Multifidelity approaches for optimization under uncertainty

It is important to design robust and reliable systems by accounting for uncertainty and variability in the design process. However, performing optimization in this setting can be computationally expensive, requiring many evaluations of the numerical model to compute statistics of the system performance at every optimization iteration. This paper proposes a multifidelity approach to optimization under uncertainty that makes use of inexpensive, low‐fidelity models to provide approximate information about the expensive, high‐fidelity model. The multifidelity estimator is developed based on the control variate method to reduce the computational cost of achieving a specified mean square error in the statistic estimate. The method optimally allocates the computational load between the two models based on their relative evaluation cost and the strength of the correlation between them. This paper also develops an information reuse estimator that exploits the autocorrelation structure of the high‐fidelity model in the design space to reduce the cost of repeatedly estimating statistics during the course of optimization. Finally, a combined estimator incorporates the features of both the multifidelity estimator and the information reuse estimator. The methods demonstrate 90% computational savings in an acoustic horn robust optimization example and practical design turnaround time in a robust wing optimization problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Phase‐I monitoring of log‐linear model‐based processes (a case study in health care: Kidney patients)

Processes with multiple correlated categorical quality characteristics are called multivariate categorical processes. These processes are usually shown by contingency tables and are characterized by log‐linear models. In this paper, two monitoring approaches including likelihood ratio test (LRT) and F test are developed to monitor multivariate categorical processes based on the contingency table in Phase I. In addition, a change point estimator for multivariate categorical processes is developed in Phase I. The performances of the two proposed approaches are evaluated in terms of probability of signal, while the performance of the proposed change point estimator is evaluated in terms of accuracy and precision criteria through simulation experiments. Meanwhile, we compare the performance of the two proposed control charts with an existing control chart called “?2LRT” control chart for multivariate categorical processes. In the end, a typical application of the proposed methods is illustrated in a real‐world health care system.     

Robust identification of transfer functions in the s- andz-domains

A frequency-domain maximum-likelihood estimator (MLE) for estimating the transfer function of linear continuous-time systems developed by J. Schoukens et al. (1988) assumes independent Gaussian noise on both the input and the output coefficients. A Gaussian frequency-domain MLE for transfer functions of linear continuous or discrete time invariant systems in an errors-in-variables model is presented. It is demonstrated that most of the properties of the estimator remain unchanged when it is applied to measured input and output Fourier coefficients corrupted with non-Gaussian errors. The result is a robust Gaussian frequency-domain estimator that is very useful for the practical identification of linear systems. The theoretical results are verified by simulations and experiments     

噪声因子存在下的多响应参数设计的优化

采用多元损失函数法,对噪声因子存在下的多响应稳健性参数设计进行了优化.该方法考虑了噪声因子的影响,结合响应期望值和响应方差,其中响应方差结合了噪声因子产生的方差和拟合模型的预测方差,给出了综合方差的无偏估计,使解决方案对噪声因子和参数估计的不确定性都具有稳健性,避免了方差出现非正定的可能性.采用该方法对实例进行分析,得到较好的优化结果.     

不完全观测数据下混合效应模型的正交投影估计

基于矩阵的QR分解技术,对一类含有不完全观测数据的线性混合效应模型提出了一种基于正交投影的估计方法。在一些正则条件下,证明了固定效应参数的估计渐近服从标准正态分布,得到了固定效应参数的置信区间。另外,所提出的固定效应参数的估计过程不受随机效应的任何影响,具有较好的有效性和稳健性。最后,通过一些数值模拟和一个实例分析研究了所提出估计方法的有限样本性质。     

GLM profile monitoring using robust estimators

It is well known that performance of control scheme in phase II of statistical process control depends on the estimators utilized in phase I. Sometimes, outliers may be present in the data, which could seriously impact the performance of the estimators. In some practical situations, generalized linear models (GLMs) are used to model a wide class of response variables. This study deals with the robust estimation and monitoring of parameters in GLM profiles in the presence of outliers. In this study, robust estimators are used to estimate the parameters of logistic and Poisson profiles. The results are compared with the maximum likelihood estimators (MLEs). In a numerical example, the profile parameters are estimated by the MLE and robust estimators and the resulting test statistics are monitored by a control scheme. The phase II control charts are determined based on these two types of estimators and compared for different out-of-control conditions. The simulation results confirm that robust estimators in most cases lead to better estimates in comparison with the MLE estimator in terms of average run length criterion.     

