3.1. Global Sensitivity Analysis (Dakota)

Global Sensitivity Analysis in Dakota Engine is based on Eqs. (8) and (9) of [Weirs12]. Using the raw analysis samples shown in “RES” tab - “Data Values” tab after running a sensitivity analysis, one can reproduce the sensitivity indices as follows.

According to [Weirs12], the main and total Sobol indices of $i$-th random variable for a quantity of interest (QoI) are computed by

(3.1.1)$$S_i=\frac{\frac{1}{n}\sum _{j=1}^{n}f(\mathit{A}{)}_j(f({\mathit{B}}_{\mathit{A}}^{(i)}{)}_j-f(\mathit{B}{)}_j)}{\frac{1}{2n}\sum _{j=1}^{2n}f(\mathit{C}{)}_{j}f(\mathit{C}{)}_j-<f(\mathit{C}){>}^2}$$

(3.1.2)$$S_i^T=\frac{\frac{1}{2n}\sum _{j=1}^n(f(\mathit{B}{)}_j-f({\mathit{B}}_{\mathit{A}}^{(i)}{)}_j{)}^2}{\frac{1}{2n}\sum _{j=1}^{2n}f(\mathit{C}{)}_{j}f(\mathit{C}{)}_j-<f(\mathit{C}){>}^2}$$

See [Weirs12] for notation details. Note that, the second formulation looks slightly different from Eq. (9) of the original paper, becuase $\mathit{A}$ is switched with $\mathit{B}$. This is to reuse $f({\mathit{B}}_{\mathit{A}}^{(i)})$ in main index for computing the total index.

To reproduce the main and totol sensitivity indices of $i$-th random variable, one can substitute $n$ with the simulation number specified in the user interface, $f(\mathit{A})$ with the vector of first $n$ QoI sample values (sample id from $1$ to $n$), $f(\mathit{B})$ with that of subsequent $n$ QoI samples (sample id from $n+1$ to $2n$), and $f({\mathit{B}}_{\mathit{A}}^{(i)})$ with that of sample id from $(i+1)n+1$ to $(i+2)n$.

Weirs12(1,2,3)

22. 7. Weirs, J. R. Kamm, L. P. Swiler, M. Ratto, S. Tarantola, B. M. Adams, W. J. Rider, and M. S Eldred. Sensitivity analysis techniques applied to a system of hyperbolic conservation laws. Reliability Engineering and System Safety, 107:157–170, 2012