Tolerance Intervals Research Articles

Risk Assessment for Linear Regression Models in Metrology

The conformity assessment of products or a measured value with the given standards is carried out based on the global risk of producers and consumers’ calculations. A product may conform to specifications but be falsely rejected as non-conforming. This is about the producer’s risk. If a product does not meet the requirements but is falsely accepted as conforming, that poses a risk to the consumer. The conventional approach to risk assessment, which yields only a single numerical value for the global risk of producers and consumers, is naturally extended and utilized for assessing risk in measurement models with linear regression. The outcomes of the two-dimensional extension, along a moderate scale, are the parabolas with upwards openings. Risk surfaces were obtained through three-dimensional extension over the area limited by the moderate scale and guard band axes. Four models with different ranges of tolerance intervals were used to test this innovative method of risk assessment in linear regression. The corresponding standard measurement uncertainties were determined by applying a simplified measurement model with the use of comprehensive data on the measurement performance and by determining measurement uncertainty derived from consideration of the functional relationship obtained by linear regression analysis. Models that utilize information from linear regression analysis to determine measurement uncertainty are biased towards risks at the edges of the moderate scale. Testing the model’s performances with metrics related to the confusion matrix, such as the F1 score, further substantiated this assertion. The diagnostic odds ratio has been proven to be extremely effective in identifying the curve along the guard band axis, along which the global risks of producers and consumers are at their lowest.

AN EFFECTIVE APPROACH FOR POINT CLOUD DENOISING IN INTEGRATED SURVEYS

Abstract. Outliers and noise in point cloud data are unavoidable due to intrinsic and/or extrinsic survey factors. Significant errors may result from false geometry produced by a collection of anomalies, compounded by sparse structure, irregular densities, and lack of geometric cohesion typical of point clouds nature. Thus, filtering techniques on raw data are required to produce accurate point clouds suitable for further processing. This objective is pursued in the following study through a comparative analysis between two registered clouds, one obtained from TLS, used as reference dataset and the other – to be filtered – from SLAM system. Four steps make up the workflow: analysing the comparison models’ geometric attributes, specifically surface density and roughness; constructing statistical tolerance limits for the TLS cloud’s roughness distribution; cleaning the SLAM cloud; assessing the filtering outcomes. Our efforts to effectively remove and mitigate noise, while preserving the original detail features of the object surface, have been driven by the detailed articulation of point cloud denoising approaches that have been introduced in recent years. However, in this wide context, our goal is not to provide a review or to explore the details of the various methods; rather, we want to offer a simple yet efficient method for obtaining an integrated model with a uniform noise level. This can be especially useful when the data from the survey will later be used in source-based modelling.

A Systematic Method for Assessing the Machine Performance of Material Extrusion Printers

The performance assessment of additive manufacturing (AM) printers is still a challenge since no dedicated standard exists. This paper proposes a systematic method for evaluating the dimensional and geometrical performance of such machines using the concept of machine performance. The method was applied to an Ultimaker 2+ printer producing parts with polylactic acid (PLA). The X and Y axes of the printer were the most performant and led to narrower potential and real tolerance intervals than the Z axis. The proposed systematic framework can be used to assess the performance of any material extrusion printer and its achievable tolerance intervals.

A distribution-free phase II control chart for multivariate individual data with simple post signal diagnostics

Multivariate statistical process control (MSPC) charts are particularly useful when there is a need to simultaneously monitor several quality characteristics of a process. Most of the control charts in MSPC assume that the quality characteristics follow some parametric multivariate distribution, such as the normal. This assumption is almost impossible to be justified in practice. Distribution-free MSPC charts are attractive, as they can overcome this hurdle by guaranteeing a stable (or in-control (IC)) performance of the control chart without the assumption of a parametric multivariate process distribution. Utilizing an existing distribution-free multivariate tolerance interval, we propose a Phase II Shewhart-type distribution-free MSPC chart for individual observations, with control limits based on Phase I order statistics. In addition to being easy to interpret, the proposed chart preserves the original scale of measurements and can easily identify out-of-control variables after a signal. The exact in-control performance based on the conditional and unconditional perspectives is presented and examined along with the control limits determination. The out-of-control performance of the chart is studied by simulation for data from a number of multivariate distributions. Illustrative examples are provided for chart implementation, using both real and simulated data, along with a summary and conclusions.

