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

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.

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Sensitivity Analysis in Probabilistic Structural Design: A Comparison of Selected Techniques

Sustainability ◽

10.3390/su12114788 ◽

2020 ◽

Vol 12 (11) ◽

pp. 4788 ◽

Cited By ~ 5

Author(s):

Zdeněk Kala

Keyword(s):

Sensitivity Analysis ◽

Failure Probability ◽

Structural Reliability ◽

Limit State ◽

Information Value ◽

Ultimate Limit State ◽

State Function ◽

Sensitivity Indices ◽

Small Failure Probability ◽

Input Variables

Although more and more reliability-oriented sensitivity analysis (ROSA) techniques are now available, review and comparison articles of ROSA are absent. In civil engineering, many of the latest indices have never been used to analyse structural reliability for very small failure probability. This article aims to analyse and compare different sensitivity analysis (SA) techniques and discusses their strengths and weaknesses. For this purpose, eight selected sensitivity indices are first described and then applied in two different test cases. Four ROSA type indices are directly oriented on the failure probability or reliability index beta, and four other indices (of a different type) are oriented on the output of the limit state function. The case study and results correspond to cases under common engineering assumptions, where only two independent input variables with Gaussian distribution of the load action and the resistance are applied in the ultimate limit state. The last section of the article is dedicated to the analysis of the different results. Large differences between first-order sensitivity indices and very strong interaction effects obtained from ROSA are observed for very low values of failure probability. The obtained numerical results show that ROSA methods lack a common platform that clearly interprets the relationship of indices to their information value. This paper can help orientate in the selection of which sensitivity measure to use.

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Global Sensitivity Analysis of Fuzzy Distribution Parameter on Failure Probability and Its Single-Loop Estimation

Journal of Applied Mathematics ◽

10.1155/2014/490718 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Lei Cheng ◽

Zhenzhou Lu ◽

Luyi Li

Keyword(s):

Sensitivity Analysis ◽

Failure Probability ◽

Global Sensitivity Analysis ◽

Computational Cost ◽

Single Loop ◽

Global Sensitivity ◽

Sensitivity Indices ◽

Distribution Parameters ◽

Input Variables ◽

Sampling Probability

An extending Borgonovo’s global sensitivity analysis is proposed to measure the influence of fuzzy distribution parameters on fuzzy failure probability by averaging the shift between the membership functions (MFs) of unconditional and conditional failure probability. The presented global sensitivity indices can reasonably reflect the influence of fuzzy-valued distribution parameters on the character of the failure probability, whereas solving the MFs of unconditional and conditional failure probability is time-consuming due to the involved multiple-loop sampling and optimization operators. To overcome the large computational cost, a single-loop simulation (SLS) is introduced to estimate the global sensitivity indices. By establishing a sampling probability density, only a set of samples of input variables are essential to evaluate the MFs of unconditional and conditional failure probability in the presented SLS method. Significance of the global sensitivity indices can be verified and demonstrated through several numerical and engineering examples.

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Moment Matching for Sensitivity Analysis of Probabilistic Model Output

Volume 1: 30th Design Automation Conference ◽

10.1115/detc2004-57336 ◽

2004 ◽

Author(s):

Mahmoud Awad ◽

Agus Sudjianto ◽

Nanua Singh

Keyword(s):

Sensitivity Analysis ◽

Simulation Models ◽

Moment Matching ◽

Model Output ◽

Sensitivity Indices ◽

Variance Contribution ◽

Complex Engineering ◽

Input Variables ◽

The Relationship ◽

Effective Sensitivity

With the advent of highly complex engineering simulation models that describe the relationship between input variables and output response, the need for an efficient and effective sensitivity analysis is more demanding. In this article, a generalized approach that can provide efficient as well as accurate global sensitivity indices is developed. The approach consists of two steps: running an orthogonal array based experiment using moment-matched levels of the input variables and followed by a variance contribution analysis. The benefits of the approach are demonstrated through three different examples.

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Practical Experience of Sensitivity Analysis: Comparing Six Methods, on Three Hydrological Models, with Three Performance Criteria

Water ◽

10.3390/w11051062 ◽

2019 ◽

Vol 11 (5) ◽

pp. 1062 ◽

Cited By ~ 3

Author(s):

Anqi Wang ◽

Dimitri P. Solomatine

Keyword(s):

