Non-linear least squares method

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. 

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). 

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. 

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 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. 

Spreadsheet for non-linear least square method dennmiTiatnT l.OlE-07 ... 

In this chapter we will describe some of the more sophisticated uses of least squares, especially those for fitting experimental data to specific mathematical functions. First we will describe fitting data to a function of two or more independent parameters, or to a higher-order polynomial such as a quadratic. In section 3.3 we will see how to simplify least-squares analysis when the data are equidistant in the dependent variable (e.g., with data taken at fixed time intervals, or at equal wavelength increments), and how to exploit this for smoothing or differentiation of noisy data sets. In sections 3.4 and 3.5 we will use simple transformations to extend the reach of least-squares analysis to many functions other than polynomials. Finally, in section 3.6, we will encounter so-called non-linear least-squares methods, which can fit data to any computable function. 

Solver travels down a multidimensional surface in search of a minimum value of SRR, just as water runs down a mountain under the influence of gravity. Often, the water finds its way to the ocean, but sometimes it collects in a lake without an outlet, and stays there. (Here, of course, the analogy stops, because the water can get back into the cycle by evaporation. And, of course, there are also lakes below sea-level.) The point is that a non-linear least-squares method can find a false minimum, and get stuck there, in which case you must help it to get out of that minimum. In fact, we already encountered an example of such a situation in Fig. 3.6-2, and we will now take a closer look at that case. 

In the context of the analysis of enzyme kinetics it is sometimes stated that one should always use a non-linear least-squares method for such data, because the usual, unweighted least-squares fits depend on the particular analysis method (Lineweaver-Burk, Hanes, etc.) used. We have seen in section 3.5 that the latter part of this statement is correct. But how about the former ... 

The uptake curve of the amount adsorbed usually includes the amount adsorbed in the zeolite crystallite and that on the outer surface of the crystallite [10]. Hence, these two kinds of the amounts adsorbed were evaluated seperately in calculating the diffusivity from the uptake curve by a non-linear least square method. The magnitude of the amount adsorbed on the outer surface was about 5-30 % of that in the crystallite. Hence, in order to avoid several effects (such as the amount adsorbed on the outer surface of the crystallite and the temperature rise at the early period of the adsorption [11]), the diffusivity was calculated using all data of the uptake curve by use of a theoretical equation (Eq.[l]) [9]. 

Salaices et al. (2001) fitted the ac, a and be parameters to the experimental data reported in Figure 1.3 using a non-linear, least squares method. These pai ameters can also be calculated independently by using the values for P,-, P, a, and m2 (refer to Chapter IV). These results are summarized in Table 1.1. As noted, the calculated and regressed values are statistically similai, validating the applicability of the proposed model for the prediction of the /2(Cc) functionality. 

Once the selection of a possible kinetic model and suitable reactor model are complete (equation (8-1)), a non-linear, least square method can be adopted to determine the kinetic and adsorption parameters. This can be achieved by minimizing an objective function representing the sum of the differences between the model concentration estimates and the measured experimental concentrations. This non-linear, least square fit can be performed using the curve fit functions available in Matlab, as recommended by Ibrahim (2(X)1). 

When the g-factor is anisotropic this simple approach fails. A Schonland fitting procedure can be applied when the ENDOR lines for both ms = V2 and -V2 are observable, while non-linear least squares methods are more generally applicable. The latter method can be adopted e.g. to simultaneously determine hyperfine and nuclear quadrupole tensors. Detailed procedures may be obtained from the literature [12, 14]. 

Spectral fittings to the experimental data were made with the non-linear least-squares method to obtain the principal g-values gx = 2.00429(3), gy = 2.00389(3), gz =2.00216(3) yielding giso= 2.00345(3). These values differed only slightly between FAD and the protonated FADH in several investigated protein-bound... 

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