Linear least-squares method

If the rate law depends on the concentration of more than one component, and it is not possible to use the method of one component being in excess, a linearized least squares method can be used. The purpose of regression analysis is to determine a functional relationship between the dependent variable (e.g., the reaction rate) and the various independent variables (e.g., the concentrations). 

The best-fit values for kM and kd obtained by a non-linear least squares method were 3.8 x 10-4s-1 and 1.2 x 10-4M-1s-1 respectively. 

Table 4 Rate constants calculated by non-linear least square method under non-isothermal condition...

In the kinetic analysis of the experimental data with an autoclave, the non-linear least square method was used to estimate the rate constants under nonisothermal conditions. The simulation of liquefaction calculated by substituing the estimated values into the rate equations showed good agreement with experimental values. 

Later, in Chapter 4.4, General Optimisation, we discuss non-linear least-squares methods where the sum of squares is minimised directly. What is meant with that statement is, that ssq is calculated for different sets of parameters p and the changes of ssq as a function of the changes in p are used to direct the parameter vector towards the minimum. 

Only in the simplest cases—a single Gaussian component, for example— may conventional linear least-squares method be employed to solve for u. More commonly, either approximate linearized methods or nonlinear methods are employed. 

Low (<8.5) pH Solutions. The Np solution concentrations are plotted as a function of measured pe in Figure 2 for various equilibration periods. The data points for the low pH (<8.5) suspensions were fitted by a linear least squares method to a line of fixed unit slope. When the slopes were allowed to vary, their values ranged from 0.8 to 1.1. 

The constant values of /iImax and were determined as listed in Table 5 by fitting the experimental data shown in Fig. 9 using the non-linear least squares method. 

For consecutive or parallel electrode reactions it is logical to construct circuits based on the Randles circuit, but with more components. Figure 11.16 shows a simulation of a two-step electrode reaction, with strongly adsorbed intermediate, in the absence of mass transport control. When the combinations are more complex it is indispensable to resort to digital simulation so that the values of the components in the simulation can be optimized, generally using a non-linear least squares method (complex non-linear least squares fitting). 

If the function may be made linear with respect to its unknown parameters by a suitable transformation, then it may be fitted by the Linearized Least Squares method (10) so as to minimize the root mean square error in the original (untransformed) space. The essence of this technique is to use weighted (linear) least squares to effect a non-linear least squares fit. Assume that the equation has been transformed into an equal variance space and let... 

The thorough treatment of the experimental data does allow one to obtain reliable values of the reactivity ratios. The results of such a treatment are presented in Table 6.3 for some concrete system let us form a notion about an accuracy of the reactivity ratios estimations. The detailed analysis of such a significant problem in the case of the well-studied copolymerization of styrene with methyl methacrylate is reported in Ref. [227]. Important results on the comparison of the precision of rj, r2 estimates by means of different methods are presented by O Driscoll et al. [228]. Such a comparison of six well-known linear least-squares procedures [215-218,222,223] with the statistically correct non-linear least-squares method leads to the conclusion that some of them [216, 217, 222] can provide rather precise rls r2 estimates when the experiment is properly planned. 

The effects of substituents on the symmetrically disubstituted diarylethyl tosylates, [27(X = Y)j, can be described accurately in terms of the Y-T relationship with p = —4.44 and r = 0.53. The Y-T plot against the Y-T ascale with an appropriate r of 0.53 gives an excellent linear correlation for the whole set of substituents, indicating a uniform mechanism for all of them. When Y X, the overall solvolysis rate constant ki corresponds to the sum of the rate constants, k + kj, and hence k, cannot be employed directly in the Y-T analysis. The acetolysis of monosubstituted diphenylethyl tosylates gave a non-linear Y-T correlation, which is ascribed to a competitive X-substituted aryl-assisted pathway k and the phenyl-assisted k pathway. By application of an iterative non-linear least-squares method to (9), where the terms k and ks are now replaced by k and k, respectively, the substituent effect on kt can be dissected into a correlation with = —3.53, = 0.60, and an... 

Aqueous organic solvent, jcM a (v/v) mixture of (100 — x)% aqueous and x% organic solvent M, where M is E = EtOH, A = acetone, and T = TFE. Analysed by the non-linear least-squares method based on equation (9). Correlation for the substituted-aryl assisted pathway. Correlation for the unsubstituted-phenyl assisted pathway. 

Exc. X excluding substituent X. Analysed by the non-linear least-squares method by equation (23). The non-linear least-squares correlation for the a-C pathway (see text). The non-linear least-squares correlation for the bromonium ion pathway (see text). The non-linear least-squares correlation for the P-C pathway (see text), Correlation overall p for two equivalent aryl substituents. 

Numerous kinetic expressions can be placed into a form that would yield a zero y-intercept when using the linear least-squares method. A survey of a few of these models is provided in Table B.3.1. Given that the y-intercept is a known value (i.e., zero), if a perfect correlation could be achieved, the hypothesis that the true value of the parameter, Si, is equal to the specified value, a, could be tested by referring the quantity ... 

Fluorescence and affinity measurements - Peptide in 25 mM Tris, 100 mM KCl and 1 mM CaCl2 at pH 7.5 and 30 C was titrated with a stock solution of calmodulin in UV transmitting plastic cuvettes since the peptides appear to bind to glass. Fluorescence titration spectra were recorded using a SPEX FluoroMax fluorescence spectrometer with excitation at 280 nm and emission scanned from 310 to 390 nm. The value of fluorescence intensity at 330nm was plotted as a function of calmodulin concentration and fitted using standard non-linear least squares methods (6) to obtain optimal values of the dissociation constant (Kj) and the maximum fluorescence enhancement (F/F ). The detection limit under our experimental conditions was 50 nM peptide and all quoted Kj values are the average of at least 3 independent determinations. 

Both the full pattern decomposition and Rietveld refinement are based on the non-linear least squares minimization of the differences between the observed and calculated profiles. Therefore, the non-linear least squares method is briefly considered here. Assume that we are looking for the best solution of a system of n simultaneous equations with m unknown parameters (n m), where each equation is a non-linear function with respect to the unknowns, Xu X2,. .., In a general form, this system of equations can be represented as... 

The major differences between the non-linear least squares technique and the linear least squares method, described in section 5.13.1, are as follows ... 

The equation describing the variation of centre shift with temperature (Eq. 1, this Chapter) contains two adjustable parameters, m and 5q. Using a non-linear least squares method, the values of these parameters which best fit the data were determined (Table A2). The centre shift data together with the Debye model calculations are plotted in Figure A3. 

Non-Linear Least Squares Method (9). Assuming that (from equation 20)... 

The use of the non-linear least squares method does not require any derivatives, but needs an initial estimation and takes more time to compute, since several iterations (usually 3 or 4) are necessary to reduce the difference between the estimated and calculated values of the damping coefficient to within 0.1%. But since this method only requires between 100 and 150 data points without a loss in accuracy compared to as many as 1000 for the peak-finding and least squares methods, the scan rate can be reduced as much as 90% and the time required for the calculations is reduced to the order of a minute. 

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