A Bayesian Approach For The Consideration Of Measurement Errors

Metrology is a key tool for tolerancing as it is used to determine whether dimensions are within their tolerance intervals. However, measurement errors cannot be avoided and need being accounted for. The probabilistic approach is applied to both the dimensions and their measurement errors; they are modelled as random variables and characterized by their probability density function. The probability density function of the measurement error is assumed to be known; this work is included in a research project in collaboration with a metrology company, where the engineers are able to provide us with this information. This paper describes a strategy to account for such measurement errors and (partially) correct or mitigate their effects. Through Bayesian inference, the likelihood of true values given measured values is estimated, allowing for a probabilistic correction. The proposed method is applied to numerical examples with simulated data and its relevance is discussed.

Statistical Approaches for the Reduction of Measurement Errors in Metrology

Abstract Metrology is extensively used in the manufacturing industry to determine whether the dimensions of parts are within their tolerance interval. However, errors cannot be avoided. Metrology experts are of course aware of it, and able to identify the different sources that contribute to making errors. In this paper, the probability density function of measurement errors is assumed to be given as an input. Very little research has been made in metrology to develop methods that take into account such data. This work deals with a batch of measures and its statistical properties. A first method is proposed to correct the effects of the measurement errors on the distribution that characterizes the entire batch. Then a second method is proposed to estimate the true value that is hidden behind each single measure, by removing the measurement error statistically. The second method is based on the output knowledge of the first, which is integrated with Bayesian statistics. The relevance of these two methods is shown through two examples applied on simulated data.

Identifying design spaces as linear combinations of parameter ranges for biopharmaceutical control strategies

According to ICH Q8 guidelines, a biopharmaceutical manufacturer may submit a design space (DS) definition as part of the regulatory approval application, in which case process parameter (PP) deviations within this space are not considered changes and do not trigger a regulatory post approval procedure. A DS can be described by non-linear PP ranges, i.e., the range of one PP conditioned on specific values of another. However, independent PP ranges (linear combinations) are often preferred in biopharmaceutical manufacturing due to their operational simplicity, as mentioned in the guideline. While statistical software such as Modde supports the calculation of a DS comprised of linear combinations, their algorithms are generally based on discretizing the parameter space - an approach that suffers from the curse of dimensionality as the number of PPs increases. Here, we introduce a novel method for finding linear PP combinations using a numeric optimizer to calculate the largest design space within the parameter space that results in critical quality attribute boundaries within acceptance criteria, predicted by a regression model. A precomputed approximation of tolerance intervals is used in inequality constraints to facilitate fast evaluations of this boundary using a single matrix multiplication. The correctness of the method was validated by comparing results to that of a grid-based approach and, in a simple case, to an analytically defined ground truth. In the examples investigated, the volume of the resulting DS was significantly larger than that of the grid method, with the improvement being proportional to the granularity of the grid and the number of parameters involved. Furthermore, computational time for the optimization-based approach is several orders of magnitude faster in higher dimensions. In addition, a proposed weighting scheme can be used to favor certain PPs over others and therefore enabling a more dynamic approach to DS definition and exploration. The increased PP ranges of the larger DS provide greater operational flexibility for biopharmaceutical manufacturers.

