Expanding needs for metrological traceability and measurement uncertainty

In our daily life, numerical evaluation of a quantity is normally required not only for scientific decision but also for judgment of products. Under the Metre Convention, ideal system of the International System of units (SI) is globally recognized and utilized in many fields of science and technology [1]. Some documented standards such as International Vocabulary of Metrology (VIM) [2], and Guide to the expression of Uncertainty in Measurement (GUM) [3] are edited and published by the Joint Committee for Guides in Metrology (JCGM) and commonly introduced in the fields of metrology, science and conformity assessment in connection with ISO/IEC 17025. This paper describes the recent situation in measurement science, and how to obtain a reliable measurement result using the expression of metrological traceability together with measurement uncertainty. In the paper, a response to Prof. Ludwik Finkelstein’s message for ‘challenges and problems to future advancement in measurement science’ is also reported.     

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.     

“Measurement” and related concepts. Their interpretation in the VIM

The paper comments on the definitions of such terms as “measurement,” “nominal property,” “metrological traceability,” and others given in the 3-rd edition of International Vocabulary of Metrology (VIM). Proposals are offered for amendments to the terminology of the next edition of the VIM, taking into account new fields where measurements are applied.     

Use of design of experiments and Monte Carlo method for instruments optimal design

A common issue in the design of measurement instruments is the comparison between different solutions in terms of components of the measurement chain, data processing or even measurement principles; the predicted instrumental uncertainty is the driving parameter for such a comparison. While in many situations the linearization of the measuring model allows using the standard ISO GUM procedure, in complex cases it might be necessary to proceed with Monte Carlo simulations as per ISO GUM supplement 1. This paper describes a method that combines the factorial design of experiments (DOE) and the ISO GUM supplement 1 uncertainty evaluation method to guide the instrument designer in the instrument configuration optimization. The proposed approach allows estimating, in the design phase, the overall instrumental uncertainty for different configurations, the instrument sensitivity to the accuracy in the measurements of its inputs and the effects on systematic and random measurement errors deriving from the choice of all instrumental variables. The use of data populations selected with the DOE criteria allows recovering valuable parameters equivalent to the sensitivity factors of the GUM linearized approach. The data analysis allows separating the critical factors that must be accurately controlled from those only weakly affecting the measurement uncertainty. The method has been applied to a case study where the metrological performances of a system devoted to the measurement of the acoustic radiation emitted by a vibrating panel in a reverberant enclosure had to be assessed.     

Can coverage factor 2 be interpreted as an equivalent to 95% coverage level in uncertainty estimation? Two case studies

The GUM modelling, its Bayesian modification and the Monte Carlo method (MCM) to estimate the uncertainty are compared in two practical measurement situations (finding reference value of relative humidity and a generic chemical instrumental analysis procedure). The results of the three approaches agree very well when there are no dominant input quantities with type A evaluated uncertainty estimated from small number of repeated measurements. In the opposite case the GUM gives underestimated expanded uncertainties (by up to 20–25%), compared to both other approaches. Analysis of the practical measurement situations reveals that even in the case of several dominating input quantities of similar uncertainty contributions, if one of them is distributed according to the t-distribution and has a low number (3–4) of degrees of freedom, the output quantity cannot be safely assumed Normally distributed and in such a case coverage factor 2 is not an equivalent to 95% coverage level.     

Adaptive Monte Carlo and GUM methods for the evaluation of measurement uncertainty of cylindricity error

Measurement uncertainty is one of the most important concepts in geometrical product specification (GPS). The “Guide to the expression of uncertainty in measurement (GUM)” is the internationally accepted master document for the evaluation of uncertainty. The GUM method (GUMM) requires the use of a first-order Taylor series expansion for propagating uncertainties. However, when the mathematical model of measurand is strongly non-linear the use of this linear approximation may be inadequate. Supplement 1 to GUM (GUM S1) has recently been proposed based on the basis of probability density functions (PDFs) using the Monte Carlo method (MCM). In order to solve the problem that the number of Monte Carlo trials needs to be selected priori, adaptive Monte Carlo method (AMCM) described in GUM S1 is recommended to control over the quality of the numerical results provided by MCM.The measurement and evaluation of cylindricity errors are essential to ensure proper assembly and good performance. In this paper, the mathematical model of cylindricity error based on the minimum zone condition is established and a quasi particle swarm optimization algorithm (QPSO) is proposed for searching the cylindricity error. Because the model is non-linear, it is necessary to verify whether GUMM is valid for the evaluation of measurement uncertainty of cylindricity error. Then, AMCM and GUMM are developed to estimate the uncertainty. The procedure of AMCM scheme and the validation of GUMM using AMCM are given in detail. Practical example is illustrated and the result shows that GUMM is not completely valid for high-precision evaluation of the measurement uncertainty of cylindricity error if only the first-order terms in the Taylor series approximation are taken into account. Compared with conventional methods, not only the proposed QPSO method can search the minimum zone cylindricity error precisely and rapidly, but also the Monte Carlo simulation is adaptive and AMCM can provide control variables (i.e. expected value, standard uncertainty and lower and higher coverage interval endpoints) with an expected numerical tolerance. The methods can be extended to the evaluation of measurement uncertainty of other form errors such as roundness and sphericity errors.     

