Advanced Mathematical and Computational Tools in Metrology and Testing XII

Contents
Dedication
Foreword
Invited Paper
Explicit unconditionally numerically stable solution of a class of cubic equations
1. Introduction
2. Fundamentals
2.1. Primitive form
2.2. Nature of the solution
2.3. Transformations to the depressed and primitive forms
3. Traditional solution
4. Numerically stable solution
4.1. The case x3 – 3x = 2h
4.2. The case x3 + 3x = 2h
5. Simple algorithms for the traditional and stable approaches
5.1. Algorithm T: Traditional approach for the real root
5.2. Algorithm S–: Stable approach for all roots of x3 – 3x = 2h
5.4. Properties of the algorithms
6. Floating-point error analysis
6.1. Calculation of the root using the stable method
6.1.1. x3 – 3x = 2h
6.1.2. x3 + 3x = 2h
6.2. Backward error interpretation
7. Examples
8. Uncertainties
9. Applications
10. Concluding remarks and scope for further work
Appendix A. Floating-point error analysis of the discriminant calculation
Appendix B. Sum of two stably computed positive terms
Appendix C. Floating-point error analysis
Appendix C.1. Primitive cubic equation x3 – 3x = 2h
Appendix C.2. Primitive cubic equation x3 + 3x = 2h
Acknowledgments
References
Review papers
Effective number of degrees of freedom associated with regression models
1. Introduction
2. Linear regression
3. Tikhonov regularisation
4. Prior information associated with linear regression
5. Model averaging
5.1. Averaging polynomial models
5.2. Model average as a regularised solution
5.3. Weighting according to the Bayesian Information Criterion
6. Gauss-Markov regression
6.1. Calculation of the effective degrees of freedom for Gauss-Markov problems
6.2. Gauss-Markov models as regularised models
7. Gaussian processes
7.1. Approximating the eigenvalues of a spatial correlation variance matrix
7.2. Gaussian process models using reduced rank approximations
8. Numerical example: polynomial model associated with platinum resistance thermometers
9. Concluding remarks
Acknowledgements
References
Bayesian signal analysis and parameter estimation – A tutorial review
1. Introduction
2. Bayesian inference
3. Model
4. dc signal
5. Sinusoidal signal with frequency known
6. Arbitrary periodic signal with frequency known
7. Arbitrary periodic signal with frequency unknown
8. Complex generalized signals
9. Computer simulated examples
9.1. Single frequency
9.2. Two close frequencies
9.3. Two separate frequencies
9.4. Amplitude modulation
9.5. Chirp
9.6. Single frequency with exponential decay
10. Conclusions
References
Review on questionable consistency issues between the CGPM Resolution 1 on revised SI (2018) and the 9th BIPM SI Brochure (2019)
1. Introduction
2. Inconsistency of the formulation of the rSI-2018 definition
3. Lack of correct information about the reasons for the chosen base units with respect to the SI-1960
4. Lack of illustration of the dilemma between units magnitude continuity or significant discontinuity with respect to those of the SI-1960, and reasons for choosing the first
5. Need to demonstrate that the locally-realised units satisfy the requirement to be consistent with the stipulated numerical values of the relevant constants
6. Ambiguous distinction between “primary” realizations and “mise en pratique” realizations of a unit
7. Lack of information about the consequences of the circularity in the definition of some base units, in particular about the possibility to use in practice the Planck constant, h
