Comparison of ISO-GUM and Monte Carlo methods for the evaluation of measurement uncertainty: application to direct cadmium measurement in water by GFAAS

The propagation stage of uncertainty evaluation, known as the propagation of distributions, is in most cases approached by the GUM (Guide to the Expression of Uncertainty in Measurement) uncertainty framework which is based on the law of propagation of uncertainty assigned to various input quantities and the characterization of the measurand (output quantity) by a Gaussian or a t-distribution. Recently, a Supplement to the ISO-GUM was prepared by the JCGM (Joint Committee for Guides in Metrology). This Guide gives guidance on propagating probability distributions assigned to various input quantities through a numerical simulation (Monte Carlo Method) and determining a probability distribution for the measurand.In the present work the two approaches were used to estimate the uncertainty of the direct determination of cadmium in water by graphite furnace atomic absorption spectrometry (GFAAS). The expanded uncertainty results (at 95% confidence levels) obtained with the GUM Uncertainty Framework and the Monte Carlo Method at the concentration level of 3.01 μg/L were ±0.20 μg/L and ±0.18 μg/L, respectively. Thus, the GUM Uncertainty Framework slightly overestimates the overall uncertainty by 10%. Even after taking into account additional sources of uncertainty that the GUM Uncertainty Framework considers as negligible, the Monte Carlo gives again the same uncertainty result (±0.18 μg/L). The main source of this difference is the approximation used by the GUM Uncertainty Framework in estimating the standard uncertainty of the calibration curve produced by least squares regression. Although the GUM Uncertainty Framework proves to be adequate in this particular case, generally the Monte Carlo Method has features that avoid the assumptions and the limitations of the GUM Uncertainty Framework.     

Measurement uncertainty of thermodynamic data

Thermodynamic quantities of chemical reactions are commonly derived from experimental data obtained by chemical analysis. The accuracy of the evaluated thermodynamic quantities is limited by the measurement uncertainty of the analytical techniques applied. Straightforward transfer of metrological rules established for determination of single analytes to the more complex process of evaluating values of thermodynamic quantities is not possible. Computer-intensive statistical methods and Monte Carlo techniques are shown to enable integration of existing metrological concepts. An initial stage of the integration of both concepts is presented, taking solubility data for Am(III) in carbonate media as an illustrative example. A cause and effect diagram is created as a means of identification of sources of uncertainty. The uncertainties are used in a resampling-based Monte Carlo study to produce a probability distribution of the value of a quantity.     

蒙特卡洛法评定橡胶门尼黏度测量的不确定度

根据高聚物流变学原理,建立了门尼黏度测量模型,分析了模型中各个变量的概率分布,利用Mathcad软件进行了测量模型的门尼黏度模拟,给出了模拟最佳值、不确定度及其包含区间,实现了门尼黏度测量不确定度的蒙特卡洛法评定。与GUM法相比,蒙特卡洛法评定门尼黏度测量不确定度具有编程模式化,过程简单等优点,适合多变量测量模型的不确定度评价。     

化学计量中不确定度评定研究进展

测量不确定度是表征合理地赋予被测量之值的分散性的参数。本文针对化学计量不确定度评定基础模型仅适用于线性模型、概率分布为正态分布或缩放位移t分布等局限,介绍了近年来不确定度评定的研究热点:蒙特卡罗方法(Monte Carlo Method,MCM),不确定度评定的来源、评定概念、评估方法及其发展过程,扩大了测量不确定度评定与表示的适用范围。     

基于自适应蒙特卡洛法评定测量不确定度程序开发与应用

进行了基于自适应蒙特卡洛法评定测量不确定度的程序开发与应用。基于Python语言,设计开发自适应蒙特卡洛法评定测量不确定度程序,包含评定过程框架、自定义变量名称模块、过程参数关联计算模块以及蒙特卡洛法采样计算模块。程序界面简洁,操作简单,计算准确,适用于任意多个独立变量、任意多个过程参数及单一被测量的数学模型,为利用自适应蒙特卡洛法评定测量不确定度提供了方便。     

A comparison in the evaluation of measurement uncertainty in analytical chemistry testing between the use of quality control data and a regression analysis

The evaluation of measurement uncertainties has been widely applied to the calibration of measurement instruments, whereas its application to tests, despite increasing requirements, is a more recent phenomenon. The generalization of the evaluation of measurement uncertainties to tests has been a gradual process, in line with changes in the requirements of the normative framework that regulates the accreditation of tests laboratories and also as the perceived good practices have evolved. The sole identification of the relevant sources of uncertainty was followed by the requirement to provide a simplified estimate of the measurement uncertainty, and it is now an accepted requirement to properly evaluate the expanded measurement uncertainty associated with any tests. In this study, the evaluation of measurement uncertainty associated with the determination of sulfate in water will be attempted using a procedure that includes linear regression, with the regression parameters provided with associated uncertainties, and a Monte Carlo method applied as a validation tool of the conventional mainstream evaluation method, concerning the approximations in terms of linearization of the model and the assumed shape of the output distribution introduced by this approach.     

