9.9 Continuous Factors: Regression and Response Surface Methods

In many important cases the factors of an experiment are not categorical but continuous, numerical variables. When these are varied the result is often a smoothly varying response that can be described by a curve equation. With two continuous factors the response turns from a curve into a two-dimensional surface, which is often called a response surface. Such surfaces can often be described by second-degree polynomials, as the one in Equation 8.13. When more factors are investigated the dimensionality of the polynomial increases but it is still called a response surface. Fitting surface models to experimental data has many advantages over simply plotting the measurements. Firstly, the regressors, or coefficients, of the equation are very closely related to the effects previously introduced. Just looking at the equation thereby gives a quantitative measure of the importance of the factors, interactions and other terms that appear in it as predictors. Secondly, the equation is a predictive model, at least within the range where the factors have been varied. This means that we can analyze and optimize the response mathematically. We can, for instance, take the derivative of a fitted polynomial to find a potential optimum just as we can with any other equation. This makes it possible to handle data even from very complex systems analytically.

When a system is under the simultaneous influence of several factors it is often ...

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