Evaluation Of Standard Uncertainty Research Articles

Type A standard uncertainty evaluation in one measurement through uncertainty propagation from voxel values’ distribution for computed tomography metrology

According to the guide to the expression of uncertainty in measurement, ‘type A evaluation’ generally requires repeated measurements, which are time-consuming for CT scans. To solve this problem, we developed a method for estimating the standard deviation of measurement results in one measurement through uncertainty propagation, which can be regarded as repeatability standard deviation to evaluate the type A standard uncertainty. The method first fits the CT voxel value distribution, uses the ISO50 method to determine the spatial distribution of surface points from the voxel value distribution and edge shape interpolation, and then derives the measurement results by fitting geometric parameters with the least square algorithm. Finally, the standard deviation of the measurement results is evaluated according to the distribution of the surface point position through uncertainty propagation. We performed simulations and experiments using the hole-plate with 28 holes to compare the uncertainty evaluated by our method and the type A standard uncertainty evaluated on the basis of a series of observations obtained under repeatability conditions. Both simulation and experimental results show that these two uncertainties follow the same statistical variation pattern. The Pearson correlation coefficients of the two uncertainties in simulation and experiment are 0.79 and 0.33, respectively, indicating that the uncertainty evaluated by the proposed method can directly replace the type A uncertainty or provide a reference similar to type A uncertainty for the evaluation of the combined uncertainty.

Measurement science meets the reproducibility challenge

Measurement science is particularly well equipped not only to meet reproducibility challenges arising within the field of metrology, but also to suggest strategies and best practices for how such challenges can be met in other fields. This contribution illustrates three such challenges, in three different fields, and proposes ways to address them that can supplement the only way in which reproducibility challenges in science can be resolved definitively: by validated scientific advances that point toward the truth. The first example concerns a large interlaboratory, international comparison of the measurement of the mass fraction of silica in a granite reference material, using classical methods of wet analytical chemistry, carried out in the 1940s. The results delivered a shock to analysts worldwide about the state of the art at the time. The challenge was magnified by the fact that none of the measured values was qualified with an evaluation of measurement uncertainty. We offer an approach developed by Andrew Rukhin from NIST, for how to compute a meaningful consensus value in such case, and explain how the associated uncertainty can be characterized. The second example is about the currently hot topic of the Hubble tension, which refers to the mutual inconsistency of the measurement results, obtained by different methods, for the Hubble-Lemaître constant, which expresses the rate of expansion of the Universe. We suggest that such tension can be quantified in terms of the dark uncertainty that figures as a parameter in a laboratory random effects model, thus providing an objective metric whereby progress toward resolving such tension can be gauged. The third example discusses two sources of lack of reproducibility: on the one hand, the fact that different laboratories produced strikingly discrepant values for the mass fraction of arsenic in kudzu; on the other hand, that different models can be fitted to these data, each producing its own set of results. Here we use a Bayesian model selection criterion to choose one from among four models that are natural candidates to address this double reproducibility challenge. This third example also affords us the opportunity to deflate two widespread myths: that one needs at least four observations to obtain a Bayesian evaluation of standard uncertainty, and that sample standard deviations of small samples are systematically too small.

Evaluation of the Influence of Machine Tools on the Accuracy of Indoor Positioning Systems