Analysis of compositional and moment approaches to solving the calibration problem using a standard example

Moment and compositional approaches to solving the measurement problem of calibration are considered. The use of the moment approach to solving a standard example of thermometer calibration is considered. In this work, this example is solved within the framework of the compositional approach and more accurate and complete results are obtained compared to using the moment approach. This fact is consistent with the assessment of the accuracy of solutions to measurement problems carried out back in 2001 using a more rigorous compositional approach than the moment one: in the problem of calibrating measuring instruments, the moment approach does not fully take into account the inadequacy of functional models, overestimates the accuracy estimates and does not allow obtaining nonlinear models of optimal complexity. To solve the measurement problem within the framework of the compositional approach, this work uses the “MMI-Calibration 3.0” program, which allows taking into account the errors of inadequacy of mathematical models of calibration functions and obtaining tolerance intervals for a given confidence probability. Among polynomials up to the ninth degree inclusive, a three-parameter model of optimal complexity was obtained, the average module of the inadequacy error is 0.002745 °C and the tolerance interval [–0.00828; 0.00761] with a confidence probability of 0.95. Within the framework of the compositional approach, the average module of the error of inadequacy and the kappa criterion used to find a model of optimal complexity are compared. It is shown that the compositional approach fully and strictly describes the solution to the calibration problem without simplifications, allows one to avoid inflated estimates of accuracy, and therefore can serve as an exemplary approach provided that the “MMI-Calibration 3.0” program is used. The simultaneous use of the criteria of the minimum average module of the error of inadequacy and the maximum of the kappa criterion is a necessary and sufficient condition for identifying a model of optimal complexity in the class of polynomials of degree n.

TOWARDS ENERGY ATLAS OF SOFIA CITY IN BULGARIA

Abstract. This paper proposes the first energy atlas of Sofia in Bulgaria. The research uses a geographical information system (GIS) approach and a statistical tolerance methodology to estimate building energy consumption. The buildings were classified into ten categories, and tolerance intervals were computed, which provide a distribution-free summary for the consumption in each class, suitable for spatial visualisation. GIS is used to classify and visualise the results. The results show a clear contrast in the energy consumption between buildings in highly urbanised areas and those in the suburbs. It was found that the high energy consumption belongs to the areas where the shopping, commercial, industrial and sports buildings are located and already developed. The energy consumption bounds were used to enrich a semantic 3D city model of Sofia. This model can be used for further analysis of energy supply, climate change, urban heat islands, and urban health as well as for calculating the climate scenarios. An extensive outline of the utility and directions for future development of the atlas are provided.

Comparative physicochemical and biological characterization of the similar imiglucerase product Glurazyme® and the originator product Cerezyme®

A biosimilar considered as a biomolecule medicinal product that is comparable to a reference medicinal product in terms of the structural, functional, biological, and clinical attributes. Glurazyme® was developed as a biosimilar to Cerezyme® (imiglucerase) and was approved in CIS countries (Russian Federation, Belarus, Kazakhstan) and recently Algeria for the treatment of type 1 and type 3 Gaucher disease. The quality assessment of Glurazyme® was performed in accordance with the Rules for the Study of Biological Medicines of the Eurasian Economic Union harmonized with the ICH comparability guideline and the biosimilar guidelines of the European Medicines Agency and Food and Drug Administration. Extensive side-by-side comparison was employed with state-of-the-art and orthogonal assays designed to interrogate all expected physicochemical and biological activities, including those known to affect the mechanisms of action for imiglucerase. Similarity evaluation was performed on the basis of tolerance intervals determined from about 10 lots of commercial Cerezyme®. Mainly three discrepancies of quality attributes were established concerning oxidized and deamidated forms as well as phosphorylated oligomannose N-glycans reflecting the difference between cultivation and down-stream processes of both medicinal products. Nevertheless, all of them possess a little or no influence on safety and efficacy.

Simultaneous Quantification of Two Neonicotinoids Using QuEChERS-LC-MS/MS in Moroccan Spearmint (Mentha Spicata.L): Qualimetry of the Method by Uncertainty Estimation Using Generalized Pivotal Quantities Approach and Monte Carlo Simulation.

