Simplex lattice designs

When studying the properties of a q-component mixture, which are dependent on the component ratio only, the factor space is a regular, (q-1) simplex, and for the mixture the relationship holds 

Xj-is the component concentration-ratio-productions, and q-is the number of components in the mixture. 

In the concentration triangle, points lying on a straight line originating from a vertex correspond to mixtures with a constant ratio of components represented by the other two vertices. The property (y) is normally thought of as projections of lines of constant value on the plane of the concentration triangle. 

At c 4, the regular simplex is a tetrahedron where each vertex represents a straight component, an edge represents a binary system, and a face a ternary one. Points inside the tetrahedron correspond to quaternary systems. 

Scheffe [5] suggested to describe mixture properties by reduced polynomials obtainable from Eq. (3.11), which is subject to the normalization condition of Eq. (3.2) for a sum of independent variables. We shall demonstrate below how, for instance, such a reduced second-degree polynomial is derived for a ternary system. The polynomial has the general form  

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The factor space representing these components can be formed by a triangle where each vertex represents a pure component. Measurements of the criteria of mixtures of the three components are made at regular points (according to a simplex lattice design) in the factor space (Figure 4.16). 

Figure 4.16 Simplex lattice design for a special cubic model with ten...

J. W. Gorman and J.E. Hinman, Simplex lattice designs for multicomponent systems, Technometrics, 4 (1962) 463-487. 

Simplex lattice design Quantitative Mixture problems, regression models of second and higher orders... 

The most frequently used mixture-"composition-property designs of experiments belong to simplex-lattice designs suggested by Scheffe [5], The basis of this kind of designing experiments is a uniform scatter of experimental points on the so-called simplex lattice. Points, or design points form a [q,n] lattice in a (q-1) simplex, where q is the number of components in a composition and n is the degree of a polynomial. For each component there exist (n+1) similar levels Xp0,l/n,2/n.1 and all... 

Table 3.13 Number of design points of simplex lattice designs...

The number of trials of simplex lattice designs, which depend on the number of components and the degree of regression model, is given Table 3.18. 

The experiment has been realized by a simplex lattice design matrix for the fourth-degree model. This model has been chosen, for in case a lower model order is adequate, the excessive points become control points. 

N -is total number of points in a simplex lattice design, including the control point u-is current number of points in a simplex lattice design ... 

Check of lack of fit of the obtained regression model is done in additional control points, Table 3.22, since the simplex lattice design is saturated. 

Nine design points in accord with a simplex lattice design for a second-order model was done in researching for the octane number of a three-component mixture of petrol. The outcomes are given in Table 3.23. 

In solving the problem, the simplex lattice design 4.2 has been utilized. The second-order design matrix for the quaternary system and experimental results (each trial was repeated twice) are summarized in Table 3.24. By processing these outcomes, the following values of regression coefficients were obtained ... 

In order that Scheffes simplex lattice designs may be applied to this case, a renormalization is performed and compositions at vertices Aj(j=l, 2,..., q) are taken to be independent pseudocomponents so that for all the range of the local simplex the condition be met ... 

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