First-order Sensitivity Indices Research Articles

Global sensitivity analysis for the boundary control of an open channel

The goal of this paper is to solve the global sensitivity analysis for a particular control problem. More precisely, the boundary control problem of an open-water channel is considered, where the boundary conditions are defined by the position of a downstream overflow gate and an upperstream underflow gate. The dynamics of the water depth and of the water velocity are described by the Shallow Water equations, by taking into account the bottom and friction slopes. Since some physical parameters are unknown, a stabilizing boundary control is first computed for their nominal values, and then a sensitivity analysis is performed to measure the impact of the uncertainty in the parameters on a given to-be-controlled output. The unknown physical parameters are described by some probability distribution functions. Numerical simulations are performed to measure the first-order and total sensitivity indices.

Spatiotemporal pattern of the global sensitivity of the reference evapotranspiration to climatic variables in recent five decades over China

The impact of climatic variables on reference evapotranspiration (ET0) has been a critical issue for ages. The sensitivities of the FAO56 Penman–Monteith ET0 equation to climate variables in 668 stations of China from 1960 to 2009 were studied using a global sensitivity analysis method, the extended Fourier amplitude sensitivity test. The results showed a large degree of spatiotemporal variation in the sensitivity of ET0 to incoming solar radiation, temperature (T), relative humidity (RH), and wind speed, with the averaged first-order sensitivity index (Si) value of 0.41, 0.18, 0.25, and 0.15, respectively. At annul scale, solar radiation was the most sensitive parameter in southern part of China with the mean Si value reached 0.66, while T and RH were the most sensitive parameter in northeast and north China, both with mean Si of 0.37. In northwest China, the mean Si values of incoming solar radiation, T, and wind speed, were relatively constant around 0.25, while the value of RH trended to be slightly higher than other parameters (=0.28). In Tibetan Plateau, solar radiation and T were the two most sensitive parameters, with averaged Si values of 0.32 and 0.31, respectively. The spatial variation of sensitivity varied seasonally, and was closely related to the geographic location of the stations. In general, stations in the low latitude were more sensitive to solar radiation and less sensitive to T than stations in the high latitude. The merit of the sensitivity results obtained in the current study was that it estimates which climatic parameter variation has the greatest impact on ET0 variation across the entire range of the climatic parameters rather than merely based on their median values, and it is helpful to better understand the potential effects of climate changes on the water management in China.

Sensitivity of Wind Farm Output to Wind Conditions, Land Configuration, and Installed Capacity, Under Different Wake Models

In conventional wind farm design and optimization, analytical wake models are generally used to estimate the wake-induced power losses. Different wake models often yield significantly dissimilar estimates of wake velocity deficit and wake width. In this context, the wake behavior, as well as the subsequent wind farm power generation, can be expressed as functions of a series of key factors. A quantitative understanding of the relative impact of each of these key factors, particularly under the application of different wake models, is paramount to reliable quantification of wind farm power generation. Such an understanding is however not readily evident in the current state of the art in wind farm design. To fill this important gap, this paper develops a comprehensive sensitivity analysis (SA) of wind farm performance with respect to the key natural and design factors. Specifically, the sensitivities of the estimated wind farm power generation and maximum farm output potential are investigated with respect to the following key factors: (i) incoming wind speed, (ii) ambient turbulence, (iii) land area per MW installed, (iv) land aspect ratio, and (v) nameplate capacity. The extended Fourier amplitude sensitivity test (e-FAST), which helpfully provides a measure of both first-order and total-order sensitivity indices, is used for this purpose. The impact of using four different analytical wake models (i.e., Jensen, Frandsen, Larsen, and Ishihara models) on the wind farm SA is also explored. By applying this new SA framework, it was observed that, when the incoming wind speed is below the turbine rated speed, the impact of incoming wind speed on the wind farm power generation is dominant, irrespective of the choice of wake models. Interestingly, for array-like wind farms, the relative importance of each input parameter was found to vary significantly with the choice of wake models, i.e., appreciable differences in the sensitivity indices (of up to 70%) were observed across the different wake models. In contrast, for optimized wind farm layouts, the choice of wake models was observed to have marginal impact on the sensitivity indices.

