Non-linear least square fit

Figure B2.4.6. Results of an offset-saturation expermient for measuring the spin-spin relaxation time, T. In this experiment, the signal is irradiated at some offset from resonance until a steady state is achieved. The partially saturated z magnetization is then measured with a kH pulse. This figure shows a plot of the z magnetization as a fiinction of the offset of the saturating field from resonance. Circles represent measured data the line is a non-linear least-squares fit. The signal is nonnal when the saturation is far away, and dips to a minimum on resonance. The width of this dip gives T, independent of magnetic field inliomogeneity.

Figure B2.4.8. Relaxation of two of tlie exchanging methyl groups in the TEMPO derivative in figure B2.4.7. The dotted lines show the relaxation of the two methyl signals after a non-selective inversion pulse (a typical experunent). The heavy solid line shows the recovery after the selective inversion of one of the methyl signals. The inverted signal (circles) recovers more quickly, under the combined influence of relaxation and exchange with the non-inverted peak. The signal that was not inverted (squares) shows a characteristic transient. The lines represent a non-linear least-squares fit to the data.

Figure 4. Fits of lattice strain model to experimental mineral-melt partition coefficients for (a) plagioclase (run 90-6 of Blundy and Wood 1994) and (b) elinopyroxene (ran DC23 of Blundy and Dalton 2000). Different valence cations, entering the large cation site of each mineral, are denoted by different symbols. The curves are non-linear least squares fits of Equation (1) to the data for each valence. Errors bars, when larger than symbol, are 1 s.d. Ionic radii in Vlll-fold coordination are taken from Shannon (1976).

We have found that data interpretation other than simple plotting from texture studies requires a sophisticated array of software tools. These include least squares curve resolving, non-linear least squares fitting, 2-dimensional data smoothing, numerical quadrature, and high-speed interactive graphics, to mention only a few. 

The equilibrium constant and dissociation rate constant were determined simultaneously by non-linear least-squares fitting, unless the absorption signal was too low157 or no dependence of relaxation frequency on concentration was observed.159,161,162 The association rate constant was then calculated from the definition of the equilibrium constant. The equilibrium constants determined from the dynamics in this manner agree fairly well with equilibrium constants determined independently. 

In the present study we have used the phase and amplitude functions of absorber-scatterer pairs in known model compounds to fit the EXAFS of the catalysts. By use of Fourier filtering, the contribution from a single coordination shell is isolated and the resulting filtered EXAFS is then non-linear least squares fitted as described in Ref. (19, 20). 

Iqdari and Velde (unpub. data, 1992, see Table 8.2) described experiments of Ce diffusion in apatite soaked in CeCl2 with asymmetric diffusion profiles. For one of their runs carried out at 1100°C for 15 days, and described as an example of a non-linear least-square fit in Section 5.2, it has been found that the relationship between the Ce concentration CCt and the distance X to the mineral surface is described by... 

The algorithms developed in this chapter can model any situation, e.g. they can serve to demonstrate the effects of initial concentrations and rate constants in kinetics and of total concentration and equilibrium constants in equilibrium situations. Very importantly, these algorithms further form the core of non-linear least-squares fitting programs for the determination of rate or equilibrium constants, introduced and developed in Chapter 3, Model-Based Analyses. 

Chapter 4 is an introduction to linear and non-linear least-squares fitting. The theory is developed and exemplified in several stages, each demonstrated with typical applications. The chapter culminates with the development of a very general Newton-Gauss-Levenberg/Marquardt algorithm. 

This is perhaps the "best solution for the given data set, and it is certainly the most interesting. It is not offered as a rigorous solution, however, for the lack of fit (x /df -[9.64]2) implies additional sources of error, which may be due to additional scatter about the calibration curve (oy -"between" component), residual error in the analytic model for the calibration function, or errors in the "standard" x-values. (We believe the last source of error to be the most likely for this data set.) For these reasons, and because we wish to avoid complications introduced by non-linear least squares fitting, we take the model y=B+Axl 12 and the relation Oy = 0.028 + 0.49x to be exact and then apply linear WLS for the estimation of B and A and their standard errors. 

The parameters of this expansion, as well as the number N of Lorentzian functions, are determined (from the experimental data) by a non-linear least squares fit along with statistical tests. It can be noticed that this expansion has no physical meaning but is merely a numerical device allowing for smoothing and interpolation of the experimental data. Nevertheless, this procedure proves to be statistically more significant than the Cole-Cole equation and thus to account much better for the representation of experimental data. The two physically meaningful parameters, i.e., C(0) and (Xo), can then be easily deduced from the quantities involved in (71)... 

The resulting values of T and T2 are determined by means of a non-linear least squares fit, using a parameterized model based on eq. (20). For a given... 

Fig. 2. Video micrograph of exocytosis in a goblet cell grown in tissue culture. Notice that the swelling of a secretory granule has been captured at 3 consecutive times (inset). The radial expansion of the exocytosed granules follows a typical first-order kinetics. The continuous line is a non-linear least square fitting to the data points to r(t) = rf — (rf — rj e " /T, where r, and rr are the initial and final radius of the granule, and t is the characteristic relaxation time of swelling...

FRAP data were analysed by a non-linear least squares fit to an expression [8,23,29], defining the time dependence of the fluorescence recovery (F(t)). The apparatus as described above delivered a laser spot of uniform circular cross sectional intensity to the sample and the recovery curves obtained could be analysed with the expression ... 

When tr is unknown, it can be adjusted together with re and r by using non-linear least-squares fits of eqs. (15), (16) as similarly described for the treatment of NMRD profiles (Bertini et al., 1995 Ruloff et al., 1998 Toth et al., 1998), but a much simpler approach considers that both dipolar and Curie-spin contribution depend on r 6. When a nucleus for which the / -nucleus distance rref can be estimated either from crystal structure or gas-phase modeling is used as a reference, eq. (15) reduces to its simplest form (eq. (18)) and relative /Gnucleus i distances are accessible without estimations of re and rr (Barry et al., 1971 ... 

Table 2 Fit parameters for the isolated shaperesonances. Parameters obtained from non-linear least-squares fit of Fano-profile to the hydrogen-induced shape-resonance in experimental spectra. Applied experimental width Gaussian, 5 eV.

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). 

To test the validity of our model, it is necessary to determine expressions for the current from equations (2.5)-(2.8) which are applicable across the case boundaries. These expressions are given in Table 2.3. We can then analyse the experimental data using the appropriate expression. Since these equations do not include concentration polarisation in the solution, equations (2.4) and (2.15) are used to determine the concentrations of NADH and NAD+ at the solution/film interface. The case, or cases, that the experimental data span are determined by inspection and using Table 2.2. Non-linear least squares fitting was used to fit the experimental data to the appropriate equations. The resulting best-fit parameters are critically... 

Best-lit parameters from the analysis of the currents for all NADH oxidation data fitted at once using both the uninhibited and inhibited fits at poly(aniline)/poly(vinylsulfonate)-modified electrodes. The values were obtained by non-linear least squares fits of the experimental data to equations (2.4)-(2.8) for the uninhibited fit, with the addition of equations (2.15), (2.17) and (2.18) for the inhibited fit. In each case n is the total number of data points used in the fit... 

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 rapid development of computer technology has yielded powerful tools that make it possible for modem EIS analysis software not only to optimize an equivalent circuit, but also to produce much more reliable system parameters. For most EIS data analysis software, a non-linear least squares fitting method, developed by Marquardt and Levenberg, is commonly used. The NLLS Levenberg-Marquardt algorithm has become the basic engine of several data analysis programs. 

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