Least-squares surfaces

Many of the models encountered in reaction modeling are not linear in the parameters, as was assumed previously through Eq. (20). Although the principles involved are very similar to those of the previous subsections, the parameter-estimation procedure must now be iteratively applied to a nonlinear surface. This brings up numerous complications, such as initial estimates of parameters, efficiency and effectiveness of convergence algorithms, multiple minima in the least-squares surface, and poor surface conditioning. 

The common academic and commercial computer programs using the iterative fitting by the Gauss-Newton method will handle most weU-measured data with little difficulty, but in some cases where the data are bad, where preliminary estimates cannot be obtained by fitting in reciprocal form, or where the equation is simply a difficult one to fit, the program may fail to converge to a position of minimum residual sum of squares. In such cases, it is useful to examine the actual shape of the least squares surface. 

In order to constract the least squares surfaces, consider the MichaeUs-Menten equation with only two constants ... 

The construction of such a least square surface is very simple one simply progresses through a grid of V and K values and calculates... 

Examination of the least squares surfaces is useful in telling whether any minimum really exists. When no minimum is found, the contours of residual least square are of an irregular shape, forming a long shallow valley with one end closed and the another end opened. This shows that one of the constants cannot be properly estimated. The commercial statistical programs will plot the least square surfaces automatically and thus simplify the analysis. 

Solution of the above constrained least squares problem requires the repeated computation of the equilibrium surface at each iteration of the parameter search. This can be avoided by using the equilibrium surface defined by the experimental VLE data points rather than the EoS computed ones in the calculation of the stability function. The above minimization problem can be further simplified by satisfying the constraint only at the given experimental data points (Englezos et al. 1989). In this case, the constraint (Equation 14.25) is replaced by... 

Figure 1 shows the powder X-ray diffraction (XRD) pattern of the as-prepared Li(Nio.4Coo.2Mno.4)02 material. All of the peaks could be indexed based on the a-NaFeC>2 structure (R 3 m). The lattice parameters in hexagonal setting obtained by the least square method were a=2.868A and c=14.25A. Since no second-phase diffraction peaks were observed from the surface-coated materials and it is unlikely that the A1 ions were incorporated into the lattice at the low heat-treatment temperature (300°C), it is considered that the particle surface was coated with amorphous aluminum oxide. 

Consider the amount of radiation arriving on the surface of the Earth at a distance of 1 AU or 1.5 x 1011 m. The total flux of the Sun is distributed evenly over a sphere of radius at the distance of the planet, d. From the luminosity calculation of the Sun, F, the solar flux at the surface of Earth, FEarth, is F/47t(1.5 x 1011)2 = 1370 Wm-2 from the least-square law of radiation discussed in Example 2.4 (Equation 2.4). Substituting this into Equation 7.6 with the estimate of the albedo listed in Table 7.2 gives a surface temperature for Earth of 256 K. 

Terrado M, Barcelo D, Tauler R (2009) Quality assessment of the multivariate curve resolution alternating least squares method for the investigation of environmental pollution patterns in surface water. Environ Sci Technol 43 5321-5326... 

Ryason and Russel measured the temperature dependence of the IR absorption band halfwidth for valence vibrations of hydroxyl groups on the silica surface.200 At T > 325 K, the least squares method permits a straight line to be drawn through experimental points of the dependence In Avv2 (Tl), the equation of the line appearing as follows 200... 

Iqdari and Velde (unpub. data, 1992, see Table 8.2) described experiments of Ce diffusion in apatite soaked in CeCl2 with asymmetric diffusion profiles. For one of their runs carried out at 1100°C for 15 days, and described as an example of a non-linear least-square fit in Section 5.2, it has been found that the relationship between the Ce concentration CCt and the distance X to the mineral surface is described by... 

Therefore, we want to decide which direction, among all possible choices, is common to all sample subspaces, or, at least, which direction represents the best zone of the sample subspaces in a least-square sense. Since a direction can be completely described by its unit vector, we can restrict the solution set to the surface of the unit sphere centered at the origin. Let us call y the solution of unitary modulus and its projection onto the fcth sample subspace (k = 1,..., s) represented by the matrix Ak. It is a simple matter to show that... 

Figure 7. Covariability between values of C and Kd yielding best fit of diprotic surface hydrolysis model with constant capacitance model to titration data for TiC>2 in 0.1 M KNOj (Figure 5). The line is consistent with Equation 29. The crosses represent values of C and log found from a nonlinear least squares (NLLS) fit of the model to the data, with the value of capacitance imposed in all cases the fit was quite acceptable. The values of and C found by Method I (Figure 6) also fall near the line consistent with Equation 29. The agreement between these results supports the use of the linearized model (Equation 29) for developing an intuitive feel for surface reactions.

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