Robust estimation in generalized linear models: the density power divergence approach

The generalized linear model is a very important tool for analyzing real data in several application domains where the relationship between the response and explanatory variables may not be linear or the distributions may not be normal in all the cases. Quite often such real data contain a significant number of outliers in relation to the standard parametric model used in the analysis; in such cases inference based on the maximum likelihood estimator could be unreliable. In this paper, we develop a robust estimation procedure for the generalized linear models that can generate robust estimators with little loss in efficiency. We will also explore two particular special cases in detail—Poisson regression for count data and logistic regression for binary data. We will also illustrate the performance of the proposed estimators through some real-life examples.     

A multivariate spatial crash frequency model for identifying sites with promise based on crash types

Many studies have proposed the use of a systemic approach to identify sites with promise (SWiPs). Proponents of the systemic approach to road safety management suggest that it is more effective in reducing crash frequency than the traditional hot spot approach. The systemic approach aims to identify SWiPs by crash type(s) and, therefore, effectively connects crashes to their corresponding countermeasures. Nevertheless, a major challenge to implementing this approach is the low precision of crash frequency models, which results from the systemic approach considering subsets (crash types) of total crashes leading to higher variability in modeling outcomes. This study responds to the need for more precise statistical output and proposes a multivariate spatial model for simultaneously modeling crash frequencies for different crash types. The multivariate spatial model not only induces a multivariate correlation structure between crash types at the same site, but also spatial correlation among adjacent sites to enhance model precision. This study utilized crash, traffic, and roadway inventory data on rural two-lane highways in Pennsylvania to construct and test the multivariate spatial model. Four models with and without the multivariate and spatial correlations were tested and compared. The results show that the model that considers both multivariate and spatial correlation has the best fit. Moreover, it was found that the multivariate correlation plays a stronger role than the spatial correlation when modeling crash frequencies in terms of different crash types.     

Minimum density power divergence estimator for GARCH models

In this paper, we study the robust estimation for the generalized autoregressive conditional heteroscedastic (GARCH) models with Gaussian errors. As a robust estimator, we consider a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (Biometrika 85:549–559, 1998). It is shown that the MDPDE is strongly consistent and asymptotically normal. Our simulation study demonstrates that the MDPDE has robust properties in contrast to the maximum likelihood estimator. A real data analysis is performed for illustration.     

Marginal integration <Emphasis Type="Italic">M</Emphasis>-estimators for additive models

Additive regression models have a long history in multivariate non-parametric regression. They provide a model in which the regression function is decomposed as a sum of functions, each of them depending only on a single explanatory variable. The advantage of additive models over general non-parametric regression models is that they allow to obtain estimators converging at the optimal univariate rate avoiding the so-called curse of dimensionality. Beyond backfitting, marginal integration is a common procedure to estimate each component in additive models. In this paper, we propose a robust estimator of the additive components which combines local polynomials on the component to be estimated with the marginal integration procedure. The proposed estimators are consistent and asymptotically normally distributed. A simulation study allows to show the advantage of the proposal over the classical one when outliers are present in the responses, leading to estimators with good robustness and efficiency properties.     

