Second-order polynomial models, full

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. 

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 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... 

In reference 88, response surfaces from optimization were used to obtain an initial idea about the method robustness and about the interval of the factors to be examined in a later robustness test. In the latter, regression analysis was applied and a full quadratic model was fitted to the data for each response. The method was considered robust concerning its quantitative aspect, since no statistically significant coefficients occurred. However, for qualitative responses, e.g., resolution, significant factors were found and the results were further used to calculate system suitability values. In reference 89, first a second-order polynomial model was fitted to the data and validated. Then response surfaces were drawn for... 

Show that the full two-factor second-order polynomial model may be written y, - Po + Do d sd where Dq = [x, Xj,]. Show that this may be extended to full three-factor second-order polynomial models. 

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

For 176 input parameters, the full second-order HDMR expansion consists of 15,577 component functions (1 zeroth-order term-1-176 first-order terms-1-15,400 second-order terms). However, using a threshold of 1 % for the first- and second-order component functions, only five of the 176 first-order component functions and none of the 15,400 second-order component functions were approximated by optimal-order polynomials. The resulting first-order HDMR metamodel gave 99.05 % of the tested samples within the 5 % RE (relative error) range (see Eq. (5.49)) compared to a sample of 2,000 full model runs. This suggests that despite the high-dimensionality of the input space of the model, the predicted NO... 

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