General formulation of HDMR component functions with independent and correlated variables

Genyuan Li, Herschel Rabitz

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109 Scopus citations


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 languageEnglish (US)
Pages (from-to)99-130
Number of pages32
JournalJournal of Mathematical Chemistry
Issue number1
StatePublished - Jan 2012

All Science Journal Classification (ASJC) codes

  • General Chemistry
  • Applied Mathematics


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


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