Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests

The estimation of variance-based importance measures (called Sobol’ indices) of the input variables of a numerical model can require a large number of model evaluations. It turns to be unacceptable for high-dimensional model involving a large number of input variables (typically more than ten). Recently, Sobol and Kucherenko have proposed the derivative-based global sensitivity measures (DGSM), defined as the integral of the squared derivatives of the model output, showing that it can help to solve the problem of dimensionality in some cases. We provide a general inequality link between DGSM and total Sobol’ indices for input variables belonging to the class of Boltzmann probability measures, thus extending the previous results of Sobol and Kucherenko for uniform and normal measures. The special case of log-concave measures is also described. This link provides a DGSM-based maximal bound for the total Sobol indices. Numerical tests show the performance of the bound and its usefulness in practice.     

Efficient design of experiments for sensitivity analysis based on polynomial chaos expansions

Global sensitivity analysis aims at quantifying respective effects of input random variables (or combinations thereof) onto variance of a physical or mathematical model response. Among the abundant literature on sensitivity measures, Sobol indices have received much attention since they provide accurate information for most of models. We consider a problem of experimental design points selection for Sobol’ indices estimation. Based on the concept of D-optimality, we propose a method for constructing an adaptive design of experiments, effective for calculation of Sobol’ indices based on Polynomial Chaos Expansions. We provide a set of applications that demonstrate the efficiency of the proposed approach.     

Addressing factors fixing setting from given data: A comparison of different methods

This paper deals with global sensitivity analysis of computer model output. Given an independent input sample and associated model output vector with possibly the vector of output derivatives with respect to the input variables, we show that it is possible to evaluate the following global sensitivity measures: (i) the Sobol’ indices, (ii) the Borgonovo's density-based sensitivity measure, and (iii) the derivative-based global sensitivity measure of Sobol’ and Kucherenko. We compare the efficiency of the different methods to address factors fixing setting, an important issue in global sensitivity analysis. First, global sensitivity analysis of the Ishigami function is performed with the different methods. Then, they are applied to two different responses of a soil drainage model. The results show that the polynomial chaos expansion for estimating Sobol’ indices is the most efficient approach.     

Estimation of global sensitivity indices for models with dependent variables

A novel approach for estimation variance-based sensitivity indices for models with dependent variables is presented. Both the first order and total sensitivity indices are derived as generalizations of Sobol? sensitivity indices. Formulas and Monte Carlo numerical estimates similar to Sobol? formulas are derived. A copula-based approach is proposed for sampling from arbitrary multivariate probability distributions. A good agreement between analytical and numerical values of the first order and total indices for considered test cases is obtained. The behavior of sensitivity indices depends on the relative predominance of interactions and correlations. The method is shown to be efficient and general.     

基于复杂模拟的战斗机系统失效敏感性分析

战斗机的系统试验往往成本较高，随着战斗机系统的日益复杂化，基于计算机模拟的表征系统失效信息的功能响应量计算也越来越耗时费力。针对此类基于复杂计算机模拟的敏感性分析问题，提出了一种结合Sobol方法和基于主动学习的Kriging模型的敏感性分析方法，称之为AK-S（Adaptive-Kriging-based Sobol）方法，AK-S方法通过Kriging预测来代替真实响应值计算，因而可以更加高效地计算各输入变量的敏感性指标并得到重要度排序。通过与直接蒙特卡洛法（直接法，MC）和传统Sobol法对数值算例的处理结果进行对比，AK-S方法的计算效率和精度得到了证明。最后，AK-S方法被应用于基于复杂模拟的实际工程案例的失效敏感性分析，并获得了敏感性指标。AK-S算法被证明在同等计算精度的条件下，其效率大大高于MC和传统Sobol法，能很好地解决工程中基于复杂计算机模拟下的失效敏感性分析问题。     

Influence of input PDF parameters of a model on a failure probability estimation

Critical probability estimation is of major interest in safety and reliability applications. In this article, we focus on a black-box model with multidimensional random input X and one random output Y. We consider the estimation of probability P that Y exceeds a threshold S. We assume that the random input X follows a multidimensional parametric density with parameters δ and thus the probability P will depend on the values of δ. In this paper, we analyze the sensitivity of the critical probability P to the model parameters δ. We propose a methodology that estimates Sobol indices with low computation cost. This strategy enables us to determine which statistical parameters have a great influence on the value of the probability and require a valuable determination. The last part of this article applies the proposed technique on a realistic case of missile collateral damage estimation.     

