Comparison of global sensitivity analysis techniques and importance measures in PSA

This paper discusses application and results of global sensitivity analysis techniques to probabilistic safety assessment (PSA) models, and their comparison to importance measures. This comparison allows one to understand whether PSA elements that are important to the risk, as revealed by importance measures, are also important contributors to the model uncertainty, as revealed by global sensitivity analysis. We show that, due to epistemic dependence, uncertainty and global sensitivity analysis of PSA models must be performed at the parameter level. A difficulty arises, since standard codes produce the calculations at the basic event level. We discuss both the indirect comparison through importance measures computed for basic events, and the direct comparison performed using the differential importance measure and the Fussell–Vesely importance at the parameter level. Results are discussed for the large LLOCA sequence of the advanced test reactor PSA.     

Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models

Many dynamic models are used for risk assessment and decision support in ecology and crop science. Such models generate time-dependent model predictions, with time either discretised or continuous. Their global sensitivity analysis is usually applied separately on each time output, but Campbell et al. (2006 [1]) advocated global sensitivity analyses on the expansion of the dynamics in a well-chosen functional basis. This paper focuses on the particular case when principal components analysis is combined with analysis of variance. In addition to the indices associated with the principal components, generalised sensitivity indices are proposed to synthesize the influence of each parameter on the whole time series output. Index definitions are given when the uncertainty on the input factors is either discrete or continuous and when the dynamic model is either discrete or functional. A general estimation algorithm is proposed, based on classical methods of global sensitivity analysis.The method is applied to a dynamic wheat crop model with 13 uncertain parameters. Three methods of global sensitivity analysis are compared: the Sobol'-Saltelli method, the extended FAST method, and the fractional factorial design of resolution 6.     

A general first-order global sensitivity analysis method

Fourier amplitude sensitivity test (FAST) is one of the most popular global sensitivity analysis techniques. The main mechanism of FAST is to assign each parameter with a characteristic frequency through a search function. Then, for a specific parameter, the variance contribution can be singled out of the model output by the characteristic frequency. Although FAST has been widely applied, there are two limitations: (1) the aliasing effect among parameters by using integer characteristic frequencies and (2) the suitability for only models with independent parameters. In this paper, we synthesize the improvement to overcome the aliasing effect limitation [Tarantola S, Gatelli D, Mara TA. Random balance designs for the estimation of first order global sensitivity indices. Reliab Eng Syst Safety 2006; 91(6):717–27] and the improvement to overcome the independence limitation [Xu C, Gertner G. Extending a global sensitivity analysis technique to models with correlated parameters. Comput Stat Data Anal 2007, accepted for publication]. In this way, FAST can be a general first-order global sensitivity analysis method for linear/nonlinear models with as many correlated/uncorrelated parameters as the user specifies. We apply the general FAST to four test cases with correlated parameters. The results show that the sensitivity indices derived by the general FAST are in good agreement with the sensitivity indices derived by the correlation ratio method, which is a non-parametric method for models with correlated parameters.     

Sensitivity analysis of infectious disease models: methods,advances and their application

Sensitivity analysis (SA) can aid in identifying influential model parameters and optimizing model structure, yet infectious disease modelling has yet to adopt advanced SA techniques that are capable of providing considerable insights over traditional methods. We investigate five global SA methods—scatter plots, the Morris and Sobol’ methods, Latin hypercube sampling-partial rank correlation coefficient and the sensitivity heat map method—and detail their relative merits and pitfalls when applied to a microparasite (cholera) and macroparasite (schistosomaisis) transmission model. The methods investigated yielded similar results with respect to identifying influential parameters, but offered specific insights that vary by method. The classical methods differed in their ability to provide information on the quantitative relationship between parameters and model output, particularly over time. The heat map approach provides information about the group sensitivity of all model state variables, and the parameter sensitivity spectrum obtained using this method reveals the sensitivity of all state variables to each parameter over the course of the simulation period, especially valuable for expressing the dynamic sensitivity of a microparasite epidemic model to its parameters. A summary comparison is presented to aid infectious disease modellers in selecting appropriate methods, with the goal of improving model performance and design.     

