Non-linear least squares minimization

When peak shape functions and their parameters, including Bragg reflection positions, are known precisely and the background is modeled by a polynomial function with j coefficients, the solution of Eq. 6.6 is trivial because all equations are linear with respect to the unknowns (Bj, see Eq. 4.1, and / ). It facilitates the use of a linear least squares algorithm described in section 5.13.1. In practice, it is nearly always necessary to refine both peak shape and lattice parameters in addition to Bj and h to achieve a better precision of the resultant integrated intensities. Thus, a non-linear least squares minimization technique (see next section) is usually employed during full pattern decomposition using Eq. 6.6. 

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... 

During refinement using the Rietveld method, the following system of equations is solved by means of a non-linear least squares minimization ... 

A non-linear least-squares minimization technique was used to fit the filtered spectrum with a semi-empirical expression for the EXAFS as follows ... 

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... 

A quarter of a century ago Behnken [224] as well as Tidwell and Mortimer [225] pointed out that the linearization transforms the error structure in the observed copolymer composition with the result that such errors after transformation have no longer zero mean and constant variances. It means that such transformed variables do not meet the requirements for the least-squares procedure. The only statistically accurate means of estimation of the reactivity ratios from the experimental data is based on the non-linear least-squares procedure. An effective computing program for this purpose has been published by Tidwell and Mortimer (TM) [225]. Their method is considered to be such a modification of the curve-fitting procedure where the sum of the squares of the difference between the observed and computed polymer compositions is minimized. 

Figure 6.3. Two examples when the non-linear least squares technique may fail in finding the best solution of Eq, 6.9 left - the initial approximation (xq) is located near a false minimum right - the minimum is poorly defined. The arrows represent the possible outcomes of two least squares cycles. In the case on the left the minimization ends in a false minimum. In the case on the right the obtained shifts have correct signs but wrong magnitudes and instead of converging (i.e. instead of all Ar,l becoming smaller), their absolute values continue to increase. True solutions (i.e. global minima) are marked as, ne-...

Non-linear least squares technique results in finding a set of increments that are added to a set of free variables chosen to represent a certain initial approximation. Parameters, obtained in this ways, are carried over into the next refinement cycle as a more precise initial approximation. In some cases it may take a few refinement cycles to achieve the best fit, i.e. to minimize the corresponding function, while in many instances the number of required least squares steps may be quite large. Especially in Rietveld refinement, where various groups of parameters have a different and often unrelated physical origin, the ability to detect the completion of the minimization, i.e. the complete convergence of the least squares, is essential. 

Kinetic and electrochemical data, respectively, were fitted to eqs 28 and 29. Non-linear least-square fits of the observed rate constant and the formal redox potential versus [Ml were carried out using the Solver Function in Microsoft Excel-98. Sums of deviation-squared values were minimized by varying k, kivir, Ko, Kred, Eo, a and b in eqs 28-30. The ratios Yox/y ox and Yred/y rcd were assigned a value of 1. Additional limitations and constraints imposed on the adjustable parameters to improve fitting are discussed above. 

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). 

NLLSQ Non-Linear Least Squares gradient minimization ... 

The unknown values of the equivalent circuit elements are generally evaluated by a non-linear least-squares (NLS) fitting algorithm. The least-squares fitting procedure aims at finding a set of parameters which minimizes the sum [1] ... 

In a nut shell, the Rietveld method uses a non-linear least square method to minimize a function corresponding to the difference between the observed and... 

The standard error values provide only approximate ranges of confidence intervals for general non-linear least squares data fitting. They are estimated from the so called covariance matrix. In this approach one parameter at a time is allowed to vary with the other parameters allowed to change so as to minimize the square error criteria with only the one parameter changed. This is the only easy calculation one can make in the general non-linear case with an unknown distribution of errors. For a detailed discussion of standard errors the reader is referred to the literature (such as Numerical Recipes in C, Cambridge Press, 1988). 

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