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.     

Implementation of unscented transform to estimate the uncertainty of a liquid flow standard system

First-order partial derivatives of a mathematical model are an essential part of evaluating the measurement uncertainty of a liquid flow standard system according to the Guide to the expression of uncertainty in measurement (GUM). Although the GUM provides a straight-forward method to evaluate the measurement uncertainty of volume flow rate, the first-order partial derivatives can be complicated. The mathematical model of volume flow rate in a liquid flow standard system has a cross-correlation between liquid density and buoyancy correction factor. This cross-correlation can make derivation of the first-order partial derivatives difficult. Monte Carlo simulation can be used as an alternative method to circumvent the difficulty in partial derivation. However, the Monte Carlo simulation requires large computational resources for a correct simulation because it considers the completeness issue whether an ideal or a real operator conducts an experiment to evaluate the measurement uncertainty. Thus, the Monte Carlo simulation needs a large number of samples to ensure that the uncertainty evaluation is as close to the GUM as possible. Unscented transform can alleviate this problem because unscented transform can be regarded as a Monte Carlo simulation with an infinite number of samples. This idea means that unscented transform considers the uncertainty evaluation with respect to the ideal operator. Thus, unscented transform can evaluate the measurement uncertainty the same as the uncertainty that the GUM provides.     

The GUM, Bayesian inference and the observation and measurement equations

In this paper, we compare uncertainty evaluation procedures based on the measurement and observation equation approaches applied to a class of models covering many practical measuring systems. We derive general conditions for when the two approaches give the same distributions associated with the measurand and give examples of how and where they differ. We argue that while it is possible to interpret the measurement equation approach as determining a state of knowledge distribution for the measurand, for some problems there are conceptual, and for highly nonlinear models, practical difficulties with this interpretation. These conceptual difficulties do not arise if the measurement equation approach is interpreted as characterising the behaviour of a measuring system. The discussion presented here is relevant to the revision of the GUM, currently being undertaken by the Joint Committee for Guides in Metrology.     

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.     

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.     

Precision measures for processes with multiple manufacturing lines

This paper deals with measurement uncertainty of virtual instruments (VIs). First the main uncertainty sources of transducer, signal conditioning, A/D conversion and digital signal processing (DSP) are analyzed in detail. Two approaches to evaluate uncertainties of direct and indirect measurements are presented. The first approach deals with measuring an objective variable directly by application of a physical law. Its procedure includes: Step 1: the combined measurement uncertainties of transducer, signal conditioning, A/D conversion, and DSP are estimated respectively according to Type B evaluation of “guide to the expression of uncertainty in measurement (GUM)” based on Gram-Chariler series. Step 2: their corresponding relative measurement uncertainties are calculated, moreover the overall relative uncertainty of the direct measurement is evaluated in rms value. Step 3: the combined uncertainty of the direct measurement is estimated according to the measurement result and the value of overall relative uncertainty of the direct measurement. The second approach involves measurement of an objective variable that is a function of several independent variables; however these variables could be determined by direct respective measurements. The measurement uncertainty of the objective variable could be estimated by applying the “uncertainty propagation law” of GUM. Finally a case study is given to illustrate the application of these approaches.     

Monte carlo method for the uncertainty evaluation of spatial straightness error based on new generation geometrical product specification

Straightness error is an important parameter in measuring high-precision shafts. New generation geometrical product specification(GPS) requires the measurement uncertainty characterizing the reliability of the results should be given together when the measurement result is given. Nowadays most researches on straightness focus on error calculation and only several research projects evaluate the measurement uncertainty based on "The Guide to the Expression of Uncertainty in Measurement(GUM)". In order to compute spatial straightness error(SSE) accurately and rapidly and overcome the limitations of GUM, a quasi particle swarm optimization(QPSO) is proposed to solve the minimum zone SSE and Monte Carlo Method(MCM) is developed to estimate the measurement uncertainty. The mathematical model of minimum zone SSE is formulated. In QPSO quasi-random sequences are applied to the generation of the initial position and velocity of particles and their velocities are modified by the constriction factor approach. The flow of measurement uncertainty evaluation based on MCM is proposed, where the heart is repeatedly sampling from the probability density function(PDF) for every input quantity and evaluating the model in each case. The minimum zone SSE of a shaft measured on a Coordinate Measuring Machine(CMM) is calculated by QPSO and the measurement uncertainty is evaluated by MCM on the basis of analyzing the uncertainty contributors. The results show that the uncertainty directly influences the product judgment result. Therefore it is scientific and reasonable to consider the influence of the uncertainty in judging whether the parts are accepted or rejected, especially for those located in the uncertainty zone. The proposed method is especially suitable when the PDF of the measurand cannot adequately be approximated by a Gaussian distribution or a scaled and shifted t-distribution and the measurement model is non-linear.     

