Advanced Mathematical and Computational Tools in Metrology and Testing XIII

The following sections are included:

• Foreword
• Book Editors
• Contents

This paper addresses the issue of forward extrapolation in time of the trend of existing experimental datasets for forecast purposes. It aims to introduce difficulties and limitations intrinsic to that task also in science, quite studied but often ignored, with the consequent risk of propagating false information. Making a forecast is one of the most popular expectations from science in economics, politics, Society, and scientific and non-scientific Communities. Science is assumed to be the safest way to allow reliable predictions. However, science too, being based on experimental data or inference, can only provide such predictions within a limited level of confidence — according to its statistical meaning — so never provide certainty. Scientists, in fact, never ignore that uncertainty is always associated with all their current findings or thinking, including the possibility of (further) evolution in time. As a consequence, from uncertain time-series data, one should not be able to get a deterministic extrapolated trend supplying the full correct information. A few examples are reported taken from climate analysis results.

The EURAMET collaboration project 1525 on evaluation of repeatability measurement procedures in gravimetric volume calibrations of glassware was undertaken by six National Metrology Laboratories and Accredited laboratories, with the purpose of investigating the different approaches regarding repeatability measurements of glassware calibration using the procedures described in two volume standards (ISO 4787 and ASTM E542), and its influence on the determined volume and uncertainty. Six flasks of different volume capacity were calibrated. The consistency of the data was determined using several statistical methods and all results are presented in this paper, along with reproducibility analysis and uncertainty calculation.

Supervised machine learning methods were used to predict patient heart disease in this work. The dataset was extracted from an open, traceable dataset for research purposes. 1600 electrocardiogram (ECG) records were extracted, and only single label records were used. ECG analysis was performed by determining particular fiducial points for all the ECG signals from which seven intervals and five amplitudes were obtained. Furthermore, these features were individually calculated for each of the 12 ECG leads, except for the RR interval, which was calculated only once using all 12 leads. However, it was included in the feature vectors for all leads. The amplitude and interval features were used to form the feature vectors. The accuracy scores for each lead were calculated using two supervised machine learning methods and using five-fold cross-validation. Taking into account that the dataset was balanced, the results were studied using the accuracy score. The results were represented by confusion matrices and diagrams. All captured results were compared. Finally, the sensitivity of the machine learning methods was reviewed to consider variations in the classification performances.

This paper aims to address the challenges posed by the evolving landscape of measurements, especially in non-physical domains. With a growing need for informed decision-making, the landscape of measurements has expanded significantly. To address this, measurement procedures for non-physical measurements need to be developed. This paper proposes a framework for achieving traceability in non-physical measurements. A six-step process for the creation of novel measurements is suggested by the following steps. Define the underlying concept, dissect the concept into its constituent components, identify suitable measurement techniques, ensure the validity and reliability of chosen techniques, determine appropriate measurement scales and establish a reference for traceability.

This paper looks at the analysis of data from a set of experiments designed to estimate the time to failure of an engineering component. These experiments involve subjecting the component to a series of increasing loads/stresses, those greater than would be expected during the normal working life of the component, and recording the time to failure. On the basis of the data, an estimate of the time to failure of the component under typical loads is made. We consider the case in which the logarithm of the time to failure varies linearly with the applied load. Inferences are based on fitting a linear model to the data using generalised distance regression. This paper evaluates the information that can be derived if the component has yet to fail when the experiment is halted and how this information can be used in predicting time to failure.

This work introduces a data-driven credibility assessment to quantify simulation quality in industrial part re-manufacturing. The framework evaluates the dependability of sources, data, and methodologies, focusing on robustness and uncertainty for data quality and simulation confidence. A conceptual data model, designed using competency questions, maps data requirements, digitalising credibility evaluation and promoting data traceability and accessibility.

