# Response surface

A Response surface is a polynomial regression meta-model that is used to approximate functional relations with multidimensional input and singled value output. The grade of the polynomial regression used identify the grade of the response surface.

For a first degree polynomial, a first-order response surface take the form

$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_k x_k$

For a second order response surface

$y = \beta_0 + \sum^k_{i=0} \beta_i x_i + \sum^k_{i

When the βij terms are present, the response surface is said to be quadratic, while when βij = 0 the second order response surface is called pure quadratic.

Quadratic response surface of the vertical displacement of cantilever beam with a tip load, varying load P and modulus of inertia I. The figure on the left is the real function, on the right is a quadratic response surface approximation.

## Calibration of a response surface

The regression parameters β of the response surface are calibrated to fit experimental data by using a Least squares estimator. Given a set of input-output relation $(\mathbf{X},y)$, where $\mathbf{X}$ is the vector of input values and y is the corresponding output, the least square estimator of the regression parameters is defined as

$\hat{\beta} = (\mathbf{X'} \mathbf{X})^{-1} \mathbf{X'} y$

where $\mathbf{X'}$ is the transposed of the vector of input values.