Least squares polynomial response surface¶
a linear function, i.e. a polynomial of degree one;
a quadratic function, i.e. a polynomial of degree two.
where is a set of unknown coefficients.
where denotes the number of terms, which is equal to (resp. to ) when using a linear (resp. a quadratic) approximation, and the family gathers the constant monomial , the monomials of degree one and possibly the cross-terms as well as the monomials of degree two . Using the vector notation and , this rewrites:
The solution is given by:
It is clear that the above equation is only valid for a full rank information matrix. A necessary condition is that the size of the experimental design is not less than the number of PC coefficients to estimate. In practice, it is not recommended to directly invert since the solution may be particularly sensitive to an ill-conditioning of the matrix. The least-square problem is rather solved using more robust numerical methods such as singular value decomposition (SVD) or QR-decomposition.
Bjorck, 1996, “Numerical methods for least squares problems”, SIAM Press, Philadelphia, PA.