Extending a global sensitivity analysis technique to models with correlated parameters

The identification and representation of uncertainty is recognized as an essential component in model applications. One important approach in the identification of uncertainty is sensitivity analysis. Sensitivity analysis evaluates how the variations in the model output can be apportioned to variations in model parameters. One of the most popular sensitivity analysis techniques is Fourier amplitude sensitivity test (FAST). The main mechanism of FAST is to assign each parameter with a distinct integer frequency (characteristic frequency) through a periodic sampling function. Then, for a specific parameter, the variance contribution can be singled out of the model output by the characteristic frequency based on a Fourier transformation. One limitation of FAST is that it can only be applied for models with independent parameters. However, in many cases, the parameters are correlated with one another. In this study, we propose to extend FAST to models with correlated parameters. The extension is based on the reordering of the independent sample in the traditional FAST. We apply the improved FAST to linear, nonlinear, nonmonotonic and real application models. The results show that the sensitivity indices derived by FAST are in a good agreement with those from the correlation ratio sensitivity method, which is a nonparametric method for models with correlated parameters.     

Non-parametric methods for global sensitivity analysis of model output with dependent inputs

This paper addresses the issue of performing global sensitivity analysis of model output with dependent inputs. First, we define variance-based sensitivity indices that allow for distinguishing the independent contributions of the inputs to the response variance from their mutual dependent contributions. Then, two sampling strategies are proposed for their non-parametric, numerical estimation. This approach allows us to estimate the sensitivity indices not only for individual inputs but also for groups of inputs. After testing the accuracy of the non-parametric method on some analytical test functions, the approach is employed to assess the importance of dependent inputs on a computer model for the migration of radioactive substances in the geosphere.     

Understanding macroeconomic models: Structural sensitivity analysis of a medium-sized model

To get a better impression of the quantitative relationships in/of the various channels of macroeconomic models, Kuh, Neese, and Hollinger introduced the technique of systematic parameter perturbation. This technique is applied to the RWI-business cycle model, a medium sized (41 stochastic equations, 86 definitions), quarterly macroeconometric model for the FRG. The evaluation of the results concentrates on (1) the sensitivity of the model to parameter perturbations in general, and to (2) the sensitivity of policy goal variables in particular. The findings show that in the model the number of important within-block and between-block relationships is much smaller than suggested by usual incidence matrices, providing additional evidence for Simon's empty world hypothesis.Paper presented at the Annual Meeting of the Society for Economic Dynamics and Control, Anacapri, Italy, 18–20 June, 1991.     

Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the SWAT model

Global Sensitivity Analysis (GSA) is an essential technique to support the calibration of environmental models by identifying the influential parameters (screening) and ranking them.In this paper, the widely-used variance-based method (Sobol') and the recently proposed moment-independent PAWN method for GSA are applied to the Soil and Water Assessment Tool (SWAT), and compared in terms of ranking and screening results of 26 SWAT parameters. In order to set a threshold for parameter screening, we propose the use of a “dummy parameter”, which has no influence on the model output. The sensitivity index of the dummy parameter is calculated from sampled data, without changing the model equations. We find that Sobol' and PAWN identify the same 12 influential parameters but rank them differently, and discuss how this result may be related to the limitations of the Sobol' method when the output distribution is asymmetric.     

Robust global sensitivity analysis under deep uncertainty via scenario analysis

Complex social-ecological systems models typically need to consider deeply uncertain long run future conditions. The influence of this deep (i.e. incalculable, uncontrollable) uncertainty on model parameter sensitivities needs to be understood and robustly quantified to reliably inform investment in data collection and model refinement. Using a variance-based global sensitivity analysis method (eFAST), we produced comprehensive model diagnostics of a complex social-ecological systems model under deep uncertainty characterised by four global change scenarios. The uncertainty of the outputs, and the influence of input parameters differed substantially between scenarios. We then developed sensitivity indicators that were robust to this deep uncertainty using four criteria from decision theory. The proposed methods can increase our understanding of the effects of deep uncertainty on output uncertainty and parameter sensitivity, and incorporate the decision maker's risk preference into modelling-related activities to obtain greater resilience of decisions to surprise.     

A simple and efficient method for global sensitivity analysis based on cumulative distribution functions

Variance-based approaches are widely used for Global Sensitivity Analysis (GSA) of environmental models. However, methods that consider the entire Probability Density Function (PDF) of the model output, rather than its variance only, are preferable in cases where variance is not an adequate proxy of uncertainty, e.g. when the output distribution is highly-skewed or when it is multi-modal. Still, the adoption of density-based methods has been limited so far, possibly because they are relatively more difficult to implement. Here we present a novel GSA method, called PAWN, to efficiently compute density-based sensitivity indices. The key idea is to characterise output distributions by their Cumulative Distribution Functions (CDF), which are easier to derive than PDFs. We discuss and demonstrate the advantages of PAWN through applications to numerical and environmental modelling examples. We expect PAWN to increase the application of density-based approaches and to be a complementary approach to variance-based GSA.     