Second-level global sensitivity analysis of numerical simulators with application to an accident scenario in a sodium-cooled fast reactor

Numerical simulators are widely used to model physical phenomena and global sensitivity analysis (GSA) aims at studying the global impact of the input uncertainties on the simulator output. To perform GSA, statistical tools based on inputs/output dependence measures are commonly used. We focus here on the Hilbert–Schmidt independence criterion (HSIC). Sometimes, the probability distributions modeling the uncertainty of inputs may be themselves uncertain and it is important to quantify their impact on GSA results. We call it here the second-level global sensitivity analysis (GSA2). However, GSA2, when performed with a Monte Carlo double-loop, requires a large number of model evaluations, which is intractable with CPU time expensive simulators. To cope with this limitation, we propose a new statistical methodology based on a Monte Carlo single-loop with a limited calculation budget. First, we build a unique sample of inputs and simulator outputs, from a well-chosen probability distribution of inputs. From this sample, we perform GSA for various assumed probability distributions of inputs by using weighted HSIC measures estimators. Statistical properties of these weighted estimators are demonstrated. Subsequently, we define 2nd-level HSIC-based measures between the distributions of inputs and GSA results, which constitute GSA2 indices. The efficiency of our GSA2 methodology is illustrated on an analytical example, thereby comparing several technical options. Finally, an application to a test case simulating a severe accidental scenario on nuclear reactor is provided.     

Sensitivity analysis approaches applied to systems biology models

With the rising application of systems biology, sensitivity analysis methods have been widely applied to study the biological systems, including metabolic networks, signalling pathways and genetic circuits. Sensitivity analysis can provide valuable insights about how robust the biological responses are with respect to the changes of biological parameters and which model inputs are the key factors that affect the model outputs. In addition, sensitivity analysis is valuable for guiding experimental analysis, model reduction and parameter estimation. Local and global sensitivity analysis approaches are the two types of sensitivity analysis that are commonly applied in systems biology. Local sensitivity analysis is a classic method that studies the impact of small perturbations on the model outputs. On the other hand, global sensitivity analysis approaches have been applied to understand how the model outputs are affected by large variations of the model input parameters. In this review, the author introduces the basic concepts of sensitivity analysis approaches applied to systems biology models. Moreover, the author discusses the advantages and disadvantages of different sensitivity analysis methods, how to choose a proper sensitivity analysis approach, the available sensitivity analysis tools for systems biology models and the caveats in the interpretation of sensitivity analysis results.     

Contribution to the sample mean plot for graphical and numerical sensitivity analysis

The contribution to the sample mean plot, originally proposed by Sinclair, is revived and further developed as practical tool for global sensitivity analysis. The potentials of this simple and versatile graphical tool are discussed. Beyond the qualitative assessment provided by this approach, a statistical test is proposed for sensitivity analysis. A case study that simulates the transport of radionuclides through the geosphere from an underground disposal vault containing nuclear waste is considered as a benchmark. The new approach is tested against a very efficient sensitivity analysis method based on state dependent parameter meta-modelling.     

Analysis of the sensitivity properties of a model of vector-borne bubonic plague.

Model sensitivity is a key to evaluation of mathematical models in ecology and evolution, especially in complex models with numerous parameters. In this paper, we use some recently developed methods for sensitivity analysis to study the parameter sensitivity of a model of vector-borne bubonic plague in a rodent population proposed by Keeling & Gilligan. The new sensitivity tools are based on a variational analysis involving the adjoint equation. The new approach provides a relatively inexpensive way to obtain derivative information about model output with respect to parameters. We use this approach to determine the sensitivity of a quantity of interest (the force of infection from rats and their fleas to humans) to various model parameters, determine a region over which linearization at a specific parameter reference point is valid, develop a global picture of the output surface, and search for maxima and minima in a given region in the parameter space.     

基于傅里叶幅值检验扩展法的轨道车辆垂向模型全局灵敏度分析

为筛选明显影响轨道车辆垂向振动特性的设计参数,简化车辆模型参数优化设计过程,采用傅里叶幅值灵敏度检验扩展法对典型轨道车辆垂向模型进行参数灵敏度分析。研究结果表明,二系悬挂与一系悬挂阻尼变化对车体沉浮运动影响较一系悬挂刚度大;调整车辆定距也会影响车体点头运动;模型设计参数间存在微弱的交互作用。所用该研究方法及结论对轨道车辆模型的振动特性研究及参数优化设计具有一定参考价值与工程意义。     

Multiple predictor smoothing methods for sensitivity analysis: Description of techniques

The use of multiple predictor smoothing methods in sampling-based sensitivity analyses of complex models is investigated. Specifically, sensitivity analysis procedures based on smoothing methods employing the stepwise application of the following nonparametric regression techniques are described: (i) locally weighted regression (LOESS), (ii) additive models, (iii) projection pursuit regression, and (iv) recursive partitioning regression. Then, in the second and concluding part of this presentation, the indicated procedures are illustrated with both simple test problems and results from a performance assessment for a radioactive waste disposal facility (i.e., the Waste Isolation Pilot Plant). As shown by the example illustrations, the use of smoothing procedures based on nonparametric regression techniques can yield more informative sensitivity analysis results than can be obtained with more traditional sensitivity analysis procedures based on linear regression, rank regression or quadratic regression when nonlinear relationships between model inputs and model predictions are present.     

