Nonlinear least squares method, data

Data can be fit to this equation by the nonlinear least-squares method. As it turns out, the Guggenheim approach for first-order kinetics is valid, even though the reaction... 

Parameter estimation. Integral reactor behavior was used for the interpretation of the experimental data, using N2O conversion levels up to 70%. The temperature dependency of the rate parameters was expressed in the Arrhenius form. The apparent rate parameters have been estimated by nonlinear least-squares methods, minimizing the sum of squares of the residual N2O conversion. Transport limitations could be neglected. 

Nonlinear Least-Squares Methods of Data Analysis.174... 

The constants needed to obtain a value of ifl as a function of m can be obtained by fitting the enthalpy of dilution data to Equation (18.67) by a nonlinear least-squares method, (see Section A.l). 

Figure 12 Conformational transition of BpUreG as revealed by steady-state fluorescence signals, (a) Steady-state emission spectra of BpUreG at 24 C at increasing concentrations of GuHCI (from 0 M to 3 M, incubation time of 10 min), (b) Changes in emission max (black circles) and steady-state anisotropy (clear squares) as a function of denaturant concentration. The solid lines represent the fits by a nonlinear least-squares method of the experimental data. (Reprinted from Reference 187 with permission of the ACS.)...

Since the nonlinear least-squares method requires initial guesses to start the procedure, three different initial trials were performed (1) (0,0), (2) (1,1), and (3) the values obtained from the Lineweaver-Burk plot in Example 4.2.4. All three initial trials give the same result (and thus the same relative error). Note the large differences in the values obtained from the nonlinear analysis versus those from the linear regression. If the solutions are plotted along with the experimental data as shown below, it is clear that the Lineweaver-Burk analysis does not provide a good fit to the data. 

Determine the parameter values bj andZ>2 by using the data given in Example 9.1 and the nonlinear least squares method. Recall that in Example 9.1 we needed the elements of the Jacobian matrix 7 (see equation (9.142)). In this case, integrate simultaneously the time dependent sensitivity coefficients (i.e., the Jacobi matrix elements dyfdb and dyjdb2 ) and the differential equations. The needed three differential equations can be developed by taking the total derivative (as shown below) of the right hand side of equation (9.149) which we call h ... 

The amount of dead cells was constant at 11% every 24h, which corresponds to P = 0.89. The data of N72/N0 against a in Figure 35.4 were fitted by Eq. (35.3) using the nonlinear least squares method. The fitting result showed tD = 17.5 h, which was different from the value of 24 h for the doubling time for the first cell division in the model mentioned above. This might be derived from the lag time, characteristic of a first division. 

Besides the above methods, Tosi suggested a new method called method of grouping, which was claimed to have minimal computation difficulty, but only provides approximate values. Tidwell and Mortimer proposed a nonlinear least square method by minimizing the difference between the observed and calculated copolymer compositions. This method was claimed to circumvent the subjective judging of experimental data and lead to better results compared to other methods, although its computation process is quite complicated. 

The above-mentioned nonlinear least square method for the case with two parameters (a,j8) is the basic one and easily extended to the cases with more parameters. Considering the possibility of obtaining the reliable Sh of this NMR titration experiment, data treatment should be carried out with three parameters for the better regression. The programs of spreadsheet software for this three-parameters-method are developed and shown in Appendix 2.4 [24]. 

In both pulse and phase fluorometries, the most widely used method of data analysis is based on a nonlinear least-squares method. The basic principle of this method is to minimize a quantity which expresses the mismatch between data and fitted function. This quantity is the reduced chisquare defined as the weighted sum of the squares of the deviations of the experimental response R(t ) from the calculated ones... 

In order to determine the reactivity of pentachlorophenyl acrylate, 8, in radical initiated copolymerizations, its relative reactivity ratios were obtained with vinyl acetate (M2), ri=1.44 and r2=0.04 using 31 copolymerization experiments, and with ethyl acrylate (M2), ri=0.21 and r2=0.88 using 20 experiments.The composition conversion data was computer-fitted to the integrated form of the copolymer equation using the nonlinear least-squares method of Tidwell and Mortimer,which had been adapted to a computerized format earlier. 

These parameters were estimated to give the best fit to the data for unpromoted and alkali-promoted UFP catalysts by use of a nonlinear least-squares method. The fitness between simulated and experimental product distribution was... 

R was determined from the area under the breakthrough curves using a planimeter. Mass eluted compared well with mass injected, indicating that mass balance was achieved. Dispersion (D) for a conservative tracer was determined by fitting the KCl breakthrough curve to the equilibrium model the fitted parameters were R and P. A nonlinear least squares method was used for parameter estimation 7. The sum of the squares of the deviation between model and data (ssq) was used as a measure of total error in the model fit. 

From the point-by-point solution for the data provided by the nonlinear least-squares method it was estimated that the error in determining the optical rotation (a) for the foregoing run was constant and about 0-013°. This corresponds to an average observational error of 0-06% in a. Excellent agreement was achieved by three different methods of calculation, and thus the criterion suggested by Collins and Lietzke was satisfied. Two separate runs gave similar results. Of the fourteen separate kinetic runs, only one was discarded, and this was one of four determinations of 4 for 115. The results of the calculations of 4 for the discarded run are ... 

We applied a nonlinear least squares method to determined the four constants Ki, Kj, Ni, and Nj in Eq. (3). The eonstants that gave the best fit with the experimental data are listed in Fig. 3. The solid line in Fig. 3 represents the theoretical curve calculated from Eq. using the constants. Figure 4 shows a comparison of the proton-binding capacity of HA with that of some inorganic adsorbents such as silica gel (Silikagel H and Aerosol 200), hydrous titanium oxide (HTiO), y-alumina, hydrous throium oxide (HThO), and magnetite. It is evident that the number of... 

Nonlinear least squares method allows to compare the measured data N(tk) with values predicted Ndtk) from a model and the parameters of the model to be varied to yield the minimum deviation from the data through minimization of the goodness-of-fit parameter calculated from... 

A straightforward extension of the three-point technique is to utilize a larger number of measured AE-i data pairs, and to analyze the data by using some kind of numerical curve-fitting procedure, usually a nonlinear least-squares method. This increases... 

Equation (2.25) represents a nonUnear model of the measured absorbance as a function of the molar fractions and parameters of the individual bands. An optimization procedure based on a nonlinear least squares method is then performed to minimize the difference between the measured spectral data and those simulated through Eq. (2.25). 

Although the values of and Ks should be calculated from nonlinear Equation 3.2, very often the linearized form of Equation 3.2 (i.e., Equation 4.1) is used, probably because it does not require an approximation method, such as nonlinear least-squares method, and computer for data analysis and, furthermore. Equation 4.1 gives the exact solution. 

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