An MCMC algorithm based on GUM Supplement 1 for uncertainty evaluation

This paper describes a simple Markov chain Monte Carlo algorithm for evaluating measurement uncertainty according to Bayesian principles. The algorithm has two phases, the first coinciding with the Monte Carlo method described in GUM Supplement 1 (GUMS1), the second a simple Metropolis–Hastings algorithm. The second phase can be regarded as a post-processing add-on to the GUMS1 calculation and can be used whenever a GUMS1 approach is adopted. The algorithm allows users freedom to choose their preferred prior distribution for the measurand, rather than that implicitly assigned in the GUMS1 approach, thereby avoiding some of the problems that can arise when applying GUMS1 to certain types of measurement model. The post-processing can be implemented in a few lines of software, so that many of the practical difficulties in implementing Bayesian approaches to measurement uncertainty evaluation are largely removed.     

Probability distributions and coverage probability in GUM,JCGM documents,and statistical inference

We discuss the interpretations of a probability distribution to express the state of knowledge about a quantity and the resulting coverage probability in the 1993 Guide to the Expression of Uncertainty in Measurement (GUM) and the subsequent JCGM documents: JCGM 101:2008, JCGM 104:2009, and JCGM 200:2008. The JCGM 101:2008 is titled ‘Supplement 1 to the GUM’ and the JCGM 104:2009 is titled ‘Introduction to the GUM and related documents’. It is reasonable to expect that they would have followed the GUM definitions of probability distribution and coverage probability. We submit that such is not the case. We submit that a connection between the standard deviation of a JCGM probability distribution and the GUM standard uncertainty is obscure. The JCGM interpretation of a probability distribution is seemingly based on Bayesian statistics. Also coverage probability has a well-established meaning in conventional statistics. Therefore we discuss the meanings of probability distribution and coverage probability in conventional and Bayesian statistical inference.     

Comparison of GUM and Monte Carlo methods for evaluating measurement uncertainty of perspiration measurement systems

Measurement uncertainty is an important parameter to express measurement results including means and reliability. The uncertainty analysis of the biomedical measurement system needs to be established. A perspiration measurement system composed of several sensors was developed. We aim to estimate the measurement uncertainty of this system with several uncertainty sources, including airflow rate, air density, and inlet and outlet absolute humidity. Measurement uncertainty was evaluated and compared by the Guide to the expression of the uncertainty in measurement (GUM) method and Monte Carlo simulation. The standard uncertainty for the perspiration measurement system was 6.81 × 10⁻⁶ kg/s and the uncertainty percentage <10%. The major source of the uncertainty was airflow rate, and inlet and outlet absolute humidity. The Monte Carlo simulation could be executed easily with available spreadsheet software programs of the Microsoft Excel. GUM and Monte Carlo simulation did not differ in measurement uncertainty with precision to two decimal places. However, the sensitivity coefficient derived by GUM provided useful information to improve measurement performance, which was not evaluated with the Monte Carlo simulation method.     

Bayesian recursive estimation of linear dynamic system states from measurement information

The evaluation of uncertainty in dynamic measurements has recently become a demanding issue. A Bayesian approach is employed here to derive the equations required to recursively generate the solution to the problem of estimating (and predicting) the states of linear dynamic systems. It is shown that this approach allows a derivation of Kalman’s filtering algorithm which is more easily accessible to those involved with dynamic measurements. The complete time-varying Kalman filter is particularly useful when the linear dynamic system and/or signal statistics are time varying and also when optimum estimates are required from the very beginning.     

基于贝叶斯推论和自组织映射的轴承性能退化评估方法

为在线识别和评估轴承的动态运行状态,提出一种基于贝叶斯推论和自组织映射的轴承性能退化评估方法。首先运用独立成分分析算法从原始特征集提取表征轴承正常运行的特征集,建立描述轴承健康状态的基准自组织映射模型,进而提出基于负对数似然概率的设备性能量化评估指标和基于贝叶斯推论的失效概率计算方法,在线识别和评估轴承的动态运行状态。通过在轴承全寿命测试床的实验结果表明,与一些传统的特征值指标和基于支持向量数据描述的性能退化评估方法相比,提出的评估指标可有效地量化轴承的全寿命性能退化过程,为进一步制定维护计划提供重要的设备健康信息。     

A Bayesian paradox and its impact on the GUM approach to uncertainty

The paper presents ideas and observations about the use of the frequentist and the Bayesian approach to estimation and uncertainty. The merits and the pitfalls of the Bayesian approach, compared with the frequentist one, are illustrated using a simple example, which gives rise to an instructive paradox. The impact of the paradox on the GUM approach to uncertainty prescribed in Supplement 1 is highlighted and discussed.     