Influence of input PDF parameters of a model on a failure probability estimation

Critical probability estimation is of major interest in safety and reliability applications. In this article, we focus on a black-box model with multidimensional random input X and one random output Y. We consider the estimation of probability P that Y exceeds a threshold S. We assume that the random input X follows a multidimensional parametric density with parameters δ and thus the probability P will depend on the values of δ. In this paper, we analyze the sensitivity of the critical probability P to the model parameters δ. We propose a methodology that estimates Sobol indices with low computation cost. This strategy enables us to determine which statistical parameters have a great influence on the value of the probability and require a valuable determination. The last part of this article applies the proposed technique on a realistic case of missile collateral damage estimation.     

Orthogonal Array Sampling for Monte Carlo Rendering

We generalize N‐rooks, jittered, and (correlated) multi‐jittered sampling to higher dimensions by importing and improving upon a class of techniques called orthogonal arrays from the statistics literature. Renderers typically combine or “pad” a collection of lower‐dimensional (e.g. 2D and 1D) stratified patterns to form higher‐dimensional samples for integration. This maintains stratification in the original dimension pairs, but looses it for all other dimension pairs. For truly multi‐dimensional integrands like those in rendering, this increases variance and deteriorates its rate of convergence to that of pure random sampling. Care must therefore be taken to assign the primary dimension pairs to the dimensions with most integrand variation, but this complicates implementations. We tackle this problem by developing a collection of practical, in‐place multi‐dimensional sample generation routines that stratify points on all t‐dimensional and 1‐dimensional projections simultaneously. For instance, when t=2, any 2D projection of our samples is a (correlated) multi‐jittered point set. This property not only reduces variance, but also simplifies implementations since sample dimensions can now be assigned to integrand dimensions arbitrarily while maintaining the same level of stratification. Our techniques reduce variance compared to traditional 2D padding approaches like PBRT's (0,2) and Stratified samplers, and provide quality nearly equal to state‐of‐the‐art QMC samplers like Sobol and Halton while avoiding their structured artifacts as commonly seen when using a single sample set to cover an entire image. While in this work we focus on constructing finite sampling point sets, we also discuss potential avenues for extending our work to progressive sequences (more suitable for incremental rendering) in the future.     

Global Sensitivity Analysis of environmental models: Convergence and validation

We address two critical choices in Global Sensitivity Analysis (GSA): the choice of the sample size and of the threshold for the identification of insensitive input factors. Guidance to assist users with those two choices is still insufficient. We aim at filling this gap. Firstly, we define criteria to quantify the convergence of sensitivity indices, of ranking and of screening, based on a bootstrap approach. Secondly, we investigate the screening threshold with a quantitative validation procedure for screening results. We apply the proposed methodologies to three hydrological models with varying complexity utilizing three widely-used GSA methods (RSA, Morris, Sobol’). We demonstrate that convergence of screening and ranking can be reached before sensitivity estimates stabilize. Convergence dynamics appear to be case-dependent, which suggests that “fit-for-all” rules for sample sizes should not be used. Other modellers can easily adopt our criteria and procedures for a wide range of GSA methods and cases.     

Generating modified Sobol sequences for multiple run march memory tests

The generation of modified Sobol sequences for multiple run march memory tests is considered. The representation of direction numbers of a Sobol sequence in the form of a lower triangular matrix with a unit diagonal is formalized. Algorithms for generating modified Sobol sequences by formalizing the formation of a direction number matrix are proposed. The use of the Hamming distance as a metric for choosing Sobol sequences is substantiated.     

