Least squares non-linear

When the function to be fitted to data does not depend linearly on the parameters, recursive methods must be used. A slightly modified version of the Newton-Raphson method (Chapter 3) will be used (Hamilton, 1964). Let jc be the vector of the n unknowns Xj and y — f(x) the m-vector of observable functions y, = /(jc). The analytical form of the functions f(x) may be the same or not. Let the vector / represent the m observations / of these functions. A vector jc is sought which minimizes the scalar c2 such that 

Since we are dealing with a finite sample, we define, as for the linear case, the least-square estimators and / of jc and y, respectively, as the vectors such as 

From the initial guess x°, we calculate the m values of f(x°) and their derivatives relative to each Xj. Solving the least-square system, we get an improved estimate of x, that we use as the initial value for the next iteration until the values cease to change significantly. Indicating the fcth estimate by the superscript k, we can write 

In Section 8.8., we find that the advection-diffusion model of Craig (1969) amounts to a sum of exponentials. The data listed in Table 5.11 are to be fitted by 

As the initial guess, we chose 0) = —20, / 0) = +20, 0)= 1. Table 5.12 shows some results for the first iteration. For the second iteration, we take the values of aU), fSil and i(1) as the new starting point. 

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 most extensive sources of various software links are found at the International Union of Crystallography (www.iucr.org) and/or Collaborative Computational Project No. 14 (http //www.ccpl4.ac.uk) Web sites. 

Commercial manufacturers, which offer a variety of software products for processing powder diffraction data are Bruker (http //www.bruker-axs.com/production/indexnn.htm), Philips (http //www-us.analytical.philips.com/products/xrd/), Rigaku/Molecular Structure Corporation (http //www.rigaku.com/xrd/index.jsp), STOE Cie, Gmbh (http //www.stoe.com/products/index.htm) and many others. 

Obviously, a linear least squares algorithm described in section 5.13.1 is not directly applicable to find the best solution of Eq. 6.8. In some instances, it may be possible to convert each equation in (6.8) into a linear form by appropriate substitutions of variables and thus reduce the problem to a linear case. In general, the least squares solution of Eq. 6.8 is obtained by expanding the left hand side of every equation using Taylor s series and truncating the expansion after the first partial derivatives of the respective functions. Hence, Eq. 6.8 maybe converted into  

The refined parameters are computed by using both the set of the original Xi°, X2,. .., Xm, which represents the initial approximation of the unknowns, and the vector Ax, which has been obtained from least squares (Eq. 6.10), as  

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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.

To extract the agglomeration kernels from PSD data, the inverse problem mentioned above has to be solved. The population balance is therefore solved for different values of the agglomeration kernel, the results are compared with the experimental distributions and the sums of non-linear least squares are calculated. The calculated distribution with the minimum sum of least squares fits the experimental distribution best. 

These problems were addressed by Tidwell and Mortimer117 118 who advocated numerical analysis by non-linear least squares and Kelen and Tiidos110 1"0 who proposed an improved graphical method for data analysis. The Kelen-Tiidos equation is as follows (eq. 43) ... 

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).

Alcock, R. M. Hartley, F. R. Rogers, D. E., A damped non-linear least-squares computer program (dalsfek) for the evaluation of equilibrium constants from spectrophotometric and potentiometric data, J. Chem. Soc. Dalton Trans. 115-123 (1978). 

Gampp, H. Maeder, M. Zuberbiihler, A. D., General non-linear least-squares programfor the numerical treatment of spectrophotometric data on a single-precision game computer, Talanta 27, 1037-1045 (1980). 

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. 

These parameters are determined by non-linear least-squares optimization of the fit of the function to both the experimental storage and loss moduli curves. As emphasized, the two determiners of temperature-scan peak width referred to above (i.e., in terms of equation (2), activation energy AH of x0 and a ) have features that allow distinguishing... 

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. 

We are also developing an improved approach, based on probability theory, for smoothing the observed data and for describing the features in orientation distributions. Since this approach relies heavily on non-linear least squares techniques, it is best done off line. 

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. 

Figure 3 Non-linear least-squares curve fitting of the orthorhombic WAXS profile of an ethylene 1-decene random copolymer with 2.7 mol% branches. The two crystalline reflections and the amorphous halo are shown. 

The p.c.s. measurements were carried out using a Malvern multibit correlator and spectrometer together with a mode stabilized Coherent Krypton-ion laser. The resulting time correlation functions were analysed using a non-linear least squares procedure on a PDP11 computer. The latex dispersions were first diluted to approximately 0.02% solids after which polymer solution of the required concentration was added. 

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). 

Table 5.12. The first step of non-linear least square refinement of the parameters, ft, and e for selected depths in the water column. The last three columns represent the elements of the matrix...

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 collection of kinetic modelling programs will be adapted in the subsequent chapter for the non-linear least-squares analysis of kinetic data and the determination of rate constants. 

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. 

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