Advanced Mathematical and Computational Tools in Metrology and Testing XII

The following sections are included:

• Dedication
• Foreword
• Book Editors
• Contents

Explicit algebraic expressions are derived for the roots of a cubic equation having one real and two complex roots. As opposed to traditional methods for direct (that is, non-iterative) solution, evaluation of the expressions is unconditionally numerically stable. A floating-point error analysis of the computations is given, which shows that the computed roots have extremely small relative errors. The computations are proved to be backward stable, that is, the computed roots are exact for a closely neighbouring cubic equation. Applications and uncertainty propagation are also considered.

This paper is concerned with the approximation of data using linear models for which there is additional information to be incorporated into the analysis. This situation arises, for example, in Tikhonov regularisation, model averaging, Bayesian inference and approximation with Gaussian process models. In a least-squares approximation process, the m-vector of data y is approximated by a linear function ŷ = Hy of the data, where H is an m × m matrix related to the observation matrix. The effective number of degrees of freedom associated with the model is given by the sum of the eigenvalues of H. For standard linear least-squares regression, the matrix H is a projection and has n eigenvalues equal to 1 and all others zero, where n is the number of parameters in the model. Incorporating prior information about the parameters reduces the effective number of degrees of freedom since the ability of the model to approximate the data vector y is partially constrained by the prior information. We give a general approach for providing bounds on the effective number of degrees of freedom for models with prior information on the parameters and illustrate the approach on common problems. In particular, we show how the effective number of degrees of freedom depends on spatial or temporal correlation lengths associated with Gaussian processes. The correlation lengths are seen to be tuning parameters used to match the model degrees of freedom to the (generally unknown) number of degrees of freedom associated with the system giving rise to the data. We also show how Gaussian process models can be related to Tikhonov regularisation for an appropriate set of basis functions.

Bayesian parameter estimation techniques have been successfully applied to the analysis of acoustic and oceanographic chirped signals and of decaying sinusoidal signals in chemical material analysis employing nuclear magnetic resonance. It has been also applied to astronomy, radar, telecommunication, and audio testing and measurement. The method has been additionally applied to waveform metrology. The main objective of this tutorial review is to explain the essentials of the method so that it will be accessible to scientists and engineers involved in measurement and interested in disseminating and advancing best practices for measurement.

One of the major issues debated during the preparation of the 2018 Resolution 1 of the CGPM concerned the switch from the use of physical states to the use of physical constants—in both cases experiments are involved. The differences in the necessary competence and in the implementation difficulties are very considerable. As a consequence, two positions have emerged: the first, now implemented, does not recognise a two-frames structure to the revised SI and does not consider necessary to locally implement any realisation of the constants. This is the BIPM position, now implemented in the text of the BIPM 9th Edition of the SI Brochure. The second position asserts two frames in a revised SI structure, and requires a demonstration that the stipulated values are effectively locally valid. The aim of this paper is to bring evidence that, as a consequence, the last SI Brochure edition contains the above and other interpretations of the Resolution 1, the only official reference document, that are in contradiction with the prescriptions, explicit or implicit, of the latter.

Metrologists are increasingly being faced with challenges in statistical data analysis and modeling, data reduction, and uncertainty evaluation, that require an ever more demanding and comprehensive analytical and computational toolkit as well as a strategy for communication of more complex results. For example, conventional assumptions of Gaussian (or normal) measurement errors may not apply, which then necessitates alternative procedures for uncertainty evaluation.

This contribution, aimed at metrologists whose specialized knowledge is in a particular area of science, and whose prior study of topics in probability or statistics will have been merely introductory, provides illustrative examples and suggestions for self-study. These examples aim to empower metrologists to attain a working level of concepts, statistical methods, and computational techniques from these particular areas, to become self-sufficient in the statistical analysis and modeling of their measurement data, and to feel comfortable evaluating, propagating, and communicating the associated measurement uncertainty.

The contribution also addresses modern computational requirements in measurement science. Since it is becoming clear to many metrologists that tools like Microsoft Excel, Libreoffice Calc, or Apple’s Numbers often are in-sufficiently flexible to address emerging needs, or simply fail to provide required specialized tools, this contribution includes accompanying R code with detailed explanations that will guide scientists through the use of a new computing tool.

