Generalized sensitivity indices based on vector projection for multivariate output

Analyzing the sensitivity of model outputs to inputs is important to assess risk and make decisions in engineering application. However, for model with multiple outputs, it is difficult to interpret the sensitivity index since the effect of the dimension and the correlation between multiple outputs are often ignored in the existing methods. In this paper, a new kind of sensitivity analysis method is proposed by use of vector projection and dimension normalization for multiple outputs. Through the dimension normalization, the space of multiple outputs can be unified into a dimensionless one to eliminate the effect of the dimension of the different output. After an affine coordinate system is constructed by considering the correlation of the multiple normalized outputs, a total variance vector for the multiple outputs can be composed by the individual variance of each output. Then, by projecting the variance contribution vector composed by the individual variance contribution of the input to each output on the total variance vector, the new sensitivity indices are proposed for measuring the comprehensive effect of the input on the total variance vector of multiple outputs, it is defined as the ratio of the projection of the variance contribution vector to the norm of the total variance vector. We derive that the Sobol’ indices for a scalar output and the covariance decomposition based indices for multiple outputs are special cases of the proposed vector projection based indices. Then, the mathematical properties and geometric interpretation of the proposed method are discussed. Three numerical examples and a rotating shaft model of an aircraft wing are used to validate the proposed method and show their potential benefits.     

Borgonovo moment independent global sensitivity analysis by Gaussian radial basis function meta-model

Moment independent sensitivity index is widely concerned and used since it can reflect the influence of model input uncertainty on the entire distribution of model output instead of a specific moment. In this paper, a novel analytical expression to estimate the Borgonovo moment independent sensitivity index is derived by use of the Gaussian radial basis function and the Edgeworth expansion. Firstly, the analytical expressions of the unconditional and conditional first four-order moments are established by the training points and the widths of the Gaussian radial basis function. Secondly, the Edgeworth expansion is used to express the unconditional and conditional probability density functions of model output by the unconditional and conditional first four-order moments, respectively. Finally, the index can be readily computed by measuring the shifts between the obtained unconditional and conditional probability density functions of model output, where this process doesn't need any extra calls of model evaluation. The computational cost of the proposed method is independent of the dimensionality of model inputs and it only depends on the training points and the widths which are involved in the Gaussian radial basis function meta-model. Results of several case studies demonstrate the effectiveness of the proposed method.     

A variant of radial measure capable of dealing with negative inputs and outputs in data envelopment analysis

Data envelopment analysis (DEA) is a linear programming methodology to evaluate the relative technical efficiency for each member of a set of peer decision making units (DMUs) with multiple inputs and multiple outputs. It has been widely used to measure performance in many areas. A weakness of the traditional DEA model is that it cannot deal with negative input or output values. There have been many studies exploring this issue, and various approaches have been proposed.     

Moving least squares based sensitivity analysis for models with dependent variables

For models with dependent input variables, sensitivity analysis is often a troublesome work and only a few methods are available. Mara and Tarantola in their paper (“Variance-based sensitivity indices for models with dependent inputs”) defined a set of variance-based sensitivity indices for models with dependent inputs. We in this paper propose a method based on moving least squares approximation to calculate these sensitivity indices. The new proposed method is adaptable to both linear and nonlinear models since the moving least squares approximation can capture severe change in scattered data. Both linear and nonlinear numerical examples are employed in this paper to demonstrate the ability of the proposed method. Then the new sensitivity analysis method is applied to a cantilever beam structure and from the results the most efficient method that can decrease the variance of model output can be determined, and the efficiency is demonstrated by exploring the dependence of output variance on the variation coefficients of input variables. At last, we apply the new method to a headless rivet model and the sensitivity indices of all inputs are calculated, and some significant conclusions are obtained from the results.     

Studying the sensitivity of pollutants’ concentrations caused by variations of chemical rates

A systematic procedure for sensitivity analysis of a case study in the area of air pollution modeling has been performed. Contemporary mathematical models should include a large set of chemical and photochemical reactions to be established as a reliable simulation tool. The Unified Danish Eulerian Model is in the focus of our investigation as one of the most advanced large-scale mathematical models that describes adequately all physical and chemical processes.Variance-based methods are one of the most often used approaches for providing sensitivity analysis. To measure the extent of influence of the variation of the chemical rate constants in the mathematical model over pollutants’ concentrations the Sobol’ global sensitivity indices are estimated using efficient techniques for small sensitivity indices to avoid a loss of accuracy. Studying relationships between input parameters and the model’s output as well as internal mechanisms is very useful for a verification and an improvement of the model and also for development of monitoring and control strategies of harmful emissions, for a reliable prediction of the final output of scenarios when the concentration levels of pollutants are exceeded. The proposed procedure can also be applied when other large-scale mathematical models are used.     

