GUM与基于观测方程的测量不确定度评估

比较了GUM系列文件与贝叶斯方法评估不确定度的过程,说明GUM是以测量方程为基础的前向分析方法,而贝叶斯分析是以观测方程为基础、以数据分析为主的一种后向不确定度评估方法。运用概率模型的描述方法对上述两种分析过程进行了分析与比较,说明了两种方法在线性测量模型的条件下,不考虑被测量先验分布时两种结果基本一致,对于非线性的测量模型,GUM S1得到的结果与一定先验分布条件下的贝叶斯分析的结果也基本一致。最后通过实例说明了该结论。     

测量方程、观测方程与不确定度评估

GUM不确定度评估方法以测量方程为基础,通过标准不确定度传递的方法实现被测量标准不确定度的计算,是一种前向的不确定度评估方法;从观察方程入手的贝叶斯分析方法则是一种基于概率密度函数传递的后向的不确定度评估方法。该文详细说明两种方法评估测量不确定度的过程,解释两种方法的相同与差异之处。最后通过典型的不确定度评估实例,说明对于线性的测量模型,依据GUM准则评估的结果与利用被测量无信息先验的贝叶斯统计得到的结果是一致的,但在设定较强被测量先验信息或者非线性测量模型条件下,两种方法评估的结果有一定的差异。     

加速度计校准的贝叶斯不确定度评估

主要论述了贝叶斯统计用于加速度计校准结果的分析.首先介绍了对于线性测量模型,GUM、GUM S1以及基于贝叶斯统计分析测量不确定度的过程,说明三种方法分析的不同之处.然后结合实际工作中振动与冲击校准加速度计的数据,利用不同先验分布的贝叶斯统计和GUM系列方法进行了分析并对结果进行了比较.针对冲击加速度国际关键比对的部分数据建立了贝叶斯独立和层次两种不同的数据统计模型,在此基础之上结合马尔科夫链蒙特卡罗法(MCMC)对比对参考值和相应不确定度的计算,并且与通用方法的计算结果进行了比较.通过不同方法得到结果的一致性与差异性说明了贝叶斯统计用于不确定度评估的优缺点.     

测量不确定度评估及基于Monte Carlo法的数值计算

对于测量结果的不确定度评价主要依据的文件是ISO GUM,本文对GUM以及最新的利用MonteCarlo法基于概率密度函数(PDF)的传递评价测量不确定度的标准ISO GUM S1进行分析,阐述了两者评价测量不确定度的联系、基于Monte Carlo法的评价测量不确定度的数值计算方法.文中针对两个典型的例子,分别采用GUM与GUM S1规定的两种不确定度评定方法对其进行了不确定度评估,并对结果进行了比较.     

计量中回归模型参数值及其不确定度评估

针对计量领域中广泛应用的数据回归处理方法,阐述了在基于正态分布噪声条件下,最小二乘法与贝叶斯推断法用于回归模型参数估计以及相应不确定度评估的过程。GUM系列不确定度评估准则中没有明确指出如何对回归参数进行不确定度评估,同时有些回归模型也无法唯一地转化为相应的测量方程。通过计量校准的实例说明了如何处理相应参数的确定等问题,以此说明2种方法的相同与不同之处。最小二乘方法计算简单直接且便于使用;而基于贝叶斯推断的方法则能充分利用计量校准中的经验和历史数据等信息,但由于其参数后验分布计算通常较为复杂,需采用马尔科夫链-蒙特卡罗(MCMC)法通过数值计算得到关注参数的结果。     

贝叶斯定理在合并样本方差估计中的应用

合并估计是计量学中的一种常见情况,<测量不确定度表示指南>(简称GUM)推荐了在此情况下估计测量不确定度的常规方法,但它的不足之处在于没有运用先前的测量信息.另一种被称为频率方法则充分利用了先前的测量信息,但在实际中却难以操作.而根据贝叶斯定理推导出的贝叶斯方法克服了上述方法的缺陷和不足,可以给出更可靠的测量结果,并且具有较好的可操作性.     

