Simplex centroid

Just as previously, a model and design matrix can be obtained. However, the nature of die model requires some detailed thought. Consider trying to estimate model of the form 

This model consists of 10 terms, impossible if only seven experiments are performed. How can the number of terms be reduced Arbitrarily removing three terms such as die quadratic or interaction terms has little theoretical justification. A major problem with the equation above is that the value of A2 depends on x and. t 2, since it equals 1 — A — a 2 so diere are, in fact, only two independent factors. If a design madix consisting of die first four terms of the equation above was set up, it would not have an inverse, and the calculation is impossible. The solution is to set up a reduced model. Consider, instead, a model consisting only of the first three terms  

This is, in effect, equivalent to a model containing just the diree single factor terms widiout an intercept since 

It is not possible to produce a model contain bodi the intercept and the three single factor terms. Closed datasets, such as occur in mixtures, have a whole series of interesting madiematical properties, but it is primarily important simply to watch for diese anomalies. 

The two common types of model, one widi an intercept and one without an intercept term, are related. Models excluding the intercept are often referred to as Sheffe models and those with die intercept as Cox models. Normally a full Sheffe model includes all higher order interaction terms, and for diis design is given by 

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XLOW(I) = low limit component I XCEN(I) = simplex centroid value of I X(I) value of I prior to step... 

Scheffe, H. The simplex-centroid design for experiments with mixtilrtffe, Statist. Soc., B25, 235,... 

Scheffe s simplex centroid designs contain 2q-l points, q of which fall on straight components, Cq2 on binary mixtures, Cq3 on ternary mixtures, and so forth, and one observation on a q-component mixture. Simplex centroid designs, consist of the points whose coordinates are (1,0,...,0), (1/2, 1/2,0,...,0),...,(l/q,l/q,...,l/q), and of all the points that can be obtained from these by permutations of coordinates. Thus, the design contains a point at the center (centroid) of the simplex and the centroids of all the component simplexes of lesser dimension, its proper faces. 

Polynomials obtained from simplex-centroid designs contain as many coefficients as there are points in the design, and for the q-component mixture they have the form ... 

Table 3.26 Matrix of simplex centroid design for a quaternary system q=4...

In the general case, the formula for coefficients of regression equation obtained from the simplex-centroid design, takes the form [6] ... 

Adequacy of a regression equation derived by the simplex-centroid design is tested and the confidence intervals of property values, predicted by the equation, are assigned in much the same way as in the case of the simplex-lattice method. 

The simplex centroid design for q=3 is applied. The design matrix and experimental results are arrayed in Table 3.27. 

Figure 3.33a Simplex-centroid design in each point of a 2 full factorial experiment...

Table 3.54 Simplex-centroid x full factorial design...

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

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

Another standard mixture experiment strategy is the so-called simplex centroid design, where data are collected at the extremes of the experimental region and for every equal-parts two-component mixture, every equal-parts three-component mixture, and so on. Figure 5.22 identifies the blends included in a p = 3 simplex centroid design. 

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