Practical approaches to construct RS-HDMR component functions

A general set of quantitative model assessment and analysis tools, termed high-dimensional model representations (HDMR), has been introduced recently for improving the efficiency of deducing highdimensional input-output system behavior. HDMR is a particular family of representations where each term in the representation reflects the independent and cooperative contributions of the inputs upon the output. When data are randomly sampled, a RS (random sampling)-HDMR can be constructed. To reduce the sampling effort, different analytical basis functions, such as orthonormal polynomials, cubic B splines, and polynomials may be employed to approximate the RS-HDMR component functions. Only one set of random input-output samples is necessary to determine all the RS-HDMR component functions, and a few hundred samples may give a satisfactory approximation, regardless of the dimension of the input variable space. It is shown in an example that judicious use of orthonormal polynomials can provide a sampling saving of ∼10³ in representing a system compared to employing a direct sampling technique. This paper discusses these practical approaches: their formulas and accuracy along with an illustration from atmospheric modeling.