In recent years, the use of indoor localization techniques has increased significantly in a large number of areas, including industry and healthcare, primarily for monitoring and tracking reasons. From the field of radio frequency technologies, an ultra-wideband (UWB) system offers comparatively high accuracy and is therefore suitable for use cases with high precision requirements in position determination, for example for localizing an employee when interacting with a machine tool on the shopfloor. Indoor positioning systems with radio signals are influenced by environmental obstacles. Although the influence of building structures like walls and furniture was already analysed in the literature before, the influence of metal machine tools was not yet evaluated concerning the accuracy of the position determination. Accordingly, the research question for this article is defined: To what extent is the positioning accuracy of the UWB system influenced by a metal machine tool?The accuracy was measured in a test setup, which consists of a total of four scenarios in a production environment. For this purpose, the visual contact between the transmitter and the receiver modules, including the influence of further interfering factors of a commercially available indoor positioning system, was improved step by step from scenario 1 to 4. A laser tracker was used as the reference measuring device. The data was analysed based on the type A evaluation of standard uncertainty according to the guide to the expression of uncertainty in measurement (GUM). It was possible to show an improvement in standard deviation from 87.64cm±32.27cm to 6.07cm±2.24cm with confidence level 95% and thus provides conclusions about the setup of an indoor positioning system on the shopfloor.

An accurate orientation measurement system of self-propelled artillery barrel based on coordinate system mapping

Barrel orientation accuracy is an essential factor affecting the hit rate of self-propelled artillery, which comprehensively reflects the working state of the self-propelled artillery fire control system. An accurate self-propelled artillery orientation measurement system based on the coordinate system mapping is proposed to reduce the axis simulation error and introduce the north datum. The measurement system uses Radio Navigation Satellite System orientation technology to determine the measurement area’s standard north orientation. Based on the standard north orientation, a coordinate measurement system is established. The total station and self-propelled artillery positions are unified to the coordinate measurement system. The total station measures the coordinates of the marked points on the barrel. According to the coordinates of the marker points under different datums, the mapping relationship of different coordinate systems can be obtained. The total station measures the azimuth and elevation angle of the line connecting the marked points. Based on the coordinate system mapping relationship, the accurate orientation of the self-propelled artillery barrel can be calculated. The combined standard uncertainty evaluation is performed, and the uncertainties of the azimuth and elevation angles are and , respectively. A particular type of self-propelled artillery is selected for experiments. The results show that the system’s measurement accuracy can be guaranteed within .

An informed type A evaluation of standard uncertainty valid for any sample size greater than or equal to 1

An informed type A evaluation of standard uncertainty is here derived based on Bayesian analysis. The result is mathematically simple, easily interpretable, applicable both in the theoretical framework of the Guide to the Expression of Uncertainty in Measurement (propagation of standard uncertainties) and in that of the Supplement 1 of the Guide (propagation of distributions), valid for any size greater than or equal to 1 of the sample of present observations. The evaluation consistently addresses prior information in the form of the sample variance of a series of recorded experimental observations and in the form of an educated guess based on expert’s experience. It turns out that distinction between type A and type B evaluation is, in this context, contrived.

GUM method for evaluation of measurement uncertainty: BPL long wave time service monitoring

This paper constitutes the first report on the uncertainty evaluation based on Guide to the expression of Uncertainty in Measurement (GUM) method for BPL long wave time service monitoring. First, the applicability of the GUM method was considered from two perspectives: the BPL long wave time service monitoring model and the monitoring data distribution. A complete monitoring measurement model was then established. Subsequently, the uncertainty sources were analyzed via Type A and Type B evaluation. The bootstrap method was used to compare and verify the standard uncertainty evaluation by the GUM method. Thus, the uncertainty results for BPL time service monitoring were obtained under the existing monitoring model and equipment conditions. Furthermore, this paper also provides certain suggestions for improving the quality of time service monitoring. This work is expected to facilitate methods and provide guidance for evaluating the uncertainty in corresponding monitoring systems of other land-based time service systems.

A Review on the Statistical Methods and Implementation to Homogeneity Assessment of Certified Reference Materials in Relation to Uncertainty

An importance of data analysis, methods for homogeneity test and standard uncertainty evaluation associated in any measurement for exact quantification of certified value of any product is vital to be stressed in the scientific community. Herein, we have collectively summarized the detailed discussion on the basics of statistical parameters such as mean, median, mode, standard deviation, variance, range, normal distribution, and central limit theorem. Various statistical analysis methods such as z test, t test, Chi-squared test, and ANOVA including F test have also been discussed in great detail to test the homogeneity of samples for certification of the reference material. The ISO guide 35 (2006) and Guide to Uncertainty in Measurement (GUM) are primarily considered to describe the basic concept of evaluating the associated uncertainty in the light of GUM modelling approach to avoid the error in the measurement which normally occurs in many scientific reports.

