Nonlinear least squares fitting

A typical nonlinear model encountered in spectroscopy is one where the spectral line frequencies, y, are computed from differences between eigenvalues (specified by two groups of quantum numbers) of two effective Hamiltonian matrices, defined, in turn, by the molecular constants, Xj. For example, consider the 2x2 matrix of a 2H state  

The utility of Eq. (4.4.22) may be illustrated very simply. At very high J-values, a 2n state will approach the case (b) limit, at which point there is essentially no information in the spectrum from which the spin-orbit constant, A, may be determined. The U matrix for the case (a)— (b) transformation at the high-J limit is 

Equation (4.4.24a) looks exactly like Eq. (4.4.3), but the hidden difference is that the Bim coefficients depend implicitly, through the U matrix, on the unknown parameters, Xm. It will be necessary to resort to an iterative procedure whereby the approximate B matrix elements are recalculated after each adjustment of the parameter values, 

The matrix formulation of the iterative process is derived by a series of steps identical to Eqs. (4.4.5) - (4.4.12), 

The right-hand side of Eq. (4.4.26) is expressed entirely in terms of known (yobs) and calculable (W(°),B(0)) quantities, but since the model is nonlinear, the next iteration requires that the B(1) matrix and y alc be recalculated by Eqs. 

Usually, Raman spectra can be seen as the sum of single specific bands, each centered on a wavenumber v . Raman bands are also characterized by some spreading around the central wavenumber v, defined by the full width at half maximum (FWHM), w, of the band. Unlike intensity, FWHM does not generally change with the concentration of the related chemical species. Theoretically, Raman bands can be described by two kinds of disp ersion function Gaussian or Lorentzian. The most general case is a linear combination of both functions. 

Gaussian In a first attempt, the intensity of a Raman band can be modeled by the Gaussian function  

Lorentzian The other model for the Raman band shape is the Lorentzian function  

General Case The most general case of a Raman band shape bination of Gaussian and Lorentzian functions is the linear com- 

the number of parameters necessary to describe a Raman band with mixed Gaussian and Lorentzian contributions is four, namely v , w, I ax and g. 

---

Potentiometric titration curves are used to determine the molecular weight and fQ or for weak acid or weak base analytes. The analysis is accomplished using a nonlinear least squares fit to the potentiometric curve. The appropriate master equation can be provided, or its derivation can be left as a challenge. 

The coefficients listed were determined by nonlinear least-squares fitting of the data of Ref 79, and have dimensions appropriate for xj in mm, tj in usee, and Px in kbar... 

Kinetic data for the decomposition of diacetone alcohol, from Table 2-3. were obtained by dilatometry. The nonlinear least-squares fit of the data to Eq. (2-30) is shown on the left. Plots are also shown for two methods presented in Section 2.8 they are the Guggenheim method, center, and the Kezdy-Swinbourne approach, right. 

Figure 3-3 depicts the analysis of the first data set in Table 3-2 according to Eq. (3-28). The plot displays the variation of [A]r versus time. The nonlinear least-squares fit gives k = 0.407 0.002 s 1. This value and the equilibrium constant give k- = 3.8 X 102 L mol-1 s 1. (The agreement is impressive because the data were simulated from values that are essentially these—see note a to Table 3-2.)... 

The buildup of [P] is biexponential. Nonlinear least-squares fitting will provide the best solution for k and k2 (and, if they are unknown, for [A ]o and [A2]o also). Another method is to plot ln([P] - [P],) against time. At long times, the exponent with the higher value of k will have fallen off, such that the slope will approximate the slower one. Calling this ks (it might be either k or k2), we perform a subtraction, emphasizing the data at shorter times ... 

The first of these reactions was carried out in 1,4-cyclohexadiene over a temperature range of 39 to 100 °C. It is fairly slow the half-times were 20 h and 3.4 min at the extremes. Reaction (7-11) is quite fast the second-order rate constant, kn, was evaluated over the range 6.4 to 47.5 °C. Values of feio and fen are presented in Table 7-1. The temperature profiles are depicted in Fig. 7-1 from their intercepts and slopes the activation parameters can be obtained. A nonlinear least-squares fit to Eq. (7-1) or... 

This expression was numerically integrated and regressed to the experimental data using a nonlinear least-squares fitting procedure. The resulting integrated equation is... 

The fraction undissolved data until the critical time can be least-square fitted to a third degree polynomial in time as dictated by Eq. (29). The moments of distribution ij, p2, and p3 can be evaluated from Eqs. (30) through (32), with three equations used to solve for three unknowns. These values may be used as first estimates in a nonlinear least-squares fit program, and the curve will, hence, reveal the best values of both shape factor, size distribution, and A -value. 

