Least squares with weighting factors

Step 11. Fit the selected high temperature data with the modified Guggenheim equation using least squares with weighting factors r,. 

The original predictive control law is based on the solution of a quadratic cost function formulated as a least-squares problem with weighting factors on the manipulated variable moves [9]... 

The temperature-factor parameter B and the scale factor k were determined by a least-squares procedure/ with observational equations set up in logarithmic form and with weights obtained from those in equation (9) by multiplying by (G (obs.))2. Since a semi-logarithmic plot of G2 (obs.)/Gf (calc.) against B showed a pronounced deviation from linearity for the last five lines, these lines were omitted from the subsequent treatments. They were much broader than the others, and apparently their intensities were underestimated. The temperature-factor parameter B was found by this treatment to have the value 1-47 A2. 

Although satisfactory criteria for deciding whether data are better analyzed by distributions or multiexponential sums have yet to established, several methods for determining distributions have been developed. For pulse fluorometry, James and Ware(n) have introduced an exponential series method. Here, data are first analyzed as a sum of up to four exponential terms with variable lifetimes and preexponential weights. This analysis serves to establish estimates for the range of the preexponential and lifetime parameters used in the next step. Next, a probe function is developed with fixed lifetime values and equal preexponential factors. An iterative Marquardt(18) least-squares analysis is undertaken with the lifetimes remaining fixed and the preexponential constrained to remain positive. When the preexponential... 

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 calibration was represented in the computer program by a fifth-degree polynomial. The conventional method of least-squares was followed to determine the coefficients of the polynomial. The sensitivity of the normal equations made round-off error a significant factor in the calculations. The effect of round-off error was greatly reduced when the calculations were performed with double-precision arithmetic. The molecular weights corresponding to selected count numbers were calculated from the coefficients. The coefficients were input information for the data-reduction program. 

Once the phase problem is solved, then the positions of the atoms may he relined by successive structure-factor calculations (Eq. 21 and Fourier summations (Eq. 3) or by a nonlinear least-squares procedure in which one minimizes, for example, )T u ( F , - F,il(, )- with weights w lakcn in a manner appropriate to the experiment. Such a least-squares refinement procedure presupposes that a suitable calculalional model is known. 

The final least-squares refinement, with all atoms anisotropic, gave an R factor of 0.052 for all 1140 reflections, and 0.036 for the 862 (/ > 3o(/)) nonzero weighted reflections. The weighted 0.025. The standard deviation of an observation of unit weight was 1.36. The largest shift of any parameter divided by its estimated standard deviation on the last cycle of least-squares was <0.0012. 

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