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Variance-based sensitivity analysis for models with correlated inputs and its state dependent parameter solution

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Abstract

Sensitivity analysis is indispensable to structural design and optimization. This paper focuses on sensitivity analysis for models with correlated inputs. To explore the contributions of correlated inputs to the uncertainty in a model output, the universal expressions of the variance contributions of the correlated inputs are first derived in the paper based on the high dimensional model representation (HDMR) of the model function. Then by analyzing the composition of these variance contributions, the variance contributions by an individual correlated input to the model output are further decomposed into independent contribution by the individual input itself, independent contribution by interaction between the individual input and the others, contribution purely by correlation between the individual input and the others, and contribution by interaction associated with correlation between the individual input and the others. The general expressions of these components are also derived. Based on the characteristics of these general expressions, a universal framework for estimating the various variance contributions of the correlated inputs is developed by taking the efficient state dependent parameter (SDP) method as an illustration. Numerical and engineering tests show that this decomposition of the variance contributions of the correlated inputs can provide useful information for exploring the sources of the output uncertainty and identifying the structure of the model function for the complicated models with correlated inputs. The efficiency and accuracy of the SDP-based method for estimating the various variance contributions of the correlated inputs are also demonstrated by the examples.

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Abbreviations

d :

Number of input variables

X :

Vector of input variables, i.e., X = (X 1, X 2,  … , X d )T

x :

Random samples of input X

Y :

Scalar model output equal to Y = g(X)

y :

Sample of output Y

f X (x):

Joint probability density function (PDF) of X

\( {f}_{{\mathbf{X}}_{- i}\mid {X}_i}\left({\mathbf{x}}_{- i}\right) \) :

Conditional PDF of X -i given X i

\( {f}_{X_i\mid {\mathbf{X}}_{- i}}\left({x}_i\right) \) :

Conditional PDF of X i given X -i

V[•]:

Variance operator

E(•):

Expectation operator

Cov[•, •]:

Covariance of [•, •]

d :

Set of subscripts of inputs X, i.e., d = {1,2,…,d}

\( {V}_i^{\mathrm{TC}} \) :

Total correlated contribution of X i

\( {V}_i^{\mathrm{C}} \) :

Correlated contribution of X i

\( {V}_i^{\mathrm{IC}} \) :

Variance contribution by interaction associated with correlation between X i and X -i

\( {V}_i^{\mathrm{OC}} \) :

Variance contribution due to pure correlation between X i and X -i

\( {V}_i^{\mathrm{TU}} \) :

Total uncorrelated contribution of X i

\( {V}_i^{\mathrm{U}} \) :

Independent variance contribution by X i itself

\( {V}_i^{\mathrm{IU}} \) :

Independent part of the contributions by interaction between X i and X -i

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. NSFC 51505382) and the Natural Science Foundation of Shaan Xi province (Grant No. 2016JQ1034).

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Authors and Affiliations

  1. School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, People’s Republic of China

    Luyi Li & Zhenzhou Lu

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  1. Luyi Li
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  2. Zhenzhou Lu
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Correspondence to Luyi Li.

Appendices

Appendix 1 Reduction of (13)

It can be shown that

$$ E\left({g}_i\left({X}_i\right)- E\left({g}_i\left({X}_i\right)|{\mathbf{X}}_{- i}\right)\right)= E\left({g}_i\left({X}_i\right)\right)- E\left( E\left({g}_i\left({X}_i\right)|{\mathbf{X}}_{- i}\right)\right)=0 $$