To construct the Hill plot (Figure 5.10E), it was assumed that fimax was 0.654 fmol/mg dry wt., the Scatchard value. The slope of the plot is 1.138 with a standard deviation of 0.12, so it would not be unreasonable to suppose % was indeed 1 and so consistent with a simple bimolecular interaction. Figure 5.10B shows a nonlinear least-squares fit of Eq. (5.3) to the specific binding data (giving all points equal weight). The least-squares estimates are 0.676 fmol/mg dry wt. for fimax and... 

Spalek, T., Pietrzyk, P. and Sojka, Z. (2005) Application of the genetic algorithm joint with the Powell method to nonlinear least-squares fitting of powder EPR spectra,. /. Chem. Inf. Model., 45, 18. 

Another noteworthy example is x-ray absorption fine structure (EXAFS). EXAFS data contain information on such parameters as coordination number, bond distances, and mean-square displacements for atoms that comprise the first few coordination spheres surrounding an absorbing element of interest. This information is extracted from the EXAFS oscillations, previously isolated from the background and atomic portion of the absorption, using nonlinear least-square fit procedures. It is important in such analyses to compare metrical parameters obtained from experiments on model or reference compounds to those for samples of unknown structure, in order to avoid ambiguity in the interpretation of results and to establish error limits. 

An alternative to evaluating the KIE and the rate constants from the above equations is to apply nonlinear least-squares fitting to the complete kinetic set of a and t values. This latter procedure has the advantage that errors in the reaction model, e.g. an incorrect mechanism, or extraneous data points are more easily discovered. This method was applied by Bergson et al. (1977) and Matsson (1985) in the determination of both the primary deuterium and secondary a-deuterium KIEs in the 1-methylindene rearrangement to 3-methylindene (reaction (67)). For example, a secondary /3-deuterium KIE of 1.103 0.001 was determined very accurately in toluene at 20°C using this method (Bergson et al., 1977). 

Chapter 4, Model-Based Analyses, is essentially an introduction into least-squares fitting. It is crucial to clearly distinguish between linear and nonlinear least-squares fitting linear problems have explicit solutions while non-linear problems need to be solved iteratively. Linear regression forms the base for just about everything and thus requires particular consideration. 

For fitting such a set of existing data, a much more reasonable approach has been used (P2). For the naphthalene oxidation system, major reactants and products are symbolized in Table III. In this table, letters in bold type represent species for which data were used in estimating the frequency factors and activation energies contained in the body of the table. Note that the rate equations have been reparameterized (Section III,B) to allow a better estimation of the two parameters. For the first entry of the table, then, a model involving only the first-order decomposition of naphthalene to phthalic anhydride and naphthoquinone was assumed. The parameter estimates obtained by a nonlinear-least-squares fit of these data, are seen to be relatively precise when compared to the standard errors of these estimates, s0. The residual mean square, using these best parameter estimates, is contained in the last column of the table. This quantity should estimate the variance of the experimental error if the model adequately fits the data (Section IV). The remainder of Table III, then, presents similar results for increasingly complex models, each of which entails several first-order decompositions. 

The noise-free Stern-Volmer lifetime plots are clearly curved, which indicates a failure of a two discrete site model. However, this is a difficult nonlinear least-squares fitting problem, and the unquenched apparent lifetimes are within a factor of two of each other. Thus, for real data, it is much more difficult to pick up on the nonlinearities and exclude a discrete two-site model. For distributions with smaller R s, of course, fitting becomes too difficult for reliable model testing at least at 104 counts in the peak channel. 

Thus, the left side of Equation (12.23) is the result of a titration calorimetry experiment, and the right-hand side includes experimental quantities and the equilibrium constant K and AH. Therefore, the parameters AH and K can both be obtained by a nonlinear, least-squares fitting of the data to the relation that we have derived... 

At 25°C the partial molar volume of urea [CO(NH2)2] solution in water is found by a nonlinear least-squares fitting procedure to be the following function of m2 up to 17-molal concentration (with V in cm mol ), with experimental data from Gucker et al. [10] and the form of the equation from Stokes ... 

IC50 values were determined at 48 h by nonlinear least-square fit to sigmoidal functions. ... 

To quantify the measured spectra, a combination of linear and nonlinear least-squares fitting routines are used, in which the measured intensities are fit to those of scaled reference spectra while minimizing the residual absorbance. Taking the natural logarithm of Eq. (E), one obtains... 

Near the origin of the graph, the film growth rate is proportional to the sulfur incident rate, r(i, S). The slope in this region is approximately equal to the value of the sulfur reflection factor, 8(S), that was obtained from a nonlinear least-square fit of equation 40 to the experimental data. Thus, in the sulfur-limited regime, the deposition rate of ZnS is given by r(d, ZnS) 8(S) r(i, S), in which 8(S) 0.5. 

In the case of fast exchange, the NMR rate constants were determined using a computer program that calculated rates, populations, and chemical shifts by nonlinear least-squares fitting of NMR line-shape data to the GMS equation (15). 

---