Parameter identification and global sensitivity analysis of Xin'anjiang model using meta-modeling approach

Parameter identification, model calibration, and uncertainty quantification are important steps in the model-building process, and are necessary for obtaining credible results and valuable information. Sensitivity analysis of hydrological model is a key step in model uncertainty quantification, which can identify the dominant parameters, reduce the model calibration uncertainty, and enhance the model optimization efficiency. There are, however, some shortcomings in classical approaches, including the long duration of time and high computation cost required to quantitatively assess the sensitivity of a multiple-parameter hydrological model. For this reason, a two-step statistical evaluation framework using global techniques is presented. It is based on (1) a screening method (Morris) for qualitative ranking of parameters, and (2) a variance-based method integrated with a meta-model for quantitative sensitivity analysis, i.e., the Sobol method integrated with the response surface model (RSMSobol). First, the Morris screening method was used to qualitatively identify the parameters' sensitivity, and then ten parameters were selected to quantify the sensitivity indices. Subsequently, the RSMSobol method was used to quantify the sensitivity, i.e., the first-order and total sensitivity indices based on the response surface model (RSM) were calculated. The RSMSobol method can not only quantify the sensitivity, but also reduce the computational cost, with good accuracy compared to the classical approaches. This approach will be effective and reliable in the global sensitivity analysis of a complex large-scale distributed hydrological model.

Stochastic sensitivity analysis for timing and amplitude of pressure waves in the arterial system.

In the field of computational hemodynamics, sensitivity quantification of pressure and flow wave dynamics has received little attention. This work presents a novel study of the sensitivity of pressure-wave timing and amplitude in the arterial system with respect to arterial stiffness. Arterial pressure and flow waves were simulated with a one-dimensional distributed wave propagation model for compliant arterial networks. Sensitivity analysis of this model was based on a generalized polynomial chaos expansion evaluated by a stochastic collocation method. First-order statistical sensitivity indices were formulated to assess the effect of arterial stiffening on timing and amplitude of the pressure wave and backward-propagating pressure wave in the ascending aorta, at the maximum pressure and inflection point in the systolic phase. Only the stiffness of aortic arteries was found to significantly influence timing and amplitude of the backward-propagating pressure wave, whereas other large arteries in the systemic tree showed marginal impact. Furthermore, the ascending aorta, aortic arch, thoracic aorta, and infrarenal abdominal aorta had the largest influence on amplitude, whereas only the thoracic aorta influenced timing. Our results showed that the non-intrusive polynomial chaos expansion is an efficient method to compute statistical sensitivity measures for wave propagation models. These sensitivities provide new knowledge in the relative importance of arterial stiffness at various locations in the arterial network. Moreover, they will significantly influence clinical data collection and effective composition of the arterial tree for in-silico clinical studies.

Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters

We propose a stochastic multiscale method to quantify the correlated key-input parameters influencing the mechanical properties of polymer nanocomposites (PNCs). The variations of parameters at nano-, micro-, meso- and macro-scales are connected by a hierarchical multiscale approach. The first-order and total-effect sensitivity indices are determined first. The input parameters include the single-walled carbon nanotube (SWNT) length, the SWNT waviness, the agglomeration and volume fraction of SWNTs. Stochastic methods consistently predict that the key parameters for the Young’s modulus of the composite are the volume fraction followed by the averaged longitudinal modulus of equivalent fiber (EF), the SWNT length, and the averaged transverse modulus of the EF, respectively. The averaged longitudinal modulus of the EF is estimated to be the most important parameter with respect to the Poisson’s ratio followed by the volume fraction, the SWNT length, and the averaged transverse modulus of the EF, respectively. On the other hand, the agglomeration parameters have insignificant effect on both Young’s modulus and Poisson’s ratio compared to other parameters. The sensitivity analysis (SA) also reveals the correlation between the input parameters and its effect on the mechanical properties.

Sensitivity analysis of a final repository model with quasi-discrete behaviour using quasi-random sampling and a metamodel approach in comparison to other variance-based techniques

This paper contributes to the investigation of recent computationally efficient variance-based methods for sensitivity analysis and sampling schemes on the basis of a Performance Assessment (PA) model for a repository for Low- and Intermediate-Level Radioactive Waste (LILW) in an abandonned salt mine. The PA model takes account of typical characteristics of repository systems including a quasi-discrete nature.The results indicate best convergence of the State-Dependent-Parameter (SDP) metamodel method and the simple first-order sensitivity indices (SI1) calculation scheme (EASI) if combined with the LpTau low discrepancy sampling method. The SI1 indices obtained from the SDP and EASI methods are very similar. The Fourier Amplitude Sensitivity Test (EFAST) seems to have some convergence problems, likely due to the quasi-discrete behaviour of the PA model. Due to parameter interactions, none of the investigated methods, however, could directly determine all significant parameters. This is illustrated by means of a simplified way of factor fixing.

