Models full second-order

Many systems that cannot be represented by a first-order empirical model can be described by a full second-order polynomial equation, such as that for two factors. 

Fit the full second-order (quadratic) model to the data. 

One of the most useful models for approximating a region of a multifactor response surface is the full second-order polynomial model. For two factors, the model is of the form... 

In general, if k is the number of factors being investigated, the full second-order polynomial model contains V2 k -t- 1)(A -h 2) parameters. A rationalization for the widespread use of full second-order polynomial models is that they represent a truncated Taylor series expansion of any continuous function, and such models would therefore be expected to provide a reasonably good approximation of the true response surface over a local region of experiment space. 

Efficiency of full second-order polynomial models fit to data from central composite designs without replication. 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

To find the coordinates of the stationary point, we first differentiate the full second-order polynomial model with respect to each of the factors and set each derivative equal to zero. For two-factor models we obtain... 

The corresponding matrix least squares treatment for the full second-order polynomial model proceeds as follows. 

Write full second-order polynomial models for 1, 2, 3, 4, and 5 factors. 

Confidence Intervals for Full Second-Order Polynomial Models... 

In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surface itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models. 

In two-factors, the full second-order polynomial (FSOP) model is... 

Figure 13.1 Sums of squares and degrees of freedom tree for a two-factor full second-order polynomial model fitted to a central composite design with a total of four center point replicates.

Add one design point to a two-factor star design to generate a design that is sufficient to fit a full second-order polynomial model ( y, = Po + PiJCj,- + P Tj, + Pn- ii + P22 i + Pi2 iA2i + "ii)- Hint see Figure 13.11. 

In a set of experiments, x is temperature expressed in degrees Celsius and is varied between 0°C and 100 C. Fitting a full second-order polynomial in one factor to the experimental data gives the fitted model y, = 10.3 + 1.4xi, + 0.0927xf, + r,. The second-order parameter estimate is much smaller than the first-order parameter estimate h,. How important is the second-order term compared to the first-order term when the temperature changes from 0°C to 1°C How important is the second-order term compared to the first-order term when temperature changes from 99°C to 100°C Should the second-order term be dropped from the model if it is necessary to predict response near the high end of the temperature domain ... 

The design of Table 2.12 will permit the estimation of all the terms of a full-second order model... 

Shoemaker et al. [37] give several examples of the reduction in the number of experimental runs that can occur when it is assumed that some of the terms in the full second-order model are negligible. The reader is warned, however, that assuming a term is negligible is not an assurance that it can be ignored. The presence of terms in the true model that were assumed negligible will bias the estimates of the other coefficients. 

Optimal Sequential Composite Design for Two Process Variables and Two Polynomial Models Mu Full Second-Order Model and M2, Incomplete Third-Order Model... 

The value of the hyperparameter jt may be chosen by considering the prior expected number of active effects. Illustrative calculations are now given for a full second-order model with / factors, and for subsets of active effects that include linear and quadratic main effects and linear x linear interactions. Thus, the full model contains / linear effects, / quadratic effects, and ( ) linear x linear interaction effects. Prior probabilities on the subsets being active have the form of (22) and (23) above. A straightforward extension of the calculations of Bingham and Chipman (2002) yields an expected number of active effects as... 

The extremely versatile full second-order polynomial model in Equation 3.32 can also be fitted when at least three-level factorial designs are used and 3 experiments are run. Alternatively, a central composite design may be used effectively (see Figure... 

However, as discussed in chap 1.2.7, the gradient-diffusion models can fail because counter-gradient (or up>-gradient) transport may occur in certain occasions [15, 85], hence a full second-order closure for the scalar flux (1.468) can be a more accurate but costly alternative (e.g., [2, 78]). 

Block 3 Add the six star points plus 2 center points, fit the full second order model, estimate block effects. 

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