A General Probabilistic Framework for uncertainty and global sensitivity analysis of deterministic models: A hydrological case study

The present study proposes a General Probabilistic Framework (GPF) for uncertainty and global sensitivity analysis of deterministic models in which, in addition to scalar inputs, non-scalar and correlated inputs can be considered as well. The analysis is conducted with the variance-based approach of Sobol/Saltelli where first and total sensitivity indices are estimated. The results of the framework can be used in a loop for model improvement, parameter estimation or model simplification. The framework is applied to SWAP, a 1D hydrological model for the transport of water, solutes and heat in unsaturated and saturated soils. The sources of uncertainty are grouped in five main classes: model structure (soil discretization), input (weather data), time-varying (crop) parameters, scalar parameters (soil properties) and observations (measured soil moisture). For each source of uncertainty, different realizations are created based on direct monitoring activities. Uncertainty of evapotranspiration, soil moisture in the root zone and bottom fluxes below the root zone are considered in the analysis. The results show that the sources of uncertainty are different for each output considered and it is necessary to consider multiple output variables for a proper assessment of the model. Improvements on the performance of the model can be achieved reducing the uncertainty in the observations, in the soil parameters and in the weather data. Overall, the study shows the capability of the GPF to quantify the relative contribution of the different sources of uncertainty and to identify the priorities required to improve the performance of the model. The proposed framework can be extended to a wide variety of modelling applications, also when direct measurements of model output are not available.     

Global sensitivity analysis of a feedback-controlled stochastic process model

The purpose of the study was to investigate whether global sensitivity analysis can be utilised in concurrent process and control design to gain insight into the process and its control. This paper addresses the issue of sensitivity of a control law performance to its parameters in a dynamic, hybrid deterministic–stochastic process model. The control law under investigation is a collection of single-input, single-output type tower level controllers in a papermaking process. Global sensitivity analysis is shown to attribute higher importance to certain key parameters, thus providing valuable insight for the designer.     

A new derivative based importance criterion for groups of variables and its link with the global sensitivity indices

A new derivative based criterion τy for groups of input variables is presented. It is shown that there is a link between global sensitivity indices and the new derivative based measure. It is proved that small values of derivative based measures imply small values of total sensitivity indices. However, for highly nonlinear functions the ranking of important variables using derivative based importance measures can be different from that based on the global sensitivity indices. The computational costs of evaluating global sensitivity indices and derivative based measures, are compared and some important tests are considered.     

Identifying parametric controls and dependencies in integrated assessment models using global sensitivity analysis

Integrated assessment models for climate change (IAMs) couple representations of economic and natural systems to identify and evaluate strategies for managing the effects of global climate change. In this study we subject three policy scenarios from the globally-aggregated Dynamic Integrated model of Climate and the Economy IAM to a comprehensive global sensitivity analysis using Sobol' variance decomposition. We focus on cost metrics representing diversions of economic resources from global world production. Our study illustrates how the sensitivity ranking of model parameters differs for alternative cost metrics, over time, and for different emission control strategies. This study contributes a comprehensive illustration of the negative consequences associated with using a priori expert elicitations to reduce the set of parameters analyzed in IAM uncertainty analysis. The results also provide a strong argument for conducting comprehensive model diagnostics for IAMs that explicitly account for the parameter interactions between the coupled natural and economic system components.     

A new approach to visualizing time-varying sensitivity indices for environmental model diagnostics across evaluation time-scales

Assessing the time-varying sensitivity of environmental models has become a common approach to understand both the value of different data periods for estimating specific parameters, and as part of a diagnostic analysis of the model structure itself (i.e. whether dominant processes are emerging in the model at the right times and over the appropriate time periods). It is not straightforward to visualize these results though, given that the window size over which the time-varying sensitivity is best integrated generally varies for different parameters. In this short communication we present a new approach to visualizing such time-varying sensitivity across time scales of integration. As a case study, we estimate first order sensitivity indices with the FAST (Fourier Amplitude Sensitivity Test) method for a typical conceptual rainfall–runoff model. The resulting plots can guide data selection for model calibration, support diagnostic model evaluation and help to define the timing and length of spot gauging campaigns in places where long-term calibration data are not yet available.     

基于参数灵敏度分析的交通仿真模型校正研究

为了使交通仿真模型校正工作能够高效开展,提出了以参数灵敏度分析为基础的模型校正框架。通过灵敏度分析确定影响模型精度的关键参数,以简化模型;对关键参数进行标定,以校正模型。以城市快速路交织区为仿真案例,以跟车模型和换道模型为研究对象;首先进行了大量仿真实验,分析不同车流量水平下模型参数的取值特征;据此制定模型参数的区间划分规则和交叉组合规则,从而对LH-OAT算法和遗传算法（GA）进行改进;然后应用改进LH-OAT算法（ILH-OAT）对模型参数进行灵敏度分析,再应用GA对关键参数进行标定;最后依据校验指标对仿真结果进行误差分析。结果表明ILH-OAT和GA相结合,不仅简化了仿真模型,降低了仿真运行成本,仿真效果也更加接近真实的道路交通运行情况。     