Monte Carlo simulation for the evaluation of measurement uncertainty of spent fuel analytical results

The present paper describes an approach based on Monte Carlo simulation for the evaluation of uncertainty of nuclear spent fuel analysis. The mathematical model of measurement was established by examining the dissolution process step by step. The results are consistent with those obtained by the classical propagation of variance approach. This paper shows the importance of taking the process into account in order to give a more reliable uncertainty assessment to the result of a concentration ratio of two isotopes in spent fuel. Indeed, for some radionuclides, the uncertainty associated with the upstream steps of the analysis (“process” uncertainty) can represent up to 95 % of the overall uncertainty.     

On the use of the ‘uncertainty budget’ to detect dominant terms in the evaluation of measurement uncertainty

In the evaluation of measurement uncertainty, the uncertainty budget is usually used to identify dominant terms that contribute to the uncertainty of the output estimate. Although a feature of the GUF method, it is also recommended as a qualitative tool in MCM by using ‘nonlinear’ equivalents of uncertainty contributions and sensitivity coefficients. In this paper, the use of ‘linear’ and ‘nonlinear’ parameters is discussed. It is shown that when and only when the standard uncertainty of the output estimate is nearly equal to the square root of the sum of the squares of the individual uncertainty contributions, will the latter be a reliable tool to detect the degree of contribution of each input quantity to the measurand uncertainty.     

Monte Carlo computation of ground‐state energy derivatives

Monte Carlo is a simple technique, which uses random numbers to compute ground‐state energies of small molecules (and quantum systems in general). The results always have a small statistical error, which poses a major obstacle when estimating properties defined as ground‐state‐energy derivatives (such as the molecule's geometry, its vibrational frequencies, polarizabilities, etc.). In this article, we present and demonstrate an approach that makes an accurate Monte–Carlo estimation of such derivatives possible. This is achieved by realizing that the simulation constitutes an autocorrelated stochastic process, whose proper analysis then enables us to estimate various energy derivatives as a combination of total correlation between readily computable quantities. The resulting procedure is a natural extension of the usual Monte Carlo algorithm for computing the ground‐state energy, with relatively small computational overhead. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Effect of insignificant bias and its uncertainty on the coverage probability of uncertainty intervals Part 1. Evaluation for a given value of the true bias

This paper investigates the coverage probability of the uncertainty intervals determined in compliance with the GUM and EURACHEM Guide, which are defined by expanded uncertainty U about the results uncorrected with the insignificant biases and corrected with the significant biases. This coverage probability can significantly fall below the chosen level of confidence in some cases as Maroto et al. discovered by using the Monte Carlo method. Their numerical results obtained provided that only the β errors have occurred in the test significance and findings that the coverage reduction depends on the mutual proportions of the magnitudes of the systematic error, overall uncertainty and bias uncertainty are confirmed in this paper by using probability calculus and numerical integration. This problem is also studied when all possible experimental biases, both significant and insignificant, are considered. From this point of view, the reduction of the coverage probability turns out to be less severe than from the previous one. The coverage probability is also investigated for some uncertainty intervals computed in different ways than the above mentioned documents recommend. The intervals defined by U about the results corrected with both significant and insignificant bias give always the same coverage probability equalling the chosen level of confidence. The intervals with some uncertainties modified or enlarged with the insignificant biases remove or moderate the coverage reduction.     

Look at data to evaluate uncertainty: a comment on “Uranium assay determination …” by Mathew et al.

In a recent paper, Mathew et al. detailed, for a specific titration-based assay of uranium, a “step-by-step approach to calculate the GUM uncertainty of the measurand”, in which their uncertainty assessment was based solely on prior knowledge, ignoring the manifest variability in their replication data. A simple analysis of the variance from their data reveals that the uncertainty in the average of the replicated quantity (TEF) was at least 3.5 times their estimate. Since the observables that contribute most to the final uncertainty in their method were not replicated, it is unknown whether the estimates for the uncertainties of those quantities, and thus of the output quantity, were also underestimated. This comment demonstrates how a better uncertainty evaluation is possible by extracting as much knowledge as possible from the extant data.     

