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

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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Global Sensitivity Analysis Based on Entropy: From Differential Entropy to Alternative Measures

Entropy ◽

10.3390/e23060778 ◽

2021 ◽

Vol 23 (6) ◽

pp. 778

Author(s):

Zdeněk Kala

Keyword(s):

Sensitivity Analysis ◽

Global Sensitivity Analysis ◽

Differential Entropy ◽

Global Sensitivity ◽

Negative Entropy ◽

Sensitivity Measure ◽

Sensitivity Indices ◽

Scientific Disciplines ◽

Alternative Measures ◽

Open Question

Differential entropy can be negative, while discrete entropy is always non-negative. This article shows that negative entropy is a significant flaw when entropy is used as a sensitivity measure in global sensitivity analysis. Global sensitivity analysis based on differential entropy cannot have negative entropy, just as Sobol sensitivity analysis does not have negative variance. Entropy is similar to variance but does not have the same properties. An alternative sensitivity measure based on the approximation of the differential entropy using dome-shaped functionals with non-negative values is proposed in the article. Case studies have shown that new sensitivity measures lead to a rational structure of sensitivity indices with a significantly lower proportion of higher-order sensitivity indices compared to other types of distributional sensitivity analysis. In terms of the concept of sensitivity analysis, a decrease in variance to zero means a transition from the differential to discrete entropy. The form of this transition is an open question, which can be studied using other scientific disciplines. The search for new functionals for distributional sensitivity analysis is not closed, and other suitable sensitivity measures may be found.

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A Novel Global Sensitivity Measure Based on Probability Weighted Moments

Symmetry ◽

10.3390/sym13010090 ◽

2021 ◽

Vol 13 (1) ◽

pp. 90

Author(s):

Shufang Song ◽

Lu Wang

Keyword(s):

Sensitivity Analysis ◽

Global Sensitivity Analysis ◽

Small Samples ◽

Double Loop ◽

Numerical Estimation ◽

Global Sensitivity ◽

Sensitivity Measure ◽

Probability Weighted Moments ◽

Input Variables ◽

Single Set

Global sensitivity analysis (GSA) is a useful tool to evaluate the influence of input variables in the whole distribution range. Variance-based methods and moment-independent methods are widely studied and popular GSA techniques despite their several shortcomings. Since probability weighted moments (PWMs) include more information than classical moments and can be accurately estimated from small samples, a novel global sensitivity measure based on PWMs is proposed. Then, two methods are introduced to estimate the proposed measure, i.e., double-loop-repeated-set numerical estimation and double-loop-single-set numerical estimation. Several numerical and engineering examples are used to show its advantages.

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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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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 for Field Response Based on the Manifold of Feature Covariance Matrix

10.1115/detc2021-69086 ◽

2021 ◽

Author(s):

Zhouzhou Song ◽

Zhao Liu ◽

Can Xu ◽

Ping Zhu

Keyword(s):

Sensitivity Analysis ◽

Covariance Matrix ◽

Computational Models ◽

Global Sensitivity Analysis ◽

Feature Selection Method ◽

Global Sensitivity ◽

Sensitivity Indices ◽

Field Response ◽

Input Variables ◽

The Impact

Abstract In real-world applications, it is commonplace that the computational models have field responses, i.e., the temporal or spatial fields. It has become a critical task to develop global sensitivity analysis (GSA) methods to measure the effect of each input variable on the full-field. In this paper, a new sensitivity analysis method based on the manifold of feature covariance matrix (FCM) is developed for quantifying the impact of input variables on the field response. The method firstly performs feature extraction on the field response to obtain a low-dimensional FCM. An adaptive feature selection method is proposed to avoid the FCM from singularity. Thereby, the field response is represented by a FCM, which lies on a symmetric positive-definite matrix manifold. Then, the GSA technique based on the Cramér-von Mises distance for output valued on the Riemannian manifold is introduced for estimating the sensitivity indices for field response. An example of a temporal field and an example of a 2-D displacement field are introduced to demonstrate the applicability of the proposed method in estimating global sensitivity indices for field solution. Results show that the proposed method can distinguish the important input variables correctly and can yield robust index values. Besides, the proposed method can be implemented for GSA for field responses of different dimensionalities.

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A Global Sensitivity Analysis Framework for Hybrid Simulation with Stochastic Substructures

Frontiers in Built Environment ◽

10.3389/fbuil.2021.778716 ◽

2021 ◽

Vol 7 ◽

Author(s):

Nikolaos Tsokanas ◽

Xujia Zhu ◽

Giuseppe Abbiati ◽

Stefano Marelli ◽

Bruno Sudret ◽

...