Measurements are basic for all quantitative science and for many human activities. Usually numbers are considered to be basic for the description of measurement results of one-dimensional continuous quantities. By the limited accuracy of every measurement equipment the result of one measurement of a continuous quantity is not a number but a more or less vague result. This unavoidable uncertainty has to be analyzed in order to obtain realistic results. Using the concept of so-called fuzzy numbers, which are special fuzzy subsets of the set of real numbers, helps to give a more realistic description of the uncertainty of individual measurement results. Consequences of this description for the analysis of measurement data are described in this contribution.

The calibration of drug delivery devices and flow meters at microflow range is often performed by the gravimetric method, but for flow rates lower than 100 nL/min this method presents some limitations, mainly evaporation, flow instabilities and balance characteristics. In order to overcome these problems a new calibration method for microflow measurements down to 16 nL/min using interferometry was developed in a partnership between the Laboratory of Volume and Flow of the Portuguese Institute for Quality and the Department of Mechanical and Industrial Engineering of the New University of Lisbon under the EMPIR project MeDD II – Metrology for drug delivery. One of the main goals of this project is to develop a primary standard for measuring microflows down to 100 nL/min together with the appropriated uncertainty calculation. To validate the obtained results the Guide to the expression of Uncertainty in Measurement (GUM) and the Monte Carlo Method (MCM) were applied.

In many practical situations, the only information that we have about measurement errors is the upper bound on their absolute values. In such situations, the only information that we have after the measurement about the actual (unknown) value of the corresponding quantity is that this value belongs to the corresponding interval: e.g., if the measurement result is 1.0, and the upper bound is 0.1, then this interval is [1.0−0.1, 1.0+0.1] = [0.9, 1.1]. An important practical question is what is the resulting interval uncertainty of indirect measurements, i.e., in other words, how interval uncertainty propagates through data processing. There exist feasible algorithms for solving this problem when data processing is linear, but for quadratic data processing techniques, the problem is, in general, NP-hard. This means that (unless P = NP) we cannot have a feasible algorithm that always computes the exact range, we can only find good approximations for the desired interval. In this paper, we propose two new metrologically motivated approaches (and algorithms) for computing such approximations.

Data metrology – the assessment of the quality of data – particularly in scientific and industrial settings, has emerged as an important requirement for the UK National Physical Laboratory (NPL) and other national metrology institutes. Data provenance and data curation are key components for emerging understanding of data metrology. However, to date provenance research has had limited visibility to or uptake in metrology. In this work, we summarize a scoping study carried out with NPL staff and industrial participants to understand their current and future needs for provenance, curation and data quality. We then survey provenance technology and standards that are relevant to metrology. We analyse the gaps between requirements and the current state of the art.

The problem of calculating the uncertainty bands for a linear regression with correlated initial data is considered. The coverage factors for the uncertainty bands with different models of errors in the initial data are obtained analytically and by means of Monte Carlo method. The linear regression coefficients are estimated by the generalized method of least squares.

One of the basic steps in characterising the behaviour of a system that is subject to random effects is to observe a number of outputs of the system and make inferences about the system on the basis of the observed outputs. The inferences are usually made in terms of a parametric model in which the samples are modelled as being drawn from a probability distribution that depends on a finite number of parameters. An important example of this approach is in determining the repeatability of a measurement system in the construction of its uncertainty budget. Repeatability and reproducibility studies in measurement system analysis and analysis of variance studies also follow this approach.

The Guide to the Expression of Uncertainty in Measurement considers the case of repeated sampling from a Gaussian distribution under the assumption that the samples are drawn independently. In practice, we cannot be sure that the samples are drawn independently (nor can we be sure that they are drawn from a Gaussian distribution). The objectives of this paper are, firstly, to examine the consequences of the assumption of independence in evaluating measurement uncertainty and, secondly, to provide alternative models for correlated effects that are both plausible and computationally tractable.

