Parameter Standard Deviations

The parameter (regression coefficient) variances (and thus standard deviations) can be estimated from the matrix G of Equation 7.10. The following is stated without proof  

Variance [var(c)] represent the parameter variances. S is sometimes called the variance-covariance matrix. Variance is a measure of the spread of expected values of random variables belonging to a specific probability distribution. As has been mentioned previously, the validity of the least-squares method for determining regression parameters is based on errors in the data having a normal (Gaussian) distribution (the familiar bell-shaped curve) with zero mean and constant variance. The values of the parameters determined from data with such normal errors are, in a 

When the variances must be estimated, the probability distribution of the parameters is the t-distribution. The t-distribution is similar to the normal distribution, but it has an additional argument called the degrees of freedom, which is the difference between the number of data points and the number of parameters  

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It can be seen from the figures that the size distributions of spray droplets become narrower after impingement, or, in other words, the droplet sizes become more uniform than before. Such a variation is observed in most of the runs, although some exceptions also appeared (about 10% in all the runs). As is well known, the uniformity of droplet sizes, or inversely, the scattering of the size distribution can be expressed with the parameter Standard Deviation , a, which is the defined as... 

FIGURE 12.16 Correlation between mean DOC concentrations and mean salinity, using the data set compiled in Fig. 12.12. To show the variability of both parameters, standard deviations at the individual stations are given for both parameters. (Modified and completed from Nausch et al., 2003.)... 

Average number of protein-protein and protein-DNA H-bonds determined by the program HBPLUS (McDonald and Thornton, 1994) with the default geometric parameters. Standard deviations are in parentheses. 

Another method for finding the minimum is the so-called downhill simplex method [3, 619]. It requires only a function evaluation and does not use either function derivatives or matrix inversitMi. It may be relatively slow if one is trying to optimize many parameters and a shallow minimum, but it will always find a minimum (at least a local minimum). The problem with this technique is that it does not calculate the parameters standard deviations directly. In such cases, it is advisable, after finding the ntinimum by the simplex method, to use these parameters in the CNLS approximation, which should cmiverge quickly and provide standard deviations of the parameters. 

Initially, the variables are vectors from a bounded subspace of R". Afterwards, the search domain is extended to the whole space. The Meta-evolutionary programming approach also incorporates strategy parameters (standard deviations) in this representation. 

The parameter standard deviations and t-ratios are shown in Figure 7.13. 

This is an NLR problem. Solve this by minimizing the sum of squares of residuals using Solver. After you have determined the optimal values for Kj and Ajj, calculate numerically (using second-order correct formulas) the derivatives Zj and Z2 at each data point, form the G matrix, calculate the parameter standard deviations, and calculate the t-ratios for each parameter. 

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