### Abstract

The High Dimensional Model Representation (HDMR) technique decomposes an n-variate function f (x) into a finite hierarchical expansion of component functions in terms of the input variables x = (x _{1}, x _{2},..., x _{n}). The uniqueness of the HDMR component functions is crucial for performing global sensitivity analysis and other applications. When x _{1}, x _{2},..., x _{n} are independent variables, the HDMR component functions are uniquely defined under a specific so called vanishing condition. A new formulation for the HDMR component functions is presented including cases when x contains correlated variables. Under a relaxed vanishing condition, a general formulation for the component functions is derived providing a unique HDMR decomposition of f (x) for independent and/or correlated variables. The component functions with independent variables are special limiting cases of the general formulation. A novel numerical method is developed to efficiently and accurately determine the component functions. Thus, a unified framework for the HDMR decomposition of an n-variate function f (x) with independent and/or correlated variables is established. A simple three variable model with a correlated normal distribution of the variables is used to illustrate this new treatment.

Original language | English (US) |
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Pages (from-to) | 99-130 |

Number of pages | 32 |

Journal | Journal of Mathematical Chemistry |

Volume | 50 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2012 |

### All Science Journal Classification (ASJC) codes

- Chemistry(all)
- Applied Mathematics

### Keywords

- D-MORPH regression
- Extended bases
- Global sensitivity analysis
- HDMR
- Least-squares regression
- Orthonormal polynomial