Central composite designs setting

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... 

A possible disadvantage of the central composite design is that it requires five levels of each variable (0, 1, a). In some situations it might be necessary or preferable to have only three different settings of the variables. In this case a can be chosen to be 1 and the design is called a face-centered composite design. These designs are not rotatable. 

Blocking is done by running a set of experiments in balanced blocks and correcting the results for the differences (if any) between blocks prior to or in the process of analysing the results In a four-factor central composite design, the experiment may be split into three equal blocks as indicated. 

Figure 3.8. Possible configurations of nine sets of factor values (experiments) that could be used to discover information about the response of a system. Dotted ellipses indicate the (unknown) response surface, (a) A two-factor, three-level factorial design (b) a two-factor central composite design.

The operating conditions of a new blending device which mixed the mussel tissue with the extractant were set after an experimental design and optimised with a central composite design... 

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. 

Three factors, namely (1) irradiation power as a percentage, (2) irradiation time in seconds and (3) number of cycles, are used to study the focused microwave assisted Soxhlet extraction of olive oil seeds, the response measuring the percentage recovery, which is to be optimised. A central composite design is set up to perform the experiments. The results are as follows, using coded values of the variables ... 

Once the right set of parameters has been identified, computer-aided optimization using modified sequential simplex or central composite design methods can be applied to further hne-tune the separation under investigation, as has been published for the optimization of reverse-phase HPLC [17-20] and chiral separations [21-23]. 

If the experimental region is defined by maximum and minimum values of each factor, then the domain is cubic. The central composite design can be applied to such a situation, the axial points being set then at 1, coded values corresponding to the minimum and maximum allowed values. Other designs for the cubic domain are reviewed in Ref... 

In the first case, the structural description of over 100 thienyl- and furyl-benzimidazoles and benzoxazoles was multivariately characterized to identify three latent variables. A set of 16 informative molecules was derived thereafter on applying a central composite design criterion in these latent variables to all the available structures. The data were analyzed by a linear PLS model, which permitted the optimization of three structural features out of four. The fourth one, the substituent linked to the homocyclic ring of the bicyclic system was finally optimized by the CARSO procedure in terms of the substituents PPs, predicting two new compounds as possible optimal structures. Indeed, later analysis revealed the accuracy of these predictions. 

The 2 design with centre point can be extended by adding experiments along each of the axes at values a of the other coded variable (A, = a, Xj = 0 and X, = 0, 2 = a). These are called axial points. The result is the central composite design for 2 factors. If a is set equal to 1, as in figure 5.1b, the design is also a full factorial design at 3 levels (3 ), quite often used for studies on 2 factors. 

The individual results for the duplicated star points and the extra centre point were listed in table 5.2. These, when added to the factorial design of table 4.2, make up the duplicated central composite design. Analysis of variance for regression may be carried out on the complete data set of the composite design for the second degree model (20 data). The results are summarised in table 5.5. 

The response surface methodology (RSM) is a combination of mathematical and statistical techniques used to evaluate the relationship between a set of eontrollable experimental factors and observed results. This optimization proeess is used in situations where several input variables influence some output variables (responses) of the system. The main goal of RSM is to optimize the response, whieh is influenced by several independent variables, with minimum number of experiments. The central composite design (CCD) is the most common type of seeond-order designs that used in RSM and is appropriate for fitting a quadratic surface [26,27]. 

Inscribed central composite designs are a scaled down version of the circumcised version in that they use the factor settings as the star points and create factorial or fractional factorial design within those limits. In other words, inscribed designs are those where each factor of the circumcised version is divided by a for their generation. Face centered central composite designs occur when the star points are at the center of each face of the factorial space, where a= +1. 

A fractional factorial design plus two added points in the centre of the domain selected the relevant variables, which were set using an orthogonal central composite design. A desirability function was defined. 

Sample treatment was studied by the saturated fractional design considering volumes and concentrations of acids, temperatures, ramp time and hold time for the microwave heating. An optimised programme was set after the central composite study... 

To optimize the process of isomerization of sulphanylamide from Problem 2.6, a screening experiment has been done by the random balance method. Factors X1 X2 and X3 have been selected for this experiment. Optimization of the process is done by the given three factors at fixed values of other factors. To obtain the second-order model, a central composite rotatable design has been set up. Factor-variation levels are shown in Table 2.148. The design of the experiment and the outcomes of design points are in Table 2.149. 

A process having properties dependent on four factors has been tested. A full factorial experiment and optimization by the method of steepest ascent have brought about the experiment in factor space where only two factors are significant and where an inadequate linear model has been obtained. To analyze the given factor space in detail, a central composite rotatable design has been set up, as shown in Table 2.152. 

A central composite experimental design leading to a set of 13 experiments with different variable value combinations for finding the maximal region of the percentage of hydrolysis, as a function of pH, temperature, and concentration of reactants (oil-to-water ratio), is shown at Table 2 for free lipase. The percentage of hydrolysis (POH%) ranged from 0.62 to... 

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