E-optimality criterion

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). 

By minimizing the maximum eigenvalue of the information matrix, M, we will be assured that M will be invertible. This would be impossible if the design matrix X and, subsequently, the information matrix M were ill conditioned. A variant of the previously mentioned E-optimality criterion is shown in Equation 8.90,... 

The y" values in these expressions were optimized by computer, an operation which is readly done for any given set of experimental data by substituting the expressions in Eq. (22) and using for an optimization criterion the requirement that the rms value of (e — c)/(t>e should be as small as possible. 

The overall cost optimization criterion (5.4-137) to be minimized is composed of two terms. The first, which is called proportional, is related to the yield of both C and E one needs to maximize the yield of C, Yc, while minimising the ratio of yields, Y c. Yc and Kg are nc.f/nB.o and nE./nBA, respectively, n is number of moles, and y is a factor expressing the relative weight of the two terms (y was assumed to be one). The second term, called the non-proportional or fixed cost of operation, is the reciprocal of the ratio of yield of C per batch time, s, and as such it should be minimized, p is the weighing factor (equal to 174 in the process under consideration) in this term. 

On the other hand, since the model must be considered as a flexible tool that can be adapted to the experimental data by changing the values of the adjustable parameters, the method consists in computing the optimal values of the parameters, 0, on the basis of a suitable optimality criterion and submitting to a statistical analysis the residual errors %n,j, i.e., the differences between the measured data and the corresponding optimal computed values, em, = dmj - ymj(0) = dmj —ym,j-... 

The objective function to be maximized is the total probability of these errors. Since the experimental data are assumed to be independent, the total probability can be obtained by multiplying the single probabilities. Thus, the maximum likelihood optimality criterion is obtained by exploiting the probability density function (3.14), i.e.,... 

If two values for Pv for each peak are obtained, then this strategy corresponds to the use of (1 /2) ZPV or V/7Pv as the optimization criterion. This third approach appears to be the most correct one, since it creates a common basis for all sum and product criteria (i.e. those based on P, Pm, P Rs and 5), which may allow a comparison between the different propositions. 

Bayard, D. S., Yam, Y., and Mettler, E., A criterion for joint optimization of identification and robust control, IEEE Trans, on Autom. Control 37, 986 (1992). 

As a consequence of 1 and/or 2, the average excess production is no longer a nondecreasing function of time. Optimization of the average excess production may still occur, but then it is restricted to certain choices of initial conditions (Appendix 5). Jones [19] derived a more complicated function e(t) shown in Appendix 5 that represents a universal optimization criterion in the replication-mutation system, but the physical meaning of this Lyapunov function is unclear. 

E-optimality This criterion is fulfilled when the largest eigenvalue of the dispersion matrix is as small as possible. This minimizes the largest variance of the estimated model parameters. 

The coupled thermodynamic analysis, i.e. the calculation of the coefficients Gj in Eq. (3.204) is performed using the multiple linear regression analysis omitting the statistically non-important terms according to the Student test on the chosen confidence level. As the optimizing criterion for the best fit between the experimental and calculated temperatures of primary crystallization, the following condition was used for all the p measured points... 

The optimum economic characteristics of the "solar array + electrolyzer" system were computed as the third step in solving the optimization problem. The condition of the maximum plant capacity referred to the cost of the system (i.e., to the investment) was used as an optimality criterion. The cost of the system was calculated conditionally from that of the unit area of the solar cell Ug and the cost of the electrolyzer per unit area of its electrodes Ug. Then... 

The number of unique phylogenetic trees increases exponentially with the number of taxa, becoming astronomical even for, say, 50 sequences (Swofford et al., 1996 Li, 1997). In most cases, computational limitations permit exploration of only a small fraction of possible trees. The exact number will depend mainly on the nmnber of taxa, the optimality criterion (e.g., MP is much faster than ML), the parameters (e.g., unweighted MP is much faster than weighted ML with fewer preset parameters is much faster than with more and/or simultaneously optimized parameters), computer hardware, and computer software (some algorithms are faster than others some software allows multiprocessing some software limits the number and kind of trees that can be stored in memory). The search procediue is also affected by data structure poorly resolvable data produce more nearly optimal trees that must be evaluated to find the most optimal. 

The ACE model is stored in the form ofp pairs of yj,g yj)] and The transformation functions are not in closed or analytical form in contrast to parametric models. The transformation is obtained on the basis of an optimality criterion, according to which the variance of the error e in Eq. (6.122) is minimized with respect to the variance of the transformed variable y. 

The SVR algorithm attempts to place a tube around the regression function as shown in Fig. 3.19, wherein the region enclosed by the tube is called as s-insensitive zone where s represents the radins of the tube. The diameter of the tube should ideally be the amount of noise in the data. The optimization criterion in SVR penalizes those data points, whose y valnes lie more than e distance away from the fitted function (hyperplane). 

Typically, resolution diagrams in MLC are complex, with several local maxima, frequently denoting interaction between factors. For this reason, reliable optimal conditions require considering all factors simultaneously, by applying an interpretive optimization strategy (i.e., based on the description of the retention behavior and peak shape of solutes). In this task, the product of free peak areas or purities has proved to be the best optimization criterion. An interactive computer program is available to obtain the best separation conditions in... 

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