Three-Dimensional Wind Measurement Based on Ultrasonic Sensor Array and Multiple Signal Classification.

The wind power industry continues to experience rapid growth worldwide. However, the fluctuations in wind speed and direction complicate the wind turbine control process and hinder the integration of wind power into the electrical grid. To maximize wind utilization, we propose to precisely measure the wind in a three-dimensional (3D) space, thus facilitating the process of wind turbine control. Natural wind is regarded as a 3D vector, whose direction and magnitude correspond to the wind’s direction and speed. A semi-conical ultrasonic sensor array is proposed to simultaneously measure the wind speed and direction in a 3D space. As the ultrasonic signal transmitted between the sensors is influenced by the wind and environment noise, a Multiple Signal Classification algorithm is adopted to estimate the wind information from the received signal. The estimate’s accuracy is evaluated in terms of root mean square error and mean absolute error. The robustness of the proposed method is evaluated by the type A evaluation of standard uncertainty under a varying signal-to-noise ratio. Simulation results validate the accuracy and anti-noise performance of the proposed method, whose estimated wind speed and direction errors converge to zero when the SNR is over 15 dB.

Influence of sample surface height for evaluation of peak extraction algorithms in confocal microscopy.

The axial resolution of confocal microscopy is not only dependent on optical characteristics but also on the utilized peak extraction algorithms. In previous evaluations of peak extraction algorithms, sample surface height is generally assumed to be zero, and only sampling-noise-induced peak extraction uncertainty was analyzed. Here we propose a sample surface-height-dependent (SHD) evaluation model that takes the combined considerations of sample surface height and noise for comparisons of algorithms' performances. Monte Carlo simulations were first conducted on the centroid algorithm and several nonlinear fitting algorithms such as the parabola fitting algorithm, Gaussian fitting algorithm, and sinc2 fitting algorithm. Subsequently, the evaluation indicators, including mean peak extraction error and mean uncertainty were suggested for the algorithms' performance ranking. Finally, experimental verifications of the SHD model were carried out using a fiber-based chromatic confocal system. From our simulations and experiments, we demonstrate that sample surface height is a critical influencing factor in peak extraction computation in terms of both the accuracy and standard deviations. Compared to the conventional standard uncertainty evaluation model, our SHD model can provide a more comprehensive characterization of peak extraction algorithms' performance and offer a more flexible and consistent reference for algorithm selection.

Bayesian methods for type A evaluation of standard uncertainty

One of the aims of the revision of the guide to the expression of uncertainty in measurement (GUM), is to make it self-consistent as the current edition would combine frequentist and Bayesian statistical methods. The methods for type A evaluation of standard uncertainty in the GUM are considered to be ‘frequentist’ and hence would have to be replaced by their Bayesian counterparts. The aim of this paper is to propose Bayesian methods for four mainstream type A evaluations of standard uncertainty, based on the same assumptions as those underlying the current methods for type A evaluation given in the GUM. Specific attention is given to the computational aspects of the Bayesian methods. An approach involving weakly informative prior distributions is proposed to ensure a proper posterior distribution, still allowing the data to dominate it. The applications for which Bayesian methods are proposed include the calculation of a mean of a series of observations, analysis of variance, ordinary least squares and errors-in-variables regression. Examples H.3 (ordinary least squares) and H.5 (analysis of variance) from the GUM are reworked accordingly.The second aim of this paper is to propose Bayesian methods for type A evaluation that take the information typically available in metrology into account in the form of weakly informative prior distributions. These methods can applied in a wide variety of situations. By comparing the results from the proposed Bayesian methods involving weakly informative priors with their classical counterparts, it is seen that in most instances these methods give very similar estimates and standard uncertainties as classical methods do under the same conditions and assumptions. Estimates are often identical up to the last meaningful digit, and the standard uncertainties are usually slightly yet persistently larger in comparison to its classical counterpart. In many cases these differences turn out to have limited or little practical meaning, save in the case of a small series of observations, where the differences in the computed standard uncertainty of the mean is larger.

