Variance with minimization

Example 4 Calculation of Sample Weight for Surface Moisture Content An example is given with reference to material with minimal internal or pore-retained moisture such as mineral concentrates wherein physically adhering moisture is the sole consideration. With this simphfication, a moisture coefficient K is employed as miiltipher of nominal top-size particle size d taken to the third power to account for surface area. Adapting fundamental sampling theory to moisture sampling, variance is of a minimum sample quantity is expressed as... 

The semivariogram as a variance function can also be used to estimate the value and the variance for new points not sampled in the investigated area. The method applied for this purpose is termed kriging. Kriging is a special regression method for interpolation of spatially or temporally correlated data with minimization of variance. The normal distribution of the data is an important condition. If the original data are not normally distributed, which is often the case for trace components in environmental compartments, the logarithm of the data or otherwise transformed data have to be applied to obtain a normal distribution of the data (see also Section 9.4). 

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... 

Prom (2.62) it can be taken that the model s goodness of fit and the number of parameters used are counterbalanced. Among a set of model specifications, the specification with minimal IC value is recommended. In other words, information criteria aim at minimizing the residuals variance with as few parameters as possible. Often used information criteria for time series models are the Akaike information criterion (AIC)," the Schwarz information criterion (SIC) or the Hannan-Quinn information criterion (HQIC) with the following k functions ... 

Notice that if the order of summation of the product terms in the convolution summation is changed, then the order in which the h k) appear in Eq. (8.73) changes. If the order is changed so that the h k) with smallest magnitude is first, followed by the next smallest, etc., then the roundoff noise variance is minimized. Performing the convolution summation in nonsequential order, however, greatly complicates data indexing and so may not be worth the reduction obtained in roundoff noise. 

An other definition of the electron domains is based on the hypothesis that the variance a (N(Q)) of their populations N(Q) = p(r)dr should be minimal with respect to a variation of the domain boundaries. The minimization of the variance with respect to the domain volumes implies that the variational equation... 

By one time slot method, the s3mchronization can be derived from the cyclical log. The s5mchronization based on sending a synchronization signal which is cyclically received and evaluated by aU network participants. To ensure a best possible synchronization, it is required that the signal in a fixed time interval with minimal timing variances will be sent and received. 

Control system performance can be assessed by comparison with a benchmark. For example, historical data representing periods of satisfactory control can be used as a benchmark. Alternatively, the benchmark could be an ideal control system performance, such as minimum variance control. As the name implies, a minimum variance controller minimizes the variance of the controlled variable when unmeasured, random disturbances occur. This ideal performance hmit can be estimated from closed-loop operating data then the ratio of minimum variance to the actual variance is used as the measure of control system performance. This statistically based approach has been commercialized, and many successful industrial applications have been reported (Kozub, 1997 Desborough and Miller, 2002 Harris and Seppala, 2002 Hoo et al., 2003 Paulonis and Cox, 2003). 

Plots of RMSE as a functimi of number of factors (for latent variable methods) are also used to find the optimum number of model compraients. When validation residual variance is minimal, RMSEP also is, and the model with an optimal number of components will have the lowest expected prediction error. 

Find the y that minimizes this integral, subject to the constraints that y is a density function with given mean and variance, i.e.,... 

More recent publications on sulfosuccinates have confirmed the minimal or close to zero skin and eye irritation caused by these products. In a general screening of product safety evaluation methods the authors [16] rejected the sulfosuccinate from further consideration in the statistical analysis of experimental data (variance analysis) because the product had not shown any irritation in the Duhring-Chamber test. The sulfosuccinate (based on fatty alcohol ethoxy late) was tested in a screening with 14 other surfactants, namely, alkyl sulfates, sulfonates, ether sulfates, and a protein fatty acid condensation product. 

Supposing D = Cov(e) to be known, we would possibly improve our estimation procedure by weighting more those points of which we are more certain, that is, those whose associated errors have the least variance, taking also into account the correlations among the errors. We may then indicate with 0 the weighted least-squares estimator (WLSE), which is the value of 0 minimizing... 

---