Simplex Lattice

A GCS can be constructed in any number of dimensions from one upwards. The fundamental building block is a /c-dimensional simplex this is a line for k = 1, a triangle for k = 2, and a tetrahedron for k = 3 (Figure 4.2). In most applications, we would choose to work in two dimensions because this dimensionality combines computational and visual simplicity with flexibility. Whatever the number of dimensions, though, there is no requirement that the nodes should occupy the vertices of a regular lattice. 

The main idea of the simplex decomposition method is to divide the lattice, on which the scalar field is specified, into small subunits and approximate the... 

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. 

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