Which Parameters Are Important? Differential Importance Under Uncertainty

Abstract

In probabilistic risk assessment, attention is often focused on the expected value of a risk metric. The sensitivity of this expectation to changes in the parameters of the distribution characterizing uncertainty in the inputs becomes of interest. Approaches based on differentiation encounter limitations when (i) distributional parameters are expressed in different units or (ii) the analyst wishes to transfer sensitivity insights from individual parameters to parameter groups, when alternating between different levels of a probabilistic safety assessment model. Moreover, the analyst may also wish to examine the effect of assuming independence among inputs. This work proposes an approach based on the differential importance measure, which solves these issues. Estimation aspects are discussed in detail, in particular the problem of obtaining all sensitivity measures from a single Monte Carlo sample, thus avoiding potentially costly model runs. The approach is illustrated through an analytical example, highlighting how it can be used to assess the impact of removing the independence assumption. An application to the probabilistic risk assessment model of the Advanced Test Reactor large loss of coolant accident sequence concludes the work.

Divergence-Based Risk Measures: A Discussion on Sensitivities and Extensions

“…As the -divergence risk measure extends from a general divergence, the degree of complexity of its sensitivity analysis increases, adapting to multi-parameter forms, such as, for instance, Sharma-Mittal divergence [ 41 , 42 ], rather than the one-parameter cases of Tsallis- or Rényi-divergence risk measures. To handle the sensitivity analysis in relation to the multiple parameters involved, with or without different units, we refer to the framework of the differential importance measure (see [ 43 , 44 ] and references therein for details).…”

Cascade Sensitivity Measures

“…Sensitivity measures are often constructed via partial derivatives either of outputs with respect to inputs ('local' sensitivity measures, see Borgonovo and Plischke (2016) and references therein) or of the output risk measure in the direction of random inputs (Tsanakas and Millossovich, 2016;Antoniano-Villalobos et al, 2018). One drawback of such sensitivity measures is that they do not fully account for interactions among or statistical dependence between input factors.…”

“…Further work on derivatives of risk measures, and closely related to ours, is Hong (2009); Tsanakas and Millossovich (2016). Our framework also includes sensitivity of expected utilities, considered in Cao and Wan (2017); Antoniano-Villalobos et al (2018). Note that, for the trivial weight function γ ≡ 1, the distortion risk measure reduces to the expectation,…”

“…Remark. The cascade sensitivity framework also includes sensitivity to uncertain statistical parameters of input factors (Antoniano-Villalobos et al, 2018). Let F i|Θ i (•|θ i ) denote the conditional distribution of X i given parameter Θ i = θ i .…”

“…In addition, we address a potential criticism of any sensitivity measure defined via partial derivatives: that, for input factors on different scales, it is difficult to draw conclusions regarding their relative importance; in particular, such sensitivity measures are generally not invariant under monotone transformations of input factors (Antoniano-Villalobos et al, 2018). Hence, in 8, for the input factor X 1 and the distortioñ X 1 , respectively.…”