Multiple predictor smoothing methods for sensitivity analysis: Example results

The use of multiple predictor smoothing methods in sampling-based sensitivity analyses of complex models is investigated. Specifically, sensitivity analysis procedures based on smoothing methods employing the stepwise application of the following nonparametric regression techniques are described in the first part of this presentation: (i) locally weighted regression (LOESS), (ii) additive models, (iii) projection pursuit regression, and (iv) recursive partitioning regression. In this, the second and concluding part of the presentation, the indicated procedures are illustrated with both simple test problems and results from a performance assessment for a radioactive waste disposal facility (i.e., the Waste Isolation Pilot Plant). As shown by the example illustrations, the use of smoothing procedures based on nonparametric regression techniques can yield more informative sensitivity analysis results than can be obtained with more traditional sensitivity analysis procedures based on linear regression, rank regression or quadratic regression when nonlinear relationships between model inputs and model predictions are present.     

A methodology for global-sensitivity analysis of time-dependent outputs in systems biology modelling

One of the main challenges in the development of mathematical and computational models of biological systems is the precise estimation of parameter values. Understanding the effects of uncertainties in parameter values on model behaviour is crucial to the successful use of these models. Global sensitivity analysis (SA) can be used to quantify the variability in model predictions resulting from the uncertainty in multiple parameters and to shed light on the biological mechanisms driving system behaviour. We present a new methodology for global SA in systems biology which is computationally efficient and can be used to identify the key parameters and their interactions which drive the dynamic behaviour of a complex biological model. The approach combines functional principal component analysis with established global SA techniques. The methodology is applied to a model of the insulin signalling pathway, defects of which are a major cause of type 2 diabetes and a number of key features of the system are identified.     

Global sensitivity analysis for fuzzy inputs based on the decomposition of fuzzy output entropy

To analyse the component of fuzzy output entropy, a decomposition method of fuzzy output entropy is first presented. After the decomposition of fuzzy output entropy, the total fuzzy output entropy can be expressed as the sum of the component fuzzy entropy contributed by fuzzy inputs. Based on the decomposition of fuzzy output entropy, a new global sensitivity analysis model is established for measuring the effects of uncertainties of fuzzy inputs on the output. The global sensitivity analysis model can not only tell the importance of fuzzy inputs but also simultaneously reflect the structural composition of the response function to a certain degree. Several examples illustrate the validity of the proposed global sensitivity analysis, which is a significant reference in engineering design and optimization of structural systems.     

Sparse grids‐based stochastic approximations with applications to aerodynamics sensitivity analysis

This work compares sample‐based polynomial surrogates, well suited for moderately high‐dimensional stochastic problems. In particular, generalized polynomial chaos in its sparse pseudospectral form and stochastic collocation methods based on both isotropic and dimension‐adapted sparse grids are considered. Both classes of approximations are compared, and an improved version of a stochastic collocation with dimension adaptivity driven by global sensitivity analysis is proposed. The stochastic approximations efficiency is assessed on multivariate test function and airfoil aerodynamics simulations. The latter study addresses the probabilistic characterization of global aerodynamic coefficients derived from viscous subsonic steady flow about a NACA0015 airfoil in the presence of geometrical and operational uncertainties with both simplified aerodynamics model and Reynolds‐Averaged Navier‐Stokes (RANS) simulation. Sparse pseudospectral and collocation approximations exhibit similar level of performance for isotropic sparse simulation ensembles. Computational savings and accuracy gain of the proposed adaptive stochastic collocation driven by Sobol' indices are patent but remain problem‐dependent. Copyright © 2015 John Wiley & Sons, Ltd.     

