Squares Curve Fitting

A secondary but no less important goal is to obtain the standard deviations of the regression parameters, and an estimate of the goodness of fit of the data to the model equation. 

The method of least squares yields the parameters that minimize the sum of squares of the residuals (the deviation of each measurement of the dependent variable from its calculated value). 

In Chapters 11 and 12, the symbol o is used for the population standard deviation (i.e., when the sample size is large) and the symbol s for the sample standard deviation (when the sample size is small). 

Linear regression is not restricted merely to straight-line relationships, but refers to any relationship that is linear in the coefficients, that is, any relationship of the form y = ao + a xi + a i%2 + a pc +. ... The X can be different independent variables (e.g., pressure, temperature, time) or functions of a single independent variable (e.g., [H ], [H+]3). 

---

Nonlinear least squares curve fitting using the Microsoft Solver is reviewed in Appendix B. 

Solution of Sets of Simultaneous Linear Equations 71. Least Squares Curve Fitting 76. Numerical Integration 78. Numerical Solution of Differential Equations 83. 

For a function f(x) given only as discrete points, the measure of accuracy of the fit is a function d(x) = f(x) - g(x) where g(x) is the approximating function to f(x). If this is interpreted as minimizing d(x) over all x in the interval, one point in error can cause a major shift in the approximating function towards that point. The better method is the least squares curve fit, where d(x) is minimized if... 

Rao, G.R., et. al. "Factor Analysis and Least-Squares Curve- Fitting of Infrared Spectra An Application to the Study of Phase Transitions in Organic Molecules", Appl. Spec. 1984, (38) 795-803. 

B0 and B, are the amounts bound initially (at 1 = 0) and at specific times (t) after initiating dissociation. A plot of log,/l, against l is linear with a slope of -k, k may thus be estimated directly from the slope of this plot or may be obtained by nonlinear least-squares curve fitting to Eq. (5.12). It is always desirable to plot log,/) , against l to detect any nonlinearity that might reflect either the presence of multiple binding sites or the existence of more than one occupied state of the receptor. 

Figure 3 Non-linear least-squares curve fitting of the orthorhombic WAXS profile of an ethylene 1-decene random copolymer with 2.7 mol% branches. The two crystalline reflections and the amorphous halo are shown. 

Fit a polynomial curve of the desired type (degree) to the data, using least-square curve fitting. 

Learning, types of, 15 475t Least Bounding Rectangle (LBR), 18 148 Least Feret s Diameter (LFD), 18 147 Least-squares curve fitting, 14 237 Leather... 

W.R. Hruschka and K. Norris, Least squares curve fitting of near-infrared spectra predicts protein and moisture content in ground wheat, Appl. Spectrosc., 36, 261-265 (1982). 

Results of these studies are very encouraging and Indicate that a reasonably fast, accurate, and practical method for the quantitative determination of minerals In complex solids can be achieved with this approach, particularly If multivariate least squares curve fitting methods can be automated. 

After rearranging Eq. (2), the values of ixDCD and KB can be estimated by nonlinear least squares curve-fitting methods (similar to the Michaelis-Menten equation) or the expression can be rearranged under different linear forms (y = rnx I n), where y = (jueff yiD) and x = [CD], Well known are... 

In the case of a-AGT, the best fit for (RS)-carvedilol and also for the individual enantiomers was obtained via nonlinear least-squares curve fitting to the following equation (52) ... 

Least Squares Curve Fitting for Quantitative Analysis. 108... 

Figure 4-9 Least-squares curve fitting. The points (1,2) and (6,5) do not fall exactly on the solid line, but they are too close to the line to show their deviations. The Gaussian curve drawn over the point (3,3) is a schematic indication of the fact that each value of y, is normally distributed about the straight line. That is, the most probable value of y will fall on the line, but there is a finite probability of measuring y some distance from the line.

To learn how to compute variance and covariance and to see how to include weighting factors in least-squares curve fitting, see J. Tellinghuisen, Understanding Least Squares Through Monte Carlo Calculations, J.Chem. Ed. 2005,82, 157. 

To dearly distinguish between these two modes of solvent penetration of the gel, we immersed poly(acrylamide-co-sodium methacrylate) gels swollen with water and equilibrated with either pH 4.0 HQ or pH 9.2 NaOH solution into limited volumes of solutions of 10 wt % deuterium oxide (DzO) in water at the same pHs. By measuring the decline in density of the solution with time using a densitometer, we extracted the diffusion coefficient of D20 into the gel using a least squares curve fit of the exact solution for this diffusion problem to the data [121,149]. The curve fit in each case was excellent, and the diffusion coefficients obtained were 2.3 x 10 5cm2/s into the ionized pH 9.2 gel and 2.4 x 10 5 cm2/s into the nonionized pH 4.0 gel. These compare favorably with the self diffusion coefficient of D20, which is 2.6 x 10 5 cm2/s, since the presence of the polymer can be expected to reduce the diffusion coefficient about 10% in these cases [150], In short, these experiments show that individual solvent molecules can rapidly redistribute between the solution and the gel by a Fickian diffusion process with diffusion coefficients slightly less than in the free solution. 

Fig. 15. Diffusion of dilute, aqueous acetaminophen into a long, swollen cylinder of 10x4 PNIPAAm gel at 25 °C. The diffusion coefficient is extracted from a nonlinear least squares curve fit of the exact solution for diffusion into a cylinder of infinite length immersed in a well-stirred solution of finite volume to the data [123, 149]...

The XPS spectra were recorded on a Surface Science Laboratories small spot system using a monochromatized A1K X-ray radiation source. The take-off angle used for these measurements was 35°. Full details of the methods used in interpreting the XPS data have been described elsewhere [14], Data reduction was done using Surface Science Laboratories software version 8.0. This software utilizes a least squares curve fitting approach with only chi square statistics for goodness of the calculated fit to the experimental data. 

Hruschka, W.R. and Norris, K., Least Squares Curve Fitting of Near-Infrared Spectra Predicts Protein and Moisture Content in Ground Wheat Appl. Spectrosc. 1982, 36, 261-265. 

---