MATLAB与WinBUGS在贝叶斯方法测量不确定度评定中的应用

针对贝叶斯方法仪器测量不确定度评定计算的复杂性,分析基于蒙特卡罗的贝叶斯测量不确定度评定实现方法;应用MATLAB软件和WinBUGS软件实现测量不确定度分布的传递,并应用WinBUGS软件与MATLAB软件的接口程序,实现MAT LAB对WinBUGS程序的设置与控制。     

Validation of the GUM using the Monte Carlo method when applied in the calculation of the measurement uncertainty of a compact prover calibration

This article presents the calibration of a compact prover using the weighing method. An evaluation of measurement uncertainty of the prover calibration has been developed using the GUM and Monte Carlo methodologies. A water draw kit was utilized to direct the liquid flow from the compact prover to a water container in order to weigh the transferred water mass on a balance. This amount of mass was used as reference for the calculation of the prover base volume. A modeling of the flow rate into the water draw kit as a function of time was conceived. This modeling was applied for calculating the error in the liquid volume of the water container due to the switching of two solenoid valves of the water draw kit. A mathematical model of the prover base volume has been developed. This model is non-linear and the two largest sources of uncertainty are related to the balance calibration certificate that together account for 31.84% of the uncertainty budget. This work showed that the GUM approach was validated by Monte Carlo method in the calculation of the measurement uncertainty of the calibration of a compact prover. The absolute differences of the respective endpoints of the coverage intervals of these two methods are less than 0.00023% of estimate of the prover base volume whose value is 151.427 dm³. This result was obtained for a coverage probability of 95% and ${10}^6$ Monte Carlo iterations. The density of the calibration water and its uncertainty have been calculated through an innovative approach.     

The use of measurement uncertainty and precision data in conformity assessment of automotive fuel products

In order to use a test result to decide whether it indicates compliance or non-compliance, it is necessary to take into account the dispersion of the values that can be attributed to the measurand. When dealing with conformity assessment of automotive fuel samples against European Union specification limits, this dispersion may be represented by uncertainty estimates based on either standard method precision data (ISO 4259 approach) or within laboratory precision data (intermediate precision approach). The present work presents possible decision rules based on these approaches and directly related to the required or acceptable level of probability of making a wrong decision. Acceptance limits for 95% and 99% confidence levels are calculated for all the properties of automotive fuels. Moreover, the effect of different approaches for defining guard bands, different levels of confidence or different number of replicate measurements is investigated using the results of the analyses of 769 diesel fuel samples for the determination of sulfur mass concentration.     

皮托管测量风速时GUM评定不确定度的偏差修正

针对不确定度表示指南法(GUM)进行皮托管测量风速值的不确定度评定时,测量结果存在偏差的现状,为确认该方法的适用性,并获取其偏差的修正值,首先采用GUM和蒙特卡洛法(MCM)对皮托管测量风速值的不确定度进行评定;然后,通过JCGM 101:2008 GUM supplement 1给出的方法,对GUM的适用性进行验证;最后,将GUM与MCM获取的风速测量结果进行对比。结果表明,GUM方法依然适用于皮托管进行风速测量时的不确定度的评定; GUM评定的结果标准不确定度产生了负偏差,其相对于实测风速值的偏差率为-0. 611 8×10-5,对应的修正值为"0. 611 8×10-5×实测风速值",由于该值相对于实测风速值较小,因此,在应用时可根据实际情况,决定是否对其进行修正。     

Estimation of the measurement uncertainty in the anisotropy test

The evaluation of the sheet metal drawability in mechanical shaping processes depends on a large number of analysis, among which the anisotropy evaluation. Nowadays, there is no Brazilian test laboratory accredited by the Metrology, Quality and Technology National Institute (INMETRO) to perform this analysis. So, the object of the present work is to establish a procedure for the estimation of the measurement uncertainty in the plastic anisotropy ratio of sheet metals, in accordance to the Guide to the Expression of Uncertainty in Measurement (GUM), aiming at the accreditation of this test in the Physical Metallurgy Laboratory. As results, we present the calculations performed and the uncertainty form proposed, and an analysis of which sources contributed the most for the uncertainty in the execution of a test. Finally, we propose improvement actions aiming at the reduction of the calculated uncertainty and the adequacy of the Measurement System for the desired application.     