融合生产信息的产品检验不确定度评定

为合理减小测量不确定度对产品合格判定的影响,研究融合生产信息的产品检验不确定度评定。建立融合加工信息和测量信息的贝叶斯信息融合模型,基于被测参数后验分布对测量结果及其不确定度进行重新估计,依据信息融合后的结果进行产品合格判定;提出利用质量控制信息计算贝叶斯先验参数的方法。仿真实验和实例分析结果表明,在产品检验不确定度评定中融入生产信息,可有效扩大产品检验合格区,合理减小合格判定的工作量。     

坐标测量机孔径测量的不确定度评定模型研究

以坐标测量机测量孔径为例,阐述测量过程中影响测量结果的不确定度来源,根据测量模型建立孔径测量的GUM法不确定度评定模型;利用对坐标测量机的测量系统量值特性指标分析的方法 ,给出基于量值特性分析法的各标准不确定度分量的评定模型。通过对汽车空调压缩机后缸体的孔径测量,比较两种方法评定的扩展不确定度。实例分析可以看出:对于坐标测量机复杂的非线性测量模型,GUM法在计算灵敏系数时,运算量较大且获得的是近似结果,因此其可操作性不强;量值特性分析法通过对测量系统整体的分析,基于大量的实验数据对测量结果进行测量不确定度评定,其流程和操作性更为便捷、有效。     

电压驻波比测量不确定度的评定和表述

电压驻波比是无线电领域表征反射特性的传统参量,目前仍有应用场景,然而一些实验室对电压驻波比测量不确定度的报告是不完善的。比较了GUM法线性模型、GUM法非线性模型和蒙特卡洛法对电压驻波比测量不确定度的评定结果,给出了一种在常规测量条件下可代替蒙特卡洛法的快速估算方法,并将该方法推广到一元非线性测量模型。     

钢筋混凝土深受弯构件受剪承载力概率模型研究

贝叶斯理论具有充分利用模型信息和数据信息且考虑先验分布等优点,已被广泛应用于各个领域。通过贝叶斯多元线性参数估计方法,建立基于影响参数的钢筋混凝土深受弯构件贝叶斯概率抗剪模型。基于该模型和271组钢筋混凝土深受弯构件试验结果,完成了模型参数计算及基于贝叶斯理论的受剪承载力计算,同时,利用贝叶斯参数剔除法对抗剪模型进行简化,并与我国混凝土结构设计规范(GB50010-2010)、ACI318-08、CSA和EC2等现有规范计算结果进行了对比分析。研究表明：利用贝叶斯方法建立的基于影响参数的深受弯构件抗剪模型计算结果与试验吻合良好,采用贝叶斯动态更新后结果较规范值更接近试验值,简化后模型能较为合理的进行深受弯构件受剪承载力计算。     

Measurement Uncertainty of Dew-Point Temperature in a Two-Pressure Humidity Generator

This article describes the measurement uncertainty evaluation of the dew-point temperature when using a two-pressure humidity generator as a reference standard. The estimation of the dew-point temperature involves the solution of a non-linear equation for which iterative solution techniques, such as the Newton?CRaphson method, are required. Previous studies have already been carried out using the GUM method and the Monte Carlo method but have not discussed the impact of the approximate numerical method used to provide the temperature estimation. One of the aims of this article is to take this approximation into account. Following the guidelines presented in the GUM Supplement 1, two alternative approaches can be developed: the forward measurement uncertainty propagation by the Monte Carlo method when using the Newton?CRaphson numerical procedure; and the inverse measurement uncertainty propagation by Bayesian inference, based on prior available information regarding the usual dispersion of values obtained by the calibration process. The measurement uncertainties obtained using these two methods can be compared with previous results. Other relevant issues concerning this research are the broad application to measurements that require hygrometric conditions obtained from two-pressure humidity generators and, also, the ability to provide a solution that can be applied to similar iterative models. The research also studied the factors influencing both the use of the Monte Carlo method (such as the seed value and the convergence parameter) and the inverse uncertainty propagation using Bayesian inference (such as the pre-assigned tolerance, prior estimate, and standard deviation) in terms of their accuracy and adequacy.     