An Improved Measurement Uncertainty Calculation Method of Profile Error for Sculptured Surfaces

The current researches mainly adopt "Guide to the expression of uncertainty in measurement(GUM)" to calculate the profile error. However, GUM can only be applied in the linear models. The standard GUM is not appropriate to calculate the uncertainty of profile error because the mathematical model of profile error is strongly non-linear. An improved second-order GUM method(GUMM) is proposed to calculate the uncertainty. At the same time, the uncertainties in different coordinate axes directions are calculated as the measuring points uncertainties. In addition, the correlations between variables could not be ignored while calculating the uncertainty. A k-factor conversion method is proposed to calculate the converge factor due to the unknown and asymmetrical distribution of the output quantity. Subsequently, the adaptive Monte Carlo method(AMCM) is used to evaluate whether the second-order GUMM is better. Two practical examples are listed and the conclusion is drawn by comparing and discussing the second-order GUMM and AMCM. The results show that the difference between the improved second-order GUM and the AMCM is smaller than the difference between the standard GUM and the AMCM. The improved second-order GUMM is more precise in consideration of the nonlinear mathematical model of profile error.     

基于蒙特卡罗法与GUM法的直线度测量不确定度评定

由于形位误差测量的复杂性和测量结果评定的多样性,导致在实际测量结果中形位误差的不确定度评定成了难题。通过GUM法和蒙特卡罗法对直线度的测量不确定度进行评定。首先,根据最小二乘法得到直线度的误差模型;然后采用GUM方法对测量结果进行不确定度评定,采用蒙特卡罗仿真方法对测量值进行模拟仿真,从而得到直线度误差的不确定度;设置实验对比,通过数据分析验证了蒙特卡罗方法评定的可行性,为形位误差测量结果不确定度评定提供了更加简便的方法。     

激光跟踪仪多边测量的不确定度评定

激光跟踪仪多边测量是大型高端装备制造现场溯源的重要手段,正确评定其不确定度是确保制造过程量值统一、结果可靠的关键。本文提出了一种准确、快速的激光跟踪仪多边测量的不确定度评定方法。从仪器误差、环境干扰及靶球制造误差等方面分析激光跟踪仪多边测量的不确定度来源。针对多边测量的输出量为多维向量的特点,重点研究基于多维不确定度传播律(GUM法)的不确定度合成方法,同步评定目标点坐标和跟踪仪站位的不确定度。最后,介绍了点到点长度的不确定度计算方法。实验表明:GUM法评定的不确定度结果与蒙特卡洛法(MCM法)的结果相比,坐标不确定度偏差小于0.000 2 mm,相关系数偏差小于0.01,满足数值容差,且GUM法用时仅为MCM法的0.08%;点到点长度测试的En值均小于1。因此,基于GUM法评定激光跟踪仪多边测量的不确定度具有可行性及高效性,且评定结果正确、可靠。     

Uncertainty analysis of positional deviations of CNC machine tools

The assessment of the measurement uncertainty is an indispensable task in all calibration procedures. By international accord, the evaluation is to be done in accordance with the ISO Guide to the Expression of Uncertainty in Measurement (GUM). To calibrate the positional deviations of computer numerically controlled (CNC) machine tools, calibration laboratories will usually follow the guidelines in ISO 230-2 International Standard. However, that standard does not address uncertainty. In this paper, we present an uncertainty evaluation scheme that is firmly grounded in the GUM, and can therefore be of use as a guide to develop appropriate uncertainty calculations in this and similar types of calibrations.     

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.     

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.     

Prediction of data variability in hand-arm vibration measurements

The long-term exposure to vibrations transmitted to the human upper limb by hand-held powered tools leads to a group of diseases commonly known as hand-arm vibration (HAV) syndrome. The risk deriving from the vibration exposure can be assessed with direct measurements, using data declared by the tool manufacturer or by retrieving measurements from specific databases. The discrepancies between data belonging to each of these three groups were evidenced in several studies. This paper analyzes the causes of the HAV measurements variability following the ISO GUM approach. The work process was modeled with a lumped parameter scheme of the tool-operator-measurement chain interactions. The measurement uncertainty has been identified propagating the uncertainties of the influencing parameters through the model. The soundness of the approach was verified by comparing the predicted and the observed variability in a specific case study. The major outcome of the proposed method is that the uncertainty budget allows understanding which parameters have to be controlled to limit data dispersion.