考虑边信息的多层贝叶斯需求预测模型

随着工业互联网经济的发展,需求的不确定性日益增大,为提高需求预测的准确性,提出一个考虑边信息的多层贝叶斯需求预测模型(DFSI)。DFSI模型通过构造隐层的网络结构以实现对客户需求更加精确的刻画,该隐层结构主要包含两组参数:一组用于描述客户需求在时间上固有的连续性特征,另一组则用于融合相关的边信息特征。进一步,以贝叶斯推断为理论基础,以最大化后验概率为目标,推导出了DFSI的优化目标,并基于梯度下降方法设计了相应的求解算法。使用京东商城及某制造企业的真实销售数据对提出的模型进行了检验。结果显示,与常用的需求预测模型相比,DFSI能获得更好的预测结果。     

Laser tracker performance quantification for the measurement of involute profile and helix measurements

Standard-conforming measurements for a large involute gear were performed with a manually operated laser tracker system and the corresponding task-specific measurement uncertainties were estimated. Especially, readers using laser trackers for inspecting large involute gears will get information of a taskspecific measurement performance for the first time, which significantly differs from the laser tracker machine specification. To ensure unambiguous and repeatable measurement results, user-friendly auxiliary tools are used, which allows the operator to probe the measurement points according to existing guidelines and standards. Measurements were taken on a robust and highly accurate large involute gear measurement standard of the Physikalisch-Technische Bundesanstalt (PTB) under laboratory conditions. The size of this gear measurement standard complies with those gears used in wind power plants. The external gear materializes a left and a right hand gear as well as a spur gear. The obtained results of profile, helix and surface measurements are presented. These research activities were carried out at the PTB in the department of coordinate metrology.     

Repeatability and Reproducibility techniques for the analysis of measurement systems

This research was conducted with the aim of analyzing two of the main metrological characteristics of any measurement system: Repeatability and Reproducibility. Both of these features play an important role in the analysis of the measurements and they can give us a lot of information about who and what influences any measuring system. The analysis of Repeatability and Reproducibility is generally carried out through the use of the study Gage R&R. This study permits to understand which are the decisive factors in a measurement system, and, definitively, if the process is stable, that is under statistical control or out of statistical control.     

Evaluation of the uncertainty of an optical machine with a vision system for contact-less three-dimensional measurement

Optical machines with a vision system containing a video camera are designed to perform contact-less three-dimensional measurements. They are becoming more widely used in the industrial sector because of the many possibilities for automation they permit and because of the speed and economies in measurement tasks that can consequently be obtained. Within dimensional control these machines introduce significant characteristics of flexibility, savings and reliability. Therefore the purpose is to identify the principal metrological characteristics of such machines and so characterise the metrological traceability of the measurements obtained from them. The aim is to qualify each part of the optical machine as well as on the whole. Previously, no metrological traceability of this equipment has been provided in primary metrology, either at national (SIT—Sistema di Taratura in Italia—Calibration System in Italy) or at international level (EA–European co-operation for Accreditation). For this reason we develop a mathematical model that permits the evaluation of the measurement uncertainties in the use of such a device. The purpose is to obtain the certification of the measurement results furnished by the optical machine in as broad a context as possible. The experimental results of the tests are introduced for validating the proposed method. This paper provides the basis of the expression of the uncertainty of a measurement result obtained using the optical measurement machines and it shows the necessary requirements for the numerical evaluation of such uncertainty.     

