Modified screening-based Kriging method with cross validation and application to engineering design

In this paper, a basis screening Kriging method using cross validation error is proposed to alleviate computational burden of the dynamic Kriging while maintaining its accuracy. Metamodeling is widely used for design optimization of complex engineering applications where considerable computation time is required. The Kriging method is one of popular metamodeling methods due to its accuracy and efficiency. There have been many attempts to improve accuracy of the Kriging method, and the dynamic Kriging method using cross-validation error, which selects adequate basis functions to best describe the mean structure of a response using a genetic algorithm, achieves outstanding performance in terms of accuracy. However, despite its accuracy, the dynamic Kriging requires very large amounts of computation because of the genetic algorithm and no limitation for order of basis functions. In the proposed method, a basis function set is determined by screening each basis function instead of using the genetic algorithm, which has advantages in computation for high dimensional metamodels or repeated metamodel generation. Numerical studies with four mathematical examples and two engineering applications verify that the proposed basis screening Kriging significantly reduces computation time with similar accuracy as the dynamic Kriging.     

基于粒子群优化Kriging模型的边坡可靠度分析

采用传统极限平衡法进行边坡可靠度分析时,不可避免会遇到一个问题,即边坡功能函数形式的高度非线性以及隐含性.对于隐式功能函数,传统的求解方法是通过对功能函数进行多次迭代,从而得到安全系数值.但是由于功能函数的形式较为复杂,导致迭代计算的过程变得尤为繁琐且效率低下.鉴于传统边坡可靠度分析中存在的安全系数计算繁琐耗时的问题,...     

Kriging surrogate model with coordinate transformation based on likelihood and gradient

The Kriging surrogate model, which is frequently employed to apply evolutionary computation to real-world problems, with a coordinate transformation of the design space is proposed to improve the approximation accuracy of objective functions with correlated design variables. The coordinate transformation is conducted to extract significant trends in the objective function and identify the suitable coordinate system based on either one of two criteria: likelihood function or estimated gradients of the objective function to each design variable. Compared with the ordinary Kriging model, the proposed methods show higher accuracy in the approximation of various test functions. The proposed method based on likelihood shows higher accuracy than that based on gradients when the number of design variables is less than six. The latter method achieves higher accuracy than the ordinary Kriging model even for high-dimensional functions and is applied to an airfoil design problem with spline curves as an example with correlated design variables. This method achieves better performances not only in the approximation accuracy but also in the capability to explore the optimal solution.     

A novel single-loop procedure for time-variant reliability analysis based on Kriging model

This paper proposes a novel single-loop procedure for time-variant reliability analysis based on a Kriging model. A new strategy is presented to decouple the double-loop Kriging model for time-variant reliability analysis, in which the extreme value response in double-loop procedure is replaced by the best value in the current sampled points to avoid the inner optimization loop. Consequently, the extreme value response surface for time-variant reliability analysis can be directly established through a single-loop Kriging surrogate model. To further improve the accuracy of the proposed Kriging model, two methods are provided to adaptively choose a new sample point for updating the model. One method is to apply two commonly used learning functions to select the new sample point that resides as close to the extreme value response surface as possible, and the other is to apply a new learning function to select the new point. Synchronously, the corresponding different stopping criteria are also provided. It is worth nothing that the proposed single-loop Kriging model for time-variant reliability analysis is for a single time-variant performance function. To verify the proposed method, it is applied to four examples, two of which have with random process and others have not. Other popular methods for time-variant reliability analysis including the existing single-loop Kriging model are also used for the comparative analysis and their results testify the effectiveness of the proposed method.     

Diseño óptimo robusto utilizando modelos Kriging: aplicación al diseño óptimo robusto de estructuras articuladas

Conventional methods addressing the robust design optimization problem of structures usually require high computational requirements due to the nesting of uncertainty quantification within the optimization process. In order to address such a problem, this work proposes a methodology, based on Kriging models, to efficiently assess the uncertainty quantification in the optimization process. The Kriging model approximates the structural performance both in the design domain and in the stochastic domain, which allows to decouple the uncertainty quantification process and the optimization process. In addition, an infill criterion based on the variance of the Kriging prediction is included to update the Kriging model towards the global Pareto front. Three numerical examples show the applicability and the accuracy of the proposed methodology. The results show that the proposed method is appropriate to solve the robust design optimization problem with reasonable accuracy and a considerably lower number of function calls than required by conventional methods.     

A note on estimating the bent line quantile regression model

This paper considers a new estimating method for the bent line quantile regression model. By a simple linearization technique, the proposed method can simultaneously obtain the estimates of the regression coefficients and the change-point location. Moreover, it can be readily implemented by current software. Simulation studies demonstrate that the proposed method has good finite sample performance. Two empirical applications are also presented to illustrate the method.     

