Content
Overview
The reliability of measurement results is a crucial prerequisite for today’s worldwide economy, e.g. when checking the conformity of products with an agreed quality standard. A complete measurement result hence requires a quantitative statement describing its associated uncertainty. Measurement uncertainty is particularly relevant in metrology, for example when the result of a measurement is traced back to an SI unit.
The uncertainty about a measurement result is often due to random variations in measured data. Systematic deviations which remain unchanged in repeated measurements are another source of uncertainty. In order to evaluate measurement uncertainties, statistical methods are employed. Classical statistics, for example, can be used to determine confidence intervals for the sought quantities that account for observed random variations in the measured data. Bayesian statistics provides an alternative that also allows further information to be taken into account. This approach treats random variability and systematic deviations in a consistent way, and it results in probability statements about the quantities of interest. In particular, the Bayesian approach enables to be taken into account prior knowledge, e.g., when negative results can be ruled out for physical reasons. As a result probability statements can then be made about the sought measureand.
GUM: Guide to the Expression of Uncertainty in Measurement
In metrology the uncertainty of a measurement result is often dominated by systematic deviations. The “Guide to the Expression of Uncertainty in Measurement“ (GUM) enables systematic deviations and random variations to be coherently taken into account. The GUM can be seen as the de facto standard for the evaluation of measurement uncertainty in metrology. An important concept of the GUM methodology is that of a model which relates the measurand to socalled input quantities. In using information about the input quantities, this model is utilized to determine an estimate and its associated uncertainty for the measurand according to the rule of “propagation of uncertainties”.
Supplement 1 to the GUM (GUM S1) proposes a Monte Carlo Method (MCM) for the calculation of uncertainties. Similarl to the GUM, a model relating the measurand and input quantities is taken as the basis. By using probability density functions (PDFs) expressing the knowledge about the input quantities, MCM is then used to determine the PDF associated with the measurand by “propagation of distributions”. There is a certain relationship between GUM S1 and a Bayesian uncertainty analysis. A further supplement to the GUM deals with uncertainty evaluation for multivariate (e.g., complex valued) quantities (GUM S2).
Software
In order to facilitate the application of the methods developed in the working group, the following software implementations are made available free of charge.
 MCMC implementation for the analysis of magnetic field fluctuation thermometry
Bayesian approaches to performing regression often require numerical methods such as Markov Chain Monte Carlo (MCMC) sampling. 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. Related publication
G. Wübbeler, F. Schmähling, J. Beyer, J. Engert, and C. Elster (2012). Analysis of magnetic field fluctuation thermometry using Bayesian inference. Meas. Sci. Technol. 23, 125004 (9pp), [DOI: 1018088/09570233/23/12/125004].
 WinBUGS software for the analysis of immunoassay data
The Bayesian approach enables the inclusion of additional prior knowledge in regression problems, but often requires numerical methods such as Markov Chain Monte Carlo (MCMC) sampling. 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. Related publications
K. Klauenberg, M. Walzel, B. Ebert, and C. Elster (2015). Informative prior distributions for ELISA analyses. Biostatistics 16, 454464, [DOI: 10.1093/biostatistics/kxu057].
C. Elster, K. Klauenberg, M. Walzel, G. Wübbeler, P. Harris, M. Cox, C. Matthews, I. Smith, L. Wright, A. Allard, N. Fischer, S. Cowen, S. Ellison, P. Wilson, F. Pennecchi, G. Kok, A. van der Veen, and L. Pendrill (2015). A Guide to Bayesian Inference for Regression Problems Deliverable of EMRP project NEW04 “Novel mathematical and statistical approaches to uncertainty evaluation”, [download (pdf)].
 Rejection sampling for the flow meter calibration problem
Bayesian approaches to Normal linear regression problems yield analytical solutions under certain circumstances. Nevertheless, accounting for constraints on the values of the regression curve when calibrating flow meters requires a Monte Carlo procedure combined with an accept/reject algorithm to obtain samples from the posterior distribution.
MATLAB source code implementing this algorithm is available in A Guide to Bayesian Inference for Regression Problems Related publications
G. J. P. Kok, A. M. H. van der Veen, P. M. Harris, I.M. Smith, C. Elster (2015). Bayesian analysis of a flow meter calibration problem. Metrologia 52, 392399, [DOI: 10.1088/00261394/52/2/392].
C. Elster, K. Klauenberg, M. Walzel, G. Wübbeler, P. Harris, M. Cox, C. Matthews, I. Smith, L. Wright, A. Allard, N. Fischer, S. Cowen, S. Ellison, P. Wilson, F. Pennecchi, G. Kok, A. van der Veen, and L. Pendrill (2015). A Guide to Bayesian Inference for Regression Problems Deliverable of EMRP project NEW04 “Novel mathematical and statistical approaches to uncertainty evaluation”, [download (pdf)].
 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 highdimensional 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.
 Related Publication
K. Klauenberg und C. Elster Markov chain Monte Carlo methods: an introductory example. Metrologia, 53(1), S32, 2016. [DOI: 10.1088/00261394/53/1/S32]
Workshops
 315. PTBSeminar: Berechnung der Messunsicherheit, 30 September  1 October 2020
 308. PTBSeminar: Berechnung der Messunsicherheit, 1516 March 2018
 293. PTBSeminar: Berechnung der Messunsicherheit, 1718 March 2016
 MATHMET 2016
 BIPM "Workshop on Measurement Uncertainty" (June 2015)
 Workshop and training course: Novel mathematical and statistical approaches to uncertainty evaluation
 MATHMET 2014
 277. PTB Seminar zur Messunsicherheit
 266. PTB Seminar zur Messunsicherheit
 260. PTB Seminar zur Messunsicherheit
 MATHMET 2010
Research
While the available guidelines are appropriate in many metrological applications, the development of procedures for the evaluation of measurement uncertainty in more involved applications is a topic of ongoing research. Our current research interests are
 Development and application of procedures based on Bayesian statistics
 (Markov Chain) Monte Carlo methods for the numerical calculation of posterior PDFs. (MFFT, ELISA, introductory example)
 Regression
 Analysis of dynamic measurements
 Virtual experiments (Form measurement of curved optical surfaces)
Publications
• 
G. Bartl, C. Elster, J. Martin, R. Schödel, M. Voigt and A. Walkov
Measurement Science and Technology, 31(6),
2020.
[DOI: 10.1088/13616501/ab7359]