一种面向可靠性的改进不确定性重要度分析方法

作为识别和评估系统薄弱环节的有效工具之一的重要度理论,一直都被广泛应用于系统可靠性和安全性工程。而作为其主要环节的不确定性重要度分析更是扮演着不可或缺的角色。因此,为有效表征多态系统的可靠性,本文给出了一种变换数据处理方式的改进不确定性重要度分析方法。该模型是以两个累积分布函数之间的空间几何距离为基础,通过改变随机输入变量的不确定性范围和对应分布情况,模拟和描述输入对系统输出变量的相对影响趋势,并以两个累积分布函数之间的定积分面积表示。最后利用系统故障树对其进行不确定性重要度分析。结果表明,空间几何距离是一种用来表示输入变化对输出分布变化的相对影响的不确定性重要度度量的有效工具。     

Global sensitivity analysis for multivariate outputs using polynomial chaos-based surrogate models

We propose an efficient global sensitivity analysis method for multivariate outputs that applies polynomial chaos-based surrogate models to vector projection-based sensitivity indices. These projection-based sensitivity indices, which are powerful measures of the comprehensive effects of model inputs on multiple outputs, are conventionally estimated by the Monte Carlo simulations that incur prohibitive computational costs for many practical problems. Here, the projection-based sensitivity indices are efficiently estimated via two polynomial chaos-based surrogates: polynomial chaos expansion and a proper orthogonal decomposition-based polynomial chaos expansion. Several numerical examples with various types of outputs are tested to validate the proposed method; the results demonstrate that the polynomial chaos-based surrogates are more efficient than Monte Carlo simulations at estimating the sensitivity indices, even for models with a large number of outputs. Furthermore, for models with only a few outputs, polynomial chaos expansion alone is preferable, whereas for models with a large number of outputs, implementation with proper orthogonal decomposition is the best approach.     

Using artificial reaction force to design compliant mechanism with multiple equality displacement constraints

Monolithic compliant mechanisms are elastic workpieces which transmit force and displacement from an input position to an output position. Continuum topology optimization is suitable to generate the optimized topology, shape and size of such compliant mechanisms. The optimization strategy for a single input single output compliant mechanism under volume constraint is known to be best implemented using an optimality criteria or similar mathematical programming method. In this standard form, the method appears unsuitable for the design of compliant mechanisms which are subject to multiple outputs and multiple constraints. Therefore an optimization model that is subject to multiple design constraints is required. With regard to the design problem of compliant mechanisms subject to multiple equality displacement constraints and an area constraint, we here present a unified sensitivity analysis procedure based on artificial reaction forces, in which the key idea is built upon the Lagrange multiplier method. Because the resultant sensitivity expression obtained by this procedure already compromises the effects of all the equality displacement constraints, a simple optimization method, such as the optimality criteria method, can then be used to implement an area constraint. Mesh adaptation and anisotropic filtering method are used to obtain clearly defined monolithic compliant mechanisms without obvious hinges. Numerical examples in 2D and 3D based on linear small deformation analysis are presented to illustrate the success of the method.     

Classic Kriging <Emphasis Type="Italic">versus</Emphasis> Kriging with bootstrapping or conditional simulation: classic Kriging’s robust confidence intervals and optimization

Kriging is a popular method for estimating the global optimum of a simulated system. Kriging approximates the input/output function of the simulation model. Kriging also estimates the variances of the predictions of outputs for input combinations not yet simulated. These predictions and their variances are used by ‘efficient global optimization’ (EGO), to balance local and global search. This article focuses on two related questions: (1) How to select the next combination to be simulated when searching for the global optimum? (2) How to derive confidence intervals for outputs of input combinations not yet simulated? Classic Kriging simply plugs the estimated Kriging parameters into the formula for the predictor variance, so theoretically this variance is biased. This article concludes that practitioners may ignore this bias, because classic Kriging gives acceptable confidence intervals and estimates of the optimal input combination. This conclusion is based on bootstrapping and conditional simulation.     

