Modeling Test Responses by Multivariable Polynomials of Higher Degrees

Engineering tests generally acquire a large set of response data at once while focusing on a small number of variables. Higher-order response models are better for capturing and displaying the characteristics such data can reveal, but response surface models traditionally are linear or quadratic (first or second order) only. If attempted for a higher order, the matrix X of the test inputs formed for fitting the test responses often immediately stops satisfying the full-rank assumption on X traditionally made for computing the regression coefficients. The resulting product matrix XT X is singular and hence cannot be inverted, directly or indirectly, for obtaining the desired least-squares regression coefficients from the system of normal equations. Ill-conditioning of X when near rank deficiency, not unusual even in the traditional second-order response modeling, can also render the computed coefficients unreliable. This paper incorporates singular value decomposition (SVD) to handle the rank deficiency and ill-conditioning problems directly. The proposed SVD-based fitting algorithm enables response modeling to the desirable high-order reliably and efficiently with no difficulty despite rank deficiency in X. Two proposed trimming algorithms enable reduction of redundant terms or insignificant coefficients of the fitted polynomials on a sound mathematical base.