Non-Probabilistic Structural Reliability Analysis Integrating the PSO and the Kriging Model北大核心CSCD

针对复杂结构可靠性分析中面临的隐式功能函数和小样本问题,提出了一种粒子群优化和Kriging模型相结合的结构非概率可靠性分析方法。采用多维椭球描述结构不确定参数,运用粒子群优化对模型相关参数进行求解,并构建隐式功能函数的Kriging模型进行可靠性分析。三个算例结果表明所提方法有效可行,精度和效率均优于基于Kriging模型的非概率可靠性分析方法。     

Preliminary test and Stein estimations in simultaneous linear equations

In modeling of an economic system, there may occur some stochastic constraints, that can cause some changes in the estimators and their respective behaviors. In this approach we formulate the simultaneous equation models into the problem of estimating the regression parameters of a multiple regression model, under elliptical errors. We define five different sorts of estimators for the vector-parameter. Their exact risk expressions are also derived under the balanced loss function. Comparisons are then made to clarify the performance of the proposed estimators. It is shown that the shrinkage factor of the Stein estimator is robust with respect to departures from normality assumption.     

Support vector quantile regression with varying coefficients

Quantile regression has received a great deal of attention as an important tool for modeling statistical quantities of interest other than the conditional mean. Varying coefficient models are widely used to explore dynamic patterns among popular models available to avoid the curse of dimensionality. We propose a support vector quantile regression model with varying coefficients and its two estimation methods. One uses the quadratic programming, and the other uses the iteratively reweighted least squares procedure. The proposed method can be applied easily and effectively to estimating the nonlinear regression quantiles depending on the high-dimensional vector of smoothing variables. We also present the model selection method that employs generalized cross validation and generalized approximate cross validation techniques for choosing the hyperparameters, which affect the performance of the proposed model. Numerical studies are conducted to illustrate the performance of the proposed model.     

纵向数据下半参数回归模型的统计分析

对于纵向数据下半参数回归模型,基于广义估计方程和一般权函数方法构造了模型中参数分量和非参数分量的估计.在适当的条件下证明了参数估计量具有渐近正态性,并得到了非参数回归函数估计量的最优收敛速度.通过模拟研究说明了所提出的估计量在有限样本下的精确性.     

基于Kriging模型和提升小波变换的随机模型修正北大核心CSCD

为提高随机模型修正效率,减小计算量,提出了一种基于Kriging模型和提升小波变换的随机模型修正方法.首先,对加速度频响函数进行提升小波变换,提取第5层近似系数代替原频响函数.其次,采用拉丁超立方抽样抽取待修正样本,将其作为Kriging模型的输入,对应的近似系数作为输出,构建Kriging模型.提出了一种引入莱维飞行(Lévy flight)的蝴蝶优化算法(LBOA),并将其应用于Kriging模型相关参数的寻优中,提高Kriging模型的精度.最后,以最小化Wasserstein距离为目标,通过鲸鱼优化算法求解待修正参数的均值.测试函数结果表明,LBOA在寻优能力、收敛精度和稳定性等方面有了很大的提升.数值算例的修正误差均低于0.4%,验证了所提模型修正方法具有较高的修正精度和效率.     

基于偏最小二乘变换与Kriging模型的全局优化方法

针对传统Kriging模型在多变量(高维)输入全局优化中因超参数过多而引发收敛速度慢,精度低,建模效率不高问题,提出了基于偏最小二乘变换技术和Kriging模型的有效全局优化方法.首先,构造偏最小二乘高斯核函数;其次,借助差分进化算法寻找满足期望改进准则最大化条件的新样本点;然后,将不同核函数和期望改进准则组合,构建四种有效全局优化算法并进行比较;最后,数值算例结果表明,基于偏最小二乘变换的Kriging全局优化方法在解决高维全局优化问题方面相比于标准的全局优化算法在收敛精度及收敛速度方面更具优势.     

Testing in linear composite quantile regression models

Composite quantile regression (CQR) can be more efficient and sometimes arbitrarily more efficient than least squares for non-normal random errors, and almost as efficient for normal random errors. Based on CQR, we propose a test method to deal with the testing problem of the parameter in the linear regression models. The critical values of the test statistic can be obtained by the random weighting method without estimating the nuisance parameters. A distinguished feature of the proposed method is that the approximation is valid even the null hypothesis is not true and power evaluation is possible under the local alternatives. Extensive simulations are reported, showing that the proposed method works well in practical settings. The proposed methods are also applied to a data set from a walking behavior survey.     

A screening-based gradient-enhanced Kriging modeling method for high-dimensional problems

By exploring the auxiliary information from gradients, the accuracy of Kriging model can be improved. However, the dramatically increased time for model training tends to be unaffordable. Therefore, a novel gradient-enhanced Kriging modeling method which utilizes only a partial set of gradients, is developed in this article. Within the framework of this method, a balance between model accuracy and modeling efficiency can be achieved. More specifically, the influence of each input variable on output is estimated and ranked by feature selection technique. Then an empirical evaluation rule is proposed to facilitate the selection of gradients. Five representative numerical benchmarks from 10-D to 30-D and an airfoil optimal shape design with 18 variables are used for validation. Results show that when compared with the conventional Gradient-enhanced Kriging, the modeling time of the proposed method is significantly reduced, while the loss of accuracy is negligible. As a consequence, the proposed surrogate modeling method can provide an alternative way for approximating high-dimensional problems.     