The symbolic representation of quantities, units, and dimensions is reviewed with regard to information carried by conventional notation. The formal structure of unit systems like the International System of Units (SI) has been an efficient way for scientists to share data about the magnitudes of quantities, but it is an unsatisfactory basis for digitalisation. SI notation does not convey information about kind of quantity, which must be obtained from some other source, and the use of dimensional expressions in the SI also loses information about kinds of quantity. It is argued that digitalisation of quantities, units, and dimensions should focus on the more fundamental measurement concepts of quantity kinds and scales of measurement to provide strong foundations on which to build support for quantities, including features familiar from SI unit notation.

AI techniques have been actively and successfully used in data processing. This tendency started with fuzzy techniques, now neural network techniques are actively used. With each new technique comes the need for the corresponding uncertainty quantification (UQ). In principle, for both fuzzy and neural techniques, we can use the usual UQ methods — however, these techniques often require an unrealistic amount of computation time. In this paper, we show that in both cases, we can use specific features of the corresponding techniques to drastically speed up the corresponding computations.

Two worldwide events opened novel reflections in a highly vast scientific literature concerning a universal concept like that of “data”, namely as the results of observations, either experimental or human-mind generated: the “Big Data” and the universal use of informatics, related to each other and both in an exponential increase.

The capacity and speed of modern computers allow us to obtain such immense amounts of data, both from experimental setups or from the elaboration of algorithms, either human-built or through AI. Their handling too is necessarily operated via informatics means, considered distinct from mathematical means. Apparently, these facts have disconnected data evaluation from traditional fields, not only science but also the way to make decisions based on them.

This paper contains reflections on the properties and meaning of the data in themselves, specifically as the vehicle of information almost universally necessary to make decisions, the latter being a frame that is mixed and often prevalent in the literature about big data.

Measurement service provision is an established process of measurement quality management systems, which includes the exchange of information between service providers and consumers. It is an informative use case for modeling digital systems for quality management. This paper describes a conceptual model of quality system processes that govern traceable measurement services. The model identifies metrology concepts, measurement processes, and their relationship with measurement data. The model demonstrates applicability for machine-interpretable digital calibration certificates.

This paper presents a procedure for measuring a quantity when there is serially correlated data. In this situation, the statistical method of determining the standard uncertainty that assumes independence is inaccurate and, usually, anti-conservative. The procedure presented here involves calculating (a) an alternative estimate of the standard deviation of the sample mean for use as the standard uncertainty and (b) an associated number of degrees of freedom. Simulations for large and small samples show the procedure to substantially improve on the method that assumes independence, with the amount of improvement increasing with sample size.

This paper focuses on parameter estimation within the Errors-in-Variables (EIV) model with nonlinear constraints and the determination of associated uncertainties. This is motivated by a collaborative project in nanoscale material characterization, particularly the assessment of material properties like hardness and elasticity through instrumented indentation testing. Here we confront the challenge of precisely fitting curves specified by nonlinear constraints of the parameters to measurement results which come with associated uncertainties, while simultaneously managing their correlations. Such challenges are prevalent in numerous statistical and metrological applications. Our proposed approach is founded on iterative linearizations of the EIV model with nonlinear parameter constraints specified in implicit form, employing the Locally Best Linear Unbiased Estimation (LBLUE) method. LBLUE is a locally optimal technique known for providing an efficient and robust solution, particularly when dealing with weakly nonlinear models. We refer to this approach as the Optimum Estimate of Function Parameters by Iterated Linearization (OEFPIL). This method has been effectively implemented in various computing environments, including R and C, and is here introduced within the context of MATLAB, complete with illustrative examples. Compared to alternative estimation approaches in EIV models with nonlinear constraints, OEFPIL stands out for its simplicity, versatility, practicality, and computational efficiency, making it an excellent choice for applications in metrology and beyond.