Probabilistic measurement evaluation for the implementation of the Measuring Instrument Directive

The approval of the European Measuring Instrument Directive (2004) constitutes a noteworthy novelty in legal metrology, since this Directive implements the so called “new approach” to technical normalization and harmonization and moves toward a “global approach” to evaluation and conformity. According to the principles of the new approach regulatory model, the Directive establishes some essential requirements for a wide class of measuring devices, subject to legal control, whilst leaving manufacturer free to develop proper technical solutions for meeting them. Moreover, the conformity of instruments to prescribed requirement may now be ensured through quality assurance procedures, in the context of a quality system. This increase in technical and metrological freedom on the manufactures side may promote innovation, social benefit and costumers’ satisfaction, provided that some recent advances in measurement theory are properly employed.In this paper, after a brief discussion regarding the philosophy of the Directive, a probabilistic approach to conformity assessment is presented. The proposed approach concerns some currently open issues such as measurement uncertainty due to the influence of operating conditions, the assessment and reduction of the risk related to measurement uncertainty in conformity decision and the costs of instrument uncertainty and of non-conformity, for both the manufacturer and the user. The approach is supported by a package of software codes that assists the application of verification procedures, with a user-friendly approach.All these aspects are discussed with reference to the case of water meters, considering both end-of-production acceptance tests and the performance of meters under operation.     

Design and analysis of experiments in CMM measurement uncertainty study

In applying a coordinate measuring machine to measure a mechanical object, many factors affect the measurement uncertainty. Although a number of studies have been reported in evaluating measurement uncertainty, few have applied the factorial design of experiments (DOE) to examine the measurement uncertainty, as defined in the ISO Guide to the Expression of Uncertainty in Measurement (GUM). This research applies the DOE approach to investigate the impact of the factors and their interactions on the uncertainty while following the fundamental rules of the GUM. The measurement uncertainty of the location of a hole measured by a coordinate measuring machine is used to demonstrate our methodology.     

Evaluation of the standard measurement uncertainty due to the ISO50 surface determination method for dimensional computed tomography

Surface determination is the process by which a CT volume is converted from an image-based representation of an object to a surface-based representation. The process of surface determination relies on well-established image processing algorithms, but these algorithms are not exact, surface determination therefore contributes a component of uncertainty to X-ray CT based dimensional measurements. The purpose of this work is to develop a method for quantifying the standard uncertainty due to surface determination such that this standard uncertainty can be combined with other standard uncertainties in order to calculate the combined standard uncertainty for X-ray CT based dimensional measurements. The proposed method relies on evaluating the mode and the 68% dispersion of the air and material grey values of a given CT data-set. These values are propagated through the ISO50 threshold calculation to give the standard uncertainty of the ISO50 threshold value from which the standard uncertainty due to the ISO50 surface determination method can be evaluated. The proposed method is verified numerically and then demonstrated experimentally for dimensional measurements of an aluminium workpiece, a polymer workpiece, and a steel additively manufactured workpiece. The results show that the measurement uncertainty due to the ISO50 surface determination method is largest for dimensions that have both internal and external surfaces such as wall thicknesses, followed by dimensions such as internal and external diameters, whilst dimensions that are not sensitive to surface determination, for example centre-to-centre distances have the smallest uncertainty due to the ISO50 surface determination method.     

Mechanics analysis and simulation of material Brinell hardness measurement

Material Brinell hardness values are usually measured by force loading Brinell hardness testers. In this paper, firstly, based on experiment data and theoretical analysis, not only the relationship between Brinell hardness and mechanical properties of the measured material is discussed, but also all measurement uncertainty sources are precisely calibrated. Secondly, force loading characteristics of the Brinell hardness measurement are described and a finite element analysis for the whole Brinell hardness measurement process is introduced. Thirdly, to acquire theoretical measurement uncertainties and the reliable Brinell hardness values of the material under analysis, Monte Carlo method is applied to simulate Brinell hardness experiments. At last, several sets of experiments are carried out on HB-3000C, the Brinell hardness results acquired by this proposed novel mechanics analysis method and practical experiments shown to be consistent, and the validity of the proposed method is attained.     

Comparising measuring methods of pitch diameter of thread gauges and analysis of influences on the measurement results

In this article, the differences between three calibration methods of the thread ring pitch diameter are being presented. Two of these methods are classical methods (executed on a 1D-length measuring machine with a two-ball stylus for mechanical probing and on a coordinate measuring machine also with a two-ball stylus for mechanical probing); the third method is a modern one, executed on a profile scanner to calculate the pitch diameter with an adjustment technique (least square fit).