PTB Working Group 8.42 provides software implementations within the scope of scientific publications. The software can be downloaded free of charge for noncommercial use. The corresponding publications should be cited when publishing results produced by the software.
Disclaimer: All applications were developed at Physikalisch-Technische Bundesanstalt (PTB). PTB assumes no responsibility whatsoever for its use by other parties, and makes no guarantees, expressed or implied, about its quality, reliability, safety, suitability or any other characteristic. In no event will PTB be liable for any direct, indirect or consequential damage arising in connection with the use of this software.
Measurement uncertainty
A number of software applications are available for evaluating uncertainties according to the GUM. An overview of existing software and their characteristics is offered by the consortium of the EMN Mathmet activity "MU Training" and is freely available 
here.
Rejection sampling for Bayesian uncertainty evaluation using the Monte Carlo techniques of GUM-S1
Supplement 1 to the GUM (GUM-S1) extends the GUM uncertainty framework to nonlinear functions and non-Gaussian distributions. For this purpose, it employs a Monte Carlo method that yields a probability density function for the measurand. This Monte Carlo method has been successfully applied in numerous applications throughout metrology. However, considerable criticism has been raised against the type A uncertainty evaluation of GUM-S1. Most of the criticism could be addressed by including prior information about the measurand which, however, is beyond the scope of GUM-S1. We propose an alternative Monte Carlo method that will allow prior information about the measurand to be included. The proposed method is based on a Bayesian uncertainty evaluation and applies a simple rejection sampling approach using the Monte Carlo techniques of GUM-S1. Software support in Python, Matlab® and even as a spreadsheet example is provided.
Related Publication: Manuel Marschall Gerd Wübbeler and Clemens Elster (2021). Rejection sampling for Bayesian uncertainty evaluation using the Monte Carlo techniques of GUM-S1. Metrologia, (2021) [DOI: 10.1088/1681-7575/ac3920].
Expert: Manuel Marschall
Bayesian Uncertainty Quantification for Magnetic Resonance Fingerprinting
Magnetic Resonance Fingerprinting (MRF) is a promising method in quantitative magnetic resonance (MR) imaging. MRF is based on a sequence of highly undersampled MR images which are analyzed with a pre-computed dictionary leading to estimates of the relaxation times T1 and T2. To calculate uncertainties for these dictionary-based MRF estimates, the Working Group 8.42 has developed a Bayesian uncertainty quantification, which assumes a homoscedastic Gaussian model for the reconstructed magnetization images. The proposed approach was validated by simulated as well as phantom and in vivo data.
Related Publication: Selma Metzner, Gerd Wübbeler, Sebastian Flassbeck, Constance Gatefait, Christoph Kolbitsch and Clemens Elster (2021). Bayesian Uncertainty Quantification for Magnetic Resonance Fingerprinting. Physics in Medicine & Biology (2021), 66(7), 075006 [DOI: 10.1088/1361-6560/abeae7].
Expert: Gerd Wübbeler
Bayesian sample size planning in type A uncertainty evaluation
An easy-to-use GUI for sample size planning and inference based on Bayesian statistics and a method developed in the working group 8.42. No foreknowledge in Bayesian statistics is required.
Related Publication: J.Martin and C. Elster. GUI for Bayesian sample size planning in type A uncertainty evaluation. Measurement Science and Technology (accepted manuscript).
Expert: Jörg Martin
A simple method for Bayesian uncertainty evaluation in linear models
Metrologists often possess useful prior knowledge about the measurand which cannot be accounted for by the GUM uncertainty framework. PTB Working Group 8.42 has developed a simple Bayesian uncertainty evaluation scheme which accounts for prior knowledge about the measurand and the employed measurement device. The proposed approach can be applied to a class of linear models as often encountered in metrology, and it provides a coherent treatment of small numbers of observations, including the case of a single observation only. The calculation of the posterior distribution is carried out by means of a simple Monte Carlo sampling scheme avoiding the application of Markov Chain Monte Carlo methods. In order to ease the application of the proposed Bayesian uncertainty evaluation, corresponding Python software is made available.
