Response Surface Analysis

Definition

The goal of response surface analysis is finding the optimal condition for the response. It employs Polynomial Regression model to model response surface.

Consider explanatory variables $X_i$ and a response variable $Y$. The second-order response surface model is defined as $Y={\beta }_0+{\mathbf{x}}^{⊺}\mathbf{\beta }+{\mathbf{x}}^{⊺}\mathbf{Bx}+ϵ$ where $\mathbf{x}=(X_1,X_2,\dots ,X_k{)}^{⊺}$, $\mathbf{\beta }=({\beta }_1,{\beta }_2,\dots ,{\beta }_k{)}^{⊺}$, $\mathbf{B}=\left[\begin{array}{cccc}{\beta }_{11}& \frac{{\beta }_{12}}{2}& \dots & \frac{{\beta }_{1k}}{2}\\ \frac{{\beta }_{12}}{2}& {\beta }_{22}& \dots & \frac{{\beta }_{2k}}{2}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{{\beta }_{1k}}{2}& \frac{{\beta }_{2k}}{2}& \dots & {\beta }_{kk}\end{array}\right]$ Let the fitted model is $\hat{Y}={\hat{\beta }}_0+{\mathbf{x}}^{⊺}\hat{\mathbf{\beta }}+{\mathbf{x}}^{⊺}\hat{\mathbf{B}}\mathbf{x}$. Then, the stationary point is obtained by ${\mathbf{x}}_s=-\frac{1}{2}{\hat{\mathbf{B}}}^{-1}\hat{\mathbf{\beta }}$ and the fitted value at the stationary point is ${\hat{Y}}_s={\hat{\beta }}_0+{\mathbf{x}}_s^{⊺}\hat{\mathbf{\beta }}+{\mathbf{x}}_s^{⊺}\hat{\mathbf{B}}{\mathbf{x}}_s$. If $\hat{\mathbf{B}}$ is Positive-Definite Matrix, then ${\hat{Y}}_s$ is minimum If $\hat{\mathbf{B}}$ is Negative-Definite Matrix, then ${\hat{Y}}_s$ is maximum If $\hat{\mathbf{B}}$ is neither Positive-Definite Matrix nor Negative-Definite Matrix, then ${\hat{Y}}_s$ is saddle point.