Sensitivity Analysis ◽

Practical Experience ◽

Distribution Functions ◽

Ease Of Use ◽

Cumulative Distribution ◽

Performance Criteria ◽

Hydrological Models ◽

Sensitivity Indices ◽

Kolmogorov Smirnov

Currently, practically no modeling study is expected to be carried out without some form of Sensitivity Analysis (SA). At the same time, there is a large number of various methods and it is not always easy for practitioners to choose one. The aim of this paper is to briefly review main classes of SA methods, and to present the results of the practical comparative analysis of applying them. Six different global SA methods: Sobol, eFAST (extended Fourier Amplitude Sensitivity Test), Morris, LH-OAT, RSA (Regionalized Sensitivity Analysis), and PAWN are tested on three conceptual rainfall-runoff models with varying complexity: (GR4J, Hymod, and HBV) applied to the case study of Bagmati basin (Nepal). The methods are compared with respect to effectiveness, efficiency, and convergence. A practical framework of selecting and using the SA methods is presented. The result shows that, first of all, all the six SA methods are effective. Morris and LH-OAT methods are the most efficient methods in computing SI and ranking. eFAST performs better than Sobol, and thus it can be seen as its viable alternative for Sobol. PAWN and RSA methods have issues of instability, which we think are due to the ways Cumulative Distribution Functions (CDFs) are built, and using Kolmogorov–Smirnov statistics to compute Sensitivity Indices. All the methods require sufficient number of runs to reach convergence. Difference in efficiency of different methods is an inevitable consequence of the differences in the underlying principles. For SA of hydrological models, it is recommended to apply the presented practical framework assuming the use of several methods, and to explicitly take into account the constraints of effectiveness, efficiency (including convergence), ease of use, and availability of software.

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New Importance Measures Based on Failure Probability in Global Sensitivity Analysis of Reliability

Mathematics ◽

10.3390/math9192425 ◽

2021 ◽

Vol 9 (19) ◽

pp. 2425

Author(s):

Zdeněk Kala

Keyword(s):

Sensitivity Analysis ◽

Global Sensitivity Analysis ◽

Binary Outcome ◽

Global Sensitivity ◽

First Order ◽

Sensitivity Measure ◽

Sensitivity Indices ◽

Alternative Measures ◽

Input Variables ◽

Dome Shape

This article presents new sensitivity measures in reliability-oriented global sensitivity analysis. The obtained results show that the contrast and the newly proposed sensitivity measures (entropy and two others) effectively describe the influence of input random variables on the probability of failure Pf. The contrast sensitivity measure builds on Sobol, using the variance of the binary outcome as either a success (0) or a failure (1). In Bernoulli distribution, variance Pf(1 − Pf) and discrete entropy—Pfln(Pf) − (1 − Pf)ln(1 − Pf) are similar to dome functions. By replacing the variance with discrete entropy, a new alternative sensitivity measure is obtained, and then two additional new alternative measures are derived. It is shown that the desired property of all the measures is a dome shape; the rise is not important. Although the decomposition of sensitivity indices with alternative measures is not proven, the case studies suggest a rationale structure of all the indices in the sensitivity analysis of small Pf. The sensitivity ranking of input variables based on the total indices is approximately the same, but the proportions of the first-order and the higher-order indices are very different. Discrete entropy gives significantly higher proportions of first-order sensitivity indices than the other sensitivity measures, presenting entropy as an interesting new sensitivity measure of engineering reliability.

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Reduced Partial Factors for Assessment in Uk Assessment Standards

IABSE Conference, Copenhagen 2018: Engineering the Past, to Meet the Needs of the Future ◽

10.2749/copenhagen.2018.312 ◽

2018 ◽

Author(s):

C. R. Hendy ◽

L. S. Man ◽

R. P. Mitchell ◽

H. Takano

Keyword(s):

Structural Reliability ◽

Distribution Model ◽

Reference Period ◽

Minimum Level ◽

Design Standards ◽

Reliability Level ◽

Assessment Standards ◽

Input Variables ◽

Distribution Type ◽

The Cost

Design standards are based upon a range of input variables for resistance, action and modelling. The distribution type and parameters for each determine the partial factors appropriate to achieve a defined reliability level over a specified reference period. For assessment a reduced reliability level may be accepted due to the greater cost of providing reliability through strengthening when compared to the cost of providing it at design. This would allow the use of lower partial factors, although they are still limited by the need to provide a minimum level for human safety. Adoption of this approach for assessment would have significant benefits for an ageing UK infrastructure by reducing the need to carry out costly strengthening and retrofitting schemes whilst still ensuring appropriate structural reliability levels are maintained. This paper presents a study investigating appropriate reduced partial factors to be applied through UK assessment standards, the sensitivity of these values to input distribution model assumptions, and how they could be implemented in industry.

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Variable Selection in Regression Models Using Global Sensitivity Analysis

Journal of Time Series Econometrics ◽

10.1515/jtse-2018-0025 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

William Becker ◽

Paolo Paruolo ◽

Andrea Saltelli

Keyword(s):

Sensitivity Analysis ◽

Variable Selection ◽

Regression Models ◽

Global Sensitivity Analysis ◽

Physical Models ◽

Model Fit ◽

Global Sensitivity ◽

Testing Sequence ◽

Input Variables

Abstract Global sensitivity analysis is primarily used to investigate the effects of uncertainties in the input variables of physical models on the model output. This work investigates the use of global sensitivity analysis tools in the context of variable selection in regression models. Specifically, a global sensitivity measure is applied to a criterion of model fit, hence defining a ranking of regressors by importance; a testing sequence based on the ‘Pantula-principle’ is then applied to the corresponding nested submodels, obtaining a novel model-selection method. The approach is demonstrated on a growth regression case study, and on a number of simulation experiments, and it is found competitive with existing approaches to variable selection.