Related Publication: G. Wübbeler, M. Marschall and C. Elster (2020). A simple method for Bayesian uncertainty evaluation in linear models. Metrologia [DOI: 10.1088/1681-7575/aba3b8]
Experts: Gerd Wübbeler, Manuel Marschall
An introductory example for Markov chain Monte Carlo (MCMC)
When the Guide to the Expression of Uncertainty in Measurement (GUM) and methods from its supplements are not applicable, the Bayesian approach may be a valid and welcome alternative. Evaluating the posterior distribution, estimates or uncertainties involved in Bayesian inferences often requires numerical methods to avoid high-dimensional integrations. Markov chain Monte Carlo (MCMC) sampling is such a method—powerful, flexible and widely applied. PTB Working Group 8.42 has developed a concise introduction, illustrated by a simple, typical example from metrology. Accompanied with few lines of software code to implement the most basic and yet flexible MCMC method, interested readers are invited to get started. MATLAB as well as R source code are available in the related publication.
Software (in publication)
Related Publication: K. Klauenberg und C. Elster Markov chain Monte Carlo methods: an introductory example. Metrologia, 53(1), S32, 2016. [DOI: 10.1088/0026-1394/53/1/S32]
Expert: Katy Klauenberg
WinBUGS software for the analysis of immunoassay data
For the analysis of immunoassay data, PTB Working Group 8.42 has developed WinBUGS software code to perform MCMC sampling from the posterior distribution for the calibration parameters and the unknown concentration. This software is available in A Guide to Bayesian Inference for Regression Problems.
Software (in publication)
Expert: Katy Klauenberg
MCMC implementation for the analysis of magnetic field fluctuation thermometry
For the analysis in magnetic field fluctuation thermometry, PTB Working Group 8.42 has developed a MATLAB software package to perform MCMC sampling from the posterior distribution of the calibration parameters and to subsequently estimate temperatures. This software is available in the electronic supplement to the related publication.
Software (electronic supplement to the paper)
Related Publication: G. Wübbeler, F. Schmähling, J. Beyer, J. Engert and C. Elster Analysis of magnetic field fluctuation thermometry using Bayesian inference. Measurement Science and Technology, 23(12), 125004, 2012. [DOI: 10.1088/0957-0233/23/12/125004]
Expert: Gerd Wübbeler
Large-scale data analysis
A smoothness informed Low-Rank reconstruction method
Performing high-resolution measurements is an important and challenging task, which often involves long measurement times. To reduce the time-consuming measurement process, the Working Group 8.42 has developed a new “Low-Rank” reconstruction method, which incorporates smoothness information of the dataset, to recover the full data from a smaller set of observations. In a cooperation with Working Group 7.11 and the FU-Berlin, the algorithm was applied to FTIR measurements of complex biological specimens. With only 5% of the original data, a successful reconstruction of the complete dataset was observed. In order to make this algorithm accessible, a corresponding Python software is made available, which can be easily adjusted to various types of data and application.
Related Publication: M. Marschall, Andrea Hornemann, Gerd Wübbeler, Arne Hoehl, Eckart Rühl, Bernd Kästner and Clemens Elster (2020). Compressed FTIR spectroscopy using low-rank matrix reconstruction. Optics Express, Vol. 28, Issue 26, pp. 38762-38772 (2020) [DOI: 10.1364/OE.404959].
Expert: Manuel Marschall
Bayesian Uncertainty Quantification for Magnetic Resonance Fingerprinting
Magnetic Resonance Fingerprinting (MRF) is a promising method in quantitative magnetic resonance (MR) imaging. MRF is based on a sequence of highly undersampled MR images which are analyzed with a pre-computed dictionary leading to estimates of the relaxation times T1 and T2. To calculate uncertainties for these dictionary-based MRF estimates, the Working Group 8.42 has developed a Bayesian uncertainty quantification, which assumes a homoscedastic Gaussian model for the reconstructed magnetization images. The proposed approach was validated by simulated as well as phantom and in vivo data.
Related Publication: Selma Metzner, Gerd Wübbeler, Sebastian Flassbeck, Constance Gatefait, Christoph Kolbitsch and Clemens Elster (2021). Bayesian Uncertainty Quantification for Magnetic Resonance Fingerprinting. Physics in Medicine & Biology (2021), 66(7), 075006 [DOI: 10.1088/1361-6560/abeae7].
Expert: Gerd Wübbeler
Deep learning
Inspecting adversarial examples using the fisher information
This software implements a method for the detection of adversarial examples based on the Fisher information, presented in the Neurocomputing article "Inspecting adversarial examples using the fisher information". Written in python.
Related Publication: J. Martin and C. Elster Inspecting adversarial examples using the fisher information. Neurocomputing, 382 80--86, 2020. [DOI: 10.1016/j.neucom.2019.11.052].
