Overall variance, minimization

More attention to selecting and obtaining a representative sample. The design of a statistically based sampling plan and its implementation are discussed earlier, and in more detail than in other textbooks. Topics that are covered include how to obtain a representative sample, how much sample to collect, how many samples to collect, how to minimize the overall variance for an analytical method, tools for collecting samples, and sample preservation. 

Now consider the optimal placement of intermediates leading to the least overall variance. The criterion of minimal variance is obtained by inserting (6.45) into the right-hand side of (6.47) for each stage, and minimizing the overall error subject to... 

For coal that is sampled in accordance with standard methods (ASTM D-2234 ASTM D-4596 ASTM D-4916 ASTM D-6315 ASTM D-6518 ISO 13909) and with the standard preparation of the samples for analysis (ASTM D-346 ASTM D-2013), the overall variance of the final analytical data is minimized and falls within the limits of anticipated experimental difference. 

In conventional variance minimization calculations (14) (i.e. the unconstrained A = 1 case), the above property is used to find an overall fit to IP (we drop the A superscript for simplicity). The procedure is to determine the parameters a in the trial function vPt(R-5 a ) by minimizing the variance of local energy a2... 

The term tr(M)-1 designates the trace of the dispersion matrix. Because the diagonal elements of M 1 present the variances of the regression coefficients, the trace (e.g., their sum) is a measure of the overall variance of the regression coefficients. The minimization of this measure ensures better precision in the estimation of the regression coefficients. 

To obtain estimates of the relative contribution to overall variance, Jones et al. used a voltage interruption/polarity reversal method. Huang et al. attribute part of the excessive peak width observed in their experiments to analyte-wall interaction. One may minimize such interactions by the appropriate choice of buffer pH, ionic strength, or buffer additives (see Section 1.3). 

Although in this formula it would be ideal to eliminate as many variances as possible, there are inherent limitations. Variances can be minimized, but cannot be eliminated. While instrumental variance is a characteristic feature of instrumentation involved, minimal sample preparation and non-destructive sample participation in the experiment will stron y affect overall variance. In view of these conaderations it has been always one of the highest priorities to design a hyphenated instrument that would be non-intrusive and non-destructive to a specimen, yet allowing full characterization. 

A criterion that is closely related to D-optimality is E-optimality. The D-optimality criterion minimizes the volume of the confidence ellipsoid of the regression coefficients. Hence, it minimizes the overall uncertainty in the estimation of the regression coefficients. The E-optimality criterion minimizes the length of the longest axis of the same confidence ellipsoid. It minimizes the uncertainty of the regression coefficient that has the worst estimate (highest variance). 

The difficulty of separation is highly dependent on peak spreading, as shown in Fig. 6.5. It is therefore critical to minimize the peak width as far as possible. This would be difficult for cell display methods if only single color fluorescent labeling were used, because the primary source of variability is biological. Flow cytometry instrumentation point spread functions generally contribute below 2 % to the overall coefficient of variance (CV = standard deviation/mean), but typical overall CVs for yeast display are approximately 50 - 100 % for the logarithmic fluorescence intensity. 

As can be seen from Table 10 and Fig. 7, the spinning riffler is the best of the five methods in minimizing overall sampling variance and therefore provides a more representative sample. A commercially available apparatus is shown in Fig. 8 and consists of a discharge hopper which holds about 25 g of material... 

In this relation, the first term on the right-hand side is the contribution of Taylor dispersion to the overall band broadening. The second is due to molecular diffusion. A key point is that the contribution to the variance due to Taylor dispersion increases with nominal electric field, Eo, whereas variance due to molecular diffusion decreases with electric field, I/Eq. This is a consequence of the Uj and therefore, Eq scaling of Deft on the one hand, and the Lju and therefore, I/Aq scaling of time. Therefore, there exists an optimum electric field that will minimize the peak broadening. This optimum electric field can be derived by minimizing the variance with respect to field to obtain... 

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