Computing first-order sensitivity indices with contribution to the sample mean plot

In this paper, we investigate the use of the contribution to the sample mean plot (CSM plot) as a graphical tool for sensitivity analysis (SA) of computational models. We first provide an exact formula that links, for each uncertain model input Xj, the CSM plot Cj(·) with the first-order variance-based sensitivity index Sj. We then build a new estimate for Sj using polynomial regression of the CSM plot. This estimation procedure allows the computation of Sj from given data, without any SA-specific design of experiment. Numerical results show that this new Sj estimate is efficient for large sample sizes, but that at small sample sizes it does not compare well with other Sj estimation techniques based on given data, such as the effective algorithm for computing global sensitivity indices method or metamodel-based approaches.

Quantitative Sensitivity Analysis of Surface Attached Optical Fiber Strain Sensor

Optical fiber strain sensors, in particular, the fiber Bragg grating (FBG) type, are widely applied in different applications. The most common installation method is surface-attached. In principle, the optical fiber strain sensor with adequate sampling and signal processing techniques is usually more accurate than electrical resistive strain gauge. However, the strain of the surface of structure may not transfer to the sensing element perfectly. The ratio between the measured and actual strain can be correlated by a strain transfer factor (STF). However, it depends on the material and geometrical properties of the optical fiber and adhesive. It is noneconomical and impractical to measure the STF for every installed sensor. It is desirable to identify the most of the sensitive parameters on the variation of the STF so that the quality control and assurance procedure can be performed more efficiently. In this paper, a quantitative global sensitivity analysis, called extended Fourier amplitude sensitivity test will be performed to compute the first-order and total sensitivity indexes based on a well-established semi-analytical/empirical mechanical model of three material and five geometrical parameters of both integral and optical FBG type optical fiber strain sensor with two different kinds of polymeric coating under three types of strain field in 16 different configurations. From the detail analysis, the most of the sensitive parameters on the STF are bond length, the thickness of adhesive beneath the optical fiber and the deviation of grating position, which are related to workmanship instead of the material properties of the optical fiber and adhesive.

Uncertainty quantification of dry woven fabrics: A sensitivity analysis on material properties

Based on sensitivity analysis, we determine the key meso-scale uncertain input variables that influence the macro-scale mechanical response of a dry textile subjected to uni-axial and biaxial deformation. We assume a transversely isotropic fashion at the macro-scale of dry woven fabric. This paper focuses on global sensitivity analysis; i.e. regression- and variance-based methods. The sensitivity of four meso-scale uncertain input parameters on the macro-scale response are investigated; i.e. the yarn height, the yarn spacing, the yarn width and the friction coefficient. The Pearson coefficients are adopted to measure the effect of each uncertain input variable on the structural response. Due to computational effectiveness, the sensitivity analysis is based on response surface models. The Sobol’s variance-based method which consists of first-order and total-effect sensitivity indices are presented. The sensitivity analysis utilizes linear and quadratic correlation matrices, its corresponding correlation coefficients and the coefficients of determination of the response uncertainty criteria. The correlation analysis, the response surface model and Sobol’s indices are presented and compared by means of uncertainty criteria influences on MataBerkait-dry woven fabric material properties. To anticipate, it is observed that the friction coefficient and yarn height are the most influential factors with respect to the specified macro-scale mechanical responses.

A comprehensive evaluation of various sensitivity analysis methods: A case study with a hydrological model

Sensitivity analysis (SA) is a commonly used approach for identifying important parameters that dominate model behaviors. We use a newly developed software package, a Problem Solving environment for Uncertainty Analysis and Design Exploration (PSUADE), to evaluate the effectiveness and efficiency of ten widely used SA methods, including seven qualitative and three quantitative ones. All SA methods are tested using a variety of sampling techniques to screen out the most sensitive (i.e., important) parameters from the insensitive ones. The Sacramento Soil Moisture Accounting (SAC-SMA) model, which has thirteen tunable parameters, is used for illustration. The South Branch Potomac River basin near Springfield, West Virginia in the U.S. is chosen as the study area. The key findings from this study are: (1) For qualitative SA methods, Correlation Analysis (CA), Regression Analysis (RA), and Gaussian Process (GP) screening methods are shown to be not effective in this example. Morris One-At-a-Time (MOAT) screening is the most efficient, needing only 280 samples to identify the most important parameters, but it is the least robust method. Multivariate Adaptive Regression Splines (MARS), Delta Test (DT) and Sum-Of-Trees (SOT) screening methods need about 400–600 samples for the same purpose. Monte Carlo (MC), Orthogonal Array (OA) and Orthogonal Array based Latin Hypercube (OALH) are appropriate sampling techniques for them; (2) For quantitative SA methods, at least 2777 samples are needed for Fourier Amplitude Sensitivity Test (FAST) to identity parameter main effect. McKay method needs about 360 samples to evaluate the main effect, more than 1000 samples to assess the two-way interaction effect. OALH and LPτ (LPTAU) sampling techniques are more appropriate for McKay method. For the Sobol' method, the minimum samples needed are 1050 to compute the first-order and total sensitivity indices correctly. These comparisons show that qualitative SA methods are more efficient but less accurate and robust than quantitative ones.