• 
G. Wübbeler and C. Elster
Metrologia, 57(1),
2020.
[DOI: 10.1088/16817575/ab50d6]

• 
K. Klauenberg, G. Wübbeler and C. Elster
Measurement Science Review, 19(5),
204208,
2019.
[DOI: 10.2478/msr20190026]

• 
J. Martin, G. Bartl and C. Elster
Measurement Science and Technology, 30
045012,
2019.
[DOI: 10.1088/13616501/ab094b]

• 
O. Bodnar and C. Elster
AStA Advances in Statistical Analysis, 102(1),
120,
2018.
[DOI: 10.1007/s1018201602797]

• 
G. Wübbeler, O. Bodnar and C. Elster
Metrologia, 55(1),
20,
2018.
[DOI: 10.1088/16817575/aa98aa]

• 
F. Schmähling, G. Wübbeler, U. Krüger, B. Ruggaber, F. Schmidt, R. D. Taubert, A. Sperling and C. Elster
Color, Research and Application, 43(1),
616,
2017.
[DOI: 10.1002/col.22185]

• 
M. Reginatto, M. Anton and C. Elster
Metrologia, 54(4),
S74S82,
2017.
[DOI: 10.1088/16817575/aa735b]

• 
S. Eichstädt and V. Wilkens
J. Acoust. Soc. Am., 141(6),
41554167,
2017.
[DOI: 10.1121/1.4983827]

• 
O. Bodnar, R. Behrens and C. Elster
Metrologia, 54(3),
S29S33,
2017.
[DOI: 10.1088/16817575/aa69ad]

• 
S. Eichstädt, C. Elster, I. M. Smith and T. J. Esward
J. Sens. Sens. Syst., 6
97105,
2017.
[DOI: 10.5194/jsss6972017]

• 
O. Bodnar, A. Link, B. Arendacká, A. Possolo and C. Elster
Statistics in Medicine, 39(2),
378399,
2017.
[DOI: 10.1002/sim.7156]

• 
K. Klauenberg and C. Elster
Metrologia, 54(1),
5968,
2017.
[DOI: 10.1088/16817575/54/1/59]

• 
G. Wübbeler, J. Campos Acosta and C. Elster
Color Research & Application,
2016.
[DOI: 10.1002/col.22109]

