Orthogonal plan

Table 4. Example of experimental runs added at each stage and used in each analysis multiple stage with orthogonal plans at each stage and no reuse of runs...

The decision to use orthogonal plans or to allow nonorthogonal plans, and the decision to allow or disallow the reuse of runs from stage to stage, are related operational issues. They individually and jointly affect the performance of a screening plan and the complexity of calculations required to assess analytically that performance. Depending on characteristics of the application, such as the degree of measurement error and the need to account for block effects in sequential experimentation, either or both may be important considerations. 

Table 5.24 Computed a values for a second order orthogonal plan.

The relation (5.104) can be particularized to the general case of the second order orthogonal plan when we obtain the following relation for coefScients variances ... 

Second Order Orthogonal Plan, Example of the Nitration of an Aromatic Hydrocarbon... 

The presentation of this example has two objectives (i) to solve a problem where we use a second order orthogonal plan in a concrete case (ii) to prove the power of statistical process modelling in the case of the non-continuous nitration of an aromatic hydrocarbon. 

To solve this problem we have to use a second order orthogonal plan based on a 2 CFE plan. According to Table 5.24, we can establish that, for a dimensionless values of factors, we can use the numerical value a = 1.414. Table 5.27 contains all the data that are needed for the statistical calculation procedure of the coefficients, variances, confidence, etc., including the data of the dependent variables of the process (response data). 

Even though the second order orthogonal plan is not a rotatable plan (for instance see Eqs. (5.114) and (5.115)), the errors of the experimental responses (from the response surface) are smaller than those coming from the points computed by regression. It is possible to carry out a second order rotatable plan using the Box and Hunter [5.23, 5.27] observation which stipulates that the conditions to transform a sequential plan into a rotatable plan are concentrated in the dimensionless a value where a = for k<5 and a = for k>5 respectively. Simulta-... 

In the previous sections we have shown that the variances relative to the Pj coefS-cients for the orthogonal plans are Sp = s / > and that, for a simple regu-... 

The effect of different pai ameters such as temperature, pressure, modifier volume, dynamic and static extraction time on the SFE of the plant were investigated. The orthogonal array experimental design method was chosen to determine experimental plan, (5 ). In this design the effect of five parameters and each at five levels were investigated on the extraction efficiency and selectivity [4]. 

Because variables in models are often highly correlated, when experimental data are collected, the xrx matrix in Equation 2.9 can be badly conditioned (see Appendix A), and thus the estimates of the values of the coefficients in a model can have considerable associated uncertainty. The method of factorial experimental design forces the data to be orthogonal and avoids this problem. This method allows you to determine the relative importance of each input variable and thus to develop a parsimonious model, one that includes only the most important variables and effects. Factorial experiments also represent efficient experimentation. You systematically plan and conduct experiments in which all of the variables are changed simultaneously rather than one at a time, thus reducing the number of experiments needed. 

The sample quantities to be collected should be sufficient for a replicate analysis. However, alternative plans for sampling can be implemented. For example, a homogenized sample can be divided for on-site analysis, archival, and also to have a portion sent for orthogonal analysis to an off-site lab. 

Therefore the held results from the addition of two distinct helds a polar held and an axial held . In the available literature, the expressions longitudinal and transverse, which respectively refer to the and vectors, do not necessarily imply that these vectors are projections of on orthogonal direchons. Indeed (if we leave Beltrami helds apart that fulfill the condition A VA 0), for A VA A = 0, the vector B = VA A is normal to if A is parallel to . If not, that is, when A L , both and have the same direchon. This analysis demonstrates the V A VA operator, as applied to the vector A, is itself a vector having either the same direction as A or having components in direchons parallel to A and perpendicular to the plan originating from vectors A and V A A. 

Upon completion of these experiments it became apparent that the differential pressure in the gap was a more important variable than the water repellency of the powder. Hence a 3 X 3 orthogonal set of experiments was planned, using nylon hair nets as the spacer instead of silicone-coated pumice. It was believed that this design would be more reproducible. 

Pulsed discharges may be utilized to analyze nonconductive samples. Steiner et al. described the use of a pin-type pulsed rf-GD in conjunction with a linear orthogonal extraction TOF-MS [58]. Figure 12.24 depicts differences in the spectra obtained 4.0 msec after the power plateau and 7.0 msec after the plateau. As expected, the sputtered species (Si) in the discharge are found in the afterpeak at 7.0 msec (see Fig. 12.24). The resolution and sensitivity of this early system were limited and improvements were planned in the input optical system of the instrument. 

Voss, D. T. (1999). Analysis of orthogonal saturated designs. Journal of Statistical Planning and Inference, 78, 111-130. 

From a theoretical point of view, if we transform the matrix according to the 2 experimental plan, we obtain the state form shown in Table 5.14. This matrix has two important properties the first is its orthogonality, the mathematical expression of which is ... 

The orthogonality of the planning matrix, results in an easier computation of the matrix of regression coefficients. In this case, the matrix of the coefficients of the normal equation system (X X) has a diagonal state with the same value N for all diagonal elements. As a consequence of the mentioned properties, the elements of the inverse matrix (X X) i have the values djj = 1/N, dj] = 0, j / k. 

When we select the good value of the dimensionless a, then the corresponding sequential plan remains orthogonal like its CFE basic plan. At the same time, if we do not have any special request concerning a sequential plan, the number of experiments to determine fundamental factors can be drastically reduced to... 

For k = 2, a second order orthogonal matrix plan is the state shown in Table 5.25. Due to the orthogonality of the matrix plan, the regression coefficients will... 

It is important to emphasize that in the case of an orthogonal composition plan, as shown by relation (5.122), the various regression coefficients are not calculated with similar precisions. 

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