A new uncertainty importance measure

Uncertainty in parameters is present in many risk assessment problems and leads to uncertainty in model predictions. In this work, we introduce a global sensitivity indicator which looks at the influence of input uncertainty on the entire output distribution without reference to a specific moment of the output (moment independence) and which can be defined also in the presence of correlations among the parameters. We discuss its mathematical properties and highlight the differences between the present indicator, variance-based uncertainty importance measures and a moment independent sensitivity indicator previously introduced in the literature. Numerical results are discussed with application to the probabilistic risk assessment model on which Iman [A matrix-based approach to uncertainty and sensitivity analysis for fault trees. Risk Anal 1987;7(1):22–33] first introduced uncertainty importance measures.     

基于统计分析技术的有限元模型修正研究

采用统计学的方差分析和回归分析技术研究模型修正的有关问题,主要包括基于方差分析的参数筛选、基于回归分析的响应面拟合和利用响应面进行模型修正三个方面。目前工程上采用的基于灵敏度分析的参数挑选方法根据参数在某设计点处的灵敏度进行挑选,而基于方差分析的参数筛选是从全局的角度,在整个设计空间上挑选对特征量有显著影响的设计参数。基于响应面的修正方法,首先在参数的整个设计空间范围内利用回归分析技术,以显式的响应面模型逼近特征量与设计参数间复杂的隐式函数关系,然后在其基础上进行迭代修正。提出的方法不但可以应用于线性、低频等现有的模型修正方法适用的范围,而且易于推广到非线性、冲击等现有修正方法较少涉及的领域。此外,现有的方法由于每次迭代都需要调用有限元分析软件进行计算,在缺少软件接口的情况下,较难实现工程应用。这种方法只在准备样本数据时需要进行有限元分析,修正过程中无需调用,因而利于工程应用。GARTEUR飞机模型有限元模型的修正结果验证了方法的有效性     

Global sensitivity analysis for calculating the contribution of genetic parameters to the variance of crop model prediction

Dynamic models are often used to predict the effects of farmers’ practices on crop yield, crop quality, and environment. These models usually include many parameters that must be estimated from experimental data before practical use. Parameter estimation is a difficult problem especially when some of the parameters vary across genotypes. These genetic parameters may be estimated from plant breeding experiments but this is very costly and requires a lot of experimental work. Moreover, some of the genetic parameters may account for only a very small part of the output variance and, so, do not deserve an accurate determination. This paper shows how methods of global sensitivity analysis can be used to evaluate the contributions of the genetic parameters to the variance of model prediction. Two methods are applied to a complex crop model for estimating the sensitivity indices associated to 13 genetic parameters. The results show that only five genetic parameters have a significant effect on crop yield and grain quality.     

A fatigue model with local sensitivity analysis

The goal of this paper is two fold. First, it introduces a general parametric lifetime model for high‐cycle fatigue regime derived from physical, statistical, engineering and dimensional analysis considerations. The proposed model has two threshold parameters and three Weibull distribution parameters. A two‐step procedure is presented to estimate the parameters. In the first step, the two threshold parameters are estimated by minimizing a least squares regression function. In the second step, the parameters are estimated by the maximum likelihood method after pooling together the data from different stress levels. Since parameter estimation should always be accompanied by a sensitivity analysis of the fitted model, the second goal of this paper is to propose a method for sensitivity analysis for fatigue models. We show that the proposed sensitivity analysis methods are general and can be applied to any fatigue or lifetime model, not just to the one proposed here. Although several fatigue models have been proposed in the literature, to our knowledge this is the first attempt to produce methods for sensitivity analysis for fatigue models. The proposed method makes use of the well‐known duality property of mathematical programming, which states that the partial derivatives of the primal objective function with respect to the constraints right hand side parameters are the optimal values of the negative of the dual problem variables. For the parameters or data, for which sensitivities are sought, to appear on the right hand side, they are converted into artificial variables and set to their actual values, thus obtaining the desired constraints. Both the estimation and sensitivity analysis methods are illustrated by two examples, one application using real fatigue data and the other using simulated data. In addition, the sensitivity proposed method is also applied to an alternative fatigue model. Finally, some specific conclusions and recommendations are also given.     

Calculating first-order sensitivity measures: A benchmark of some recent methodologies

This work compares three different global sensitivity analysis techniques, namely the state-dependent parameter (SDP) modelling, the random balance designs, and the improved formulas of the Sobol’ sensitivity indices. These techniques are not yet commonly known in the literature. Strengths and weaknesses of each technique in terms of efficiency and computational cost are highlighted, thus enabling the user to choose the more suitable method depending on the computational model analysed. Two test functions proposed in the literature are considered. Computational costs and convergence rates for each function are compared and discussed.