A new proposed model of restricted data envelopment analysis by correlation coefficients

The concept of efficiency in data envelopment analysis (DEA) is defined as weighted sum of outputs/weighted sum of inputs. In order to calculate the maximum efficiency score, each decision making unit (DMU)’s inputs and outputs are assigned to different weights. Hence, the classical DEA allows the weight flexibility. Therefore, even if they are important, the inputs or outputs of some DMUs can be assigned zero (0) weights. Thus, these inputs or outputs are neglected in the evaluation. Also, some DMUs may be defined as efficient even if they are inefficient. This situation leads to unrealistic results. Also to eliminate the problem of weight flexibility, weight restrictions are made in DEA. In our study, we proposed a new model which has not been published in the literature. We describe it as the restricted data envelopment analysis ((ARIII(COR))) model with correlation coefficients. The aim for developing this new model, is to take into account the relations between variables using correlation coefficients. Also, these relations were added as constraints to the CCR and BCC models. For this purpose, the correlation coefficients were used in the restrictions of input–output each one alone and their combination together. Inputs and outputs are related to the degree of correlation between each other in the production. Previous studies did not take into account the relationship between inputs/outputs variables. So, only with expert opinions or an objective method, weight restrictions have been made. In our study, the weights for input and output variables were determined, according to the correlations between input and output variables. The proposed new method is different from other methods in the literature, because the efficiency scores were calculated at the level of correlations between the input and/or output variables.     

Parameter uncertainty, sensitivity analysis and prediction error in a water-balance hydrological model

Analysis of uncertainty is often neglected in the evaluation of complex systems models, such as computational models used in hydrology or ecology. Prediction uncertainty arises from a variety of sources, such as input error, calibration accuracy, parameter sensitivity and parameter uncertainty. In this study, various computational approaches were investigated for analysing the impact of parameter uncertainty on predictions of streamflow for a water-balance hydrological model used in eastern Australia. The parameters and associated equations which had greatest impact on model output were determined by combining differential error analysis and Monte Carlo simulation with stochastic and deterministic sensitivity analysis. This integrated approach aids in the identification of insignificant or redundant parameters and provides support for further simplifications in the mathematical structure underlying the model. Parameter uncertainty was represented by a probability distribution and simulation experiments revealed that the shape (skewness) of the distribution had a significant effect on model output uncertainty. More specifically, increasing negative skewness of the parameter distribution correlated with decreasing width of the model output confidence interval (i.e. resulting in less uncertainty). For skewed distributions, characterisation of uncertainty is more accurate using the confidence interval from the cumulative distribution rather than using variance. The analytic approach also identified the key parameters and the non-linear flux equation most influential in affecting model output uncertainty.     

Assessment and propagation of input uncertainty in tree‐based option pricing models

This paper aims to provide a practical example of assessment and propagation of input uncertainty for option pricing when using tree‐based methods. Input uncertainty is propagated into output uncertainty, reflecting that option prices are as unknown as the inputs they are based on. Option pricing formulas are tools whose validity is conditional not only on how close the model represents reality, but also on the quality of the inputs they use, and those inputs are usually not observable. We show three different approaches to integrating out the model nuisance parameters and show how this translates into model uncertainty in the tree model space for the theoretical option prices. We compare our method with classical calibration‐based results assuming that there is no options market established and no statistical model linking inputs and outputs. These methods can be applied to pricing of instruments for which there is no options market, as well as a methodological tool to account for parameter and model uncertainty in theoretical option pricing. Copyright © 2008 John Wiley & Sons, Ltd.     

Measuring environmental efficiency of thermoelectric power plants: a common equilibrium efficient frontier DEA approach with fixed-sum undesirable output

China’s rapid development in economy has intensified many problems. One of the most important issues is the problem of environmental pollution. In this paper, a new DEA approach is proposed to measure the environmental efficiency of thermoelectric power plants, considering undesirable outputs. First, we assume that the total amount of undesirable outputs of any particular type is limited and fixed to current levels. In contrast to previous studies, this study requires fixed-sum undesirable outputs. In addition, the common equilibrium efficient frontier is constructed by using different input/output multipliers (or weights) for each different decision making unit (DMU), while previous approaches which considered fixed-sum outputs assumed a common input/output multiplier for all DMUs. The proposed method is applied to measure the environmental efficiencies of 30 thermoelectric power plants in mainland China. Our empirical study shows that half of the plants perform well in terms of environmental efficiency.

