Simplex centroid designs model

In the case of constraints on proportions of components the approach is known, simplex-centroid designs are constructed with coded or pseudocomponents [23]. Coded factors in this case are linear functions of real component proportions, and data analysis is not much more complicated in that case. If upper and lower constraints (bounds) are placed on some of the X resulting in a factor space whose shape is different from the simplex, then the formulas for estimating the model coefficients are not easily expressible. In the simplex-centroid x 23 full factorial design or simplex-lattice x 2n design [5], the number of points increases rapidly with increasing numbers of mixture components and/or process factors. In such situations, instead of full factorial we use fractional factorial experiments. The number of experimental trials required for studying the combined effects of the mixture com-... 

The common structure of the regression model applicable to all simplex-centroid designs is shown in Equation 8.20. 

The reduced cubic model and the simplex centroid design... 

The simplex centroid design (figure 9.3) is the best design by all the criteria for determining the reduced cubic model. In the case of 3 components, a single point is added to the second-order design. It is that of the ternary mixture, with equal proportions of each component, each diluent being present at a level of Va or 33.3% (test point 7, in table 9.2). 

Table 9.5 ANOVA of the Regression on the Reduced Cubic Model Simplex Centroid Design with Test Points...

Fig. 7.6. Simplex centroid design, with average responses for the mixtures represented by the points and contour curves for the special cubic model, Eq. (7.20). The three boxed response values close to each vertex were used to test the quality of the fit.

The special cubic model for four-component mixtures has 14 terms, and its coefficients can be estimated from the design shown in Fig. 7.10b. The points on each face now reproduce the arrangement corresponding to the simplex centroid design, which we used to fit the special cubic model for three-component mixtures. 

As shown, mixture components are subject to the constraint that they must equal to the sum of one. In this case, standard mixture designs for fitting standard models such as simplex-lattice and simplex-centroid designs are employed. When mixtures are subject to additional constraints, constrained mixture designs (extreme-vertices) are then appropriate. Like the factorial experiments discussed above, mixture experimental errors are independent and identically distributed with zero mean and common variance. In addition, the true response surface is considered continuous over the region being studied. Overall, the measured response is assumed to depend only on the relative proportions of the components in the mixture and not on the amount. 

Figure 12 Simplex centroid design. The first seven points are used to estimate the model, the three following to validate it...

The given designs are used for fitting a three-component simplex-centroid (or an incomplete cube model) with main effects of process factors ... 

Like the centroid designs, the Scheff6 simplex lattice designs, described below, are easy to constmct. Their R-efficiencies are equal to 100%, as the number of experimental points is equal to the number of coefficients in the corresponding mathematical model. They may be resolved directly, that is the coefficients may be calculated directly without using a computer for least squares regression. They can sometimes be built up sequentially. The precision of the calculated response is optimal, that is, the variance over the design space is minimal for the number of experiments. 

The experimental design normally used to determine the values of the coefficients of the special cubic model is called the simplex centroid, which we obtain by simply adding a center point, corresponding to a 1 1 1 ternary mixture, (xi,X2,X3) = (5,5,5) to the simplex lattice design. The coefficient of the cubic term is given by... 

A so-called centroid simplex mixture design was used to optimise the acid proportions to extract the metals. After fitting a quadratic model to the data, ANOVA was used to assess the significance of each variable. 

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