The motivation for the alternative models comes from the fact that in an uncertainty budget it is possible to associate an approximate time scale with each component of the budget. We show that such models can often be approximated by an autoregressive model that can also be expressed in terms of a Gaussian process model of temporal correlation. In this paper, we describe the models, analysis methods and algorithms and report on metrology applications.

This work specifies general methods for the calibration of rheometers and evaluates the measurement uncertainty associated with the complex mathematical expression of viscosity derived for rotational rheometer equipment using an alternative method to that in the GUM. This method is a Monte Carlo implementation of the propagation of distributions (GUM-S1), which can treat measurement models having any degree of non-linearity and possibly large input uncertainties. It is therefore particularly suited to handling the complexity of the applicable measurement model, with the advantage of avoiding the need to calculate partial derivatives. In this work the difference between both approaches proved negligible for all practical purposes using baseline conditions. However further simulation showed how changing input conditions could alter this conclusion.

The problem of measuring instrument calibration and check by its comparison with other instrument used as a reference standard is discussed. The typical tasks of data processing are formulated and solved. The results being published earlier and new ones are presented.

The discrete wavelet transform has a huge number of applications in science and engineering. It allows to extract features and obtain information often hidden in raw signals. For applications in metrology the propagation of uncertainties for this transform is relevant. By using and extending existing techniques for propagating uncertainties in the context of discrete filtering, we propose an efficient implementation of the discrete wavelet transform that enables the online propagation of uncertainties. The proposed approach is illustrated by its application to simulated data and to data from an experimental testbed.

Widespread use of digital systems provides an opportunity to improve the quality of information reported for traceable measurements. By using digital technology to pass information along a traceability chain, the effects of common influences can be rigorously accounted for, so the provenance of any influence becomes traceable. This can be achieved by using a simple form of chain rule for differentiation to evaluate and propagate components of uncertainty. The method can achieve the goal of internal consistency identified in the Guide to the expression of uncertainty in measurement, thereby improving on current common practice.

Relationships between physical quantities are often expressed as mathematical equations, but evaluation of these equations is subject to restrictions related to the quantities involved and the scales of measurement. Calculations will be prone to error if quantity values are assigned to standard numeric data types in digital systems. To address this, the digital representation of quantities must capture the semantics of quantity calculations. The paper shows that simple dimensional analysis must be used to identify all quantities of interest in specific problems. This provides sufficient information for a framework to monitor computations and detect illegal operations. The approach can monitor calculations with so-called ‘dimensionless’ quantities and could also allow systems to resolve issues related to incompatible measurement units.

With the advent of the Internet of Things, sensors have become almost ubiquitous. From monitoring air quality to optimising manufacturing processes, networks of sensors collect vast quantities of data every day, creating multiple data streams. While some applications may warrant the use of very high quality sensors that are regularly calibrated, this may not always be the case. In most applications, sensors may have been calibrated prior to installation but their performance in the field may be compromised, or a combination of low and high quality sensors may be used. Thus, the uncertainty of the data from a sensor operating in the field may be unknown, or the quality of an estimate of the uncertainty from a prior calibration of the sensor may not be known. Taking into account the unknown or doubtful provenance of the data is critical in its subsequent analysis in order to derive meaningful insights from it.

Many data analysis problems arising in metrology involve fitting a model to observed or measured data. This paper addresses the issue of how to make inferences about model parameters based on data from multiple streams, each of doubtful or unknown provenance. It is assumed the data arises in a system modelled as a linear response subject to Gaussian noise. In order to account for the lack of perfect knowledge of data quality, Bayesian hierarchical models are used. Hyper-parameters of a gamma distribution encode the degree of belief in uncertainty estimates associated with each data stream. In this way, information on the uncertainties can be learnt or updated based on the data. The methodology can be extended to data streams for which uncertainty statements are missing. The Bayesian posterior distribution for such hierarchical models cannot be expressed analytically in closed form but approximate inferences can be made in terms of the Laplace approximation to the posterior distribution determined by finding its mode i.e., the maximum a posteriori (MAP) estimate. The Laplace approximation can in turn be used in a Metropolis-Hastings algorithm to sample from the posterior distribution. The methodology is illustrated on metrological examples, including the case of inter-laboratory comparisons.