Keyword(s):

Sensitivity Analysis ◽

Hybrid Model ◽

Surrogate Model ◽

Global Sensitivity Analysis ◽

Hybrid Simulation ◽

Hybrid Models ◽

Analysis Framework ◽

Global Sensitivity ◽

Sensitivity Indices ◽

Realistic Situation

Hybrid simulation is an experimental method used to investigate the dynamic response of a reference prototype structure by decomposing it to physically-tested and numerically-simulated substructures. The latter substructures interact with each other in a real-time feedback loop and their coupling forms the hybrid model. In this study, we extend our previous work on metamodel-based sensitivity analysis of deterministic hybrid models to the practically more relevant case of stochastic hybrid models. The aim is to cover a more realistic situation where the physical substructure response is not deterministic, as nominally identical specimens are, in practice, never actually identical. A generalized lambda surrogate model recently developed by some of the authors is proposed to surrogate the hybrid model response, and Sobol’ sensitivity indices are computed for substructure quantity of interest response quantiles. Normally, several repetitions of every single sample of the inputs parameters would be required to replicate the response of a stochastic hybrid model. In this regard, a great advantage of the proposed framework is that the generalized lambda surrogate model does not require repeated evaluations of the same sample. The effectiveness of the proposed hybrid simulation global sensitivity analysis framework is demonstrated using an experiment.

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A Global Sensitivity Analysis Framework for Hybrid Simulation

10.31224/osf.io/wnqgb ◽

2021 ◽

Author(s):

Giuseppe Abbiati ◽

Stefano Marelli ◽

Nikolaos Tsokanas ◽

Bruno Sudret ◽

Bozidar Stojadinovic

Keyword(s):

Sensitivity Analysis ◽

Hybrid Model ◽

Polynomial Chaos ◽

Global Sensitivity Analysis ◽

Hybrid Simulation ◽

Polynomial Chaos Expansion ◽

Chaos Expansion ◽

Analysis Framework ◽

Global Sensitivity ◽

Sensitivity Indices

Hybrid Simulation is a dynamic response simulation paradigm that merges physical experiments and computational models into a hybrid model. In earthquake engineering, it is used to investigate the response of structures to earthquake excitation. In the context of response to extreme loads, the structure, its boundary conditions, damping, and the ground motion excitation itself are all subjected to large parameter variability. However, in current seismic response testing practice, Hybrid Simulation campaigns rely on a few prototype structures with fixed parameters subjected to one or two ground motions of different intensity. While this approach effectively reveals structural weaknesses, it does not reveal the sensitivity of structure's response. This thus far missing information could support the planning of further experiments as well as drive modeling choices in subsequent analysis and evaluation phases of the structural design process.This paper describes a Global Sensitivity Analysis framework for Hybrid Simulation. This framework, based on Sobol' sensitivity indices, is used to quantify the sensitivity of the response of a structure tested using the Hybrid Simulation approach due to the variability of the prototype structure and the excitation parameters. Polynomial Chaos Expansion is used to surrogate the hybrid model response. Thereafter, Sobol' sensitivity indices are obtained as a by-product of polynomial coefficients, entailing a reduced number of Hybrid Simulations compared to a crude Monte Carlo approach. An experimental verification example highlights the excellent performance of Polynomial Chaos Expansion surrogates in terms of stable estimates of Sobol' sensitivity indices in the presence of noise caused by random experimental errors.

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A general first-order global sensitivity analysis method

Reliability Engineering & System Safety ◽

10.1016/j.ress.2007.04.001 ◽

2008 ◽

Vol 93 (7) ◽

pp. 1060-1071 ◽

Cited By ~ 53

Author(s):

Chonggang Xu ◽

George Zdzislaw Gertner

Keyword(s):

Sensitivity Analysis ◽

Global Sensitivity Analysis ◽

Analysis Method ◽

Global Sensitivity ◽

First Order ◽

Sensitivity Analysis Method

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Comprehensive global sensitivity analysis of a repository model using different types of transformations and metamodeling techniques

10.5194/egusphere-egu21-6131 ◽

2021 ◽

Author(s):

Sabine M. Spiessl ◽

Dirk-A. Becker ◽

Sergei Kucherenko

Keyword(s):