Probabilistic analysis of numerical simulated railway track global stiffness

The aim of this work is to assess numerically the influence of track geomaterials variability on the railway track stiffness. A non-intrusive probabilistic methodology based on in situ cone resistance tests is implemented in a 2D bidimensional finite element model with a modified plane strain condition. This model is used to estimate the track response to a moving load, which is characterized by the track stiffness measurement rolling stock. Spatial variability is taken into account by considering the cone resistance of each track layer as independent random fields, each one characterized by a marginal probability density function obtained from a statistical description of measured in situ data and a theoretical autocorrelation function. Despite input data variability, results presented much less variability than the input, which could be explained by both: load repartition over steppers, i.e. homogenization of the track stiffness measure, and deterministic characteristic of rail pads. Moreover, reduction of variance is observed, which means that less variance is observed for smaller correlation distances. In addition, a sensitivity analysis is also performed based on the Fourier Amplitude Sensitivity Test (FAST) and it showed, for the present case study, that the platform presents the highest first-order sensitivity index for all analyzed cases.

Application of a combined sensitivity analysis approach on a pesticide environmental risk indicator

Sensitivity analysis aims to characterize factors (i.e., model inputs) accounting for the amount of uncertainty in model output. Input factors are usually assumed to be independent, which may lead to incorrect conclusions. In this study, a combined sensitivity analysis approach, composed of the Sobol' and Importance Measurement (IM) methods, is applied on a pesticide environmental risk indicator (called PURE), where main, interaction, and correlation effects (i.e., the effects of factor correlations on sensitivity indices) are all addressed. PURE calculates pesticide risk scores for air, soil, groundwater, and surface water based on pesticide properties and surrounding environmental conditions. The Sobol' method calculates the first-order sensitivity index (Si) and the total-effect sensitivity index (STi) in noncorrelated-factor setting to address the main and interaction effects; while the IM method calculates Si in both noncorrelated-factor and correlated-factor settings to show the correlation effects. In the tested case, the Si estimations in noncorrelated-factor setting by the Sobol' and IM methods are very similar, which not only cross-validates the main effect estimations by the two different methods, but also provides the common ground for combining the two methods to address both interaction and correlation effects. In addition, the Si estimations in correlated-factor setting are relatively different from the ones in noncorrelated-factor setting, which demonstrates that it is cautious to assume all factors are independent in sensitivity analysis. Take the soil risk evaluation as an example, the positive correlation between the chronic no-observed-effect concentration and acute 50%-lethal concentration to earthworms largely increases the Si of the latter factor. The results of Si estimations show that the risk scores for air, soil, groundwater, and surface water are most sensitive to the application rate of pesticide product, the application rate of pesticide active ingredient, the organic carbon sorption constant, and the monthly maximum daily water input, respectively. In summary, while this study enhances the understanding of PURE, it also provides an option for investigating both interaction and correlation effects, and hence promotes sensitivity analysis with factor-correlation structures in environmental modeling.

Stochastic predictions of bulk properties of amorphous polyethylene based on molecular dynamics simulations

The effect of the chain length, the temperature and the strain rate on the yield stress and the elastic modulus of glassy polyethylene is systematically studied using united-atom molecular dynamics (MD) simulations. Based on our MD results, a sensitivity analysis (SA) is carried out in order to quantify the influence of the uncertain input parameters on the predicted yield stress and elastic modulus. The SA is based on response surface (RS) models (polynomial regression and moving least squares). We use partial derivatives (local SA) and variance-based methods (global SA) where we compute first-order and total sensitivity indices. In addition, we use the elementary effects method on the mechanical model. All stochastic methods predict that the key parameter influencing the yield stress and elastic modulus is the temperature, followed by the strain rate.