Neonicotinoids (NEOs) are used for the phytosanitary treatment of Mentha Spicata.L crops, and this practice requires precise control of these harmful substances at very low concentrations. The objective of this study is to apply an approach allowing simultaneously validation and evaluation of measurement uncertainty based on total error methodology, in order to accurately quantify the presence of two NEOs in Mentha Spicata.L utilizing a Quick, Easy, Cheap, Effective, Rugged, and Safe (QuEChERS)-LC-MS/MS methodology. The quantification of imidacloprid and acetamiprid employing a QuEChERS extraction method, coupled with LC-MS/MS, ensuring the accuracy of the analytical method and managing the risks associated with its routine use.A complete and exhaustive validation approach based on the "β-content, γ-confidence" tolerance interval was used for the uncertainty assessment, using the generalized pivot quantity (GPQ) concept and Monte Carlo simulation, which avoids the need for additional data while achieving intermediate precision for each concentration level within predetermined acceptable limits. The validation procedure is based on the choice of a quadratic model for the two NEOs, allowing the validation of acetamiprid and imidacloprid by LC-MS/MS assay within the range of working concentration. The flexibility of the uncertainty profile intervals was demonstrated with a variation in β-content values (66.7, 80, and 90%) and risk values (10 and 5%), which remained within the acceptability limits of 20%, and the relative expanded uncertainty did not exceed 15 and 11%. A QuEChERS-LC-MS/MS method for the analysis of two NEOs has been successfully fully validated using the uncertainty profile strategy. Implementation of an overall validation strategy, which involves both the validation and uncertainty assessment known as the uncertainty profile, for the quantification of two important NEOs in Mentha Spicata.L using QuEChERS-LC-MS/MS. This qualimetric approach has been conducted by computing the measurement uncertainty of the method utilizing data from analytical validation under conditions of intermediate precision at each level of concentration without additional effort. After that we have demonstrated the flexibility of this strategy for the LC-MS/MS quantification of acetamiprid and imidacloprid, using a decision tool that enables the choice and modification of β-content and γ-confidence values.

Robust Passivity and Control for Directed and Multiweighted Coupled Dynamical Networks.

In this article, the passivity and control issues for directed uncertain coupled dynamical networks are solved. The presented model is directly coupled with multiple coupling matrices and parametric uncertainty, while previous literatures of multiweighted networks usually suppose that outer coupling matrices (OMs) are connected, undirected, and certain. The viewpoint of inner coupling matrices (IMs) in this article is added and OMs can be directed and not connected, which is a great improvement on the existing results. First, for all diagonal IMs, considering each dimension separately, we can derive if the weighted combination of multiple OMs for each dimension is strongly connected, then passivity and pinning control rules can be established. In addition, we also discuss the situation that IMs are positive definite but not diagonal. By means of the weighted combination of normalized left eigenvectors (NLEVec) corresponding to zero eigenvalue for multiple coupling matrices, we prove if the Chebyshev distance (Cheb-Dist) among these NLEVec is less than a tolerant deviation interval, then passivity, synchronization, and pinning control criteria are acquired. Moreover, a matter of adaptive coupling strengths is also settled. Examples are provided to verify the validity of established results.

Methodology for Reliable Assessment of Water Quality. V. Statistical Methods for Water Quality Research: Parametric Approach

The possibility of using a parametric tolerance interval to improve the accuracy of studying water quality indicators was analyzed. It was demonstrated that the proposed approach provides advantages over non-parametric interval rank, but not in all cases. The advantages of the new method due to the use of more complete experimental information were substantiated, and the reasons for the decrease in the accuracy of water quality assessment at its application were analysed.