7.1. Case of the temperature unit
7.2. Case of the mass unit
7.3. Case of the length unit
7.4. Measurement equations of primary methods
8. Conclusions
References
Appendix
Statistical and computational tools for metrologists
1. Introduction
2. Replicated Observations
3. Calibration
4. Consensus Building Model
5. Conclusions and Recommendations
Acknowledgments
References
Measurement results and fuzzy models
1. Introduction
2. What is the result of a measurement?
3. Fuzzy numbers
4. Fuzzy vectors
5. Arithmetic operations for fuzzy numbers
5.1. Translation of a fuzzy number
5.2. Scalar multiplication of fuzzy numbers
5.3. Sum of fuzzy numbers
5.4. Difference of fuzzy numbers
5.5. Product of fuzzy numbers
6. Variation and vagueness of measurement results
7. Analysis of measurement data
8. Vector-valued quantities
References
Papers
Uncertainty calculation in nanoflow measurements using interferometry
1. Introduction
2. Interferometry applied to microflow measurements
3. Uncertainty calculation
3.1. Measurement model
3.2. Uncertainty evaluation
4. Results
5. Conclusions
Acknowledgements
References
What if we use almost-linear functions instead of linear ones as a first approximation in interval computations
1. Why Interval Computations
2. Interval Computations – Successes and Challenges: A Very Brief Overview
3. First Approach: Taking Major Inputs into Account
4. Second Approach: Taking Major Combinations of Inputs into Account
5. Conclusions
Acknowledgments
References
Data provenance, curation and quality in metrology
1. Introduction
2. Overview of provenance and digital curation
2.1. Provenance requirements and other considerations
2.2. Provenance content
2.3. Digital Curation Lifecycle Model
2.3.1. Digital Curation Lifecycle Components
3. Existing resources and standards
3.1. Standardization of provenance concepts
3.2. Available tools and techniques
4. Summary of study: interviews and use cases
4.1. Recommendations
5. Conclusion
References
On estimation of linear regression confidence bands: Analytical solution and Monte Carlo simulation
1. Introduction
2. Analytical solution for normally distributed errors
3. Monte Carlo simulation
4. Conclusion
Appendix A. Outline of the proof of the formula (4)
References
Repeatability, reproducibility and resampled effects models
1. Introduction
2. Linear models, Gaussian random eects
2.1. Independent sampling from a univariate Gaussian
2.2. Prior information about ϕ
2.3. Linear regression associated with Gaussian random effects
2.4. Using the Cholesky factor of V to determine a and s2
2.5. Making inferences about and ϕ
3. Autoregressive model of order 1, AR(1)
3.1. Single parameter case
3.2. AR(1) model as a Gaussian process
4. Resampled effects model
4.1. Test of AR(1) correlation in fitted residuals
5. Numerical examples
5.1. Simulated linear response
5.2. Drift in coordinate measurement
5.3. Computational requirements
6. Concluding remarks
Acknowledgements
References
Uncertainty evaluation in rheology measurements
1. Metrology traceability of rheological determinations
2. Rheometer calibration methods
3. Evaluation of measurement uncertainty in rheometer calibration
4. Uncertainty assessment of the direct calibration method
5. Conclusions
Acknowledgments
References
Measuring instruments comparison for calibration and check: Four typical tasks of data processing
1. Introduction
2. Data general character analysis
3. Analysis of data structure and probabilistic nature
4. Formulation and solution of data processing task
5. Conclusions
References
Discrete wavelet transform on uncertain data: Efficient online implementation for practical applications
1. Introduction
2. Methods
2.1. Online Discrete Filter
2.2. Single-level DWT
2.3. Multi-level DWT
3. Results
3.1. Application to Simulated Data
3.2. Application to Experimental Data
4. Conclusion and Outlook
5. Acknowledgments
References
Digital representation of measurement uncertainty for metrological traceability
1. Introduction
2. Two views of a measurement scenario
2.1. GUM analysis of measurement uncertainty
2.2. Components of uncertainty for composed models
2.3. Maintaining internal consistency
3. Discussion
3.1. Recommendations
4. Conclusions
Acknowldgement
References
Software representation of measured physical quantities
1. Introduction
2. Background
2.1. Dimensions
2.2. Dimensional analysis
2.3. Quantity calculus
3. Dimensional signatures
3.1. Summary of signature operations
3.2. Other examples
3.2.1. Electrical dimensions
3.2.2. Angle
3.2.3. Fuel consumption
4. Discussion
5. Conclusion
Acknowldgement
References
Combining data streams of doubtful provenance
1. Introduction
2. Basic model for multi-sensor systems
2.1. Motivation for the hierarchical model for lack of provenance
3. All uncertainties known exactly
3.1. Weighted least squares
3.2. Gauss-Markov problem
4. Partial information about σk
4.1. Posterior distribution for α and ϕ
4.2. Marginalising with respect to ϕk
4.3. Posterior point estimate for ϕ
4.4. Approximate sampling from the joint posterior distribution p(σ, ϕ|y)
5. Partial information about σ2k of doubtful provenance
5.1. Posterior distribution for α, ϕ, ν
5.2. Approximate inferences based on point estimates of ϕ and ν
5.3. Sampling from p(α, ϕ, ν|y)
6. Numerical examples
6.1. Laser tracker data
6.2. Inter-laboratory comparison data
6.3. Discussion: relationship to the GUM and measurement system analysis
7. Concluding remarks
Using measurement uncertainty in a risk-based decision-making framework for clinical diagnosis
1. Introduction and conformity assessment
2. Background to the clinical application
3. Risk-based clinical guideline
3.1. Conformance probability
3.2. Specific risk
3.3. Global risk
4. Summary and concluding remarks
Acknowledgements
References
Evaluation of measurement uncertainty in SBI – Single Burning Item reaction to fire test