Sensitivity Analysis ◽

Global Sensitivity Analysis ◽

Higher Order ◽

Model Parameters ◽

High Dimensional Model Representation ◽

Model Output ◽

Global Sensitivity ◽

Sparse Polynomial ◽

Sensitivity Indices ◽

Highly Nonlinear

Due to their highly nonlinear, non-monotonic or even discontinuous behavior, sensitivity analysis of final repository models can be a demanding task. Most of the output of repository models is typically distributed over several orders of magnitude and highly skewed. Many values of a probabilistic investigation are very low or even zero. Although this is desirable in view of repository safety it can distort the evidence of sensitivity analysis. For the safety assessment of the system, the highest values of outputs are mainly essential and if those are only a few, their dependence on specific parameters may appear insignificant. By applying a transformation, different model output values are differently weighed, according to their magnitude, in sensitivity analysis. Probabilistic methods of higher-order sensitivity analysis, applied on appropriately transformed model output values, provide a possibility for more robust identification of relevant parameters and their interactions. This type of sensitivity analysis is typically done by decomposing the total unconditional variance of the model output into partial variances corresponding to different terms in the ANOVA decomposition. From this, sensitivity indices of increasing order can be computed. The key indices used most often are the first-order index (SI1) and the total-order index (SIT). SI1 refers to the individual impact of one parameter on the model and SIT represents the total effect of one parameter on the output in interactions with all other parameters. The second-order sensitivity indices (SI2) describe the interactions between two model parameters.In this work global sensitivity analysis has been performed with three different kinds of output transformations (log, shifted and Box-Cox transformation) and two metamodeling approaches, namely the Random-Sampling High Dimensional Model Representation (RS-HDMR) [1] and the Bayesian Sparse PCE (BSPCE) [2] approaches. Both approaches are implemented in the SobolGSA software [3, 4] which was used in this work. We analyzed the time-dependent output with two approaches for sensitivity analysis, i.e., the pointwise and generalized approaches. With the pointwise approach, the output at each time step is analyzed independently. The generalized approach considers averaged output contributions at all previous time steps in the analysis of the current step. Obtained results indicate that robustness can be improved by using appropriate transformations and choice of coefficients for the transformation and the metamodel.[1] M. Zuniga, S. Kucherenko, N. Shah (2013). Metamodelling with independent and dependent inputs. Computer Physics Communications, 184 (6): 1570-1580.[2] Q. Shao, A. Younes, M. Fahs, T.A. Mara (2017). Bayesian sparse polynomial chaos expansion for global sensitivity analysis. Computer Methods in Applied Mechanics and Engineering, 318: 474-496.[3] S. M. Spiessl, S. Kucherenko, D.-A. Becker, O. Zaccheus (2018). Higher-order sensitivity analysis of a final repository model with discontinuous behaviour. Reliability Engineering and System Safety, doi: https://doi.org/10.1016/j.ress.2018.12.004.[4] SobolGSA software (2021). User manual https://www.imperial.ac.uk/process-systems-engineering/research/free-software/sobolgsa-software/.

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Sensitivity analysis and the challenges posed by multiple approaches: a multifaceted mess

10.5194/egusphere-egu2020-6626 ◽

2020 ◽

Author(s):

Monica Riva ◽

Aronne Dell'Oca ◽

Alberto Guadagnini

Keyword(s):

Sensitivity Analysis ◽

Model Calibration ◽

Global Sensitivity Analysis ◽

System Model ◽

Model Parameters ◽

Model Output ◽

Multiple Model ◽

Total Uncertainty ◽

Global Sensitivity ◽

Sensitivity Indices

Modern models of environmental and industrial systems have reached a relatively high level of complexity. The latter aspect could hamper an unambiguous understanding of the functioning of a model, i.e., how it drives relationships and dependencies among inputs and outputs of interest. Sensitivity Analysis tools can be employed to examine this issue.Global sensitivity analysis (GSA) approaches rest on the evaluation of sensitivity across the entire support within which system model parameters are supposed to vary. In this broad context, it is important to note that the definition of a sensitivity metric must be linked to the nature of the question(s) the GSA is meant to address. These include, for example: (i) which are the most important model parameters with respect to given model output(s)?; (ii) could we set some parameter(s) (thus assisting model calibration) at prescribed value(s) without significantly affecting model results?; (iii) at which space/time locations can one expect the highest sensitivity of model output(s) to model parameters and/or knowledge of which parameter(s) could be most beneficial for model calibration?The variance-based Sobol&#8217; Indices (e.g., Sobol, 2001) represent one of the most widespread GSA metrics, quantifying the average reduction in the variance of a model output stemming from knowledge of the input. Amongst other techniques, Dell&#8217;Oca et al. [2017] proposed a moment-based GSA approach which enables one to quantify the influence of uncertain model parameters on the (statistical) moments of a target model output.Here, we embed in these sensitivity indices the effect of uncertainties both in the system model conceptualization and in the ensuing model(s) parameters. The study is grounded on the observation that physical processes and natural systems within which they take place are complex, rendering target state variables amenable to multiple interpretations and mathematical descriptions. As such, predictions and uncertainty analyses based on a single model formulation can result in statistical bias and possible misrepresentation of the total uncertainty, thus justifying the assessment of multiple model system conceptualizations. We then introduce copula-based sensitivity metrics which allow characterizing the global (with respect to the input) value of the sensitivity and the degree of variability (across the whole range of the input values) of the sensitivity for each value that the prescribed model output can possibly undertake, as driven by a governing model. In this sense, such an approach to sensitivity is global with respect to model input(s) and local with respect to model output, thus enabling one to discriminate the relevance of an input across the entire range of values of the modeling goal of interest. The methodology is demonstrated in the context of flow and reactive transport scenarios.&#160;ReferencesSobol, I. M., 2001. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Sim., 55, 271-280.Dell&#8217;Oca, A., Riva, M., Guadagnini, A., 2017. Moment-based metrics for global sensitivity analysis of hydrological systems. Hydr. Earth Syst. Sci., 21, 6219-6234.

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