General formulation of HDMR component functions with independent and correlated variables |
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Authors: | Genyuan Li Herschel Rabitz |
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Institution: | (1) Structural Engineering Division, Department of Civil Engineering, Indian Institute of Technology Madras, Chennai, 600 036, India |
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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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Keywords: | |
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