# 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

For a second order response surface

When the β_{ij} terms are present, the response surface is said to be *quadratic*, while when β_{ij} = 0 the second order response surface is called *pure quadratic*.

## 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 , where is the vector of input values and *y* is the corresponding output, the least square estimator of the regression parameters is defined as

where is the transposed of the vector of input values.