基于自适应蒙特卡罗方法的测量不确定度评定

在GUM Sup.1提出基于分布传播的不确定度评定基础上,系统地介绍了自适应蒙特卡罗方法进行不确定度评定的原理和步骤,并以此对线性测量模型和非线性测量模型进行了仿真,得出自适应蒙特卡罗方法对两类测量模型不确定度都有较好的评定效果,同时指出自适应蒙特卡罗方法中仿真次数M和数值容差δ的合理选择都需要进一步研究。     

Robust Bayesian estimators in a one-way ANOVA model

Summary Motivated by the attractive features of robust priors, we develop Bayesian estimators for the parameters in a one-way ANOVA model using mixed priors, which are formed by incorporating at density into the usual conjugate priors to independently describe prior knowledge regarding the overall mean or regarding the factor effects. The effect of the independentt prior component is greatly different from that of the conjugate prior. The Bayesian estimators arising from such mixed priors are non-linear functions of the least squares estimators and adjust automatically to the value of the sum of squared errors. In this sense, they are adaptive and rather insensitive to extreme observations. The proposed estimators are clearly superior to the usual Bayesian estimators and to the traditional unbiased estimators, and may be practicable when the error terms are Cauchy distributed.     

Bayesian design of experiments for generalized linear models and dimensional analysis with industrial and scientific application

ABSTRACT

The design of an experiment can always be considered at least implicitly Bayesian, with prior knowledge used informally to aid decisions such as the variables to be studied and the choice of a plausible relationship between the explanatory variables and measured responses. Bayesian methods allow uncertainty in these decisions to be incorporated into design selection through prior distributions that encapsulate information available from scientific knowledge or previous experimentation. Further, a design may be explicitly tailored to the aim of the experiment through a decision-theoretic approach using an appropriate loss function. We review the area of decision-theoretic Bayesian design, with particular emphasis on recent advances in computational methods. For many problems arising in industry and science, experiments result in a discrete response that is well described by a member of the class of generalized linear models. Bayesian design for such nonlinear models is often seen as impractical as the expected loss is analytically intractable and numerical approximations are usually computationally expensive. We describe how Gaussian process emulation, commonly used in computer experiments, can play an important role in facilitating Bayesian design for realistic problems. A main focus is the combination of Gaussian process regression to approximate the expected loss with cyclic descent (coordinate exchange) optimization algorithms to allow optimal designs to be found for previously infeasible problems. We also present the first optimal design results for statistical models formed from dimensional analysis, a methodology widely employed in the engineering and physical sciences to produce parsimonious and interpretable models. Using the famous paper helicopter experiment, we show the potential for the combination of Bayesian design, generalized linear models, and dimensional analysis to produce small but informative experiments.     

Data assimilation and uncertainty analysis of environmental assessment problems—an application of Stochastic Transfer Function and Generalised Likelihood Uncertainty Estimation techniques

Stochastic Transfer Function (STF) and Generalised Likelihood Uncertainty Estimation (GLUE) techniques are outlined and applied to an environmental problem concerned with marine dose assessment. The goal of both methods in this application is the estimation and prediction of the environmental variables, together with their associated probability distributions. In particular, they are used to estimate the amount of radionuclides transferred to marine biota from a given source: the British Nuclear Fuel Ltd (BNFL) repository plant in Sellafield, UK. The complexity of the processes involved, together with the large dispersion and scarcity of observations regarding radionuclide concentrations in the marine environment, require efficient data assimilation techniques. In this regard, the basic STF methods search for identifiable, linear model structures that capture the maximum amount of information contained in the data with a minimal parameterisation. They can be extended for on-line use, based on recursively updated Bayesian estimation and, although applicable to only constant or time-variable parameter (non-stationary) linear systems in the form used in this paper, they have the potential for application to non-linear systems using recently developed State Dependent Parameter (SDP) non-linear STF models. The GLUE based-methods, on the other hand, formulate the problem of estimation using a more general Bayesian approach, usually without prior statistical identification of the model structure. As a result, they are applicable to almost any linear or non-linear stochastic model, although they are much less efficient both computationally and in their use of the information contained in the observations. As expected in this particular environmental application, it is shown that the STF methods give much narrower confidence limits for the estimates due to their more efficient use of the information contained in the data. Exploiting Monte Carlo Simulation (MCS) analysis, the GLUE technique is used to estimate how the errors involved in the STF model structure and observations influence the model outputs and errors. The discussion on updating information originating from different locations using GLUE procedure is also given. A final aim of the paper is to use the results obtained in this particular example to explore the differences between the GLUE and STF approaches.     