Importance analysis for models with correlated input variables by the state dependent parameters method

For clearly exploring the origin of the variance of the output response in case the correlated input variables are involved, a novel method on the state dependent parameters (SDP) approach is proposed to decompose the contribution by correlated input variables to the variance of output response into two parts: the uncorrelated contribution due to the unique variations of a variable and the correlated one due to the variations of a variable correlated with other variables. The correlated contribution is composed by the components of the individual input variable correlated with each of the other input variables. An effective and simple SDP method in concept is further proposed to decompose the correlated contribution into the components, on which a second order importance matrix can be solved for explicitly exposing the contribution components of the correlated input variable to the variance of the output response. Compared with the existing regression-based method for decomposing the contribution by correlated input variables to the variance of the output response, the proposed method is not only applicable for linear response functions, but is also suitable for nonlinear response functions. It has advantages both in efficiency and accuracy, which are demonstrated by several numerical and engineering examples.     

Global sensitivity analysis for large-scale socio-hydrological models using Hadoop

A multi-agent system (MAS) model is coupled with a physically-based groundwater model to understand the declining water table in the heavily irrigated Republican River basin. Each agent in the MAS model is associated with five behavioral parameters, and we estimate their influences on the coupled models using Global Sensitivity Analysis (GSA). This paper utilizes Hadoop-based Cloud Computing techniques and Polynomial Chaos Expansion (PCE) based variance decomposition approach for the improvement of GSA with large-scale socio-hydrological models. With the techniques, running 1000 scenarios of the coupled models can be completed within two hours with Hadoop clusters, a substantial improvement over the 42 days required to run these scenarios sequentially on a desktop machine. Based on the model results, GSA is conducted with the surrogate model derived from using PCE to measure the impacts of the spatio-temporal variations of the behavioral parameters on crop profits and the water table, identifying influential parameters.     

不确定性参数灵敏度分析的随机响应面法

动力学和控制系统中往往包含有不确定性参数,为此提出了一种基于随机响应面的不确定性参数灵敏度分析方法,以量化参数不确定性对响应变异性的影响.文中首先利用随机响应面建立不确定性参数和响应之间的表达式,然后通过求偏导方式推导参数的灵敏度系数,该系数综合反映了参数均值和标准差的影响.最后通过一根包含几何、材料不确定参数的数值梁来验证所提出方法,并与方差分析法结果进行了比较.     

Multiparametric sensitivity analysis of the additive model in data envelopment analysis

In this paper, we study multiparametric sensitivity analysis of the additive model in data envelopment analysis using the concept of maximum volume in the tolerance region. We construct critical regions for simultaneous and independent perturbations in all inputs/outputs of an efficient decision making unit. Necessary and sufficient conditions are derived to classify the perturbation parameters as “focal” and “nonfocal.” Nonfocal parameters can have unlimited variations because of their low sensitivity in practice and these parameters can be deleted from the final analysis. For focal parameters a maximum volume region is characterized. Theoretical results are illustrated with the help of a numerical example.     

Distance correlation-based method for global sensitivity analysis of models with dependent inputs

Structural and Multidisciplinary Optimization - Global sensitivity analysis (GSA) plays an important role to quantify the relative importance of uncertain parameters to the model response. However,...     

Sensitivity analysis of the additive model in data envelopment analysis while inputs and outputs are fuzzy data

Data envelopment analysis (DEA) requires input and output data to be precisely known. This is not always the case in real applications. Sensitivity analysis of the additive model in DEA is studied in this paper while inputs and outputs are symmetric triangular fuzzy numbers. Sufficient conditions for simultaneous change of all outputs and inputs of an efficient decision-making unit (DMU) which preserves efficiency are established. Two kinds of changes on inputs and outputs are considered. For the first state, changes are exerted on the core and margin of symmetric triangular fuzzy numbers so that the value of inputs increase and the value of outputs decrease. In the second state, a non-negative symmetric triangular fuzzy number is subtracted from outputs to decrease outputs and it is added to inputs to increase inputs. A numerical illustration is provided.     

Identification of tire force characteristics using a Hybrid method

The present study applies a Hybrid method for identification of unknown parameters in a semi-empirical tire model, the so-called Magic Formula. Firstly, the Hybrid method used a Genetic Algorithm (GA) as a global search methodology with high exploration power. Secondly, the results of the Genetic Algorithm were used as starting values for the Levenberg–Marquardt (LM) algorithm as a gradient-based method with high exploitation power. In this way the beneficial aspects of both methods are simultaneously exploited and their shortcomings are avoided. In order to establish the effectiveness of the proposed method, performance of the Hybrid method has been compared with other methods available in the literature. In addition, the use of GA as a Heuristic method for tire parameters identification has been discussed. Moreover, the extrapolation power of Magic Formula identified with Hybrid method has been properly investigated. Finally, the performance of the Hybrid method has been examined through tire parameter identification with priori known model. The results indicated that the Hybrid method has outstanding benefits such as high convergence speed, high accuracy, and null-sensitivity to the starting values of unknown parameters.