass distribution usually followed the total distribution for particles less than 1 ym. It was assumed that organic aerosol also followed the total submicron distribution. 

The usefulness of the LSER approach hinges on the similarity of the partitioning coefficients obtained from the sensing experiments (Ks) and the gas chromatographic experiments (Kqc)- In other words, it is assumed that the relationship Ks Kgc holds. This is how LSER is used for evaluation of a new sensing material. First, the coefficient Kqc is obtained from the tabulated database or experimentally. Second, the multiple linear regression technique (see Chapter 10) is used to obtain the best fit for the sensor test data, and the individual coefficients in (2.3) are evaluated. This approach has been used successfully in evaluation of multiple materials for gas sensors (Abraham et al., 1995 Grate et al., 1996). 

As stated above, the utility of the ML estimators derives essentially from their asymptotic properties of consistency and optimality (i.e., cov(0ml) — CRB). When the data exhibits significant departures from theoretical pdf (Gaussian or Rice) owing to acquisition artifacts, it may be judicious to use robust non-linear regression techniques,62 as in parametric diffusion-tensor imaging reconstructed from echo-planar data.63... 

A value of 5.2A for the ion-size parameter a yielded straight-line plots of E0 vs. m at each temperature and solvent composition. The intercepts were obtained by standard linear regression techniques. A graphical representation of the data for each of the H20/NMA solvent mixtures at 25°C is shown in Figure 1. The calculations were performed with the aid of a PDP-11 computer with a teletype output. The intercepts (E°) and the standard deviations of the intercepts are summarized in Table III. 

In organic chemistry, decomposition of molecules into substituents and molecular frameworks is a natural way to characterize molecular structures. In QSAR, both the Hansch-Fujita " and the Free-Wilson classical approaches are based on this decomposition, but only the second one explicitly accounts for the presence or the absence of substituent(s) attached to molecular framework at a certain position. While the multiple linear regression technique was associated with the Free-Wilson method, recent modifications of this approach involve more sophisticated statistical and machine-learning approaches, such as the principal component analysis and neural networks. ... 

Eq. (6.12) describes a non-linear model. The term non-linear is used here, and only in this section, in a statistical sense. In that sense an equation such as Eq. (6.2) is a linear regression model although it describes a quadratic, and therefore curved, relationship. It is however linear in the h parameters and standard linear regression techniques can be applied to obtain them. In Eq. (6.12) the parameters are part of the exponent and transformations such as the log transform cannot help that. The regression model is then called non-linear. Non-linear regression is less evident than linear regression. Software is available but it turns out that non-linear regression can lead to unstable numerical results. How to avoid this is described in Ref. [69]. 

In the past various attempts have been made to determine the property functions for all kinds of products. Almost all of these use linear regression techniques (see for example Powers and Moskovitz, 1974) to deal with the measured data. By the fact that most of the property function is highly non-linear, these techniques fail. In addition to the linear regression, a linear regression on non-linear functions of the attributes can be used. The drawback is that the non-linear functions of the attributes have to be defined by the user. At present this have to be done by traH-and-error and turns out to be a very tedious. Based on the above observations, a successful approach to determine the property function has to be based on a generic non-linear function description. One such approach is to use neural networks as a non-linear describing function. 

Once p is calculated, equation 2 can be used to estimate the dependent variable or activity for other compounds. Multiple linear regression techniques have proven to be very successful in QSAR studies. 

It is important to bear in mind that none of these subset multiple linear regression techniques are guaranteed or even expected to produce the best possible regression equation. The user of commercial software products is encouraged to experiment. 

Cross correlations and colinearity among process variables severely limit the use of traditional linear regression techniques. PLS, as a projection method, offers a suitable solution for modeling such data. 

Quantitative structure-activity relationships QSAR. The QSAR approach pioneered by Hansch and co-workers relates biological data of congeneric structures to physical properties such as hydrophobicity, electronic, and steric effects using linear regression techniques to estimate the relative importance of each of those effects contributing to the biological effect. The molecular descriptors used can be 1-D or 3-D (3D-QSAR). A statistically sound QSAR regression equation can be used for lead optimization. 

Equation 20 is nonlinear with respect to two rate parameters, but it is readily transformed into a linear form. Thus a linear regression technique can be used to obtain initial estimates of these two parameters. Equation 21 cannot be linearized. However it possesses only two unknown parameters instead of eight parameters of Equation 19. The estimates obtained from Equations 20 and 21 supply reasonably accurate initial estimates of these four parameters. Hence, the subsequent nonlinear estimation of all eight parameters of Equation 19 can be simplified substantially. 

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