Parametric identification of two-port models in the frequencydomain

The authors treat the problem of parametric estimation of linear time-invariant dynamic two-port models (e.g. the short-circuit admittance matrix) from experimental data. A multivariate frequency-domain Gaussian maximum likelihood estimator is proposed to estimate the unknown coefficients occurring in the rational two-port model. It takes the perturbing noise of all the measured voltages and currents into account. The covariance matrix of the noise is assumed to be known, e.g. from measurements. The estimates and their covariance matrix are obtained as the result of an optimization procedure. The value of the minimized loss function and the covariance matrix of the estimates can be used to determine the model structure. The ability of the estimator to handle real measurement problems is demonstrated by means of experimental results. Using the estimated two-part parameters of an unloaded band-pass filter, it was possible to predict the transfer function of the loaded filter within an error of ±0.01 dB on the magnitude and ±0.1° on the phase     

Robust deblurring of random blur

A robust method for deblurring random blur is introduced. The image is assumed to be distorted by a linear system whose impulse response function is itself random and by additive noise whose spectral density is known only to be in the neighborhood of some specified spectral density. The estimate is motivated from the generalized least-squares and robust statistical methods. Our robust deblurring is considered in the frequency domain and is of the form of weighted least squares, with the most prominent frequencies of the random impulse response being downweighted in a way similar to Huber's robust estimator.     

Discrete Multivariate Analysis

A simple heuristic is proposed for constructing robust experimental designs for multivariate generalized linear models. The method is based on clustering a set of local optimal designs. A method for finding local D-optimal designs using available resources is also introduced. Clustering, with its simplicity and minimal computation needs, is demonstrated to outperform more complex and sophisticated methods.     

A robust multivariate Shewhart chart for contaminated normal environments

Lately, the multivariate setup of control charts, especially the memory-less chart has received less attention of researchers as compared to the univariate setup. However, the multivariate setup is of paramount importance in this big-data era. In this research work, we study the multivariate Shewhart chart for monitoring location parameter by examining the robustness of this scheme with the mean estimator. We also explored the scheme with some other robust parametric estimators in different process environments. The multivariate estimators such as median, midrange, tri-mean (TM), and Hodges–Lehmann (HL) estimators were examined under uncontaminated, location contaminated, variance contaminated, and both location–variance contaminated normal environments. Through a synthetic Monte Carlo simulation and application of the schemes on a real-life dataset, the findings suggest that the proposed estimators outperform the default estimator of the multivariate scheme (mean). The performance measures of comparing these estimators through the charts are the average run length, standard deviation run length, extra-quadratic loss, and relative average run length. The charts resulting from applying the schemes on real-life dataset recorded from glass manufacturing process also buttresses the simulation findings.     

Nonstationary multivariate process modeling through spatially varying coregionalization

Models for the analysis of multivariate spatial data are receiving increased attention these days. In many applications it will be preferable to work with multivariate spatial processes to specify such models. A critical specification in providing these models is the cross covariance function. Constructive approaches for developing valid cross-covariance functions offer the most practical strategy for doing this. These approaches include separability, kernel convolution or moving average methods, and convolution of covariance functions. We review these approaches but take as our main focus the computationally manageable class referred to as the linear model of coregionalization (LMC). We introduce a fully Bayesian development of the LMC. We offer clarification of the connection between joint and conditional approaches to fitting such models including prior specifications. However, to substantially enhance the usefulness of such modelling we propose the notion of a spatially varying LMC (SVLMC) providing a very rich class of multivariate nonstationary processes with simple interpretation. We illustrate the use of our proposed SVLMC with application to more than 600 commercial property transactions in three quite different real estate markets, Chicago, Dallas and San Diego. Bivariate nonstationary process inodels are developed for income from and selling price of the property. The work of the first and second authors was supported in part by NIH grant R01ES07750-06.     

