Global Sensitivity Analysis with Mixtures: A Generalized Functional ANOVA Approach

Abstract

This work investigates aspects of the global sensitivity analysis of computer codes when alternative plausible distributions for the model inputs are available to the analyst. Analysts may decide to explore results under each distribution or to aggregate the distributions, assigning, for instance, a mixture. In the first case, we lose uniqueness of the sensitivity measures, and in the second case, we lose independence even if the model inputs are independent under each of the assigned distributions. Removing the unique distribution assumption impacts the mathematical properties at the basis of variance-based sensitivity analysis and has consequences on result interpretation as well. We analyze in detail the technical aspects. From this investigation, we derive corresponding recommendations for the risk analyst. We show that an approach based on the generalized functional ANOVA expansion remains theoretically grounded in the presence of a mixture distribution. Numerically, we base the construction of the generalized function ANOVA effects on the diffeomorphic modulation under observable response preserving homotopy regression. Our application addresses the calculation of variance-based sensitivity measures for the well-known Nordhaus' DICE model, when its inputs are assigned a mixture distribution. A discussion of implications for the risk analyst and future research perspectives closes the work.

Keywords: D-MORPH regression; mixture distributions; risk analysis; uncertainty analysis.

Fig 1

Upper panel: Ishigami output densities under $F_{\mathbf{X}}^1$, $F_{\mathbf{X}}^2$, $F_{\mathbf{X}}^3$, $P_{\mathbf{X}}$. Lower panel: Corresponding variances: ${\mathbb{V}}^1[G]=13.82$, ${\mathbb{V}}^2[G]=7.07$, ${\mathbb{V}}^3[G]=10.30$, ${\mathbb{V}}^{P_{\mathbf{X}}}[G]=11.42$.

Fig 2

Variance decompositions (classical ANOVA) under $F_{\mathbf{X}}^1$, $F_{\mathbf{X}}^2$, $F_{\mathbf{X}}^3$.

Fig 3

The comparison of the ANOVA effect functions of $g_z^{P_{\mathbf{X}}}$ constructed from 8,000 points with the effect functions of ${\stackrel{\sim }{g}}_z$.

Fig 4

Graph (a): ${\stackrel{\sim }{BT}}_i$ and $S{T}_i$; graph (b) corresponding input ranks.

Fig 5

The boxplots of $S_i^a$ and the total sensitivity indices $S{T}_i^a$ under $P_{\mathbf{X}}$ obtained from 100 replicates.

Fig 6

Selection path for sensitivity analysis when the analyst is uncertain about the input distribution.