Expert: Jörg Martin
Sampling procedures in legal metrology
Hypothesis-based acceptance sampling for the MID
This R Shiny app is an interactive web-application to calculate optimal sampling plans for statistical conformity assessment according to the modules F and F1 of the European Measuring Instruments Directive (MID).
Related Publication: K. Klauenberg, C. A. Müller and C. Elster Hypothesis-based acceptance sampling for modules F and F1 of the European Measuring Instruments Directive. Statistics and Public Policy, 2021. [DOI: 10.1080/2330443X.2021.1900762].
Expert: Katy Klauenberg
Shapiro-Wilk test
To support the statistical verification of the Normal distribution, PTB working group 8.42 has developed an Excel application, which implements the Shapiro-Wilk test for multiple samples. This application was designed specifically for the qualification procedure in legal metrology and includes the so-called Bonferroni correction.
Software (Microsoft Excel (created under Office 365 ProPlus Version 16.0.11328.20420))
Related Publication: K. Klauenberg and C. Elster Testing normality - An introduction with sample size calculation in legal metrology. tm - Technisches Messen,(online), 2019. [DOI: 10.1515/teme-2019-0148]
Expert: Katy Klauenberg
Regression
Calibration of a torque measuring system - GUM uncertainty evaluation for least-squares versus Bayesian inference
This R Markdown code shall enable metrologists to perform straight-line regression of measurement data by applying (i) ordinary and weighted least-squares estimation in combination with an uncertainty evaluation following the GUM and (ii) Bayesian inference. This is done for a show case example, but can be adapted to new measurement data.
Software (R markdown)
Related Publication: Steffen Martens; Katy Klauenberg and Clemens Elster (2020). EMUE-D6-2-Calibration Uncertainty GUM vs Bayesian. [DOI: 10.5281/zenodo.3858120].
Expert: Katy Klauenberg
Quantifying uncertainty when comparing measurement methods – Haemoglobin concentration as an example of correlation in straight-line regression
In metrology, often two methods measuring the same quantity are to be judged whether or not they are in agreement. For measurements across a whole range of values, this can be done by comparing their straight-line fit to the identity line. Such a comparison is only meaningful, when uncertainties are available. Furthermore, the estimates of the straight-line fit and their uncertainties are only reliable when all sources of uncertainty have been accounted for. When fitting a straight-line relation, the weighted total least-squares (WTLS) method accounts for correlation and uncertainties in both variables. This R code does so for a show case example, but can be adapted to new measurement data.
Software (R markdown)
Related Publication: Steffen Martens; Katy Klauenberg; Jörg Neukammer; Simon Cowen; Stephen L. R. Ellison and Clemens Elster (2020). EMUE-D5-4-Method Comparison With Correlation. [DOI: 10.5281/zenodo.3911583].
Ansprechperson: Katy Klauenberg
Calibration of a sonic nozzle as an example for quantifying all uncertainties involved in straight-line regression
When calibrating a sonic nozzle, it is recommended to estimate the straight-line relationship between the discharge coefficient of the nozzle and the square root of the inverse Reynolds number for a gas. This example emphasizes the importance of accounting for correlation for a reliable uncertainty evaluation. To show this, a measurement model based on the weighted total least-squares (WTLS) method is applied and its input quantities are fully characterized. In particular, we demonstrate in detail how to jointly evaluate the correlation, uncertainties and estimates for the input quantities of least-squares methods applying the Monte Carlo method. This R code does so for a show case example, but can be adapted to new measurement data.
Software (R markdown)
Related Publication: Steffen Martens; Katy Klauenberg; Bodo Mickan; Catherine Yardin; Nicolas Fischer and Clemens Elster (2020). EMUE-D4-3-Quantify Uncertainties In Calibration. [DOI: 10.5281/zenodo.4016915].
Ansprechperson: Katy Klauenberg
Bayesian Uncertainty Quantification for Magnetic Resonance Fingerprinting
Magnetic Resonance Fingerprinting (MRF) is a promising method in quantitative magnetic resonance (MR) imaging. MRF is based on a sequence of highly undersampled MR images which are analyzed with a pre-computed dictionary leading to estimates of the relaxation times T1 and T2. To calculate uncertainties for these dictionary-based MRF estimates, the Working Group 8.42 has developed a Bayesian uncertainty quantification, which assumes a homoscedastic Gaussian model for the reconstructed magnetization images. The proposed approach was validated by simulated as well as phantom and in vivo data.
