Data fitting examples

All of the examples so far considered in this chapter involve one independent variable and one dependent variable. While this is a very common engineering problem, many measurement problems and parameter estimation problems involve physical situations with more than one independent variable. This type of problem was anticipated when the nlstsq() routine was developed and such problems are easily handled in essentially the same manner as the previous examples. To illustrate such problems, two data fitting examples and parameter estimation examples are given in this section. Both of these involve only two independent... 

Listing 9.19. Code segment for breakdown voltage data fitting example with two independent variables. 

Result of curve-fitting for the kinetic data in Example 13.4. 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

G. Vizkelethy. NticL Instr. Meth. B45,1, 1990. Description of the program SENRAS. used in fitting NRA spectra includes examples of data fitting. 

Different C-values are needed to correlate all the data. For example, when the 100 pm micro-channel was connected to a reducing inlet section, the data could be fit by a single value of C = 0.24. 

At a later stage, the basic model was extended to comprise several organic substrates. An example of the data fitting is provided by Figure 8.11, which shows a very good description of the data. The parameter estimation statistics (errors of the parameters and correlations of the parameters) were on an acceptable level. The model gave a logical description of aU the experimentally recorded phenomena. 

Figure 8.11 Example of data fitting results - concentrations of sitosterol, campesterol, and the products in the hydrogenation of sitosterol to sitostanol (increasing curve) on Pd/Sibunit. The lower curves represent campesterol and campestanol.

Figure 3.10 Comparison of residual values, cA,ca/c CA,exP for first- and second-order fits of data in Example 3-8...

Using (5.14) and the determined value of aabs, we can estimate 8 if ab is known, and vice versa. Two examples of thermal bistability data, fit to a calculated tuning curve based on (5.12), are shown below. Figure 5.7 is for the bare sphere, and Fig. 5.8 is for the PDDA-coated sphere. In the figures, the laser scans slowly across a TM-polarized WGM dip (taking several thermal relaxation times to scan Av), first down in frequency, then reversing at the vertical dashed line, and scanning back up in frequency across the same mode. The continuous smooth lines are the theoretical fits. 

At low surface coverages (high SOHj.) only the RO--CdOH+ surface species is required to fit the data. For example, decreasing SOH. from 7.4 x 10 to 2.9 x 10 M increases the Cd(II) adsorption... 

Solver for non-linear data fitting tasks. Several examples are based on the fitting tasks already solved by the Newton-Gauss-Levenberg/Marquardt method in the earlier parts of this chapter. 

The mathematical model may not closely fit the data. For example. Figure 1 shows calibration data for the determination of iron in water by atomic absorption spectrometry (AAS). At low concentrations the curve is first- order, at high concentrations it is approximately second- order. Neither model adequately fits the whole range. Figure 2 shows the effects of blindly fitting inappropriate mathematical models to such data. In this case, a manually plotted curve would be better than either a first- or second-order model. 

Range finding v versus [S] data fitted to O Eq. 4 yields a sigmoidal curve. The estimated /Cm value, the concentration of substrate at which v = 10 luimol/h/mg, is 5 iulM in this example. At least one, and usually two data points will lie on the steep portion of a sigmoidal curve if each [S] is tenfold higher than the last... 

It is important to realize that an / or r value (instead of an or value) might give a false sense of how well the factors explain the data. For example, the R value of 0.956 arises because the factors explain 91.4% of the sum of squares corrected for the mean. An R value of 0.60 indicates that only 36% of 55 has been explained by the factors. Although most regression analysis programs will supply both R (or r) and R (or r ) values, researchers seem to prefer to report the coefficients of correlation R and r) simply because they are numerically larger and make the fit of the model look better. 

A recent example of the continuing use of this equation for data fitting is Bouyer, R. Roussel, F. Monchicourt, P. Perdrix, M. Pradel, P.J. Chem. Phys. 1994,100, 8912. 

In Eq. 13.15, the squared standard deviations (variances) act as weights of the squared residuals. The standard deviations of the measurements are usually not known, and therefore an arbitrary choice is necessary. It should be stressed that this choice may have a large influence of the final best set of parameters. The scheme for appropriate weighting and, if appropriate, transformation of data (for example logarithmic transformation to fulfil the requirement of homoscedastic variance) should be based on reasonable assumptions with respect to the error distribution in the data, for example as obtained during validation of the plasma concentration assay. The choice should be checked afterwards, according to the procedures for the evaluation of goodness-of-fit (Section 13.2.8.5). 

Try to fit an nth-order rate equation to the concentration vs. time data of Example 3.1. 

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