Then the first covariance term in Eq. (13) can be transformed into the following form

$$ \begin{array}{l}\mathrm{Cov}\left[{g}_i\left({X}_i\right),{g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right]- Cov\left[{\widehat{g}}_i\left({\mathbf{X}}_{- i}\right),{g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right]\\ {}=\mathrm{Cov}\left[{g}_i\left({X}_i\right)-{\widehat{g}}_i\left({\mathbf{X}}_{- i}\right),{g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right]\\ {}=\mathrm{Cov}\left[{g}_i\left({X}_i\right)- E\left({g}_i\left({X}_i\right)|{\mathbf{X}}_{- i}\right),{g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right]\\ {}= E\left(\left({g}_i\left({X}_i\right)- E\left({g}_i\left({X}_i\right)|{\mathbf{X}}_{- i}\right)\right){g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right)\\ {}= E\left({g}_i\left({X}_i\right){g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right)- E\left( E\left({g}_i\left({X}_i\right)|{\mathbf{X}}_{- i}\right){g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right)\end{array} $$

Considering that the second term in right hand side of above equation can be simplified as follows,

$$ E\left( E\left({g}_i\left({X}_i\right)|{\mathbf{X}}_{- i}\right){g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right)=\underset{d-1}{\int } E\left({g}_i\left({X}_i\right)|{\mathbf{X}}_{- i}\right){g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right){f}_{{\mathbf{X}}_{- i}}\left({\mathbf{x}}_{- i}\right) d{\mathbf{x}}_{- i}=\underset{d-1}{\int }{g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\underset{1}{\int }{g}_i\left({X}_i\right){f}_{X_i\mid {\mathbf{X}}_{- i}}\left({x}_i\right){ d x}_i{f}_{{\mathbf{X}}_{- i}}\left({\mathbf{x}}_{- i}\right) d{\mathbf{x}}_{- i}=\underset{d}{\int }{g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right){g}_i\left({X}_i\right){f}_{\mathbf{X}}\left(\mathbf{x}\right) d\mathbf{x}= E\left({g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right){g}_i\left({X}_i\right)\right) $$

it is clear that

$$ \mathrm{Cov}\left[{g}_i\left({X}_i\right),{g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right]-\mathrm{Cov}\left[{\widehat{g}}_i\left({\mathbf{X}}_{- i}\right),{g}_{\mathbf{u}}\left({\mathbf{X}}_{\mathbf{u}}\right)\right]=0 $$

Similarly, since the third covariance term in (13) can be transformed into

$$ \begin{array}{l}\mathrm{Cov}\left[{g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right),{g}_{i\mathbf{u}}\Big({X}_i,{\mathbf{X}}_{\mathbf{u}}\Big)\right]-\mathrm{Cov}\left[{g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right),{\widehat{g}}_{i\mathbf{u}}\left({\mathbf{X}}_{- i}\right)\right]\\ {}=\mathrm{Cov}\left[{g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right),{g}_{i\mathbf{u}}\Big({X}_i,{\mathbf{X}}_{\mathbf{u}}\Big)-{\widehat{g}}_{i\mathbf{u}}\left({\mathbf{X}}_{- i}\right)\right]\\ {}=\mathrm{Cov}\left[{g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right),{g}_{i\mathbf{u}}\Big({X}_i,{\mathbf{X}}_{\mathbf{u}}\Big)- E\left({g}_{i\mathbf{u}}\left({X}_i,{\mathbf{X}}_{\mathbf{u}}\right)|{\mathbf{X}}_{- i}\right)\right]\\ {}= E\left({g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right)\left({g}_{i\mathbf{u}}\left({X}_i,{\mathbf{X}}_{\mathbf{u}}\right)- E\left({g}_{i\mathbf{u}}\left({X}_i,{\mathbf{X}}_{\mathbf{u}}\right)|{\mathbf{X}}_{- i}\right)\right)\right)\\ {}= E\left({g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right){g}_{i\mathbf{u}}\Big({X}_i,{\mathbf{X}}_{\mathbf{u}}\Big)\right)- E\left({g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right) E\left({g}_{i\mathbf{u}}\left({X}_i,{\mathbf{X}}_{\mathbf{u}}\right)|{\mathbf{X}}_{- i}\right)\right)\end{array} $$