Evaluation of uncertainty in measurement of high voltage pulse

An uncertainty model has been proposed which is based on the black-box concept to evaluate the uncertainty of measurement and calibration in high voltage pulse divider. The initial model was proposed in principle, unnecessary variables in the formula could be omissible in the calibration experiment. For the model absent of uncertainty originating from calibration document, and resolving power of oscilloscope, the correction components which belong to the black-box model would be introduced to the formula to perfect the model. The uncertainties were combined after perfection of model according to the law of propagation of uncertainty. The results are compared with that combined from relative uncertainty. It is shown that the uncertainty combined from relative uncertainty will be correct if this model is only composed of multiplication and division of different variables, or addition and subtraction of different variables are also permitted if the mathematical expectation values of these items equal to zero. This is proven by calculating the voltage potential between two different positions according to the measured voltage pulse at those positions and analyzing the attenuation represented by dB scales. In the calibration experiment of high voltage divider, the combined standard uncertainty of dividers scaling factor is obtained by evaluating Type A standard uncertainty of the ratio of output to input. The influence of dispersion from signal source output on evaluation of standard uncertainty can be weakened with this method.

Analytical Standard Uncertainty Evaluation Using Mellin Transform

Uncertainty evaluation plays an important role in ensuring that a designed system can indeed achieve its desired performance. There are three standard methods to evaluate the propagation of uncertainty: 1) analytic linear approximation; 2) Monte Carlo (MC) simulation; and 3) analytical methods using mathematical representation of the probability density function (pdf). The analytic linear approximation method is inaccurate for highly nonlinear systems, which limits its application. The MC simulation approach is the most widely used technique, as it is accurate, versatile, and applicable to highly nonlinear systems. However, it does not define the uncertainty of the output in terms of those of its inputs. Therefore, designers who use this method need to resimulate their systems repeatedly for different combinations of input parameters. The most accurate solution can be attained using the analytical method based on pdf. However, it is unfortunately too complex to employ. This paper introduces the use of an analytical standard uncertainty evaluation (ASUE) toolbox that automatically performs the analytical method for multivariate polynomial systems. The backbone of the toolbox is a proposed ASUE framework. This framework enables the analytical process to be automated by replacing the complex mathematical steps in the analytical method with a Mellin transform lookup table and a set of algebraic operations. The ASUE toolbox was specifically designed for engineers and designers and is, therefore, simple to use. It provides the exact solution obtainable using the MC simulation, but with an additional output uncertainty expression as a function of its input parameters. This paper goes on to show how this expression can be used to prevent overdesign and/or suboptimal design solutions. The ASUE framework and toolbox substantially extend current analytical techniques to a much wider range of applications.

Standard uncertainty evaluation of multivariate polynomial

The uncertainty propagation law helps to infer the uncertainty of unobservable variables from known or assumed relationship with observable variable. Currently only analytical linear approximation or Monte Carlo simulation methods is widely adopted. The former method is limited to weakly nonlinear systems while the latter does not provide any analytical expression that links the uncertainties of input to uncertainties of output quantities. This paper proposes procedures to evaluate the standard uncertainty of multivariate polynomial using basic algebraic manipulation and tabulated Mellin transform. Case studies are presented whereby the effectiveness and practicality of the proposed method is demonstrated. The proposed method can be readily automated using computer algebra systems, thus negating the need for practitioners to perform the actual complex computation. The work theoretically enriches and extends the validity of the analytic approach in the existing uncertainty evaluation framework, thus enabling the analytic evaluation of uncertainty for many nonlinear cases which was previously an impossible task.

