General foundations of high-dimensional model representations

A family of multivariate representations is introduced to capture the input-output relationships of high-dimensional physical systems with many input variables. A systematic mapping procedure between the inputs and outputs is prescribed to reveal the hierarchy of correlations amongst the input variables. It is argued that for most well-defined physical systems, only relatively low-order correlations of the input variables are expected to have an impact upon the output. The high-dimensional model representations (HDMR) utilize this property to present an exact hierarchical representation of the physical system. At each new level of HDMR, higher-order correlated effects of the input variables are introduced. Tests on several systems indicate that the few lowest-order terms are often sufficient to represent the model in equivalent form to good accuracy. The input variables may be either finite-dimensional (i.e., a vector of parameters chosen from the Euclidean space R-n) or may be infinite-dimensional as in the function space C-n[0,1]. Each hierarchical level of HDMR is obtained by applying a suitable projection operator to the output function and each of these levels are orthogonal to each other with respect to an appropriately defined inner product. A family of HDMRs may be generated with each having distinct character by the use of different choices of projection operators. Two types of HDMRs are illustrated in the paper: ANOVA-HDMR is the same as the analysis of variance (ANOVA) decomposition used in statistics. Another cut-HDMR will be shown to be computationally more efficient than the ANOVA decomposition. Application of the HDMR tools can dramatically reduce the computational effort needed in representing the input-output relationships of a physical system. In addition, the hierarchy of identified correlation functions can provide valuable insight into the model structure. The notion of a model in the paper also encompasses input-output relationships developed with laboratory experiments, and the HDMR concepts are equally applicable in this domain. HDMRs can be classified as non-regressive, non-parametric learning networks. Selected applications of the HDMR concept are presented along with a discussion of its general utility.