High dimensional model representations generated from low dimensional data samples. I. mp-Cut-HDMR

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input-output system behavior. For a high dimensional system, an output f(x) is commonly a function of many input variables x = {x₁, x₂, ..., x_n} with n ∼ 10² or larger. HDMR describes f(x) by a finite hierarchical correlated function expansion in terms of the input variables. Various forms of HDMR can be constructed for different purposes. Cut- and RS-HDMR are two particular HDMR expansions. Since the correlated functions in an HDMR expansion are optimal choices tailored to f(x) over the entire domain of x, the high order terms (usually larger than second order, or beyond pair cooperativity) in the expansion are often negligible. When the approximations given by the first and the second order Cut-HDMR correlated functions are not adequate, this paper presents a monomial based preconditioned HDMR method to represent the higher order terms of a Cut-HDMR expansion by expressions similar to the lower order ones with monomial multipliers. The accuracy of the Cut-HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input-output samples without directly invoking the determination of higher order terms. The mathematical foundations of monomial based preconditioned Cut-HDMR is presented along with an illustration of its applicability to an atmospheric chemical kinetics model.