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A Vine Copula-Based Global Sensitivity Analysis Method for Structures with Multidimensional Dependent Variables

Mathematics ◽

10.3390/math9192489 ◽

2021 ◽

Vol 9 (19) ◽

pp. 2489

Author(s):

Zhiwei Bai ◽

Hongkui Wei ◽

Yingying Xiao ◽

Shufang Song ◽

Sergei Kucherenko

Keyword(s):

Sensitivity Analysis ◽

Global Sensitivity Analysis ◽

Joint Probability ◽

Copula Functions ◽

Global Sensitivity ◽

Sensitivity Indices ◽

Sensitivity Analysis Method ◽

Input Variables ◽

Vine Copula ◽

Bivariate Copula

For multidimensional dependent cases with incomplete probability information of random variables, global sensitivity analysis (GSA) theory is not yet mature. The joint probability density function (PDF) of multidimensional variables is usually unknown, meaning that the samples of multivariate variables cannot be easily obtained. Vine copula can decompose the joint PDF of multidimensional variables into the continuous product of marginal PDF and several bivariate copula functions. Based on Vine copula, multidimensional dependent problems can be transformed into two-dimensional dependent problems. A novel Vine copula-based approach for analyzing variance-based sensitivity measures is proposed, which can estimate the main and total sensitivity indices of dependent input variables. Five considered test cases and engineering examples show that the proposed methods are accurate and applicable.

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

Symmetry ◽

10.3390/sym13020263 ◽

2021 ◽

Vol 13 (2) ◽

pp. 263

Author(s):

Zdeněk Kala

Keyword(s):

Sensitivity Analysis ◽

Structural Reliability ◽

Sensitivity Analyses ◽

Importance Measure ◽

Mathematical Function ◽

Output Resistance ◽

Global Sensitivity ◽

Sensitivity Indices ◽

The Difference ◽

Comparison Of The Results

The article introduces quantile deviation l as a new sensitivity measure based on the difference between superquantile and subquantile. New global sensitivity indices based on the square of l are presented. The proposed sensitivity indices are compared with quantile-oriented sensitivity indices subordinated to contrasts and classical Sobol sensitivity indices. The comparison is performed in a case study using a non-linear mathematical function, the output of which represents the elastic resistance of a slender steel member under compression. The steel member has random imperfections that reduce its load-carrying capacity. The member length is a deterministic parameter that significantly changes the sensitivity of the output resistance to the random effects of input imperfections. The comparison of the results of three types of global sensitivity analyses shows the rationality of the new quantile-oriented sensitivity indices, which have good properties similar to classical Sobol indices. Sensitivity indices subordinated to contrasts are the least comprehensible because they exhibit the strongest interaction effects between inputs. However, using total indices, all three types of sensitivity analyses lead to approximately the same conclusions. The similarity of the results of two quantile-oriented and Sobol sensitivity analysis confirms that Sobol sensitivity analysis is empathetic to the structural reliability and that the variance is one of the important characteristics significantly influencing the low quantile of resistance.

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A Sampling-Based Sensitivity Analysis Method Considering the Uncertainties of Input Variables and Their Distribution Parameters

Mathematics ◽

10.3390/math9101095 ◽

2021 ◽

Vol 9 (10) ◽

pp. 1095

Author(s):

Xiang Peng ◽

Xiaoqing Xu ◽

Jiquan Li ◽

Shaofei Jiang

Keyword(s):

Sensitivity Analysis ◽

Simulation Method ◽

Transformation Method ◽

Unscented Transformation ◽

Performance Function ◽

Gaussian Integration ◽

Sensitivity Indices ◽

Sensitivity Analysis Method ◽

Distribution Parameters ◽

Input Variables

For engineering products with uncertain input variables and distribution parameters, a sampling-based sensitivity analysis methodology was investigated to efficiently determine the influences of these uncertainties. In the calculation of the sensitivity indices, the nonlinear degrees of the performance function in the subintervals were greatly reduced by using the integral whole domain segmentation method, while the mean and variance of the performance function were calculated using the unscented transformation method. Compared with the traditional Monte Carlo simulation method, the loop number and sampling number in every loop were decreased by using the multiplication approximation and Gaussian integration methods. The proposed algorithm also reduced the calculation complexity by reusing the sample points in the calculation of two sensitivity indices to measure the influence of input variables and their distribution parameters. The accuracy and efficiency of the proposed algorithm were verified with three numerical examples and one engineering example.

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