### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 197-233 |

Number of pages | 37 |

Journal | Journal of Mathematical Chemistry |

Volume | 25 |

Issue number | 2-3 |

State | Published - Oct 1 1999 |

### All Science Journal Classification (ASJC) codes

- Chemistry(all)
- Applied Mathematics

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## Cite this

*Journal of Mathematical Chemistry*,

*25*(2-3), 197-233.