Nonzero intercept model

FIGURE 5.2 Diagram of three different types of linear models with n standards. Left the simplest model has a slope and no intercept. The center model adds a nonzero intercept. The right model is typically noted in the literature as the multiple linear regression (MLR) model because it uses more than one response variable, and n>(m+ 1) with an intercept term and n> m without an intercept term. This model is shown with a nonzero intercept. 

In the absence of mean centering, it is possible to include a nonzero intercept, b0, in a calibration model by expressing the model as... 

Univariate calibration is specific to situations where the instrument response depends only on the target analyte concentration. With multivariate calibration, model parameters can be estimated where responses depend on the target analyte in addition to other chemical or physical variables and, hence, multivariate calibration corrects for these interfering effects. For the ith calibration sample, the model with a nonzero intercept can be written as... 

FIGURE 5.6 Graphical displays for the methanol model at 2274 nm with a nonzero intercept using all 11 calibration samples. The RMSEC is 2.37% methanol, (a) Actual calibration model (-) and measured values ( ). (b) Calibration residual plot, (c) A plot of estimated values against the actual values for the calibration samples the drawn line is the line of equality. 

In all these derivations, the role of the slurry chemicals during the polish process is not apparent. Even under static conditions, some of the chemicals can dissolve the material as in the case of ferric nitrate and copper or even H202/glycine and copper. This effect can, in principle, be easily included in a model description by adding a nonzero, velocity and pressure independent, intercept to the polish rate expression. In practice, it is more complicated since the relation between this nonzero intercept and static dissolution rates is not simple and is unknown due to, among other things, the effects of the polishing pad. In such cases, the role of a threshold pressure, while perhaps obvious when mechanical abrasion is the only mechanism for material removal, is not evident unless the removal rate can be broken neatly into two independent terms, one for the mechanical abrasion and the second for the chemical removal. Such is the case for the... 

The graph of 7p versus v1 2 (Figure 8.10) gives a straight line that does not pass through the origin as expected from the mathematical expression derived for the model [43,44], The nonzero intercept results from the presence of dissolution current, not considered in the model, where only the actions of Ip versus v were taken into account. 

If material is neo-Hookean, its Mooney-Rivlin plot ought to give a horizontal line and hence yield C2 = 0. Thus one is tempted to consider that nonzero C2 must be associated in one way or another with the deviation of a given material from the idealized network model, and it is understandable why so many rubber scientists have concerned themselves with evaluating the C2 term from the Mooney-Rivlin plot of uniaxial extension data. However, the point is that a linear Mooney-Rivlin plot, if found experimentally, does not always warrant that its intercept and slope may be equated to 2(9879/,) and 2(91V/9/2), respectively. This fact is illustrated below with actual data on natural rubber (NR) and styrene-butadiene copolymer rubber (SBR). 

Figure 21.5 Potential errors introduced by the assumption that Ar2+ is constant in the derivation of the linkage Eq. (21.30). The dependence of Ar2+ on [Mg2 1J was modeled by a polynomial fit to the solid gray data points in Fig. 21.4D. The polynomial was used in the integration of Eq. (21.30) to give expressions for In Kobs and 0 with [Mg2+]-dependent AT2 (in contrast to Eqs. (21.33) and (21.34), which assume AT2+ is constant). 9 is plotted for the calculated titration curve when the midpoint of the titration, [Mg2+]0, is 10 jiM (circles), 30 uA / (squares), or 100 li.M (diamonds). AT2+ (as used to calculate the displayed curves) at the titration midpoints (9 = 0.5) is 1.45, 2.30, and 2.73, respectively. The simulated data points have been fit to either a modified version of Eq. (21.34) that assumes the y-intercept of the curve has the value 6 = 0, 9 = 90 + (1 - 0o)([Mg2+]/[Mg2+]o)"/[l + ([Mg2+]/[Mg2+j0)"], or to an equation that allows a nonzero y-intercept, Eq. (21.35). The residuals of the fits are shown in the lower three panels closed symbols correspond to Eq. (21.34) and open symbols to Eq. (21.35). For the curve with a midpoint of [Mg2+]0 = 100 fiM, Eq. (21.35) could not be fit to the data because the value of Co became vanishingly small. The values of A IT. at the titration curve midpoints obtained from the modified Eq. (21.34) are 1.99, 2.39, and 2.73, in order ofincreasing [Mg2+]0. AT2+ obtained by fitting ofEq. (21.35) and application of Eq. (21.36) are 1.43 and 2.29 (10 and 30 fiM transition midpoints, respectively).

These simple one-wavelength calibration models with no intercept term are severely limited. Spectral data is used from only one wavelength, which means a lot of useful data points recorded by the instrument are thrown away. Nonzero baseline offsets cannot be accommodated. Worst of all, because spectral data from only one wavelength is used, absorbance signals from other constituents in the mixtures can interfere with analysis. Some of the problems revealed for models without an intercept term can be reduced when an intercept term is incorporated. 

---