The House of Quality (HoQ) is the key module of Quality Function Deployment (QFD), which aims to support the design of new products or services by translating customer requirements (CRs) into engineering characteristics (ECs). The independent scoring method is the most widely used technique for the prioritisation of ECs; despite this, it has two conceptual shortcomings: (i) arbitrary numerical conversion of (ordinal) relationships between CRs and ECs (i.e., null, weak, medium, and high), and (ii) “promotion” of these relationships from an ordinal to a cardinal scale. This paper introduces a novel procedure for prioritizing ECs, inspired by the Thurstone’s Law of Comparative Judgment (LCJ), which overcomes the conceptual shortcomings of the ISM while maintaining its flexibility and ease of implementation. A realistic application example illustrates the potential of the proposed procedure.

Integrated inspection system (IIS) enriches digital metrology by decreasing uncertainty due to closed-loop implementation of the cyber-components of digital metrology including planning the measurement of points, fitting the best Substitute Geometry (SGE) and evaluating the actual geometry of the surface. Optical devices are favored over contact metrology devices as they offer faster, fuller, and non-invasive measurement, but they are also susceptible to noise and outliers that may affect the accuracy and quality of the inspection. Principal Component Analysis (PCA) has been presenting high efficiency in attaining transformation matrix of the large number of discrete points given by the optical devices based on the point’s distribution; however, it is not robust to noise. Noise removal prior to SGE provides pure information feedbacked to the other cyber-components of IIS, and finally correctly identifies the actual geometry of the inspected surface. In this paper, in order to obtain the robust principal component of the unorganized point cloud of the planar surfaces, a subset of the data, which is least affected by noise, is found with Minimum Covariance Determinant (MCD), and then PCA is conducted on the obtained MCD-subset. Due to the MCD’s robustness, the noises can be detected by their large robust distances based on the MCD location and scatter matrix. The results of implementation on two case studies show the efficiency of MCD in simultaneous noise removal and robust surface fitting. Variety of inspection tools can utilize the methodology for point cloud filtration.

This research describes a novel method to correct for the biased measurement of contact area due to edge-effect phenomena in Instrumented Indentation Test (IIT) process. IIT is a widely used depth-sensing technique to evaluate the mechanical characteristics of materials e.g., Indentation modulus, Indentation hardness, residuals stresses, damping, creep. The proposed procedure utilizes the Electrical Contact Resistance approach to augment the data on contact area. Monte Carlo Markov Chain model was used to randomly generate the relatively small dataset to robustly predict projected area. The advantage of the method is demonstrated through the application for different conductive metals.

In many practical situations, to get the desired estimate or prediction, we need to process existing data. This data usually comes from measurements, and measurements are never 100% accurate. Because we only know the input values with uncertainty, the results of processing this data also comes with uncertainty. To make an appropriate decision, we need to know how accurate is the resulting estimate, i.e., how the input uncertainty “propagates” through the data processing algorithm. In the ideal case, when we know the probability distribution of each measurement error, we can, in principle, use Monte-Carlo simulations to describe the uncertainty of the data processing result. In practice, however, we often only have partial information about the measurement uncertainty: for example, instead of the exact values of the cumulative distribution function F (X), we only know bounds on F (X). Such information is known as the probability box (p-box, for short). In this paper, we provide feasible algorithms for propagating p-box uncertainty.

Dosimetry audits are carried out to determine how well radiotherapy is delivered to the patient. It is also used to understand the uncertainty introduced into the measurement result when using different computational models. As measurement procedures are becoming increasingly complex with technological advancements, it is harder to establish sources of variability in measurements and understand if they stem from true differences in measurands or in the measurement pipelines themselves. The gamma index calculation is a widely accepted metric used for the comparison of measured and predicted doses in radiotherapy. However, various steps in the measurement pipeline can introduce variation in the measurement result. In this paper, we perform a sensitivity and correlation analysis to investigate the influence of various input factors (i.e. setting) in gamma index calculations on the uncertainty introduced in dosimetry audits. We identify a number of factors where standardization will improve measurements by reducing variability in outputs. Furthermore, we also compare gamma index metrics and similarities across audit sites.