Clinical diagnosis can be approached as a problem of conformity assessment in which the patient takes the role of the item to be assessed and the classification of the patient as unhealthy or healthy is expressed as a requirement on the true value of a measured characteristic of the patient. The classification of patients into treatment and non-treatment groups is an important task in medicine with significant societal and personal implications. In this paper, a framework for clinical diagnosis that accounts for measurement uncertainty, based on the principles of conformity assessment, is described.

The framework is illustrated for the problem of deciding the extent of myocardial blood flow abnormalities in patients based on a predefined clinical guideline using clinical data from patients participating in a cardiac Positron Emission Tomography (PET) perfusion study at the Turku PET Centre.

By combining patient data with expert insights, the framework can help less experienced clinicians make better decisions regarding patient health, serve as a starting point for further clinical investigation, and be used as a screening to categorise patients so that the most severe cases can be prioritised on the clinical list. Furthermore, the framework could be used within a machine learning classification pipeline both to label unlabelled data and to assess the quality of labelled data.

This paper applies the Monte Carlo method (MCM) for measurement uncertainty propagation in the Single Burning Item (SBI) test, within the European normative framework of reaction to fire tests for building products: EN standard 13823:2010+A1. The use of an MCM is necessitated by (i) the multistage nature of the calculation and (ii) the multivariate, non-linear and complicated form of the functional relations, involving many input, intermediate and output quantities. The approach provided a validation of the GUM approach specified for use in the SBI uncertainty document CEN/TR 16988.

Type systems can be used for tracking dimensional consistency of numerical computations: we present an extension from dimensions of scalar quantities to dimensions of vectors and matrices, making use of dependent types from programming language theory. We show that our types are unique, and most general. We further show that we can give straightforward dimensioned types to many common matrix operations such as addition, multiplication, determinants, traces, and fundamental row operations.

Measurement, considered a pillar in modern science for acquiring new knowledge, a tool valid from nearly full ignorance up to any stage of progressed knowledge. Stage boundaries cannot be set easily. This reflects onto a wide range of features in expressing cognitive situations. For each of them, in addition to express the level of knowledge on a subject matter, one has to delimitate the areas of ignorance by identifying the limits of the knowledge level on qualifying aspects of the matter under investigation. It is a fact that knowledge is gained in steps. These steps can be classified in different ways. The paper will discuss a step that will be considered preliminary to measurement: evaluation, which is basically a qualitative one. Then, two posterior steps will be illustrated: validation, whose aim is the obtain an inter-subjective agreement about the results of measurement; prediction, that is the true practical aim of science, intending to extrapolate to future observations or behaviours the past and current effects of contingent findings or “laws”. This structure of the steps may require a better delimitation of the aims and features of measurement.

This work presents a flexible method to detect the fault of components in a diecasting machine. The core of this method is the combination of sensor-based statistical predictions with the expert knowledge using a series of weights determined in formal interviews. Each feature is extracted from the machine’s sensor time history using a least square regression and paired with an uncertainty estimator. Then, each uncertainty estimator is combined with the uncertainty of the relative transducer in order to obtain a combined uncertainty of the two contributions. The final result is a score index representing the distribution of different types of faults in the diecasting machine. A dataset of 451 injections was analyzed to test the method. The historical records of maintenance service recorded 19 events corresponding to a fault of a valve. All the events were correctly detected by the algorithm as well. The uncertainty estimators of the parameters have allowed performing an analysis of the effect of transducers’ uncertainty on the final prediction. A higher uncertainty is negligible in the final prediction of fault. This means that the method can work also with transducers with lower accuracy.

This research describes a strategy for evaluating the measurement uncertainty of the feature dimensions and forms of products using a coordinate measuring machine, which is widely used in manufacturing industry. The proposed strategy outputs a task-specific measurement uncertainty, where the task includes the specification of the distribution of measurement points. Being executable with a small number of measurement trials, the strategy is practical for industrial use.