A comparison of two sampling methods for global sensitivity analysis

We compare the convergence properties of two different quasi-random sampling designs – Sobol?s quasi-Monte Carlo, and Latin supercube sampling in variance-based global sensitivity analysis. We use the non-monotonic V-function of Sobol? as base case-study, and compare the performance of both sampling strategies at increasing sample size and dimensionality against analytical values. The results indicate that in almost all cases investigated here, the Sobol? design performs better. This, coupled with the fact that effective Latin supercube sampling requires a priori knowledge of the interaction properties of the function, leads us to recommend Sobol? sampling in most practical cases.     

基于分数阶滑模控制技术的永磁同步电机控制

针对传统整数阶滑模控制系统中存在的抖震问题,本文提出了分数阶滑模控制策略并应用到永磁同步电机的速度控制.传统滑模控制器中的开关函数由作用在切换流型或其整数阶导数面推广到其分数阶导数面,利用分数阶系统的特性,缓慢地传递系统的能量,有效地削减抖震.本文采用模糊逻辑推理算法,实现软开关切换增益的自整定.仿真和实验证明,本文提出的分数阶滑模控制系统不但能有效地削减抖震,而且能保持滑模控制器对系统参数变化和外部扰动的鲁棒性.     

Using support vector machines for the computationally efficient identification of acceptable design parameters in computer-aided engineering applications

This paper addresses the problem of estimating continuous boundaries between acceptable and unacceptable engineering design parameters in complex engineering applications. In particular, a procedure is proposed to reduce the computational cost of finding and representing the boundary. The proposed methodology combines a low-discrepancy sequence (Sobol) and a support vector machine (SVM) in an active learning procedure able to efficiently and accurately estimate the boundary surface. The paper describes the approach and methodological choices resulting in the desired level of boundary surface refinement and the new algorithm is applied to both two highly-nonlinear test functions and a real-world train stability design problem. It is expected that the new method will provide designers with a tool for the evaluation of the acceptability of designs, particularly for engineering systems whose behaviour can only be determined through complex simulations.     

Mechanisms for activity spread in a neural field model

Field models of continuous neural networks incorporate nonlocal connectivities as well as finite axonal propagation velocities and lead therefore to delayed integral equations. For special choices of the synaptic footprint it is possible to reduce the integral model to a system of partial differential equations. One example is that of the inhomogeneous damped wave equation in one space dimension derived by Jirsa and Haken for exponential synaptic footprint. We show that this equation can be put into the form of a conservation law with nonlinear source, and explore numerically this representation. We find two mechanisms for the spread of the activity from an initially excited region.     

R-K-F方法在外弹道计算与火控系统中的应用

分析了用R-K-F方法计算微分方程组的计算步骤及计算中改变积分步长的方法,给出了用R-K-F方法计算了3D、4D、简化6D及6D外弹道微分方程组的结果,它们包括积分步长、误差容限、射程、侧偏及计算时间,并与定步长四阶R-K方法进行了对比分析.给出了不同误差容限下计算6D外弹道微分方程组的求解时间,给出了积分过程中步长变化的曲线,得到的结论为R-K-F方法不但能控制外弹道计算过程中的误差,而且显著地提高了简化6D、6D外弹道微分方程组计算速度,简化6D外弹道模型的计算时间达到了0.2s,可应用于火控系统中.     

An analysis of parameter sensitivities of preference-inspired co-evolutionary algorithms

Many-objective optimisation problems remain challenging for many state-of-the-art multi-objective evolutionary algorithms. Preference-inspired co-evolutionary algorithms (PICEAs) which co-evolve the usual population of candidate solutions with a family of decision-maker preferences during the search have been demonstrated to be effective on such problems. However, it is unknown whether PICEAs are robust with respect to the parameter settings. This study aims to address this question. First, a global sensitivity analysis method – the Sobol’ variance decomposition method – is employed to determine the relative importance of the parameters controlling the performance of PICEAs. Experimental results show that the performance of PICEAs is controlled for the most part by the number of function evaluations. Next, we investigate the effect of key parameters identified from the Sobol’ test and the genetic operators employed in PICEAs. Experimental results show improved performance of the PICEAs as more preferences are co-evolved. Additionally, some suggestions for genetic operator settings are provided for non-expert users.     

