Parameter coefficients

The use of fundamental parameters is attractive for various reasons. They impose fewer restrictions on the number of standards required for analysis. This simplifies the standardisation protocol for maintaining a XRF system, and permits greater flexibility in dealing with different types of materials. Inten-sity/concentration algorithms of the fundamental type, i.e. without recourse to the use of standards, have gradually developed [238-240] and are now widely available [241]. Functionality and quality of XRF software have reached a very high level, with a large variety of evaluation procedures and correction models for quantitative analysis, and calculation of fundamental parameter coefficients for effective matrix corrections. Nevertheless, there is still a need for accuracy improvement of fundamental parameters, such as the attenuation functions. 

Table 11.9. The general model dependence of the largest normalized parameter coefficients on normalized Pt, Ba, and Fe weight loadings...

Figure 11.11. The normalized parameter coefficients and their dependence on the normalized Pt, Ba, and Fe weight loadings, following the form of Eqns (1) and (2) for NO, concentration and N2 selectivity dependence of the constant in the catalyst specific model, on the weight loadings of Pt and Ba (a) and (b) Two different views.

Parameter Coefficient when using only linear term t-value when using only linear term Coefficient using square term t-value using square term... 

Multiple regressio Parameters coefficient Regression coefficients for separate metals n ... 

Again, let X be the matrix of parameter coefficients defined by the model to be fit and the coordinates of the experiments in factor space. Let Y be the response matrix associated with those experiments. Let B be the matrix of parameters, and let a new matrix R be the matrix of residuals. Equation 5.25 may now be rewritten in matrix notation as... 

It is not possible to fit this model using matrix least squares techniques The matrix of parameter coefficients, X, does not exist - it is a 0x0 matrix and has no elements because there are no parameters in the model. However, the matrix of residuals, R, is defined. It should not be surprising that for this model, R = Y that is, the matrix of residuals is identical to the matrix of responses. 

The effect on the variance-covariance matrix of two experiments located at different positions in factor space can be investigated by locating one experiment at X, = 1 and varying the location of the second experiment. The first row of the matrix of parameter coefficients for the model y,- = + p,jc, + r, can be made to... 

Although it is true that the first three columns of plus and minus signs in Table 14.3 are equivalent to the abbreviated coded experimental design matrix D, the signs in Table 14.3 are used for a slightly different purpose than they were Table 14.2. In fact, as we will see, the eight columns of signs in Table 14.3 are equivalent to the matrix of parameter coefficients, X. 

Landau and Lifshitz, 1980), where J is the conjugate field of the order parameter, coefficient 4 is a function of temperature ... 

Let us fit the probabilistic model, = P0 + ru, to the same data (see Figure 5.10). If the least squares approach to the fitting of this model is employed, the appropriate matrices and results are exactly those given in Section 5.2 where the same model was fit to the different factor levels xu = 3, yn = 3, xl2 = 6, yl2 = 5. This identical mathematics should not be surprising the model does not include a term for the factor xx and thus the matrix of parameter coefficients. A", should be the same for both sets of data. The parameter / 0 is again estimated to be 4, and ar2 is estimated to be 2. 

Then one has to utilize special algorithms (software) to calculate reliable coefficients. The appropriate nonlinear parameter (coefficient) estimation should be preferred. Sometimes again it is possible to linearize a nonlinear model. For the above function, e.g., using a logarithmic transformation will give ... 

Such transformations should, however, be carried out with caution, because one has to ensure that the statistical properties of the variables are not changed. The creation of het-eroscedastic behavior may be mentioned as one possible problem. Proper nonlinear parameter (coefficient) estimation is, therefore, preferable. 

Charge carrier Parameter/coefficient Relation Unit... 

The remaining (and more complicated) cases from Table 6 are described by complicated differential equations [212], Their complexity rests in the number of parameters, coefficients, and constants rather than in the mathematical expression. They are therefore difficult to verify. 

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