Standard Uncertainty Evaluation for Dynamic Tensile Properties of Auto-Body Steel-Sheets

This paper is concerned with the standard uncertainty of the true stress–true strain curve as the tensile properties of auto-body steel sheets at intermediate strain rates ranged from 1 to 100 s−1. A procedure to obtain true stress–true strain data is properly designed for the experiment and data acquisition. An analytic model is then established to evaluate the standard uncertainty of the measurand. The measurand in this case is the true stress which is a function of the input quantities: the tensile load; the initial length, the thickness and the width of a specimen; and the deformed length of a specimen. Sources of uncertainties of the input quantities are evaluated for the high speed tensile test with their associated sensitivity coefficients. Uncertainty of the stress data acquired is also considered in the procedure of the fast Fourier transform (FFT) smoothing process used to remove unnecessary signals acquired from experiments. Image analysis using a high speed camera is carried out to measure deformation of the specimen during high speed tensile tests with proper uncertainty evaluation. A combined standard uncertainty is evaluated from the uncertainties of the input quantities as well as the influence factor for the true stress of auto-body steel sheets at intermediate strain rates. Consequently, the true stress–true strain data are obtained with proper standard uncertainty evaluation.

Evaluation of the systematic shifts and absolute frequency measurement of a single Ca+ ion frequency standard

This paper provides a detailed description of the 40Ca+ optical frequency standard uncertainty evaluation and the absolute frequency measurement of the clock transition, as a summary and supplement for the published papers of Yao Huang et al. (Phys Rev A 84:053841, 1) and Huang et al. (Phys Rev A 85:030503, 2). The calculation of systematic frequency shifts, expected for a single trapped Ca+ ion optical frequency standard with a “clock” transition at 729 nm is described. There are several possible causes of systematic frequency shifts that need to be considered. In general, the frequency was measured with an uncertainty of 10−15 level, and the overall systematic shift uncertainty was reduced to below a part in 10−15. Several frequency shifts were calculated for the Ca+ ion optical frequency standard, including the trap design, optical and electromagnetic fields geometry and laboratory conditions, including the temperature condition and the altitude of the Ca+ ion. And we measured the absolute frequency of the 729-nm clock transition at the 10−15 level. An fs comb is referenced to a hydrogen maser, which is calibrated to the SI-second through the Global Positioning System (GPS). Using the GPS satellites as a link, we can calculate the frequency difference of the two hydrogen masers with a long distance, one in WIPM (Wuhan) and the other in National Institute of Metrology (NIM, Beijing). The frequency difference of the hydrogen maser in NIM (Beijing) and the SI-second calculated by BIPM is published on the BIPM web site every 1 month, with a time interval of every 5 days. By analyzing the experimental data obtained within 32 days of a total averaging time of >2 × 106 s, the absolute frequency of the 40Ca+ 4 s 2 S 1/2–3d 2D5/2 clock transition is measured as 411 042 129 776 393.0 (1.6) Hz with a fractional uncertainty of 3.9 × 10−15.

Evaluation of systematic shifts of the88Sr+single-ion optical frequency standard at the10−17level

