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General formulation of HDMR component functions with independent and correlated variables

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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.

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Authors and Affiliations

  1. Department of Chemistry, Princeton University, Princeton, NJ, 08544, USA

    Genyuan Li & Herschel Rabitz

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  1. Genyuan Li
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  2. Herschel Rabitz
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Correspondence to Herschel Rabitz.

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Li, G., Rabitz, H. General formulation of HDMR component functions with independent and correlated variables. J Math Chem 50, 99–130 (2012). https://doi.org/10.1007/s10910-011-9898-0

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  • DOI: https://doi.org/10.1007/s10910-011-9898-0

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