Central composite design matrix

Central composite design matrix of experiments generated by the DOE tool... 

The lower left panel in Figure 13.2 shows the central composite design in the two factors X, and X2. The factor domain extends from -5 to +5 in each factor dimension. The coordinate axes in this panel are rotated 45° to correspond to the orientation of the axes in the panel above. Each black dot represents a distinctly different factor combination, or design point. The pattern of dots shows a central composite design centered at (Xj = 0, Xj = 0). The factorial points are located 2 units from the center. The star points are located 4 units from the center. The three concentric circles indicate that the center point has been replicated a total of four times. The experimental design matrix is... 

Figure 13.3 shows a similar set of four panels for a slightly different central composite design. The lower left panel shows the placement of experiments in factor space (i.e., it shows the experimental design). The upper left panel shows the normalized uncertainty as a function of factors x, and x. The upper right panel shows the normalized information as a function of factors x, and Xj. The lower right panel plots normalized information as a function of factor x, for X2 = -5, -4, -3, -2, -1, and 0. The experimental design matrix is... 

Figure 13.5 shows still another central composite design. The experimental design matrix is... 

The rotatable central composite design in Figure 13.7 is related to the rotatable central composite design in Figure 13.3 through expansion by a factor of V2 the square points expand from 2 to 2 2 from the center the star points expand from 2 2 to 4 from the center. The experimental design matrix is... 

Once the variables having the greatest influence on the responses were identified, a 20-run central composite design was used to optimize the levels of these variables (18). A design matrix was developed (Table 2) and the true values for the variables were determined (Table 3). 

The design matrix is a key concept. A design may consist of a series of experiments performed under different conditions, e.g. a reaction at differing pHs, temperatures, and concentrations. Table 2.6 illustrates a typical experimental set-up, together with an experimental response, e.g. the rate constant of a reaction. Note the replicates in the final five experiments in Section 2.4 we will discuss such an experimental design commonly called a central composite design. 

TABLE 14 Matrix of Central Composite Design and Results... 

Table 1 Experimental matrix for the central composite design (experiment 1).

It is identical to the 2-factor central composite design. The matrix is "almost orthogonal" if no extra centre-points are added. It is not rotatable, nor is its precision uniform in the experimental domain. 

The central composite design avoids the problem of higher-order factorial designs, by judiciously selecting additional parameter points. The generic regression matrix, A, for k factors consists of three different parts ... 

The Doehlert Design [6], like the Central Composite Design, allows estimating the coefficients of a model containing linear terms, interactions and quadratic terms. Table 13 reports the experimental matrix for two and three variables, and Figure 14 shows their graphical representation. 

Design matrices of central composite rotatable designs (CCRD) for k=2, k=3 and k=5 are shown in Tables 2.138 - 2.140. By using relation (2.59), which connects coded and real factor values, we switch from design matrix to operational matrix, Table 2.138. 

TABLE 4.3. Central composite face-centered design matrix for method optimization for escitalopram and related substances showing the factors snlfated p-cyclodextrin (S-p-CD) concentration, buffer concentration, voltage and temperature, and the results for the g responses resolution, Rs, between the dtalopram enantiomers and between oxalate and (R)-citalopram as well as migration time, MT, and electric current (Modified from Reference 27 with permission)... 

The effect of several variables on the efficiency of the hydride generation procedure employed to extract the analytes was studied using experimental designs. Optimum conditions were found after a central composite matrix was deployed. 

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