Local response-surface models

Fix an integer $k\ge 1$ and a declared factor region $\mathcal{R}\subseteq {\mathbb{R}}^k$. A response-surface model on $\mathcal{R}$ is a specified polynomial $m:\mathcal{R}\to \mathbb{R}$ used as a local approximation to the mean response as the $k$ factors vary over $\mathcal{R}$. The region is part of the model contract. This definition does not assert that the physical response is polynomial, that the approximation is adequate, or that it remains valid outside $\mathcal{R}$.

A second-order response-surface model has the form $m(x)={\beta }_0+x^{\mathsf{T}}\beta +x^{\mathsf{T}}Bx,$ where ${\beta }_0\in \mathbb{R}$, $\beta \in {\mathbb{R}}^k$, and $B\in {\mathbb{R}}^{k\times k}$ is symmetric. Coordinates may be coded by a declared affine rescaling of the physical factor settings; all points and regions in a calculation use the same coding.

The stationary-point and eigenvalue classification is proved in Stationary point and classification of a nonsingular quadratic response surface.