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)\[S_i=\frac{\frac{1}{n}\sum^n_{j=1} f(\boldsymbol{A})_j ( f(\boldsymbol{B_A}^{(i)})_j - f(\boldsymbol{B})_j ) }{ \frac{1}{2n}\sum^{2n}_{j=1} f(\boldsymbol{C})_j f(\boldsymbol{C})_j - <f(\boldsymbol{C})>^2}\]

(3.1.2)\[S^T_i=\frac{\frac{1}{2n}\sum^n_{j=1} ( f(\boldsymbol{B})_j - f(\boldsymbol{B_A}^{(i)})_j )^2 }{ \frac{1}{2n}\sum^{2n}_{j=1} f(\boldsymbol{C})_j f(\boldsymbol{C})_j - <f(\boldsymbol{C})>^2}\]

See [Weirs12] for notation details. Note that, the second formulation looks slightly different from Eq. (9) of the original paper, becuase \(\boldsymbol{A}\) is switched with \(\boldsymbol{B}\). This is to reuse \(f(\boldsymbol{B_A}^{(i)})\) 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, \(f(\boldsymbol{A})\) with the vector of first \(n\) QoI sample values (sample id from \(1\) to \(n\)), \(f(\boldsymbol{B})\) with that of subsequent \(n\) QoI samples (sample id from \(n+1\) to \(2n\)), and \(f(\boldsymbol{B_A}^{(i)})\) 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