Transmission accuracy–axial backlash–fatigue life-driven tolerance optimization of planetary roller screw mechanism

Abstract The planetary roller screw mechanism (PRSM) is an advanced linear transmission device. The relationship between tolerance allocation and performance risk still remains elusive, which is a challenge for its future applications. This work proposes a novel transmission accuracy–axial backlash–fatigue life-driven tolerance optimization method for the screw, roller, and nut of PRSM. A computational framework for PRSM transmission accuracy, axial backlash, and fatigue life calculation is developed to work on the parametric variation of design parameters including the eccentric, pitch, nominal diameter, and flank angle. Combinations of parametric variation are obtained by the Latin hypercube sampling-based tolerance statistical model to rapidly evaluate PRSM performance risk under operation conditions and tolerance parameters. The optimal tolerance parameters with the expanded width of tolerance interval and the minimum PRSM performance risk probability are obtained using the non-dominated sorting genetic algorithm. Results reveal that PRSM performance risk probabilities change from 89.25 to 68.72% and 58.1 to 56.86%, with 29.94 and 17.38% tolerance interval width increase under the high-precision and heavy-loading operation cases studied, respectively.

Frequentist and Bayesian tolerance intervals for setting specification limits for left-censored gamma distributed drug quality attributes.

Tolerance intervals from quality attribute measurements are used to establish specification limits for drug products. Some attribute measurements may be below the reporting limits, that is, left-censored data. When data has a long, right-skew tail, a gamma distribution may be applicable. This paper compares maximum likelihood estimation (MLE) and Bayesian methods to estimate shape and scale parameters of censored gamma distributions and to calculate tolerance intervals under varying sample sizes and extents of censoring. The noninformative reference prior and the maximal data information prior (MDIP) are used to compare the impact of prior choice. Metrics used are bias and root mean square error for the parameter estimation and average length and confidence coefficient for the tolerance interval evaluation. It will be shown that Bayesian method using a reference prior overall performs better than MLE for the scenarios evaluated. When sample size is small, the Bayesian method using MDIP yields conservatively too wide tolerance intervals that are unsuitable basis for specification setting. The metrics for all methods worsened with increasing extent of censoring but improved with increasing sample size, as expected. This study demonstrates that although MLE is relatively simple and available in user-friendly statistical software, it falls short in accurately and precisely producing tolerance limits that maintain the stated confidence depending on the scenario. The Bayesian method using noninformative prior, even though computationally intensive and requires considerable statistical programming, produces tolerance limits which are practically useful for specification setting. Real-world examples are provided to illustrate the findings from the simulation study.

Optimization and validation of a bioanalytical HPLC-UV technique for simultaneous determination of underivatized phenylalanine and tyrosine in the blood for phenylketonuria diagnosis and monitoring.

This study aimed to develop a fast, accurate, and precise high-performance liquid chromatography with UV detection method for simultaneous analysis of underivatized phenylalanine (Phe) and tyrosine (Tyr) in biological samples. Separation of the analytes was accomplished using a Discovery HS F5-3 column, which offered better retention and peak symmetry for the tested analytes. Chromatographic conditions were optimized using central composite experimental design, and three factors were investigated: the concentration of ammonium acetate (A), the acetonitrile proportion in the mobile phase (B) and the column oven temperature (C). The approach was verified using β-expectation tolerance intervals for total error measurement that did not exceed 15%. Optimal settings were A = 50mm, B = 24% and C = 28°C. The method applicability was determined using human plasma from 75 volunteers. The limits of detection and quantification of the technique were satisfactory at 9 and 29 μm for Phe and 4 and 13 μm for Tyr. The mean analytical bias in spiking levels was acceptable, ranging from -1.649 to +1.659% for both substances, with RSD <5% in all instances. The suggested approach was successfully used to analyze Phe and Tyr in human blood samples and calculate the Phe/Tyr ratio.