Related Publication: Selma Metzner, Gerd Wübbeler, Sebastian Flassbeck, Constance Gatefait, Christoph Kolbitsch and Clemens Elster (2021). Bayesian Uncertainty Quantification for Magnetic Resonance Fingerprinting. Physics in Medicine & Biology (2021), 66(7), 075006 [DOI: 10.1088/1361-6560/abeae7].
Expert: Gerd Wübbeler
A smoothness informed Low-Rank reconstruction method
Performing high-resolution measurements is an important and challenging task, which often involves long measurement times. To reduce the time-consuming measurement process, the Working Group 8.42 has developed a new “Low-Rank” reconstruction method, which incorporates smoothness information of the dataset, to recover the full data from a smaller set of observations. In a cooperation with Working Group 7.11 and the FU-Berlin, the algorithm was applied to FTIR measurements of complex biological specimens. With only 5% of the original data, a successful reconstruction of the complete dataset was observed. In order to make this algorithm accessible, a corresponding Python software is made available, which can be easily adjusted to various types of data and application.
Related Publication: M. Marschall, Andrea Hornemann, Gerd Wübbeler, Arne Hoehl, Eckart Rühl, Bernd Kästner and Clemens Elster (2020). Compressed FTIR spectroscopy using low-rank matrix reconstruction. Optics Express, Vol. 28, Issue 26, pp. 38762-38772 (2020) [DOI: 10.1364/OE.404959].
Expert: Manuel Marschall
An introductory example for Markov chain Monte Carlo (MCMC)
When the Guide to the Expression of Uncertainty in Measurement (GUM) and methods from its supplements are not applicable, the Bayesian approach may be a valid and welcome alternative. Evaluating the posterior distribution, estimates or uncertainties involved in Bayesian inferences often requires numerical methods to avoid high-dimensional integrations. Markov chain Monte Carlo (MCMC) sampling is such a method—powerful, flexible and widely applied. PTB Working Group 8.42 has developed a concise introduction, illustrated by a simple, typical example from metrology. Accompanied with few lines of software code to implement the most basic and yet flexible MCMC method, interested readers are invited to get started. MATLAB as well as R source code are available in the related publication.
Software (in publication)
Related Publication: K. Klauenberg und C. Elster Markov chain Monte Carlo methods: an introductory example. Metrologia, 53(1), S32, 2016. [DOI: 10.1088/0026-1394/53/1/S32].
Expert: Katy Klauenberg
Bayesian Normal linear regression
In connection with a tutorial, PTB Working Group 8.42 provides software to calculate the posterior distribution of all regression parameters, the regression curve, predictions as well as for estimates, most uncertainties and credible intervals, and also graphically represent these quantities. [more]
Related Publication: K. Klauenberg, G. Wübbeler, B. Mickan, P. Harris and C. Elster A tutorial on Bayesian Normal linear regression. Metrologia, 52(6), 878--892, 2015. [DOI: 10.1088/0026-1394/52/6/878]
Experts: Katy Klauenberg, Gerd Wübbeler
WinBUGS software for the analysis of immunoassay data
For the analysis of immunoassay data, PTB Working Group 8.42 has developed WinBUGS software code to perform MCMC sampling from the posterior distribution for the calibration parameters and the unknown concentration. This software is available in A Guide to Bayesian Inference for Regression Problems.
Software (in publication)
Expert: Katy Klauenberg
MCMC implementation for the analysis of magnetic field fluctuation thermometry
For the analysis in magnetic field fluctuation thermometry, PTB Working Group 8.42 has developed a MATLAB software package to perform MCMC sampling from the posterior distribution of the calibration parameters and to subsequently estimate temperatures. This software is available in the electronic supplement to the related publication.
Software (electronic supplement to the paper)
Related Publication: G. Wübbeler, F. Schmähling, J. Beyer, J. Engert and C. Elster Analysis of magnetic field fluctuation thermometry using Bayesian inference. Measurement Science and Technology, 23(12), 125004, 2012. [DOI: 10.1088/0957-0233/23/12/125004]
Expert: Gerd Wübbeler
Analysis of key comparisons
On modeling of artefact instability in interlaboratory comparisons
In this work, we present several mathematical models for the treatment of non-negligible artefact instability effects in bilateral comparisons. We highlight the underlying model assumptions and derive analytical formulas for the estimate and standard uncertainty of the instability effect. Moreover, we derive the bilateral degree of equivalence (DoE) by applying the models in a treatment essentially based on the GUM (JCGM-100). For a given 3-stage measurement scenario, the software provided visualises the distribution of the instability, the standard uncertainty of the DoE and the En value for the models proposed in the manuscript.