Splines from a Bayesian point of view

Summary Procedures using splines for estimating values of linear functionals of an unknown function based on finitely many possibly noisy observations of function values are reviewed taking a Bayesian point of view. Interpolation with splines is emphasized as an example of Bayesian numerical analysis, smoothing with splines is presented as interpolation in estimated function values. Extensions of the approach to estimating values of non-linear functionals applied to the unknown function and to estimation subject to linear constaints on the unknown function are discussed. Furthermore, invariance of Bayesian inference to modifications of the prior distribution, resulting from alternative choices of an appropriate function space for estimation, is addressed, too.     

A sequential constant‐stress accelerated life testing scheme and its Bayesian inference

In the analysis of accelerated life testing (ALT) data, some stress‐life model is typically used to relate results obtained at stressed conditions to those at use condition. For example, the Arrhenius model has been widely used for accelerated testing involving high temperature. Motivated by the fact that some prior knowledge of particular model parameters is usually available, this paper proposes a sequential constant‐stress ALT scheme and its Bayesian inference. Under this scheme, test at the highest stress is firstly conducted to quickly generate failures. Then, using the proposed Bayesian inference method, information obtained at the highest stress is used to construct prior distributions for data analysis at lower stress levels. In this paper, two frameworks of the Bayesian inference method are presented, namely, the all‐at‐one prior distribution construction and the full sequential prior distribution construction. Assuming Weibull failure times, we (1) derive the closed‐form expression for estimating the smallest extreme value location parameter at each stress level, (2) compare the performance of the proposed Bayesian inference with that of MLE by simulations, and (3) assess the risk of including empirical engineering knowledge into ALT data analysis under the proposed framework. Step‐by‐step illustrations of both frameworks are presented using a real‐life ALT data set. Copyright © 2008 John Wiley & Sons, Ltd.     

Gaussian process regression for multivariate spectroscopic calibration

Traditionally multivariate calibration models have been developed using regression based techniques including principal component regression and partial least squares and their non-linear counterparts. This paper proposes the application of Gaussian process regression as an alternative method for the development of a calibration model. By formulating the regression problem in a probabilistic framework, a Gaussian process is derived from the perspective of Bayesian non-parametric regression, prior to describing its implementation using Markov chain Monte Carlo methods. The flexibility of a Gaussian process, in terms of the parameterization of the covariance function, results in its good performance in terms of the development of a calibration model for both linear and non-linear data sets. To handle the high dimensionality of spectral data, principal component analysis is initially performed on the data, followed by the application of Gaussian process regression to the scores of the extracted principal components. In this sense, the proposed method is a non-linear variant of principal component regression. The effectiveness of the Gaussian process approach for the development of a calibration model is demonstrated through its application to two spectroscopic data sets. A statistical hypothesis test procedure, the paired t-test, is used to undertake an empirical comparison of the Gaussian process approach with conventional calibration techniques, and it is concluded that the Gaussian process exhibits enhanced behaviour.     

空气焓值法测量全年能源消耗效率不确定度评定方法研究

以焓值法的直接测试量表征的制冷量和制热量计算公式作为基础数学模型,采用GUM法和蒙特卡洛法相结合的方法来评定全年能源消耗效率(APF)的不确定度。由于焓值法数学模型呈现出明显的非线性,首先使用蒙特卡洛法来验证额定制冷量和额定制热量GUM法评定结果, 验证结果显示2种方法偏差不超过2‰。然后,给出了5个工况下换热量的不确定度评定结果,以此作为APF蒙特卡洛模拟的输入量,并给出了自适应蒙特卡洛法评定APF不确定度模拟流程,得到某空调的APF扩展不确定度评定结果为0.09 kW·h/(kW·h)（k=2）。