and

$$ E\left({g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right) E\left({g}_{i\mathbf{u}}\left({X}_i,{\mathbf{X}}_{\mathbf{u}}\right)|{\mathbf{X}}_{- i}\right)\right)=\underset{d-1}{\int }{g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right) E\left({g}_{i\mathbf{u}}\left({X}_i,{\mathbf{X}}_{\mathbf{u}}\right)|{\mathbf{X}}_{- i}\right){f}_{{\mathbf{X}}_{- i}}\left({\mathbf{x}}_{- i}\right) d{\mathbf{x}}_{- i}=\underset{d-1}{\int }{g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right)\underset{1}{\int }{g}_{i\mathbf{u}}\left({X}_i,{\mathbf{X}}_{\mathbf{u}}\right){f}_{X_i\mid {\mathbf{X}}_{- i}}\left({x}_i\right){ d x}_i{f}_{{\mathbf{X}}_{- i}}\left({\mathbf{x}}_{- i}\right) d{\mathbf{x}}_{- i}=\underset{d}{\int }{g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right){g}_{i\mathbf{u}}\left({X}_i,{\mathbf{X}}_{\mathbf{u}}\right){f}_{\mathbf{X}}\left(\mathbf{x}\right) d\mathbf{x}= E\left({g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right){g}_{i\mathbf{u}}\Big({X}_i,{\mathbf{X}}_{\mathbf{u}}\Big)\right) $$

it can be readily obtained that

$$ \mathrm{Cov}\left[{g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right),{g}_{i\mathbf{u}}\Big({X}_i,{\mathbf{X}}_{\mathbf{u}}\Big)\right]-\mathrm{Cov}\left[{g}_{{\mathbf{u}}^{\mathbf{\prime}}}\left({\mathbf{X}}_{{\mathbf{u}}^{\mathbf{\prime}}}\right),{\widehat{g}}_{i\mathbf{u}}\left({\mathbf{X}}_{- i}\right)\right]=0 $$

Appendix 2 Analytical variance contributions for correlated inputs in Example 1 of subsection 4.1

According to (1), the HDMR of the model \( Y=5+8{X}_1+{X}_2^2 \) in Example 1 is

$$ Y= g\left(\mathbf{X}\right)={g}_0+{g}_1\left({X}_1\right)+{g}_2\left({X}_2\right) $$

where

$$ {g}_0=5+8{\mu}_1+{\sigma}_2^2+{\mu}_2^2 $$
$$ {g}_1\left({X}_1\right)=8\left({X}_1\hbox{-} {\mu}_1\right) $$
$$ {g}_2\left({X}_2\right)={X}_2^2\hbox{-} {\sigma}_2^2\hbox{-} {\mu}_2^2 $$

Then, the total variance V(Y) of the model can be obtained as

$$ \begin{array}{l} V(Y)=\mathrm{Var}\left[{g}_1\left({X}_1\right)\right]+\mathrm{Var}\left[{g}_2\left({X}_2\right)\right]+2\mathrm{Cov}\left({g}_1\left({X}_1\right),{g}_2\left({X}_2\right)\right)\\ {}=64{\sigma}_1^2+2{\sigma}_2^4+4{\mu}_2^2{\sigma}_2^2+32{\rho}_{12}{\mu}_2{\sigma}_1{\sigma}_2\end{array} $$

