The Robustness Coefficient

The Robustness Coefficient (RC) has been developed especially to handle the kind of robustness problem as described in part 4.3.2 of this Chapter. The RC is extensively described in reference [20], the behaviour of the RC 

It would now be most logical to let this probability between a and b be the RC, but in case of more than one independent variable with a multivariate error distribution it is a very complicated problem to calculate an almost always asymmetrical part of this distribution. To handle this problem the 

however, the real error distribution of x is not known (but the variance/covariance structure is), then the RC is represented by the smallest Mahalanobis distance (which is directly related to probability in case of a known error distribution) between point c and a point with p = nig (here from c to d). In this case only differences in variances and covariances of the independent variables are taken into account in the RC value. 

Here the Mahalanobis distance is used because the errors in the fractions of a mixture are not normally distributed. 

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For the sake of illustration the constant relative standard deviation (cvj for the three variable components was set to 5%. The demands concerning the calculation of the robustness coefficient were to have a deviation in the crushing strength (6> ) of ION at most with a reliability (m ) of 0.95. 

The demands concerning the decision making method were a maximal value for the crushing strength and a maximal value for the robustness coefficient. 

Combining the behaviour of both criteria, maximisation of the crushing strength and the maximisation of the robustness coefficient, results in a PO-plot, as shown in Figure 4.21. 

Every point in figure 4.21 corresponds directly to a predicted value of the crushing strength and the calculated value of the robustness coefficient, at one mixture composition. These points are called Pareto Optimal (PO) points, which are listed in Table 4.5. The PO points can also be placed in the corresponding mixture triangle, which is presented in Figure 4.22. 

Oils showing a median of the bitter and/or pungent attribute of more than 5.0 should be addressed to blending. The calculation of the robust coefficient of variation provides a measure of the reliability of panel tasters according to data reported in Table 2.6. 

While over the past ten years, our ability to measure U-series disequilibria and interpret this data has improved significantly it is important to note that many questions still remain. In particular, because of uncertainties in the partition coefficients, fully quantitative constraints can only be obtained when more experimental data, as a function of P and T as well as source composition, become available. Furthermore, the robustness of the various melting models that are used to interpret the data needs to be established and 2D and 3D models need to be developed. However, full testing of these models will only be possible when more comprehensive data sets including all the geochemical parameters are available for more locations and settings. 

The robust estimator still provides a correct estimation of the covariance matrix on the other hand, the estimate J>C> provided by the conventional approach, is incorrect and the signs of the correlated coefficients have been changed by the outliers. 

A robust coefficient of variation can be calculated by using, VjqR or, vMad instead of, v, and the median instead of the mean. 

Regression M-estimates minimize 11p(ei/cr) to obtain the estimated regression coefficients, where the choice of the function p determines the robustness of the estimation, and cr is a robust estimation of the standard deviation of the residuals (Maronna et al. 2006). Note that both the residuals e, and the residual scale cr depend on the regression coefficients, and several procedures have been proposed to estimate both quantities. 

A. SCALAR SISO SYSTEMS. Remember in the scalar SISO case we looked at the closedloop servo transfer function G B/ll + GuB). The peak in this curve, the maximum closedloop log modulus L (as shown in Fig. 16.9a), is a measure of the damping coefficient of the system. The higher the peak, the more underdamped die system and the less margin for changes in parameter values. Thus, in SISO systems the peak in the closedloop log modulus curve is a measure of robustness. 

Notice that A and B contain coefficients that change up to 10% with respect to the nominal values used in the previous example. In this example, the robust controller design is based on nominal matrices defined in Example 1. The exosystem defined in Example 1 produces the immersion... 

The three robustness criteria that are explained here (Weighted-Jones (WJ), Projected-Variance (PV) and Robustness-Coefficient (RC)) describe all three the robustness of a certain mixture composition in direct relation to the response to be optimised. All three express the concept of robustness into a numerical value that can be calculated for each mixture setting (composition) of interest. So each of the criteria can be calculated as a function of the mixture composition and belongs directly to a certain response of interest. In this way a robustness criterion can be dealt with in a normal way in a mixture optimisation strategy. 

The PO points are not directly comparable to each other. Moving from point 1 to 2 in the PO plot results in a higher crushing strength but a lower robustness coefficient. Between points 7 and 6 the reverse is valid, moving... 

Considering the set PO points presented in Table 4.5, the following statement can be made there exists no mixture composition with a higher robustness coefficient and, simultaneously, a higher crushing strength than a PO mixture composition. 

In the right part of Figure 7.1, the maximal minimal partition coefficient is found in O,. Here, the composition where the ratio of the partition coefficients of compounds i and j reaches its optimum, Oo, is also the composition that gives a robust selectivity. Optima with respect to maximal partition coefficients of both compounds are obtained with different compositions. The response surfaces are not completely parallel. Little variation in the composition of the extraction liquid does influence... 

Summarising, the robustness of the partition coefficient Cp) and the robustness of the selectivity (C ) are functions of the variance in mixture composition and the shape of the response surfaces ... 

Using the rules of the propagation of errors [33,34] a measure of the robustness of the partition coefficient (C, ) and the robustness of the selectivity (C ) can be obtained. Below, a derivation of robustness of the partition coefficient P, of a compound i and the selectivity Uij for two compounds i and j with respect to variation in extraction liquid composition is given. The general form of a (Special Cubic) mixture model for three-component mixtures is given by ... 

In other words, the robustness of this ratio is a function of the robustness of the individual partition coefficients P, and Pj and of the parallelism of the tangent planes of the response surfaces in mixture composition M. 

Now, the criteria for the robustness of the partition coefficient of single compounds and of the selectivity of two compounds have been derived. The following part of this paper deals with the practical use of the criteria. 

The first part uses the robustness algorithm (i.e. the algorithm introduced and described in the preceding part of this paper) to calculate the variance of the partition coefficients P of two solutes i and j and the selectivities a j using two preselected models of InP and seven preselected extraction liquid compositions. 

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