An efficient document classification model using an improved back propagation neural network and singular value decomposition

This paper proposed a new improved method for back propagation neural network, and used an efficient method to reduce the dimension and improve the performance. The traditional back propagation neural network (BPNN) has the drawbacks of slow learning and is easy to trap into a local minimum, and it will lead to a poor performance and efficiency. In this paper, we propose the learning phase evaluation back propagation neural network (LPEBP) to improve the traditional BPNN. We adopt a singular value decomposition (SVD) technique to reduce the dimension and construct the latent semantics between terms. Experimental results show that the LPEBP is much faster than the traditional BPNN. It also enhances the performance of the traditional BPNN. The SVD technique cannot only greatly reduce the high dimensionality but also enhance the performance. So SVD is to further improve the document classification systems precisely and efficiently.     

Efficient sampling methods for global reliability sensitivity analysis

An important problem in structure reliability analysis is how to reduce the failure probability. In this work, we introduce a main and total effect indices framework of global reliability sensitivity. By decreasing the uncertainty of input variables with high main effect indices, the most reduction of failure probability can be obtained. By decreasing the uncertainty of the input variables with small total effect indices (close to zero), the failure probability will not be reduced significantly. The efficient sampling methods for evaluating the main and total effect indices are presented. For the problem with large failure probability, a single-loop Monte Carlo simulation (MCS) is derived for computing these sensitivity indices. For the problem with small failure probability, the single-loop sampling methods combined with the importance sampling procedure (IS) and the truncated importance sampling procedure (TIS) respectively are derived for improving the calculation efficiency. Two numerical examples and one engineering example are introduced for demonstrating the efficiency and precision of the calculation methods and illustrating the engineering significance of the global reliability sensitivity indices.     

A modified Sobol′ sensitivity analysis method for decision-making in environmental problems

Sensitivity analysis is an important component of environmental modelling and in recent years, variance-based, global sensitivity analysis techniques, such as Sobol′, have been a preferred approach for achieving this. However, these techniques are generally only applicable to simulation models and not to models used to rank alternative options, such as multi-criteria decision analysis (MCDA) methods. In order to overcome this limitation, a modified Sobol′ method for MCDA (Sobol′-MCDA) is introduced in this paper. The method has the following features: (i) it enables the stability or robustness of the relative ranking of two alternatives to be assessed in the light of changes in assessment criteria and stakeholder preferences; and (ii) it enables the sensitivity of the ranking of two alternatives to changes in assessment criteria and stakeholder preferences to be assessed. The approach is demonstrated for a water resources case study from the literature consisting of seven alternatives and ten assessment criteria.     

基于代理模型的XAI可解释性量化评估方法

可解释人工智能(explainable artificial intelligence, XAI)近年来发展迅速,已出现多种人工智能模型的解释技术,但是目前缺乏XAI可解释性的定量评估方法.已有评估方法大多需借助用户实验进行评估,这种方法耗时长且成本高昂.针对基于代理模型的XAI,提出一种可解释性量化评估方法.首先,针对这类XAI设计一些指标并给出计算方法,构建包含10个指标的评估指标体系,从一致性、用户理解性、因果性、有效性、稳定性5个维度来评估XAI的可解释性;然后,对于包含多个指标的维度,将熵权法与TOPSIS相结合,建立综合评估模型来评估该维度上的可解释性;最后,将该评估方法用于评估6个基于规则代理模型的XAI的可解释性.实验结果表明,所提出方法能够展现XAI在不同维度上的可解释性水平,用户可根据需求选取合适的XAI.