Weighted least-squares method

X-ray diffraction (XRD) and magnetic susceptibility measurements were carried out on polycrystalline materials that had been calcined in oxygen. XRD powder patterns were recorded with Cu Ka radiation using a Ni filter on a Rigaku diffractometer. For reference all patterns were recorded with an internal NBS Si standard. After correcting the peak position based on the observed silicon line positions, the lattice constants were then refined by a least squares method weighted proportional to the square root of the height and inversely with the square of the width of the peak. 

Many computer libraries contain programs that perform the necessary statistical calculations and relieve the engineer of this burden. For discussions of the use of weighted least squares methods for the analysis of kinetic data, see Margerison s review (8) on the treatment of experimental data and the treatments of Kittrell et al. (9), and Peterson (10). 

Ordinary least squares regression requires constant variance across the range of data. This has typically not been satisfied with chromatographic data ( 4,9,10 ). Some have adjusted data to constant variance by a weighted least squares method ( ) The other general adjustment method has been by transformation of data. The log-log transformation is commonly used ( 9,10 ). One author compares the robustness of nonweighted, weighted linear, and maximum likelihood estimation methods ( ). Another has... 

In the ordinary weighted least squares method, the most probable values of source contributions are achieved by minimizing the weighted sum of squares of the difference between the measured values of the ambient concentration and those calculated from Equation 1 weighted by the analytical uncertainty of those ambient measurements. This solution provides the added benefit of being able to propagate the measured uncertainty of the ambient concentrations through the calculations to come up with a confidence interval around the calculated source contributions. 

The mathematical procedure used to maximize the configurational similarity in Figure 1 with a space generated by the weighted physicochemical parameters is based on a least-squares method in which the basic matrix equations are ... 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

The values of kinetic parameters (pre-exponential factors k0j and activation energies Ej of rate constants k and inhibition constant Kg) can for a particular catalyst be determined by weighted least squares method, Eq. (35), from the light-off or complete ignition-extinction curves measured in experiments with slowly varying one inlet gas variable—temperature or concentration of one component (cf., e.g., Ansell et al., 1996 Dubien et al., 1997 Dvorak et al., 1994 Kryl et al, 2005 Koci et al., 2004c, 2007b Pinkas et al., 1995). 

The value of reaction rate Eq. (43) can be negative when N02 present in the mixture is transformed to NO via backward reaction, typically at higher temperatures. A comparison of measured and simulated outlet N02 concentrations in dependence on temperature can be seen for two different space velocities in Fig. 13. The pre-exponential factor k j and activation energy Ej of the kinetic constant no/no2 in the global rate law were evaluated by the weighted least squares method, Eq. (35). 

Examples of light-off experimental data for CO and C3H6 oxidation are given in Fig. 22a b, together with the simulated outlet concentrations. The respective kinetic parameters—the rate constants kj(T) and the inhibition constants ( )—were evaluated from experimental data by the weighted least squares method, Eq. (35). 

The uncertainty estimation algorithm adopted [2, 3], based on the weighted least squares method, takes into consideration ... 

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... 

Fig. 23. Molecular weight dependence of the mean-square weight average radius of gyration in the unperturbed state relative to Mw (

Comparing the last formula with the corresponding equation for the weighted least,-squares method (3.i56), we see that we have obtained exactly the same result, if we substitute the matrix W for [Pg.73]

A different modification was proposed by Cammarata and Yau (213). They arbitrarily assigned the observed biological activity of the unsubstituted compound to be the constant term. By this they assumed that there is no experimental error in determining the biological activity of the unsubstituted compound. However, since all the measured values contain experimental error, the weights given to the different activity contributions,, obtained from a least-squares procedure will not be equal. JThis is inconsistent with the definition of the least-squares method (214). 

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... 

Other methods may be more appropriate for equations with particular mathematical characteristics or when more accurate, robust, stable and efficient solutions are required. The alternative spectral methods can be classified as sub-groups of the general approximation technique for solving differential equations named the method of weighted residuals (MWR) [51]. The relevant spectral methods are called the collocation Galerkin, Tan- and Least squares methods. These methods can also be applied to subdomains. The subdomain... 

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