Distance correlation-based method for global sensitivity analysis of models with dependent inputs

Abstract

Global sensitivity analysis (GSA) plays an important role to quantify the relative importance of uncertain parameters to the model response. However, performing quantitative GSA directly is still a challenging problem for complex models with dependent inputs. A novel method is proposed for screening dependent inputs in the study. The proposed method inherits the capability of easily handing multivariate dependence from the distance correlation. With the help of a projection operator in the Hilbert space, it can work without knowing the specific conditional distribution of inputs. The advantages of the proposed method are discussed and demonstrated through applications to numerical and environmental modeling examples containing many dependent variables. Compared to classical GSA methods with dependent variables, the proposed method can be easily used, while the accuracy of inputs screening is well maintained.

Appendix

If none of the X_s, Y, and X_~s are constant convergence and $\mid {P}_{{X_{\sim s}}^{\perp }}\left({X}_s\right)\mid >0$. We have

$$ {\displaystyle \begin{array}{c}\left({P}_{{X_{\sim s}}^{\perp }}\cdot {B}_Y\right)=\left({A}_{X_s}-\alpha {C}_{{X_{\sim s}}^{\perp }}\cdot {B}_Y\right)\\ {}=\left({A}_{X_s}\cdot {B}_Y\right)-\alpha \left({C}_{{X_{\sim s}}^{\perp }}\cdot {B}_Y\right)\\ {}={\mathcal{V}}^2\left({\mathrm{X}}_s,\mathrm{Y}\right)-\frac{{\mathcal{V}}^2\left({\mathrm{X}}_s,{\mathrm{X}}_{\sim s}\right){\mathcal{V}}^2\left(\mathrm{Y},{\mathrm{X}}_{\sim s}\right)}{{\mathcal{V}}^2\left({\mathrm{X}}_{\sim s},{\mathrm{X}}_{\sim s}\right)}\end{array}} $$

(A.1)

Similarly,

$$ {\displaystyle \begin{array}{c}\mid {P}_{{X_{\sim s}}^{\perp }}\mid \cdot \mid {P}_{{X_{\sim s}}^{\perp }}\mid ={\mathcal{V}}^2\left({X}_s,{X}_s\right)-\frac{{\mathcal{V}}^2\left({X}_s,{X}_{\sim s}\right){\mathcal{V}}^2\left({X}_s,{X}_{\sim s}\right)}{{\mathcal{V}}^2\left({X}_{\sim s},{X}_{\sim s}\right)}\\ {}={\mathcal{V}}^2\left({X}_s,{X}_s\right)\left(1-\frac{{\left({\mathcal{V}}^2\left({X}_s,{X}_{\sim s}\right)\right)}^2}{{\mathcal{V}}^2\left({X}_s,{X}_s\right){\mathcal{V}}^2\left({X}_{\sim s},{X}_{\sim s}\right)}\right)\\ {}={\mathcal{V}}^2\left({X}_s,{X}_s\right)\left(1-{\mathcal{R}}^4\left({X}_s,{X}_{\sim s}\right)\right)\end{array}} $$

(A.2)

Combining (A.1) and (A.2), we have

$$ {\displaystyle \begin{array}{c}{\mathcal{R}}^2\left({\mathrm{X}}_s|{\mathrm{X}}_{\sim s},\mathrm{Y}\right)=\frac{\left({P}_{{X_{\sim s}}^{\perp }}\left({X}_s\right)\cdot {B}_Y\right)}{\mid {P}_{{X_{\sim s}}^{\perp }}\left({\mathrm{X}}_s\right)\Big\Vert {B}_Y\mid}\\ {}=\frac{{\mathcal{V}}^2\left({\mathrm{X}}_s,\mathrm{Y}\right)-\frac{{\mathcal{V}}^2\left({\mathrm{X}}_s,{\mathrm{X}}_{\sim s}\right){\mathcal{V}}^2\left(\mathrm{Y},{\mathrm{X}}_{\sim s}\right)}{{\mathcal{V}}^2\left({\mathrm{X}}_{\sim s},{\mathrm{X}}_{\sim s}\right)}}{\sqrt{{\mathcal{V}}^2\left({\mathrm{X}}_s,{\mathrm{X}}_s\right)\left(1-{\mathcal{R}}^4\left({\mathrm{X}}_s,{\mathrm{X}}_{\sim s}\right)\right)}\mathcal{V}\left(\mathrm{Y},\mathrm{Y}\right)}\\ {}=\frac{\frac{{\mathcal{V}}^2\left({\mathrm{X}}_s,\mathrm{Y}\right)}{\mathcal{V}\left({\mathrm{X}}_s,{\mathrm{X}}_s\right)\mathcal{V}\left(\mathrm{Y},\mathrm{Y}\right)}-\frac{{\mathcal{V}}^2\left({\mathrm{X}}_s,{\mathrm{X}}_{\sim s}\right){\mathcal{V}}^2\left(\mathrm{Y},{\mathrm{X}}_{\sim s}\right)}{\mathcal{V}\left({X}_s,{X}_s\right)\mathcal{V}\left(\mathrm{Y},\mathrm{Y}\right){\mathcal{V}}^2\left({\mathrm{X}}_{\sim s},{\mathrm{X}}_{\sim s}\right)}}{\sqrt{\left(1-{\mathcal{R}}^4\left({X}_s,{X}_{\sim s}\right)\right)}}\\ {}=\frac{{\mathcal{R}}^2\left({\mathrm{X}}_s,\mathrm{Y}\right)-{\mathcal{R}}^2\left({\mathrm{X}}_s,{\mathrm{X}}_{\sim s}\right){\mathcal{R}}^2\left(\mathrm{Y},{\mathrm{X}}_{\sim s}\right)}{\sqrt{1-{\mathcal{R}}^4\left({\mathrm{X}}_s,{\mathrm{X}}_{\sim s}\right)}}\end{array}} $$

(A.3)