Usually, measurement errors contain both absolute and relative components. To correctly gauge the amount of measurement error for all possible values of the measured quantity, it is important to separate these two error components. For probabilistic uncertainty, this separation can be obtained by using traditional probabilistic techniques. The problem is that in many practical situations, we do not know the probability distribution, we only know the upper bound on the measurement error. In such situations of interval uncertainty, separation of absolute and relative error components is not easy. In this paper, we propose a technique for such a separation based on the maximum entropy approach, and we provide feasible algorithms – both sequential and parallel – for the resulting separation.

Proficiency testing is carried out to evaluate the performance of participating laboratories by summarizing and comparing the measurement results. This paper discusses methodologies related to measurements with ordinal categorical data. It investigates in detail three methods of statistical significance test with ordinal categorical data, from the viewpoint of whether they can be applied to proficiency testing procedures. These methods are: the cumulative test proposed by Taguchi, the cumulative chi-square test proposed by Takeuchi and Hirotsu, and ORDANOVA proposed by Bashkansky, Gadrich, and Kuselman. Our results suggest that the ORDANOVA method is applicable to proficiency testing and proposes a procedure using these statistics to analyze proficiency testing with ordinal categorical data.

When precise measurement instruments are designed, designers try their best to decrease the effect of the main factors leading to measurement errors. As a result of this decrease, the remaining measurement error is the joint result of a large number of relatively small independent error components. According to the Central Limit Theorem, under reasonable conditions, when the number of components increases, the resulting distribution tends to Gaussian (normal). Thus, in practice, when the number of components is large, the distribution is close to normal – and normal distributions are indeed ubiquitous in measurements. However, in some practical situations, the distribution is different from Gaussian. How can we describe such distributions? In general, the more parameters we use, the more accurately we can describe a distribution. The class of Gaussian distributions is 2-dimensional, in the sense that each distribution from this family can be uniquely determined by 2 parameters: e.g., mean and standard deviations. Thus, when the approximation of the measurement error by a normal distribution is insufficiently accurate, a natural idea is to consider families with more parameters. What are 3-, 4-, 5-, n-dimensional limit families of this type? Researchers have considered 3-dimensional classes of distributions, which can – under weaker assumptions – be used to describe similar limit cases; distributions from these families are known as infinitely divisible ones. A natural next question is to describe all possible n-dimensional families for all n. Such a description is provided in this paper.

Permeation is a technique for realising a primary measurement standard for gas composition. A permeation tube is suspended in a permeation chamber and emits a constant mass flow of the nominally pure substance contained in it. This flow is combined with a flow of a carrier gas to obtain a calibration gas mixture with known composition. We used an automated weighing system to monitor the mass loss of the tube, and prepared mixtures of ammonia (NH3) in nitrogen. The advantage of such dynamic gas standards is that unlike static standards they do not have stability issues and the composition of the calibration gas mixture can be chosen more rapidly than with static mixtures.

We revisited and extended existing measurement models describing the standard. We show how dependencies between the input quantities in the extended model can be taken into account. We describe the effect of temperature fluctuations on the permeation rate, temperature and pressure effects on the dispersion of the weighing data. For evaluating the linearity of the balance, a simple Bayesian model was established that takes into account the repeatability and resolution of the balance. We also show that the use of ordinary least squares regression to obtain the permeation rate is justified.

A problem of the dynamic measurement error estimation and correction is considered. The adaptive measuring system with the dynamic measurement error estimation is developed. Correction of the dynamic error is achieved by simultaneous restoring and filtering of the sensor input signal being measured. Signal processing consists in determining of the FIR-filter optimal order by the standard deviation minimization of the dynamic measurement error estimation. Computer simulation of the proposed measuring system for the first-order sensor is carried out. The purpose of the research is to demonstrate the approach efficiency. A good proximity between input and restored sensor signals as well as between true and estimated dynamic measurement errors is shown. Future works consist in the approach expansion on high-order measuring systems.

The new extended mathematical model for evaluation uncertainties of the multivariable indirect measurements, which upgrades the method of GUM-Supplement 2 is presented. In this model the uncertainties and correlations of the processing function parameters are also taken into account. It can be used for multivariable measurements and also to describe the accuracy of instruments and systems that perform such measurements. An example is included on estimated uncertainties of indirectly-measured output voltage and current of a twoport circuit, by considering uncertainties and correlations of its elements.

