Coded experimental design matrix

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

Although it is true that the first three columns of plus and minus signs in Table 14.3 are equivalent to the abbreviated coded experimental design matrix D, the signs in Table 14.3 are used for a slightly different purpose than they were Table 14.2. In fact, as we will see, the eight columns of signs in Table 14.3 are equivalent to the matrix of parameter coefficients, X. 

Conditions (2.76) and (2.77) define independence of the design from rotation of coordinates. When selecting the null/centerpoints points (points in experimental center) take into consideration a check of lack of fit of the model, an estimate of experimental error and conditions of uniformity [37]. Centerpoints are created by setting all factors at their midpoints. In coded form, centerpoints fall at the all-zero level. The centerpoints act as a barometer of the variability in the system. All the necessary data for constructing the rotatable design matrix for k<7 are in Table 2.137. This kind of designing is called central, because all experimental points are symmetrical with reference to the experimental center. This is shown graphically for k=2 and k=3 in Fig. 2.40. 

The equiradial designs are useful when two experimental variables are studied. The coded experimental settings are evenly distributed on the periphery of the unit circle, and with at least one experiment at the center point. Without the center point the X X matrix will be singular. The designs are defined by the following relations ... 

Regression leads to a model estimating the relation between the Ai x 1 response vector y, and the Ai x r model matrix X (7,17,116) (Eq. 2.27). N is the number of design experiments, and t the number of terms included in the model. For example, in Equation 2.26, the number of terms equals six, since one intercept, two main effect terms, one interaction term, and two quadratic effect terms were included. The model matrix X is obtained by adding a row of ones before the Aix (r-1) design matrix, which consists of the coded factor levels and columns of contrast coefficients, as defined by the chosen experimental design. 

The model matrix X is derived from the iV x (t - 1) design matrix by adding a column of I s as a first column and, therefore, columns are defined as contrast effects in the experimental design by multiplying the relevant coded factor levels (see Figure 3.23). Thus, the response vector y is related to the N x 1 model matrix X (N is the number of design experiments and t the number of terms included in the proposed model). 

TABLE 3 Experimental Matrix for Investigating the Influence of Three Factors on the Ti02 Photocatalyzed Oxidative Degradation of Waste Water Pollutants in a Pilot Reactor Complete Factorial Design 23, Control Experiments 1 (Natural and Coded Variables are Indicated), and Values of the Experimental Response Y... 

Besides these three problems, one should also know how to switch from simplex to a second-order design that may describe the optimum area. This is the subject of sect. 2.5.4. The first problem in simplex optimization consists of constructing the matrix of a design of experiments for initial simplex where coordinates of experimental points-vertices are given. In solving this problem, different orientations of initial simplex to the coordinate system are possible. A simplex center is mostly set in the coordinate beginning, while the distance between simplex vertices (simplex sides) has a coded value of one. Simplex is, as a rule, oriented in a factor space in such a way that vertex l>k+I lies on the xk axis, while other vertices are distributed symmetrically with respect to coordinate axes. Simplexes of such a construction are shown in Figs. 2.50 and 2.51. 

---