A mathematical model of measurement uncertainty of single substrate enzyme assays

Clinical laboratory tests provide critical information at every stage of the medical decision‐making process, and measurement of the activity levels of enzymes such as alkaline phosphatase, lactate dehydrogenase, etc. provide information regarding various body functions such as the liver and gastrointestinal tract. The uncertainty associated with these enzyme measurement processes describes the quality of the measurement process, and therefore methods to improve the quality of the measurement process require minimizing the measurement uncertainty of the enzyme assay. In this study, we develop a mathematical model of the lactate dehydrogenase measurement process, with uncertainty introduced into its parameters that represent the sources of variation in the different components and stages of the measurement process. The Monte Carlo method is then utilized to estimate the uncertainty associated with the model, and therefore the measurement process. An empirical function used to generate estimates of uncertainty for patient samples with unknown activity levels is constructed using the model. The model is then used to quantify the contributions of the individual sources of uncertainty to the net measurement uncertainty and also quantify the effect of uncertainty within the calibration process on the distribution of the measurement result. Copyright © 2014 John Wiley & Sons, Ltd.     

Modeling of branching ratio uncertainty in chemical networks by Dirichlet distributions

Validation of complex chemical models relies increasingly on uncertainty propagation and sensitivity analysis with Monte Carlo sampling methods. The utility and accuracy of this approach depend on the proper definition of probability density functions for the uncertain parameters of the model. Taking into account the existing correlations between input parameters is essential to a reliable uncertainty budget for the model outputs. We address here the problem of branching ratios between product channels of a reaction, which are correlated by the unit value of their sum. We compare the uncertainties on predicted time-dependent and equilibrium species concentrations due to input samples, either uncorrelated or explicitly correlated by a Dirichlet distribution. The method is applied to the case of Titan ionospheric chemistry, with the aim of estimating the effect of branching ratio correlations on the uncertainty balance of equilibrium densities in a complex model.     

Approach to accuracy assessment of the glass-electrode potentiometric determination of acid-base properties

Thermodynamic data are suitable subject for investigating strategies and concepts for the evaluation of complete measurement uncertainty budgets in situations where the measurand cannot be expressed in a mathematical formula. Some suitable approaches are the various forms of Monte Carlo simulations in combination with computer-intensive statistical methods that are directed to an evaluation of empirical distribution curves for the uncertainty budget. Basis of the analysis is a cause-and-effect diagram. Some experience is available with cause-and-effect analysis of thermodynamic data derived from spectrophotometric data. Another important technique for the evaluation of thermodynamic data is glass-electrode potentiometry. On basis of a newly derived cause-and-effect diagram, a complete measurement uncertainty budget for the determination of the acidity constants of phosphoric acid by glass-electrode potentiometry is derived. A combination of Monte Carlo and bootstrap methods is applied in conjunction with the commercially available code SUPERQUAD. The results suggest that glass-electrode potentiometry may achieve a high within-laboratory precision because major uncertainty contributions become evident via interlaboratory comparisons. This finding is further underscored by analysing available literature data.     

Full optimization of Jastrow-Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules

We pursue the development and application of the recently introduced linear optimization method for determining the optimal linear and nonlinear parameters of Jastrow-Slater wave functions in a variational Monte Carlo framework. In this approach, the optimal parameters are found iteratively by diagonalizing the Hamiltonian matrix in the space spanned by the wave function and its first-order derivatives, making use of a strong zero-variance principle. We extend the method to optimize the exponents of the basis functions, simultaneously with all the other parameters, namely, the Jastrow, configuration state function, and orbital parameters. We show that the linear optimization method can be thought of as a so-called augmented Hessian approach, which helps explain the robustness of the method and permits us to extend it to minimize a linear combination of the energy and the energy variance. We apply the linear optimization method to obtain the complete ground-state potential energy curve of the C(2) molecule up to the dissociation limit and discuss size consistency and broken spin-symmetry issues in quantum Monte Carlo calculations. We perform calculations for the first-row atoms and homonuclear diatomic molecules with fully optimized Jastrow-Slater wave functions, and we demonstrate that molecular well depths can be obtained with near chemical accuracy quite systematically at the diffusion Monte Carlo level for these systems.     

