From Probabilistic to Quantile-Oriented Sensitivity Analysis: New Indices of Design Quantiles

Abstract

In structural reliability analysis, sensitivity analysis (SA) can be used to measure how an input variable influences the failure probability Pf of a structure. Although the reliability is usually expressed via Pf, Eurocode building design standards assess the reliability using design quantiles of resistance and load. The presented case study showed that quantile-oriented SA can provide the same sensitivity ranking as Pf-oriented SA or local SA based on Pf derivatives. The first two SAs are global, so the input variables are ranked based on total sensitivity indices subordinated to contrasts. The presented studies were performed for Pf ranging from 9.35 × 10−8 to 1–1.51 × 10−8. The use of quantile-oriented global SA can be significant in engineering tasks, especially for very small Pf. The proposed concept provided an opportunity to go much further. Left-right symmetry of contrast functions and sensitivity indices were observed. The article presents a new view of contrasts associated with quantiles as the distance between the average value of the population before and after the quantile. This distance has symmetric hyperbola asymptotes for small and large quantiles of any probability distribution. Following this idea, new quantile-oriented sensitivity indices based on measuring the distance between a quantile and the average value of the model output are formulated in this article.

Global Sensitivity Analysis of Quantiles: New Importance Measure Based on Superquantiles and Subquantiles

“…A reliability-oriented SA concept based on design quantiles was introduced in [62]. SA was performed using the total indices of the design quantiles of resistance and load without having to evaluate the SA of failure probability.…”

“…This article builds on [62] by introducing more general quadratic forms of quantileoriented sensitivity indices, which are compared with quantile-oriented sensitivity indices subordinated to contrasts [63]. Two SAs are compared with the classical Sobol SA in a case study using a non-linear function of the elastic static resistance of a compressed steel structural member.…”

“…All three types of SA have the ability to measure sensitivity across the input space (i.e., they are global methods), are capable of dealing with non-linear responses, and can quantify the influence of interactions in a non-additive systems. The first two SAs [62,63] are quantile-oriented, the third is the classical Sobol SA [2,3]. Both quantile-oriented SAs can study structural reliability, the assessment of which is based on limit states and design quantiles.…”

“…In the context of SA, l was first introduced as a new sensitivity measure in [62]. A property of the quantile deviation l is that it is expressed in the same unit as the data.…”

“…Contrast Q indices expressed using the quantile deviation l are the same as the indices based on contrasts defined in [63], but are obtained in a different way. Contrast Q indices can also be written in an asymptotic form [62] (p. 15), which is based on measuring the distance between an α-quantile θ* and the mean value µ of the model output l ≈ ±(θ* − µ), but limited to only large and small quantiles.…”