Curve-fitting

Curve fitting is an important tool for obtaining band shape parameters and integrated areas. Spectroscopic bands are typically modeled as Lorenzian distributions in one extreme and Gaussian distributions in the other extreme [69]. Since many observable spectroscopic features lie in between, often due to instrument induced signal convolution, distributions such as the Voight and Pearson VII have been developed [70]. Many reviews of curve fitting procedures can be found in the literature [71]. 

It should be noted that for many spectroscopies, the integrated peak area rather than peak height remains proportional to the moles of solute present, even over large regions of composition space. This quantitative feature of integrated areas has a precise analytical form in vibrational spectroscopy [72, 73]. 

The Pearson VII model contains four adjustable parameters and is particularly well suited for the curve fitting of large spectral windows containing numerous spectral features. The adjustable parameters a, p, q and v° correspond to the amplitude, line width, shape factor and band center respectively. As q —the band reduces to a Lorenzian distribution and as q approaches ca. 50, a more-or-less Gaussian distribution is obtained. If there are b bands in a data set and 

The peak map from Section 4.4.4 helps to jump start the calculations since it fixes the value of b and provides an excellent first approximation of the band center positions v°. With the first approximation of r°, the three remaining unknowns a, p, q can be optimized. At this point, the value of each r° can be relaxed within a small tolerance, and then all a, p, q and r° can be re-determined. The execution of Eq. (13) for a full series of k sequential spectra in provides not only a set 

1) Briefly, it can be mentioned that Eq. (2) can be written in an integral rather than decadic form, and that one can solve for integrated 

Even with the Kelen-Tudos refinement there are statistical limitations inherent in the linearization method. It has been shown [18] that the independent variable in any form of the linear equation is not really independent while the dependent variable does not have a constant variance. The most statistically sound method of analyzing the experimental composition data is the nonlinear method which involves direct curve fitting to the copolymer composition equation. 

From the slope of the experimental F versus fi plot an estimate of T i and T2 can be made by comparison with curves based on Eq. (7.18). The best values of r and V2 can be selected by determining which theoretical curve best fits the data by trial and error. A limitation of the method is the relative insensitivity of the curves to small changes in r and r2- The pros and cons of both the linearization and nonlinear methods have been discussed in detail, along with approaches for the best choice of feed compositions to maximize the accuracy of the r and T2 values [17,19,20]. 

A serious drawback in the use of a differential form of the copolymerization equation [Eq. (7.11) or (7.18)] is the assumption that the feed composition does not change during the experiment, which is obviously not true. One carries out the polymerization to as low a conversion as possible, but there are limitations since one must be able to isolate a suflicient sample of the copolymer for direct analysis, or, if copolymer analysis is done 

Tkble 7.1 Monomer Reactivity Ratios in Radical Copolymerization 

Data mostly from R. Z. Greenley, Free Radical Copolymerization Reactivity Ratios, pp 153-266 in Chap. II in Polymer Handbook, 3rd ed., (J. Brandrup and E. H. Immergut, eds.), Wiley Interscience, New York (1989). 

The results of most experiments consist of a finite number of values (and their errors) of a dependent variable y measured as a function of the independent variable x (Fig. 11.1). The objective of the measurement of y = y(x) may be one 

To prove that y = follows a theoretically derived function 

To use the finite number of measurements of y(jc) for the evaluation of the same function at intermediate points or at values of x beyond those measured 

These objectives could be immediately achieved if the function y(x) were known. Since it is not, the observer tries to determine it with the help of the experimental data. The task of obtaining an analytic function that represents y(x) is called curve fitting. 

After the data are plotted and a smooth curve is drawn, the observer has to answer two questions  

Unfortunately, there are not many cases in analytical chemistry where the exact mathematical form of a non-linear regression equation is known with certainty (see below), so this approach is of restricted value. 

It should also be noted that, in contrast to the situation described in the previous paragraph, results can be transformed to produce data that can be treated by unweighted methods. Data of the form y = bx with y-direction errors strongly dependent on X are sometimes subjected to a log-log transformation the errors in log y then vary less seriously with log x, so the transformed data can reasonably be studied by unweighted regression equations. 

The mathematical problems to be solved are then (i) how many terms should be included in the polynomial, and (ii) what values must be assigned to the coefficients a, b, etc. Computer software packages which address these problems are normally iterative they fit first a straight line, then a quadratic curve, then a cubic curve, and so on, to the data, and present to the user the information needed to decide which of these equations is the most suitable. In practice quadratic or cubic equations are often entirely adequate to provide a good fit to the data polynomials with many terms are 

In an instrumental analysis the following data were obtained (arbitrary units). 