Advances in algorithms and computing capacities continue to increase the popularity of neural networks. In particular, their use in safety-critical applications (e.g. aviation, transport, telecommunications) poses new challenges with regard to the consideration of uncertainties in their input data. If these consist of sensor measurements, metrology provides a standard reference for dealing with such uncertainties in the internationally harmonised and recognised Guide to the expression of uncertainty in measurements (GUM). In this work, a solid mathematical foundation is established for applying the principles of the GUM to simple neural networks, so-called multilayer perceptrons.

Constructing cruise-ship hull is complex and requires precise metalwork, welding and assembly. Early anomaly detection and correction are vital for cost and time management. This paper presents a novel statistical quality control (SQC) methodology for monitoring the panel line in cruise-ship shipyards, i.e., one of the initial manufacturing workshops, with a relatively high level of automation. The proposed methodology adopts a standardized p control chart with samples of variable size, incorporating two elements: (i) it accounts for the high level of customization of panels, and (ii) it takes into consideration the measurement uncertainty associated with the large-volume metrology instrument employed for conformity verification (such as a state-of-the-art total station), following the ISO 14253-1:2017 standard. A real-world case study demonstrates its practical application.

LNE Uncertainty allows for the evaluation of measurement uncertainties by propagating variances or distributions in accordance with the GUM and GUM Supplement 1. LNE Uncertainty is capable of handling multiple output quantities, perform sensitivity analysis and goodness of fit tests. The software has been developed following the recommendations of European Metrology Network for Mathematics and Statistics (Mathmet) to ensure reliability. The software, a standalone application, is freely available. Its user-friendly interface, guiding through the evaluation process, democratises access to statistical methods, requiring no programming skills, yet demanding experience in the measurement uncertainty evaluation process.

The expression of measurement quality is generally solved by measurement uncertainties. In addition to uncertainties, capability indexes are increasingly used to measure measurement quality, especially in industrial measurements. Although the first generation capability indices are predominantly used for measurement processes, second and third generation indices can also be used. The problem of measuring processes might be the uncertainty of so-called check standard on which the measurements in the capability assessment are being executed. The present paper addresses the issue of the influence of the uncertainty of the check standard on the capability indices.

The interpolation of the correction values of a thermometer calibration certificate is presented following literal H3 of JCGM 100:2008 for an actual number of calibration points. From JCGM 100 text is very non usual to have eleven points of calibration, so, its literal H3 methodology is applied for a typical number of points of calibration, up to 3. For each case, the analysis for correlated input quantities: the intercept and slope of the calibration curve is developed to predict the value and standard uncertainty of the correction to be applied to the thermometer for any value t of the temperature.

In this paper, we consider a computational modelling approach to gain some insight into the factors that strongly influence atmospheric pollution concentration and how uncertainties associated with these factors contribute to the uncertainties in pollution estimates. We consider a parametric advection-diffusion pollution dispersion model to estimate pollutant concentration over a spatial domain, given values for a set of model parameters and hyper-parameters. A global sensitivity analysis (GSA) is conducted to assess how the variability in the pollutant concentration can be attributed to the variability in the model parameters. We use a variance-based GSA technique based on the estimation of first order and total Sobol indices across the spatial domain using Quasi-Monte Carlo sampling. A possible use for the resulting spatial sensitivity maps is in the design of an optimal placement strategy for sensors.

We have developed a system in which numerical values can be verified to assure the management and measurement processes in calibrations and testings. The system was developed in accordance with standard operation procedure, and it works with an online application run by a third-party organization to examine data consistency, thus certifying calibration data. An example of the calibration of a vernier caliper is presented.

The application of the method for choosing a distribution law in relation to a family of two-sided power distributions for further use in metrological applications is considered. An efficiency of the method is assessed by statistical modeling based on the Monte Carlo method. Comparison is made with the maximum likelihood method when estimating a single parameter of distribution law (power). For the considered family of distributions, the coverage factor and coverage interval are estimated.