1. Introduction
2. Main measurands of the SBI test
2.1. Heat release rate measurement model
2.2. Smoke production rate measurement model
3. Uncertainty propagation
4. Results
5. Conclusions
Acknowledgment
References
Type systems for programs respecting dimensions
1. Introduction
2. Units of measure, mathematically
3. Dependent type systems
4. Multidimensional units of measure
4.1. Monoids and semirings
4.2. Dimensioned scalars
4.3. Dimensioned matrices
4.4. Dimension-aware matrix algebra
4.5. Elementary row operations
5. Conclusions and Future Work
Acknowledgments
References
Measurement in science: Between evaluation and prediction
1. Introduction
2. Evaluation (ex-ante)
3. Measurement
4. Evaluation (ex-post) and validation
5. Prediction
6. Computer simulation
7. Final remarks
References
A method based on combinations of forecaster and weighing matrix to detect fault of components in diecasting process
1. Introduction
2. Material and methods
2.1. Experts interviews
2.2. Productions detection
2.3. Predictions analysis
3. Results
4. Conclusion
Acknowledgments
References
Practical experiment design of task-specific uncertainty evaluation for coordinate metrology
1. Introduction
2. Uncertainty contributors of a CMM measurement task
2.1. Randomizing the unknown systematic-error sources
3. Design of experiment
4. Simulation
5. Discussion
6. Summary
Acknowledgments
References
How to separate absolute and relative error components: Interval case
1. Need to Separate Absolute and Relative Error Components: General Case
3. How to Separate Absolute and Relative Error Components in the Interval Case: Analysis of the Problem
4. How to Separate Absolute and Relative Error Components in the Interval Case: Algorithm
5. Numerical Example
Acknowledgments
References
Proficiency testing with ordinal categorical data
1. Introduction
2. Statistical Methods for Analyzing the Variation of Ordinal Categorical Data
2.1. Data format
2.2. ORDANOVA
3. Searching for Measures of Proficiency Testing
3.1. Measures of proficiency testing
3.2. a) Sum of cumulative proportions
3.3. b) Adjusted value of category where cumulative proportion becomes 0.5
3.4. c) Laboratory component of the total variation given by ORDANOVA
3.5. Notes on the proposed measures
4. Simulation
4.1. Procedures
4.2. Results
5. Conclusions
Acknowledgments
References
How to describe measurement errors: A natural generalization of the Central Limit Theorem beyond normal (and other infinitely divisible) distributions
1. Central Limit Theorem and Distributions of Measurement Error: A Brief Reminder and Formulation of the Problem
2. Definition and the Main Result
Acknowledgments
References
Modelling of the dynamic gravimetric preparation of calibration gas mixtures using permeation for trace gas analysis
1. Introduction
2. Principle
3. Dynamic gas mixture preparation
3.1. Weighing
3.2. Uncertainty of the mass ow rate of the dilution gas
3.3. Permeation rate
3.4. Temperature dependence of the permeation rate
4. Composition of the calibration gas mixture
4.1. Purity of the materials
4.2. Molar masses
4.3. Composition
5. Conclusions
References
Adaptive measuring system with dynamic error estimation of the first ordersensor
1. Introduction
2. Background
3. Measuring System in Adaptive Mode
4. Simulation Study
5. Summary
References
Method of estimation uncertainties of indirect multivariable measurements including the accuracy of processing function as extension of GUM-S2
1. Introduction – actual state
2. Basic formulas of the method extended the GUM-S2
3. Uncertainties of indirect measurement of twoport variables
3.1 Covariance matrix ?? when UP = 0 (GUM-S2 case)
3.3 Matrix UY for correlated impedances and uncorrelated input variables X
3.4. Numerical example
Summary and conclusions
References
Incompatibility of types of measurement uncertainty: A digital paradox and other examples
1. Introduction
2. Three measurement problems
2.1. A digital paradox
2.2. Doubt about the standard deviation of the error
2.3. Doubt about the sample size
2.4. A common feature
3. Different types of uncertainty
3.1. The natures of probability
3.2. Bayes' Theorem as frequency and belief
3.3. Explanations
3.4. Re-examining Bayes' Theorem
4. Another phenomenon
5. Principles and conclusions
5.1. General principles
5.2. Nuisance parameters
5.3. Bayes' Theorem
5.4. Bayesian statistics
5.5. Final comments
References
Invertible calibration curves: Hyperbolic segments and hyperbolic splines
1. Introduction
2. A hyperbolic segment
3. The interpolating hyperbolic segment
4. The least-squares hyperbolic segment
5. An interpolating hyperbolic spline
6. The least-squares hyperbolic spline
7. Error analysis – for interpolating functions
8. A practical study – inversion of a high-order polynomial
9. Concluding comments
References
A logical contradiction in the representation of constants by probability distributions: Premises and implications
1. Introduction
2. Combining unrelated probability distributions for a constant
2.1. Example
2.2. Proof that (1) is correct if the premise is correct
2.3. Some background
3. A contradiction
3.1. An implication
4. Discussion
4.1. Weakening the premise
4.2. Probability and belief
5. Conclusion
References
Polycal - MATLAB algorithm for comparative polynomial calibration and its applications
1. Introduction
2. Estimation of the parameters in the calibration model
3. State-of-knowledge distributions
4. Conclusion
References
Statistical models for similarity score comparisons in firearm evidence identification
1. Introduction
2. Statistical models for CMC measurements
2.1. Binomial distribution
2.2 Correlated binomial distribution
2.3 Beta-binomial distribution
2.4 Beta-correlated binomial distribution
3. Estimating the parameters of the statistical models
4 Discussions and conclusions
References
Keyword index
Author index