Canonical Analysis of Response Surface Models

In the example above, a maximum point was found within the explored domain. This is, however, not often encountered. Most frequently, the response surface is either monotonous in the explored domain or describes saddle-shaped surfaces or ridge systems. In such cases, it is not easy to comprehend the shape of the response surface from the algebraic expression of the model. A transformation to 

The constant term, ys, is the predicted response at the stationary point on the response surface. As the square variables z cannot be negative it is seen that the shape of the response surface is determined by the signs and magnitudes of the coefficients X.  

Response surface models are local Taylor expansion models which are valid only in the explored domain. It is often found that the stationary point on the response surface is remote from the explored domain and in the model may not describe any real phenomenon around the stationary point. Mathematically, a stationary point can be a maximum, a minimum, or a saddle point but it sometimes corresponds to unrealistic reponses (e.g. yield 100%) or unattainable experimental conditions (e.g. negative concentrations of reactants). When the stationary point is outside the explored domain, the response surface is monotonous in the explored experimental domain and zx directions which correspond to small coefficients will describe rising or falling ridges. Exploring such ridges offers a means for optimizing the response even if the response surface should be oddly shaped. 

The mathematics involved in canonical analysis can be explained as follows. 

In matrix notation, the fitted quadratic response surface model can be written 

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This is the case in canonical analysis of response surface model, where the coefficient matrix is symmetric. 

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