Least squares method nonlinear fits

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. 

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... 

The Bird-Carreau model is an integral model which involves taking an integral over the entire deformation history of the material (Bistany and Kokini, 1983). This model can describe non-Newtonian viscosity, shear rate-dependent normal stresses, frequency-dependent complex viscosity, stress relaxation after large deformation shear flow, recoil, and hysteresis loops (Bird and Carreau, 1968). The model parameters are determined by a nonlinear least squares method in fitting four material functions (aj, 2, Ai, and A2). 

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... 

Kinetic analysis usually employs concentration as the independent variable in equations that express the relationships between the parameter being measured and initial concentrations of the components. Such is the case with simultaneous determinations based on the use of the classical least-squares method but not for nonlinear multicomponent analyses. However, the problem is simplified if the measured parameter is used as the independent variable also, this method resolves for the concentration of the components of interest being measured as a function of a measurable quantity. This model, which can be used to fit data that are far from linear, has been used for the resolution of mixtures of protocatechuic... 

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). 

Least-squares methods are usually used for fitting a model to experimental data. They may be used for functions consisting of square sums of nonlinear functions. The well-known Gauss-Newton method often leads to instabilities in the minimization process since the steps are too large. The Marquardt algorithm [9 1 is better in this respect but it is computationally expensive. 

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. 

Full profile refinement is computationally intense and employs the nonlinear least squares method (section 6.6), which requires a reasonable initial approximation of many fi ee variables. These usually include peak shape parameters, unit cell dimensions and coordinates of all atoms in the model of the crystal structure. Other unknowns (e.g. constant background, scale factor, overall atomic displacement parameter, etc.) may be simply guessed at the beginning and then effectively refined, as the least squares fit converges to a global minimum. When either Le Bail s or Pawley s techniques were employed to perform a full pattern decomposition prior to Rietveld refinement, it only makes sense to use suitably determined relevant parameters (background, peak shape, zero shift or sample displacement, and unit cell dimensions) as the initial approximation. 

Most determination methods finally lead to discrete loading versus concentration data that have to be fitted to a continuous isotherm equation. For this purpose it is advised to use a least-squares method to obtain the parameters of the isotherm. Nonlinear optimization algorithms for such problems are implemented in standard spreadsheet programs. To select an isotherm equation and obtain a meaningful fit,... 

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. 

The treatment of statistics is focused on explicit applications of both linear and nonlinear least-squares methods, rather than on the alphabet soup (F, Q, R, T, etc.) of available tests. However, within that rather narrow framework, many practical aspects of error analysis and curve fitting are considered. They are chosen to illustrate the now almost two centuries old dictum of de Laplace that the theory of probability is merely common sense confirmed by calculation. 

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. 

A linear fit of I versus cos 0 or I versus sin 0 will reveal whether orientation is consistent for the collected spectra. If no systematic deviation from linearity is observed, there are probably no gross experimental artifacts. The sample charging artifact described in Section 4.2.2, however, can sometimes result in a linear fit because the illuminated spot size increases trigonometrically with incident angle. The standard deviations of the fit parameters provide some statistical uncertainty of the orientation measurement. With a sufficient number of points, a confidence interval can be determined. Small differences in orientation can then be judged on a statistical confidence basis. The parameters A and a can also be fit directly using nonlinear least-squares methods. 

It is obvious that, from Fig. 2, cannot be calculated with precision. An approximate estimation of Ki can be obtained only with very low concentrations of B, in a situation when the secondary graph of Slopei/ versus l/B represents an almost linear function. However, as usu, the best method for obtaining the kinetic parameters is to fit the rate equation by nonlinear least square methods (Ratkowsky, 1983 Johnson Faunt, 1992 Johnson, 1994 Watts, 1994). 

In addition, we also conducted Weibull fits on the nonparametric subsystems reliability using a nonlinear least squares method. The results proved significantly close to the ones obtained with the MLE method, increasing the confidence in these results. 

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... 

The pH dependence of the number of cadmium and lead ions complexed with HA is presented in Figs. 5 and 6, respectively. A nonlinear least squares method was applied to find and Am2- The constants that gave the best fit... 

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... 

Non-linear methods are used not only for the optimization as illustrated above but also in regression analysis when fitting functions which are nonlinear with respect to their coefficients. For example, an application of the least squares method for the estimation of coefficients a and b of the function y = a(l-exp(-bx)) leads to NLP. In Section 1.4.3 NLP has been used for the estimation of unknown parameters of a fibre migration model. 

A sample set of data is shown in Fig. 3B. We use a nonlinear least squares algorithm to fit the data to Eq. (2). The determined is nearly identical to the value obtained by other methods (Jacques et al., 2002). 

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