Deep learning is a versatile tool for condition monitoring that provides accuracy at the cost of explainability. In this contribution we explore the use of graph neural networks (GNN) for predicting the remaining useful lifetime (RUL) of an electromechanical cylinder. Through a pre-processing scheme based on a priori knowledge of the constituent sensors and the corresponding dataset, the time-series data is embedded into a graph structure. The preprocessed data is used to train a GNN to correctly predict the RUL of the cylinder. Explainable AI (xAI) methods are then used to assess the importance of nodes and edges in making predictions for the specific case of a wrongly predicted value.

This paper presents a proposed new method of determining parameters and uncertainty bands of a specific non-linear function fitted to given measured data of examined points. One or both of the variables of this non-linear function are changed so as to linearize it. Using the linear regression method, fined are the most favorable parameters of this straight line for its adjustment to the measured values of the coordinates of points tested according to the weighted total mean square WTLS criterion. Their autocorrelation and cross-correlation coefficients as well as uncertainties estimated according to the rules of the GUM guide [1] are considered. The parameters and the uncertainty band of the non-linear function result from the parameters of this straight line and its uncertainty band. Numerical examples of determining the parameters and uncertainty bands for the branch of a 2nd degree parabola (two methods) and for the complex exponential function are given.

Two possible objectives for a laboratory participating in a measurement comparison are (i) to test the reliability of its submission (x, u) and (ii) to estimate the systematic, i.e., enduring, error or bias in the technique that generated the estimate x. The first of these objectives relates to a statistical hypothesis test and the second relates to a statistical estimation procedure. With regard to the first objective, the usual way in which the results are graphed appears to confuse the two statistical tasks, but the conduct of the test itself remains appropriate. However, with regard to the second objective, careful consideration shows that — if the measurand has true value θ and reference estimate $\widehat{\theta }$ — the best estimate of the bias is not the conventional difference $d=x-\widehat{\theta }$ that forms part of the degree of equivalence. In fact, it is numerically smaller than d, meaning that — on average — the laboratory is measuring more accurately than might previously have been thought. Similarly, the standard uncertainty of the best estimate of the bias is generally smaller than the figure often denoted u(d).

Consider the estimation of a tail percentile or tail quantile from the moments of a continuous random variable. If a normal approximation to the distribution of the variable is applicable then an adequate result is found easily using the mean and variance. However, the use of higher moments would be expected to give a better result. We describe and evaluate two different procedures for estimating tail quantiles, the first of which uses the Pearson system of distributions and moments up to order 4, and the second of which uses the Cornish–Fisher expansion and moments up to order 8. Neither method is uniformly superior.

We consider the statistical straight-line calibration model, which, as it turns out, is a nonlinear regression errors-in-variables model. Since the related measurement model to determine the straight-line parameters is generally nonlinear, the derived covariance matrix of the parameter estimators only provides approximations of the uncertainties. To address this, we introduce a statistical test that helps us determine whether the linearized statistical straight-line calibration model is suitable for the measured data. If the model demonstrates linearity, we can utilize the optimal estimators, the best linear unbiased estimators for the model parameters, along with their covariance matrix. Moreover, by including additional normality assumptions, we can calculate confidence intervals for any feasible linear combination of parameters. In particular, here we identify the linearization region of the straight-line calibration model and provide a simple sufficient condition for the model to be considered linearizable.

Using the numerical methods, coverage factors were calculated for the composition of two Student distribution laws with different numbers of degrees of freedom and the ratio of standard deviations. The number of degrees of freedom corresponding to this composition was calculated. An approximating expression for the coverage coefficient of this composition is given. The results obtained by numerical methods are compared with the results obtained using the GUM method.

The following sections are included:

• Keyword Index
• Author Index