Related Publication: M. Marschall, G. Wübbeler, M. Borys and C. Elster On modelling of artefact instability in interlaboratory comparisons. Metrologia, 2023. [DOI: 10.1088/1681-7575/ace18f].
Expert: Manuel Marschall
Bayesian hypothesis testing for key comparisons
The assessment of the calibration and measurement capabilities of a laboratory based on a key comparison can often be viewed as carrying out a classical hypothesis test. PTB Working Group 8.42 has developed an alternative Bayesian approach for hypothesis testing which has the advantage that it can include prior assessment about the capabilities of the laboratories participating in the key comparison. In order to ease the application of the proposed Bayesian hypothesis testing for key comparisons, corresponding MATLAB and R software is made available. The software is able to take into account correlations within the key comparison results as well as different prior probabilities of the laboratories. The software also provides routines to enter the key comparison data as well as a graphical representation of the results.
Related Publication: G. Wübbeler, O. Bodnar and C. Elster (2016). Bayesian hypothesis testing for key comparisons. Metrologia 53(4), [DOI: 10.1088/0026-1394/53/4/1131].
Expert: Gerd Wübbeler
Analysis of dynamic measurements
Evaluation of uncertainties in the application of the DFT
The Fourier transform and its counterpart for discrete time signals, the DFT, are common tools in measurement science and application. Although almost every scientific software package offers ready-to-use implementations of the DFT, the propagation of uncertainties in line with GUM is typically neglected. This is of particular importance in dynamic metrology, when input estimation is carried out by deconvolution in the frequency domain. To this end, we present the new open-source software tool GUM2DFT, which utilizes closed formulas for the efficient propagation of uncertainties for the application of the DFT, inverse DFT and input estimation in the frequency domain. It handles different frequency domain representations, accounts for autocorrelation and takes advantage of the symmetry inherent in the DFT result for real-valued time domain signals.
The methods from this software are also part of the larger Python package PyDynamic, which is hosted on GitHub: https://github.com/PTB-PSt1/PyDynamic
Related Publication: S. Eichstädt and V. Wilkens "GUM2DFT -- A software tool for uncertainty evaluation of transient signals in the frequency domain". Meas. Sci. Technol. 27(5), 055001, 2016 [DOI: 10.1088/0957-0233/27/5/055001].
Expert: Sascha Eichstädt
PyDynamic - Software for the analysis of dynamic measurements
As part of the EMPIR project 14SIP08 Dynamic the PTB working group 8.42 together with the National Physical Laboratory (UK) develops a comprehensive Python software package, which contains a large variety of methods for the analysis of dynamic measurements.
Expert: Sascha Eichstädt
Spectral deconvolution with Richardson-Lucy
The correction of deviations in spectra measured with a spectrometer is often necessary in order to obtain accurate results. The classical approach for such a correction is based on a method by Stearns. However, it has been demonstrated that the Richardson-Lucy method can produce much better results here. Therefore, the PTB Working Group has written a software implementation of an adapted Richardson-Lucy method with an automatic stopping rule. This software includes MATLAB code, a Python code as well as a graphical user interface written in Python.
Related Publication: S. Eichstädt F. Schmähling G. Wübbeler, K. Anhalt, L. Bünger, K. Krüger and C. Elster (2013). Comparison of the Richardson-Lucy method and a classical approach for spectrometer bandpass correction. Metrologia 50, 107-118 [DOI: 10.1088/0026-1394/50/2/107].
Experts: Sascha Eichstädt, Franko Schmähling
Monte Carlo for dynamic measurements
The propagation of measurement uncertainties for dynamic measurements using the Monte Carlo propagation of distributions requires efficient implementation in order to achieve a high degree of accuracy, since the computer memory requirement for a standard implementation is very high. In the working group 8.42 a MATLAB software package has been developed to carry out Monte Carlo for dynamic measurements with high accuracy on standard desktop computers.
Related Publication: S. Eichstädt, A. Link, P. Harris and C. Elster (2012). Efficient implementation of a Monte Carlo method for uncertainty evaluation in dynamic measurements. Metrologia 49, 401-410 [DOI: 10.1088/0026-1394/49/3/401].
Expert: Sascha Eichstädt