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 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.
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.