A new method for reliability analysis of structures with mixed random and convex variables

This paper proposes a method combining projection-outline-based active learning strategy with Kriging metamodel for reliability analysis of structures with mixed random and convex variables. In this method, it is determined that the approximation accuracy of projection outlines on the limit-state surface is crucial for estimation of failure probability instead of the whole limit-state surface. To efficiently improve the approximation accuracy of projection outlines, a new projection-outline-based active learning strategy is developed to sequentially obtain update points located around the projection outlines. Taking into account the influence of metamodel uncertainty on the estimation of failure probability, a quantification function of metamodel uncertainty is developed and introduced in the stopping condition of Kriging metamodel update. Finally, Monte Carlo simulation is employed to calculate the failure probability based on the refined Kriging metamodel. Four examples including the Burro Creek Bridge and a piezoelectric energy harvester are tested to validate the performance of the proposed method. Results indicate that the proposed method is accurate and efficient for reliability analysis of structures with mixed random and convex variables.     

Optimal estimation in random coefficient regression models

In linear regression models with random coefficients, the score function usually involves unknown nuisance parameters in the form of weights. Conditioning with respect to the sufficient statistics for the nuisance parameter, when the parameter of interest is held fixed, eliminates the nuisance parameters and is expected to give reasonably good estimating functions. The present paper adopts this approach to the problem of estimation of average slope in random coefficient regression models. Four sampling situations are discussed. Some asymptotic results are also obtained for a model where neither the regressors nor the random regression coefficients replicate. Simulation studies for normal as well as non-normal models show that the performance of the suggested estimating functions is quite satisfactory.     

基于指数Laplace损失函数的回归估计鲁棒超限学习机

实际问题的数据集通常受到各种噪声的影响,超限学习机(extreme learning machine, ELM)对这类数据集进行学习时,表现出预测精度低、预测结果波动大.为了克服该缺陷,采用了能够削弱噪声影响的指数Laplace损失函数.该损失函数是建立在Gauss核函数基础上,具有可微、非凸、有界且能够趋近于Laplace函数的特点.将其引入到超限学习机中,提出了鲁棒超限学习机回归估计(exponential Laplace loss function based robust ELM for regression, ELRELM)模型.利用迭代重赋权算法求解模型的优化问题.在每次迭代中,噪声样本点被赋予较小的权值,能够有效地提高预测精度.真实数据集实验验证了所提出的模型相比较于对比算法具有更优的学习性能和鲁棒性.     

An embedded estimating equation for the additive risk model with biased-sampling data

This paper presents a novel class of semiparametric estimating functions for the additive model with right-censored data that are obtained from general biased-sampling. The new estimator can be obtained using a weighted estimating equation for the covariate coeffcients, by embedding the biased-sampling data into left-truncated and right-censored data. The asymptotic properties (consistency and asymptotic normality) of the proposed estimator are derived via the modern empirical processes theory. Based on the cumulative residual processes, we also propose graphical and numerical methods to assess the adequacy of the additive risk model. The good finite-sample performance of the proposed estimator is demonstrated by simulation studies and two applications of real datasets.     

Multivariate versus univariate Kriging metamodels for multi-response simulation models

To analyze the input/output behavior of simulation models with multiple responses, we may apply either univariate or multivariate Kriging (Gaussian process) metamodels. In multivariate Kriging we face a major problem: the covariance matrix of all responses should remain positive-definite; we therefore use the recently proposed “nonseparable dependence” model. To evaluate the performance of univariate and multivariate Kriging, we perform several Monte Carlo experiments that simulate Gaussian processes. These Monte Carlo results suggest that the simpler univariate Kriging gives smaller mean square error.     

Enhancing software reliability modeling and prediction through the introduction of time-variable fault reduction factor

Over the past three decades, many software reliability models with different parameters, reflecting various testing characteristics, have been proposed for estimating the reliability growth of software products. We have noticed that one of the most important parameters controlling software reliability growth is the fault reduction factor (FRF) proposed by Musa. FRF is generally defined as the ratio of net fault reduction to failures experienced. During the software testing process, FRF could be influenced by many environmental factors, such as imperfect debugging, debugging time lag, etc. Thus, in this paper, we first analyze some real data to observe the trends of FRF, and consider FRF to be a time-variable function. We further study how to integrate time-variable FRF into software reliability growth modeling. Some experimental results show that the proposed models can improve the accuracy of software reliability estimation. Finally, sensitivity analyses of various optimal release times based on cost and reliability requirements are discussed. The analytic results indicate that adjusting the value of FRF may affect the release time as well as the development cost.