An ion trap of the end-cap design was built recently at the National Research Council of Canada for improved control of the ${}^{88}$Sr${}^{+}$ single-ion optical frequency standard systematic shifts. The uncertainty on the micromotion-induced shifts is smaller by more than four orders of magnitude when compared to our previous trap system and reaches a fractional frequency uncertainty of $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}$. To obtain this low uncertainty level, the micromotion is minimized with trim electrodes and the trap is operated at a special frequency at which there is anticorrelation between the second-order Doppler and Stark shifts. This choice of operating frequency, determined by the differential scalar polarizability of the clock transition, yields a suppression by a factor of $\ensuremath{\approx}$28 in the combined micromotion shifts. Like many optical frequency standards, the dominant source of uncertainty in the new trap is the blackbody radiation shift. Its uncertainty has been reduced by an order of magnitude with a recent theoretical evaluation of the differential scalar polarizability of the clock transition. The fractional blackbody shift uncertainty, estimated using a model of the blackbody field at the ion, is $2.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}$. The otherwise dominant electric quadrupole shift is reduced to below the $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}19}$ level with a cancellation method based on the average frequency of several pairs of Zeeman components. This method also cancels the tensor Stark shift and simplifies the description of the frequency shifts that are quantization-axis dependent. This paper provides a detailed description of the ${}^{88}$Sr${}^{+}$ optical frequency standard uncertainty evaluation and the methods used to make the standard robust against changes in the trap environment. The total fractional frequency uncertainty of the ${}^{88}$Sr${}^{+}$ ion for our current system is estimated at $2.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}$. We also discuss the uncertainty evaluation of a recently reported measurement of the ${}^{88}$Sr${}^{+}$ $S$-$D$ clock transition made over a 2-month period by comparison with a maser referenced to the SI second. The frequency measured is $444\phantom{\rule{0.16em}{0ex}}779\phantom{\rule{0.16em}{0ex}}044\phantom{\rule{0.16em}{0ex}}095\phantom{\rule{0.16em}{0ex}}485.5(9)$ Hz, with an uncertainty limited by the evaluation of the maser frequency.

Evaluation of an optical coordinate measuring machine for measuring grated structures

The use of optical coordinate measuring machines (CMMs) in industrial metrology shows great potential due to a high measurement speed and non-contact working principle. However, at present measurements performed by different users present too large variations. The uncertainty contributions are difficult to quantify and standard uncertainty evaluation procedures are needed. In order to address this issue, we have investigated the uncertainty sources of commercial optical CMMs. We present a comparison on chessboard standards between three different vision systems. The influence of critical measurement parameters on image formation and analysis is discussed.

Trapezoidal and triangular distributions for Type B evaluation of standard uncertainty

The Guide to the Expression of Uncertainty in Measurement (GUM), published by the International Organization for Standardization (ISO), recognizes Type B state-of-knowledge probability distributions specified by scientific judgment as valid means to quantify uncertainty. The ISO-GUM discusses symmetric probability distributions only. Sometimes an asymmetric distribution is needed. We describe a trapezoidal distribution which may be asymmetric depending on the settings of its parameters. We describe the probability density function, cumulative distribution function (cdf), inverse function of the cdf, moment generating function, moments about origin (zero), expected value and variance of a trapezoidal distribution. We show that triangular and rectangular distributions are special cases of the trapezoidal distribution. Then we derive the moment generating functions, moments, expected values and variances of various special cases of the trapezoidal distribution. Finally, we illustrate through a real life example how a Type B asymmetric trapezoidal distribution may be useful in quantifying a correction for bias (systematic error) in a result of measurement and in quantifying the standard uncertainty associated with the correction.

Standard uncertainty evaluation in image-based measurements

The present paper is devoted to address the use and evaluation of measures coming from digital images in an industrial context. Starting from a discussion about the importance and role of this class of measurements, the authors show the architecture of an image-based measurement system and introduce the important issue of the related uncertainty then coming to investigate models and methods for evaluating it. After presenting an original method for evaluating the uncertainty of images deriving measurements, some examples of practical interest are introduced and evaluated leading the reader to the understanding of related problems. The paper is also enriched by the presence of an introductive section discussing a bibliography of the state of the art in the field.

The evaluation of standard uncertainty in the presence of limited resolution of indicating devices

Since the type A evaluation of standard uncertainty according to the rules given in the ISO Guide to the Expression of Uncertainty in Measurement does not take into account the uncertainty that arises from the limited resolution of indicating devices, we show in this design note that the treatment of this problem calls for a quantity about which no statistical information is available; therefore, its uncertainty has to be derived from a type B evaluation.