Probabilistic prediction on three-dimensional roughness of discontinuity based on two-dimensional traces under rock tunnel excavation based on Bayesian theory

Three-dimensional (3D) roughness of discontinuity affects the quality of the rock mass, but 3D roughness is hard to be measured due to that the discontinuity is invisible in the engineering. Two-dimensional (2D) roughness can be calculated from the visible traces, but it is difficult to obtain enough quantity of the traces to directly derive 3D roughness during the tunnel excavation. In this study, a new method using Bayesian theory is proposed to derive 3D roughness from the low quantity of 2D roughness samples. For more accurately calculating 3D roughness, a new regression formula of 2D roughness is established firstly based on wavelet analysis. The new JRC3D prediction model based on Bayesian theory is then developed, and Markov chain Monte Carlo (MCMC) sampling is adopted to process JRC3D prediction model. The discontinuity sample collected from the literature is used to verify the proposed method. Twenty groups with the sampling size of 2, 3, 4, and 5 of each group are randomly sampled from JRC2D values of 170 profiles of the discontinuity, respectively. The research results indicate that 100%, 90%, 85%, and 60% predicting JRC3D of the sample groups corresponding to the sampling size of 5, 4, 3, and 2 fall into the tolerance interval [JRCtrue–1, JRCtrue + 1]. It is validated that the sampling size of 5 is enough for predicting JRC3D. The sensitivities of sampling results are then analyzed on the influencing factors, which are the correlation function, the prior distribution, and the prior information. The discontinuity across the excavation face at ZK78 + 67.5 of Daxiagu tunnel is taken as the tunnel engineering application, and the results further verify that the predicting JRC3D with the sampling size of 5 is generally in good agreement with JRC3D true values.

Online data condensation for digitalised biopharmaceutical processes

Efficient control of a bioprocess relies on the ability to systematically capture and represent the process dynamics of critical process parameters. Multivariate monitoring techniques in biopharmaceuticals has resulted in the generation of large amounts of data comprising real-time measurements of critical quality and performance attributes. If exploited efficiently, these can provide an opportunity for developing better control action. For this, it is important to have a comprehensive view of the critical process parameter landscape, which can only be achieved by integrating both online and offline data into a single data matrix that can then be subjected to standard data analysis protocols. However, owing to the difference in the number of readings available for variables recorded online and offline, there is a need for new methods to achieve condensation capability. This paper introduces a novel methodology for condensing online data into an offline data matrix, which performed better when compared to traditionally employed averaging and helped increase the number of variables available for representing the design space of the process. The method was also used to understand how error propagates through online data, so as to identify an interval of tolerance in online monitoring of bioprocesses.

Blandaltman: A command to create variants of Bland–Altman plots

Bland–Altman plots can be useful in paired data settings such as measurement-method comparison studies. A Bland–Altman plot has differences, percentage differences, or ratios on the y axis and a mean of the data pairs on the x axis, with 95% limits of agreement indicating the central 95% range of differences, percentage differences, or ratios. This range can vary with the mean. We introduce the community-contributed blandaltman command, which uniquely in Stata can 1) create Bland–Altman plots featuring ratios in addition to differences and percentage differences, 2) allow the limits of agreement for ratios and percentage differences to vary as a function of the mean, and 3) add confidence intervals, prediction intervals, and tolerance intervals to the plots.

Quantifying agonistic interactions between group-housed animals to derive social hierarchies using computer vision: a case study with commercially group-housed rabbits

In recent years, computer vision has contributed significantly to the study of farm animal behavior. In complex environments such as commercial farms, however, the automated detection of social behavior and specific interactions between animals can be improved. The present study addresses the automated detection of agonistic interactions between caged animals in a complex environment, relying solely on computer vision. An automated pipeline including group-level temporal action segmentation, object detection, object tracking and rule-based action classification for the detection of agonistic interactions was developed and extensively validated at a level unique in the field. Comparing with observations made by human observers, our pipeline reaches 77% precision and 85% recall using a 5-min tolerance interval for the detection of agonistic interactions. Results obtained using this pipeline allow to construct time-dependent socio-matrices of a group of animals and derive metrics on the dominance hierarchy in a semi-automated manner. Group-housed breeding rabbits (does) with their litters in commercial farms are the main use-case in this work, but the idea is probably also applicable to other social farm animals.