     

考虑关键交通路段失效情景的中央储备库选址评估

我国中央储备库在应急管理中具有重要保障作用,针对现有中央储备库在应急资源保障过程中暴露出来的问题,研究了基于失效情景下的新增中央储备库选址合理性评估问题。首先进行情景分析,定义关键交通路段失效情景;其次基于失效情景设计了多级覆盖半径下的设施选址评估指标体系;最后,由于应急设施选址评估具有多影响因素特征,涉及输入和输出多个指标的测度,选取处理多输入多输出问题具有优势的评估方法—数据包络法。对新增中央储备库的不同建议选址方案,使用设计的失效情景评估指标体系进行选址合理性评估,验证了上述方法的有效性。     

Radial DEA models without inputs or without outputs

In this paper we consider radial DEA models without inputs (or without outputs), and radial DEA models with a single constant input (or with a single constant output). We demonstrate that (i) a CCR model without inputs (or without outputs) is meaningless; (ii) a CCR model with a single constant input (or with a single constant output) coincides with the corresponding BCC model; (iii) a BCC model with a single constant input (or a single constant output) collapses to a BCC model without inputs (or without outputs); and (iv) all BCC models, including those without inputs (or without outputs), can be condensed to models having one less variable (the radial efficiency score) and one less constraint (the convexity constraint).     

Neural network as a simulation metamodel in economic analysis of risky projects

An artificial neural network (ANN) model for economic analysis of risky projects is presented in this paper. Outputs of conventional simulation models are used as neural network training inputs. The neural network model is then used to predict the potential returns from an investment project having stochastic parameters. The nondeterministic aspects of the project include the initial investment, the magnitude of the rate of return, and the investment period. Backpropagation method is used in the neural network modeling. Sigmoid and hyperbolic tangent functions are used in the learning aspect of the system. Analysis of the outputs of the neural network model indicates that more predictive capability can be achieved by coupling conventional simulation with neural network approaches. The trained network was able to predict simulation output based on the input values with very good accuracy for conditions not in its training set. This allowed an analysis of the future performance of the investment project without having to run additional expensive and time-consuming simulation experiments.     

Estimating indirect allocative inefficiency and productivity change

Building on the method used in previous indirect production studies, we construct an indicator of indirect output allocative inefficiency. Our indicator equals the difference between a revenue-constrained directional input distance function and a directional input distance function that depends on outputs, rather than revenue. The indicator measures the overuse of inputs that occurs when firms do not choose a revenue maximizing mix of outputs. Adding a time dimension allows productivity change to be measured. In an empirical illustration of our method we find that Japanese banks use, between 2% and 7%, too many inputs because bank outputs are inefficiently allocated.     

Applications of sensitivity analysis for probit stochastic network equilibrium

Network equilibrium models are widely used by traffic practitioners to aid them in making decisions concerning the operation and management of traffic networks. The common practice is to test a prescribed range of hypothetical changes or policy measures through adjustments to the input data, namely the trip demands, the arc performance (travel time) functions, and policy variables such as tolls or signal timings. Relatively little use is made, however, of the full implicit relationship between model inputs and outputs inherent in these models. By exploiting the representation of such models as an equivalent optimisation problem, classical results on the sensitivity analysis of non-linear programs may be applied, to produce linear relationships between input data perturbations and model outputs. We specifically focus on recent results relating to the probit Stochastic User Equilibrium (PSUE) model, which has the advantage of greater behavioural realism and flexibility relative to the conventional Wardrop user equilibrium and logit SUE models. The paper goes on to explore four applications of these sensitivity expressions in gaining insight into the operation of road traffic networks. These applications are namely: identification of sensitive, ‘critical’ parameters; computation of approximate, re-equilibrated solutions following a change (post-optimisation); robustness analysis of model forecasts to input data errors, in the form of confidence interval estimation; and the solution of problems of the bi-level, optimal network design variety. Finally, numerical experiments applying these methods are reported.     

Interactive multiobjective optimization approach to the input–output design of opening new branches

We demonstrate a real-world application of the interactive multiple objective optimization (MOO) approach to the simultaneous setting of input and output amounts for the opening of new branches. As illustrated by the case example, all the branches of a fast-food company employ multiple inputs to generate multiple outputs. The company launches several new branches each year and, therefore, needs to plan the quantities of inputs and outputs to be used and produced before their operations. Such input–output settings are a vital practical problem that arises whenever a new branch is opened in a host of different industries. In this paper, we show in detail the entire process of the application from modeling the case problem to generating its solution. In the modeling stage, a data envelopment analysis model and a statistical method are subsequently utilized to form a nonlinear MOO problem for the input–output settings. To solve this problem, we then develop and apply an interactive MOO method, which combines the two earlier interactive methods ( and ), while compensating for their drawbacks and capturing their positive aspects.