Effects of overlying soft tissue on X-ray fluorescence bone lead measurement uncertainty

The effects of overlying soft tissue on the measurement uncertainty of the in vivo ¹⁰⁹Cd K-shell X-ray fluorescence (XRF) technique were investigated, as applied to the tibia bone site. Experimental measurements were performed on a set of nine leg phantoms of different soft tissue thickness, intended to model the lower leg at mid-tibia. A standard bone phantom made from plaster-of-Paris and having a nominal lead concentration of 25.6 μg Pb per gram was used in all trials. Monte Carlo simulations of the experimental set-up were also performed. Results indicate a strong relation between measurement uncertainty and overlying tissue thickness (OTT) for the XRF bone lead method. In increasing the OTT from 3.2 to 14.6 mm, an increase in average measurement uncertainty by a factor of 2.40 was observed experimentally. Monte Carlo simulations indicated an increase in minimum detectable limit (MDL) by a factor of 2.46 over the same interval. Experimental and Monte Carlo results were generally in strong agreement. For subject screening purposes, direct measurement of soft tissue overlying the tibia is recommended whenever practical.     

Application of cause-and-effect diagrams to the interpretation of UV-Vis spectroscopic data

The application of cause-and-effect diagrams to the evaluation of thermodynamic data from UV-Vis absorption spectroscopic analysis is demonstrated. The contributions of measurement uncertainty identified from a cause-and-effect diagram are implemented into a Monte Carlo procedure based on the threshold bootstrap computer-assisted target factor analysis (TB CAT). This algorithm aims at an improvement of data comparability and accounts for non-normality, spectral, residual and parameter correlation as well as random noise in target factor analysis. The ISO Type-B measurement uncertainties are included into the process by normally distributed random numbers with specified mean values and dispersions. The TB CAT procedure is illustrated by a flow diagram and a case study of Nd(III) complexation by picolinic acid N-oxide (pic NO) in aqueous solution. Using 12 experimental spectra as input data, the single component spectra and the formation constant 1g betaML of the Nd(pic NO)2+ species are obtained together with the respective probability density distributions. The role of the cause-and-effects approach on the further development of chemical thermodynamics is discussed.     

Estimation of uncertainty in routine pH measurement

A procedure for estimation of measurement uncertainty of routine pH measurement (pH meter with two-point calibration, with or without automatic temperature compensation, combination glass electrode) based on the ISO method is presented. It is based on a mathematical model of pH measurement that involves nine input parameters. Altogether 14 components of uncertainty are identified and quantified. No single uncertainty estimate can be ascribed to a pH measurement procedure: the uncertainty of pH strongly depends on changes in experimental details and on the pH value itself. The uncertainty is the lowest near the isopotential point and in the center of the calibration line and can increase by a factor of 2 (depending on the details of the measurement procedure) when moving from around pH 7 to around pH 2 or 11. Therefore it is necessary to estimate the uncertainty separately for each measurement. For routine pH measurement the uncertainty cannot be significantly reduced by using more accurate standard solutions than ±0.02 pH units – the uncertainty improvement is small. A major problem in estimating the uncertainty of pH is the residual junction potential, which is almost impossible to take rigorously into account in the framework of a routine pH measurement.¹ Received: 11 August 2001 Accepted: 22 February 2002     

Repeatability: some aspects concerning the evaluation of the measurement uncertainty

Various publications stress the importance of the repeatability (i.e. precision) of the calculation of the measurement of uncertainty. We reveal by detailing an example from production control in the pharmaceutical industry that the effect of other influence quantities should not be neglected, because their magnitude is even larger than the contribution of repeatability. We review the role of repeatability within the calculation of measurement uncertainty for several common validation and day-to-day measurement scenarios. They show that measurement models need to consider the measurement sequences of the various scenarios. Otherwise the size and effect of the repeatability might be overestimated. At the end Monte Carlo simulations were used to investigate the determination of the repeatability under certain restrictions. The simulation uncovered a significant bias toward the common formula for calculating the standard deviation when it is based on a duplicated measurement of a sample. Papers published in this section do not necessarily reflect the opinion of the Editors, the Editorial Board and the Publisher     

Interpolated potential energy surfaces: How accurate do the second derivatives have to be?

A global potential energy surface for the water dimer is constructed using the modified Shepard interpolation scheme of Collins et al. According to this interpolation scheme, the energy at an arbitrary geometry is expressed as a weighted sum of Taylor series expansions from neighboring data points, where the energy and derivative data required are obtained from ab initio calculations. For some ab initio methods, errors are introduced into the second derivative matrix, either by numerical differencing of ab initio energies or numerical integration during the ab initio calculation. Therefore, we test the accuracy required of the second derivative data by truncation of the exact second derivatives to a series of approximate second derivatives, and assess the effect on the results of a quantum diffusion Monte Carlo (QDMC) simulation. Our results show that the calculated zero-point energy and wave function histograms converge to within the numerical uncertainty of the QDMC simulation by inclusion of either three significant figures or three decimal places in the second derivatives.