According to the definition (9), the total correlated contributions of X 1 and X 2 are

$$ \begin{array}{l}{V}_1^{\mathrm{TC}}= V\left[ E\left( Y|{X}_1\right)\right]\\ {}=\mathrm{Var}\left[{g}_1\left({X}_1\right)\right]+\mathrm{Var}\left[ E\left(\left.{g}_2\left({X}_2\right)\right|{X}_1\right)\right]+2\mathrm{Cov}\left({g}_1\left({X}_1\right), E\left(\left.{g}_2\left({X}_2\right)\right|{X}_1\right)\right)\\ {}=64{\sigma}_1^2+2{\rho}_{12}^4{\sigma}_2^4+4{\rho}_{12}^2{\mu}_2^2{\sigma}_2^2+32{\rho}_{12}{\mu}_2{\sigma}_1{\sigma}_2\end{array} $$
$$ \begin{array}{l}{V}_2^{\mathrm{TC}}= V\left[ E\left( Y|{X}_2\right)\right]\\ {}=\mathrm{Var}\left[ E\left(\left.{g}_1\left({X}_1\right)\right|{X}_2\right)\right]+\mathrm{Var}\left[{g}_2\left({X}_2\right)\right]+2\mathrm{Cov}\left( E\left(\left.{g}_1\left({X}_1\right)\right|{X}_2\right),{g}_2\left({X}_2\right)\right)\\ {}=64{\rho}_{12}^2{\sigma}_1^2+2{\sigma}_2^4+4{\mu}_2^2{\sigma}_2^2+32{\rho}_{12}{\mu}_2{\sigma}_1{\sigma}_2\end{array} $$

According to the definition (14), the total uncorrelated contributions of X 1 and X 2 are

$$ {V}_1^{\mathrm{TU}}= V(Y)- V\left[ E\left( Y|{X}_2\right)\right]=\mathrm{Var}\left[{g}_1\left({X}_1\right)\right]-\mathrm{Var}\left[ E\left(\left.{g}_1\left({X}_1\right)\right|{X}_2\right)\right]=64{\sigma}_1^2\left(1-{\rho}_{12}^2\right) $$
$$ \begin{array}{l}{V}_2^{\mathrm{TU}}= V(Y)- V\left[ E\left( Y|{X}_1\right)\right]=\mathrm{Var}\left[{g}_2\left({X}_2\right)\right]-\mathrm{Var}\left[ E\left(\left.{g}_2\left({X}_2\right)\right|{X}_1\right)\right]\\ {}\begin{array}{rrrrrrrr}\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \end{array}=2{\sigma}_2^4\left(1-{\rho}_{12}^4\right)+4{\mu}_2^2{\sigma}_2^2\left(1-{\rho}_{12}^2\right)\end{array} $$

From the definition of the independent contribution in (16), it can be seen that

$$ {V}_1^{\mathrm{U}}=\mathrm{Var}\left[{g}_1\left({X}_1\right)\right]-\mathrm{Var}\left[{\widehat{g}}_1\left({X}_2\right)\right]=\mathrm{Var}\left[{g}_1\left({X}_1\right)\right]-\mathrm{Var}\left[ E\left({g}_1\left({X}_1\right)|{X}_2\right)\right]={V}_1^{\mathrm{TU}} $$
$$ {V}_2^{\mathrm{U}}=\mathrm{Var}\left[{g}_2\left({X}_2\right)\right]-\mathrm{Var}\left[{\widehat{g}}_2\left({X}_1\right)\right]=\mathrm{Var}\left[{g}_2\left({X}_2\right)\right]-\mathrm{Var}\left[ E\left({g}_2\left({X}_2\right)|{X}_1\right)\right]={V}_2^{\mathrm{TU}} $$

and then the independent part of the contributions by interaction between X 1 and X 2 can be obtained according to (15) as

$$ \begin{array}{l}{V}_1^{\mathrm{IU}}={V}_1^{\mathrm{TU}}-{V}_1^{\mathrm{U}}=0\\ {}{V}_2^{\mathrm{IU}}={V}_2^{\mathrm{TU}}-{V}_2^{\mathrm{U}}=0\end{array} $$