Aleatory measurement uncertainty (the uncertainty of randomness) can be distinguished from epistemic measurement uncertainty (the uncertainty of doubt). Unacceptable results in several example measurement problems show that these forms of uncertainty cannot always be combined using the idea of probability. Associated principles of Bayesian statistics are discussed. One conclusion is that a prior distribution on a nuisance parameter governing the quality of a measurement does not adequately represent its contribution to the uncertainty statement. The implications for the evaluation of measurement uncertainty and for the utility of Bayesian statistical methods are far-reaching.

A function with the form y = (a+ex)/(1+cx) is monotonic for all values of the parameters and can be inverted easily. This function describes a rectangular hyperbola, a segment of which might be used as a calibration curve for reading from x to y and, more importantly, from y to x. Methods are given for finding the hyperbolic segment that minimizes the sum of squared residuals to a set of points {(x, y)} and for finding a spline of hyperbolic segments that minimizes an objective function, e.g. the sum of squared residuals. A method is also given for constructing a spline of hyperbolic segments that interpolates a set of points. An analysis of model error is given.

A probability distribution is sometimes said to encode a set of information about a constant quantity. If that is the case then there will be some probability distribution that represents the sum of such sets of information. A derivation is given of the appropriate method for obtaining this distribution when its components represent disjoint sets of information. This derivation avoids an incomplete step that was part of an earlier analysis (Metrologia, 43, 12–20, 2006). The result leads to a logical contradiction: the combined distribution is sensitive to the arbitrary ordering of two legitimate steps in its generation. The only cause of this can be fault in the premise of the procedure, which is the premise that a set of information or ‘state of knowledge’ about a quantity can be represented by a unique probability distribution. This result serves as a conclusive counter-example to the notion of ‘logical probability’. It has implications for the development of a scientific method for the evaluation of uncertainty in measurement. The argument is extended verbally to also disqualify the notion that a distribution can represent the state of knowledge approximately.

Linear function and polynomial function are the most frequently used calibration functions in comparative calibration. Calibration is a standard tool in measurement science and applications to specify our knowledge about the parameters of the considered calibration function. This information is further used for specifying our knowledge about the unobservable stimulus from the independent future indication (measured response) received by using the calibrated measurement device. Here we present a description of the MATLAB algorithm PolyCal, which is based on the EIV (Errors-In-Variables) modelling approach and the characteristic function approach. The algorithm is available in CharFunTool – the Characteristic Functions Toolbox for MATLAB, https://github.com/witkovsky/CharFunTool. The applicability of the algorithm is illustrated by a simple example.

In the branch of forensic science known as firearm evidence identification, various similarity scores have been proposed to compare firearm marks. Some similarity score comparisons, for example, congruent matching cells (CMC) method, are based on pass-or-fail tests. The CMC method deals with pairs of topography images of breech face impressions for which the similarity has been quantified. For an image pair, the CMC method determines certain number correlated cell pairs. In addition, each correlated pair is determined to be a congruent match cell (CMC) pair, or not based on several identification parameters. The number of CMC pairs as a threshold is crucial and required so that the two images of surface topographies can be identified as matching. The threshold would be determined after carefully designed measurements and error rate estimation.

To reliably estimate error rates and the associated uncertainties, the key is to find an appropriate probability distribution for the frequency distribution of the observed CMC results. This article discusses four statistical models for CMC measurements, which are binomial and three binomial-related probability distributions. In previous studies, for a sequence of binomial distributed or other binomial-related distributed random variables (r.v.), the number of Bernoulli trials N for each r.v. is assumed to be the same. However, in practice, N (the number of cell pairs in an image pair) varies from one r.v. to another. In that case, the term of frequency function of the CMC results is not appropriate. In this article, the general frequency function is introduced to depict the behavior of the CMC values and its limiting distribution is provided. Based on that, nonlinear regression models are used to estimate the model parameters. For illustration, the methodology is applied to a set of actual CMC results of fired cartridge cases.

The following sections are included:

• Keyword Index
• Author Index