Significant figure least-square parameters

A least square error technique was used to determine values for the parameters AN and K from the experimental data. Examination of these parameters and base fuel properties yielded one significant correlation between AN and Nf. Figure 5 illustrates this straight line relationship. 

Most problems in crystallography " are not linear therefore, it is not possible to solve for the parameters directly. For example, the structure factor F hkl) is not a linear function of x, y, z, and B, because it contains cosine functions (Figure 6.21, Chapter 6). If, however,we have an initial set of parameters that are significantly better than a random guess, we can use the method of least squares to improve the parameters. The use of a Taylor s series allows certain functions, such as cosines and sines, to be expanded as a convergent power series so that the higher powers of X, the parameter, make successively smaller and smaller contributions to the total function. If/(x) is a differentiable function for x = xq -i-Ax, then the Taylor s series expression is ... 

The kinetic parameters of the equations were determined by the least square method, where the sum of Z (X exp-X modei) has teen minimized. In order to test the adequacy of models two statistica tests have been realized as it is proposed by [Froment and Bishoff, (1990)]. Lack of fit (F) test was applied as well as 95% confidence interval of the parameters was estimated. Models that show lack of fit or whose parameter(s) is non-significantly different from zero or negative were rejected A block diagram of model discrimination is shown in the Figure 3. 

Significance Tests Based on the R Values. If a parameter which is difficult to refine by the least-squares method is of particular interest, an estimate may be obtained by carrying out refinements for various fixed values of the parameter and finding the minimum in S [equation (77)] or in [equation (98)]. Let the minimum value of ifw be B. The ratios St = Ry,/R° are then calculated for other values of the parameter. An example is given in Figure 16, where the torsional an e about the S-4-B bonds in... 

Another interesting item worth commenting on is the significant difference between the Weibull modulus my for the three-point flexure bar rupture data (my = 11.96, oey = 612.7 MPa) and that of the thermal shocked disk (my = 6.91, (j0y = 345.9 MPa) experimental rupture data (shown in Figure 5) as determined by CAKES/Life maximum likelihood parameter estimation. Least squares regression (using an Excel spreadsheet) on the CARES/Life disk predictions... 

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