The correlated contribution \( {V}_1^{\mathrm{C}} \) of X 1, purely correlated contribution \( {V}_1^{\mathrm{OC}} \) and the variance contribution by interaction associated with correlation \( {V}_1^{\mathrm{IC}} \) between X 1 and X 2 can be obtained by (18–21) as follows,

$$ {V}_1^{\mathrm{C}}={V}_1^{\mathrm{TC}}-{V}_1^{\mathrm{U}}=\mathrm{Var}\left[ E\left({g}_1\left({X}_1\right)|{X}_2\right)\right]+\mathrm{Var}\left[ E\left(\left.{g}_2\left({X}_2\right)\right|{X}_1\right)\right]+2\mathrm{Cov}\left({g}_1\left({X}_1\right), E\left(\left.{g}_2\left({X}_2\right)\right|{X}_1\right)\right)=64{\sigma}_1^2{\rho}_{12}^2+4{\mu}_2^2{\rho}_{12}^2{\sigma}_2^2+2{\rho}_{12}^4{\sigma}_2^4+32{\rho}_{12}{\mu}_2{\sigma}_1{\sigma}_2{V}_1^{\mathrm{OC}}=\mathrm{Var}\left[ E\left({g}_1\left({X}_1\right)|{X}_2\right)\right]+\mathrm{Var}\left[ E\left(\left.{g}_2\left({X}_2\right)\right|{X}_1\right)\right]+2\mathrm{Cov}\left({g}_1\left({X}_1\right), E\left(\left.{g}_2\left({X}_2\right)\right|{X}_1\right)\right)=64{\sigma}_1^2{\rho}_{12}^2+4{\mu}_2^2{\rho}_{12}^2{\sigma}_2^2+2{\rho}_{12}^4{\sigma}_2^4+32{\rho}_{12}{\mu}_2{\sigma}_1{\sigma}_2{V}_1^{\mathrm{IC}}={V}_1^{\mathrm{C}}-{V}_1^{\mathrm{OC}}=0 $$

In the same way, \( {V}_2^{\mathrm{C}} \), \( {V}_2^{\mathrm{OC}} \) and \( {V}_2^{\mathrm{IC}} \) of X 2 can be obtained

$$ {V}_2^{\mathrm{C}}={V}_2^{\mathrm{TC}}-{V}_2^{\mathrm{U}}=\mathrm{Var}\left[ E\left({g}_1\left({X}_1\right)|{X}_2\right)\right]+\mathrm{Var}\left[ E\left(\left.{g}_2\left({X}_2\right)\right|{X}_1\right)\right]+2\mathrm{Cov}\left( E\left(\left.{g}_1\left({X}_1\right)\right|{X}_2\right),{g}_2\left({X}_2\right)\right)=64{\sigma}_1^2{\rho}_{12}^2+4{\mu}_2^2{\rho}_{12}^2{\sigma}_2^2+2{\rho}_{12}^4{\sigma}_2^4+32{\rho}_{12}{\mu}_2{\sigma}_1{\sigma}_2{V}_2^{\mathrm{OC}}=\mathrm{Var}\left[ E\left({g}_1\left({X}_1\right)|{X}_2\right)\right]+\mathrm{Var}\left[ E\left(\left.{g}_2\left({X}_2\right)\right|{X}_1\right)\right]+2\mathrm{Cov}\left( E\left(\left.{g}_1\left({X}_1\right)\right|{X}_2\right),{g}_2\left({X}_2\right)\right)=64{\sigma}_1^2{\rho}_{12}^2+4{\mu}_2^2{\rho}_{12}^2{\sigma}_2^2+2{\rho}_{12}^4{\sigma}_2^4+32{\rho}_{12}{\mu}_2{\sigma}_1{\sigma}_2{V}_2^{\mathrm{IC}}={V}_2^{\mathrm{C}}-{V}_2^{\mathrm{OC}}=0 $$

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Li, L., Lu, Z. Variance-based sensitivity analysis for models with correlated inputs and its state dependent parameter solution. Struct Multidisc Optim 56, 919–937 (2017). https://doi.org/10.1007/s00158-017-1699-z

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