Variance-covariance matrix parameters, calculation

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The off-diagonal elements of the variance-covariance matrix represent the covariances between different parameters. From the covariances and variances, correlation coefficients between parameters can be calculated. When the parameters are completely independent, the correlation coefficient is zero. As the parameters become more correlated, the correlation coefficient approaches a value of +1 or -1. 

Standard deviations in unit-cell parameters may be calculated analytically by error propagation. In these programs, however, the Jacobian of the transformation from Sj,. .., s6 to unit-cell parameters and volume is evaluated numerically and used to transform the variance-covariance matrix of Si,. .., s6 into the variances of the cell parameters and volume from which standard deviations are calculated. If suitable standard deviations are not obtained for certain of the unit cell parameters, it is easy to program the computer to measure additional reflections which strongly correlate with the desired parameters, and repeat the final calculations with this additional data. 

The training set will be used to calculate the multivariate mean and variance-covariance matrix however, before calculating these parameters, we will graphically examine the training set to see if it contains measurements that are approximately normally distributed. This can be accomplished by several methods, the simplest being to plot histograms of the individual variables. Use the MATLAB hist command to... 

The cell parameters of the trigonal phase can now be calculated at each pressure for which the monoclinic unit-cell was measured. The estimated uncertainties given in Table A3 were obtained from the components of the variance-covariance matrices of the fits (Table A2) through Equations (7) and (8). Note that for the unit-cell parameters, the variance-covariance matrix used in Equation (7) is that of the fit of the cubes of the unitcell parameters, yielding estimates of the uncertainty of the cubes of extrapolated unitcell... 

If one is interested in the standard uncertainties of quantities that are derived from refined parameters such as bond lengths and angles, rtab instructions (see Chapter 2) can be added to trigger the calculation of their values and their estimated standard uncertainties as derived by error propagation (which is based on the fuU variance-covariance matrix of the problem). For details see Example 10.3.2. 

More generally, the errors of parameter estimation can be calculated on the basis of the variance-covariance matrix (Eq. (5.10)). The variance-covariance matrix is computed here on the basis of the MSS for the pure experimental error as follows ... 

More difficult is the estimation of errors for the nonlinear parameters, since no variance-covariance matrix exists. Frequently, the error estimations are restricted to a locally linear range. In the linearization range, the confidence bands for the parameters are then calculated as in the linear case (Eqs. (6.25)-(6.27)). An alternative consists in error estimations on the basis of Monte Carlo simulations or bootstrapping methods (cf. Section 8.2). 

A significance test (t test) is performed, as described in Sec. 7.2.3 [(Eq. (7.102)], on the parameters to test the null hypothesis that any one of the parameters might be qual to zero. The 95% confidence intervals of each measured variable are calculated. The variance-covariance matrix and the matrix of correlation coefficients of the parameters are calculated according to Eqs. (7.135) and (7.154), respectively. The analysis of variance of the regression results is performed as shown in Table 7.2. Finally, the randomness tests are applied to the residuals to test for the randomness of the distribution of these residuals. 

Global Two-Stage Method. An extensive description of the method is provided by Steimer et al. The global two-stage (GTS) approach has been shown, through simulation, to provide unbiased estimates of the population mean parameters and their variance-covariance, whereas the estimates of the variances were upwardly biased if the STS approach was used. These simulations were done under the ideal situation that the residual error was normally distributed with a known variance. However, it is a well-known fact that the asymptotic covariance matrix used in the calculations is approximate and under less ideal conditions, the approximation can be poor. ... 

The covariance matrix of the parameters does not only provide the information on the individual significance of the parameters. A correlation coefficient between the parameter estimates can be calculated from the appropriate elements of this covariance matrix, that is, the covariance between the two parameters considered, element iij), and their corresponding variances, diagonal elements i andy ... 

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