Affine transformational HDMR and linearised rational least squares approximation

摘要

High dimensional model representation (HDMR) is a technique that is used to approximate multivariate functions with functions of less number of variables. In transformational high dimensional model representation (THDMR), the HDMR of a transformation of a given multivariate function f can be truncated at the constant term and the inverse transformation of this constant is used as an approximation to this given function. If the transformation is affine having polynomials as coefficients, then the obtained approximation to such f is a rational function.Since the computation of the best rational approximant of a function is a highly non-linear optimisation problem, the scientists have focused on linearising such minimisation problem and solve it via basic linear algebra tools. The problem of finding polynomials minimising the continuous 2-norm (-norm) of a weighted residual function is called a linearised rational least squares approximation.The major contribution of this paper is to recognise that if an affine transformation is used in transformational HDMR with a constant approximation, then this independently developed technique coincides with linearised rational least squares approximation.