Sensitivity analysis of a passive decay heat removal system under a post-loss of coolant accident condition

Passive safety features are now of interest to the design of future generation reactors because of their peculiar characteristics of simplicity, reduction of human interaction, and avoidance of failures of active components. However, the large uncertainty associated with the responses of passive systems might not be ignored. Therefore, it is necessary to identify the uncertain inputs that have the important impact on the uncertainty of the system performance. In this study, two global sensitivity measures, the first-order sensitivity index and the total-order sensitivity index, are applied to a natural circulation decay heat removal system of a gas-cooled fast reactor for identifying the important system inputs. It is found that the uncertainty in the system pressure contributes the most to the uncertainty in the system outputs. In addition, the cooler (the heat exchanger of the emergency cooling system) wall temperature, the Nusselt number in the mixed convection regime, and the friction factor in the mixed convection flow regime also have small impact on the uncertainty of the system outputs.

Bias correction for the estimation of sensitivity indices based on random balance designs

This paper deals with the random balance design method (RBD) and its hybrid approach, RBD-FAST. Both these global sensitivity analysis methods originate from Fourier amplitude sensitivity test (FAST) and consequently face the main problems inherent to discrete harmonic analysis. We present here a general way to correct a bias which occurs when estimating sensitivity indices (SIs) of any order – except total SI of single factor or group of factors – by the random balance design method (RBD) and its hybrid version, RBD-FAST. In the RBD case, this positive bias has been recently identified in a paper by Xu and Gertner [1]. Following their work, we propose a bias correction method for first-order SIs estimates in RBD. We then extend the correction method to the SIs of any order in RBD-FAST. At last, we suggest an efficient strategy to estimate all the first- and second-order SIs using RBD-FAST.

Efficient estimation of sensitivity indices

In this paper, we address the problem of efficient estimation of Sobol sensitivity indices. First, we focus on general functional integrals of conditional moments of the form 𝔼(ψ(𝔼(ϕ(Y)|X))) where (X, Y) is a random vector with joint density f and ψ and ϕ are functions that are differentiable enough. In particular, we show that asymptotical efficient estimation of this functional boils down to the estimation of crossed quadratic functionals. An efficient estimate of first-order sensitivity indices is then derived as a special case. We investigate its properties on several analytical functions and illustrate its interest on a reservoir engineering case.

An Application of Monte‐Carlo‐Based Sensitivity Analysis on the Overlap in Discriminant Analysis

Discriminant analysis (DA) is used for the measurement of estimates of a discriminant function by minimizing their group misclassifications to predict group membership of newly sampled data. A major source of misclassification in DA is due to the overlapping of groups. The uncertainty in the input variables and model parameters needs to be properly characterized in decision making. This study combines DEA‐DA with a sensitivity analysis approach to an assessment of the influence of banks’ variables on the overall variance in overlap in a DA in order to determine which variables are most significant. A Monte‐Carlo‐based sensitivity analysis is considered for computing the set of first‐order sensitivity indices of the variables to estimate the contribution of each uncertain variable. The results show that the uncertainties in the loans granted and different deposit variables are more significant than uncertainties in other banks’ variables in decision making.

Reliability of global sensitivity indices

Uncertainty and sensitivity analysis is an essential ingredient of model development and applications. For many uncertainty and sensitivity analysis techniques, sensitivity indices are calculated based on a relatively large sample to measure the importance of parameters in their contributions to uncertainties in model outputs. To statistically compare their importance, it is necessary that uncertainty and sensitivity analysis techniques provide standard errors of estimated sensitivity indices. In this paper, a delta method is used to analytically approximate standard errors of estimated sensitivity indices for a popular sensitivity analysis method, the Fourier amplitude sensitivity test (FAST). Standard errors estimated based on the delta method were compared with those estimated based on 20 sample replicates. We found that the delta method can provide a good approximation for the standard errors of both first-order and higher-order sensitivity indices. Finally, based on the standard error approximation, we also proposed a method to determine a minimum sample size to achieve the desired estimation precision for a specified sensitivity index. The standard error estimation method presented in this paper can make the FAST analysis computationally much more efficient for complex models.

Finding the Simplest Mechanistic Kinetic Model Describing the Homogeneous Catalytic Hydrogenation of Avermectin to Ivermectin

Six mechanistic kinetic models of increasing complexity are analyzed to describe the RhCl(Ph3P)3 catalyzed hydrogenation process to produce ivermectin from avermectins B1a and B1b. Global sensitivity analysis (GSA) usefulness for selecting the simplest and the most suitable model is shown. First-order and total effect sensitivity indices for model parameters computed from GSA have been used for establishing those elementary reaction steps which were the most important in an extensive reaction framework. The prediction capability of the chosen model is corroborated by comparing its predictions with experimental data from both a lab-scale reactor and an industrial-scale reactor operating under isothermal and nonisothermal conditions, respectively. The best model is simple to use while resulting in a significant computational effort saving because there is no need to perform iterative algorithms for solving model equations. Another interesting feature is that ODEs for such a model have an analytical solution...