基于有条件模拟方法的桥梁抗震分析

该文研究地震空间变异性对桥梁地震响应的影响。依据经过加速度幅值调整的EI-Centro地震波和空间相干模型,采用条件模拟算法-多变量线性预测(MLP)法合成继承选用记录物理特性的多点地震动,并通过大质量法将其施加到桥梁结构上。以一座高墩大跨曲线连续刚构桥为研究对象,分析比较地震动行波激励和完全空间变异性激励作用下的桥梁动力响应,研究结果表明:大跨度桥梁地震响应分析应考虑完全空间变异性的影响,有条件模拟技术可以应用于桥梁抗震分析。     

On the estimation of the tensile strength of carbon fibres at short lengths

Generally, to determine the fibre-matrix interfacial properties in fibre reinforced plastics, it is necessary to know the tensile strength of the fibre at very short lengths, for which direct measurements are not possible. Accordingly, in this study, the determination of the tensile strength of high strength carbon fibres and their gauge length dependence are analysed by means of the Weibull model. The influence of the estimator chosen and of the sample size on the calculated value of the tensile strength of the fibre are first determined. Secondly, the accuracy of the three- and the two-parameter Weibull distributions is examined. Finally, it is shown that the most appropriate extrapolation at short length is performed by means of a linear logarithmic dependence on gauge length of the tensile strength. This method seems to be valid for untreated as well as for surface-treated high strength carbon fibres.     

Novel algebraic meal disturbance estimation based adaptive robust control design for blood glucose regulation in type 1 diabetes patients

This study designs a robust closed‐loop control algorithm for elevated blood glucose level stabilisation in type 1 diabetic patients. The control algorithm is based on a novel control action resulting from integrating algebraic meal disturbance estimator with back‐stepping integral sliding mode control (BISMC) technique. The estimator shows finite time convergence leading to accurate and fast estimation of meal disturbance. Moreover, compensation of the estimated disturbance in controller provides significant reduction in chattering phenomenon, which is inherent drawback of sliding mode control (SMC). The controller is applied to one of the most reliable models of type 1 diabetic patients, named Bergman''s minimal model. The effectiveness and superiority of the designed controller is shown by comparing it to classical SMC and super‐twisting sliding mode control. The designed controller is subject to three different cases for detailed analysis of the controller''s robustness against meal disturbance. The three cases considered are hyperglycaemia, hyperglycaemia combined with meal disturbance and three meal disturbance. The simulation results confirm superior performance of algebraic disturbance estimator based BISMC controller for all the cases mentioned above.Inspec keywords: closed loop systems, robust control, sugar, medical control systems, variable structure systems, control system synthesis, blood, nonlinear control systems, adaptive control, diseasesOther keywords: adaptive robust control design, blood glucose regulation, type 1 diabetes patients, closed‐loop control algorithm, elevated blood glucose level stabilisation, type 1 diabetic patients, novel control action, algebraic meal disturbance estimator, mode control technique, accurate estimation, estimated disturbance, super‐twisting sliding mode control, algebraic disturbance estimator, BISMC controller, algebraic meal disturbance estimation, back‐stepping integral sliding mode control technique     

Randomized residual-based error estimators for the proper generalized decomposition approximation of parametrized problems

This article introduces a novel error estimator for the proper generalized decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: it builds on concentration inequalities of Gaussian maps and an adjoint problem with random right-hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user-defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time-harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high-dimensional parameter space.     

AC network state estimation using linear measurement functions

The real/reactive power and current magnitude measurements can be accounted for in an AC network state estimator using linear measurement functions. The nonlinearity in the conventional AC state estimator equations is transferred from the measurement functions into a system of nonlinear equality constraints which is independent of the measurement set. The new format of equations entails two advantages. First, it can be easily integrated in optimisation routines which employ first- and second-order derivative functions. Second, the linear measurement functions can benefit from scaling techniques which are well documented in the linear programming literature. This research details the implementation of a least absolute value state estimator employing the new format of equations. The optimisation method is based on a primal-dual interior-point method that can accurately account for zero injection measurements and power directions. Numerical testing is used to validate the approach for networks with measurement sets that are (i) conventional and (ii) have a high proportion of current magnitude measurements and power signs.