Fit a suitable polynomial to these results, and use it to estimate the concentrations corresponding to signal of 5, 16 and 27 units. 

We sometimes need to calculate the heights or areas of a series of peaks that severely overlap each other. One way to do this is to fit model functions to overlapped bands and allow the computer to calculate the optimum individual area of each. This approach requires some knowledge 

In many cases, experimental data points fall on either side of a straight line (Fig. 14.8). While it is often sufficient to estimate the line that best fits the data by eye, this can be done more precisely by a method of least squares. This involves determining the slope and intercept of the straight line that best fits the data analytically. The dispersion in the y direction (c/,) is determined for each point from the straight line being sought. The sum of the squares of these dispersions will be a minimum for the line that best fits the data. This procedure involves considerable calculation when the numbers of experimental points involved are large. 

software for use on a personal computer reduces the task of least squares curve fitting to a simple operation. Before the appearance of the personal computer in the early 1980s, most statistical texts contained a 

In cases where experimental points indicate a eurve, it is usually profitable to explore the use of specially ruled paper (log, semi-log, log-log, ete.) to determine whether the data gives a straight line when plotted on differently ruled papers. 

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Fig. XVin-6. Curve-fitted Mo XPS 3d spectra of a 5 wt% Mo/Ti02 catalyst (a) in the oxidic +6 valence state (b) after reduction at 304°C. Doublets A, B, and C refer to Mo oxidation states +6, +5, and +4, respectively [37]. (Reprinted with permission from American Chemical Society copyright 1974.)...

In the first pari of this project, the analytical form of the functional relationship is not used because it is not known. Integration is carried out directly on the experimental data themselves, necessitating a rather different approach to the programming of Simpson s method. In the second part of the project, a curve fitting program (TableCurve, Appendix A) is introduced. TableCurve presents the area under the fitted curve along with the curve itself. 

To anyone who has carried out curve-fitting calculations with a mechanical calculator (yes, they once existed) TableCurve (Appendix A) is equally miraculous. TableCurve fits dozens, hundreds, or thousands of equations to a set of experimental data points and ranks them according to how well they fit the points, enabling the researcher to select from among them. Many will fit poorly, but usually several fit well. 

Along with the curve fitting process, TableCurve also calculates the area under the curve. According to the previous discussion, this is the entropy of the test substance, lead. To find the integral, click on the numeric at the left of the desktop and find 65.06 as the area under the curve over the range of x. The literature value depends slightly on the source one value (CRC Handbook of Chemistry and Physics) is 64.8 J K mol. ... 

The situation is similar for a linear curve fit, except that now the data set is two-dimensional and the number of degrees of freedom is reduced to (n — 2). The analogs of the one-dimensional variance )/( 1) the standard... 

Solved using the BASIC curve fitting program QLLSQ we get as a partial output block 10 POINTS, FIT WITH STD DEV OF THE REGRESSION. 2842293... 

COMPUTER PROJECT 3-1 Linear Curve Fitting KF Solvation... 

The quadr atic curve fit leads to a number of residuals equal to the number of points in the data set. The sum of squares of residuals gives SSE by Eqs. (3-23) and MSE by Eq. (3-30), except that now the number of degrees of freedom for n points is... 

Using the expanded determinants from Problem 6, write explicit algebraic expressions for the three minimization parameters a, b, and c for a parabolic curve fit. 

Write the determinant for a sixth-degree curve fitting procedure. 

We now have two ways of inserting the correct parameters into the STO-2G calculation. We can write them out in a gen file like Input File 8-1 or we can use the stored parameters as in Input File 8-2. You may be wondering where all the parameters come from that are stored for use in the STO-xG types of calculation. They were determined a long time ago (Hehre et al, 1969) by curve fitting Gaussian sums to the STO. See Szabo and Ostlund (1989) for more detail. There are parameters for many basis sets in the literature, and many can be simply called up from the GAUSSIAN data base by keywords such as STO-3G, 3-21G, 6-31G, etc. 

Mortimer, R. G., 1999. Mathematics for Physical Chemistry, 2nd ed. Academic Press, San Diego, CA. [This book contains an introduction to computer use with brief comments, references and sources to BASIC, Excel, graphics, curve fitting, and Mathematica.]... 

One very popular technique is to use QSAR. It is, in essence, a curve-fitting... 

Understanding how the force field was originally parameterized will aid in knowing how to create new parameters consistent with that force field. The original parameterization of a force field is, in essence, a massive curve fit of many parameters from different compounds in order to obtain the lowest standard deviation between computed and experimental results for the entire set of molecules. In some simple cases, this is done by using the average of the values from the experimental results. More often, this is a very complex iterative process. 

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