• 
O. Bodnar, C. Elster, J. Fischer, A. Possolo and B. Toman
Metrologia, 53(1),
S46,
2016.
[DOI: 10.1088/00261394/53/1/S46]

• 
K. Klauenberg and C. Elster
Metrologia, 53(1),
S32,
2016.
[DOI: 10.1088/00261394/53/1/S32]

• 
C. Elster and G. Wübbeler
Metrologia, 53(1),
S10,
2016.
[DOI: 10.1088/00261394/53/1/S10]

• 
O. Bodnar, A. Link and C. Elster
Bayesian Analysis, 11(1),
2545,
2016.
[DOI: 10.1214/14BA933]
(Open Access)

• 
K. Klauenberg, G. Wübbeler, B. Mickan, P. Harris and C. Elster
Metrologia, 52(6),
878892,
2015.
[DOI: 10.1088/00261394/52/6/878]

• 
C. Elster, K. Klauenberg, M. Walzel, P. M. Harris, M. G. Cox, C. Matthews, L. Wright, A. Allard, N. Fischer, S. Ellison, P. Wilson, F. Pennecchi, G. J. P. Kok, A. Van der Veen and L. Pendrill
EMRP NEW04,
, 2015

• 
G. Wübbeler, O. Bodnar, B. Mickan and C. Elster
Metrologia, 52(2),
400405,
2015.
[DOI: 10.1088/00261394/52/2/400]

• 
G. J. P. Kok, A. M. H. van der Veen, P. M. Harris, I. M. Smith and C. Elster
Metrologia, 52(2),
392399,
2015.
[DOI: 10.1088/00261394/52/2/392]

• 
A. Possolo and C. Elster
Metrologia, 51(3),
339353,
2014.
[DOI: 10.1088/00261394/51/3/339]

• 
O. Bodnar and C. Elster
Journal of Statistical Planning and Inference, 147
106116,
2014.
[DOI: 10.1016/j.jspi.2013.11.003]

• 
O. Bodnar and C. Elster
Metrologia, 51(5),
516521,
2014.
[DOI: 10.1088/00261394/51/5/516]

• 
B. Arendacká, A. Täubner, S. Eichstädt, T. Bruns and C. Elster
Measurement Science Review, 14(2),
5261,
2014.
[DOI: 10.2478/msr20140009]

• 
S. J. Haslett, S. Puntanen and B. Arendacká
Statistical Papers, 56(3),
849861,
2014.
[DOI: 10.1007/s0036201406119]

• 
C. Elster
Metrologia, 51(4),
S159S166,
2014.
[DOI: 10.1088/00261394/51/4/S159]

• 
G. Wübbeler and C. Elster
Measurement Science and Technology, 24(11),
115004,
2013.

• 
S. Eichstädt, A. Link, P. M. Harris and C. Elster
Efficient implementation of a Monte Carlo method for uncertainty evaluation in dynamic measurements.
Metrologia, 49(3),
401,
2012.
[DOI: 10.1088/00261394/49/3/401]

• 
K. Klauenberg and C. Elster
Metrologia, 49(3),
395400,
2012.
[DOI: 10.1088/00261394/49/3/395]

• 
G. Wübbeler, F. Schmähling, J. Beyer, J. Engert and C. Elster
Measurement Science and Technology, 23(12),
125004,
2012.

• 
W. Bich, M. G. Cox, R. Dybkaer, C. Elster, W. T. Estler, B. Hibbert, H. Imai, W. Kool, C. Michotte, L. Nielsen, L. Pendrill, S. Sidney, A. M. H. van der Veen and W. Wöger
Metrologia, 49(6),
702705,
2012.
[DOI: 10.1088/00261394/49/6/702]

• 
C. Elster and I. Lira
Computational Statistics, 27(2),
219235,
2012.
[DOI: 10.1007/s0018001102517]

• 
G. Wübbeler, P. M. Harris, M. G. Cox and C. Elster
In F. Pavese, M. Bär, J.M. Limares, C. Perruchet, N.F. Zhang, editor, Volume Advanced Mathematical & Computational Tools in Metrology IXof Series on Advances in Mathematics for Applied Sciences
Chapter 54, page 434
Publisher: World Scientific New Jersey,
84 edition
, 2012

• 
S. Eichstädt and C. Elster
In F. Pavese, M. Bär, J.R. Filtz, A. B. Forbes, L. Pendrill, K. Shirono, editor, Volume Advanced Mathematical & Computational Tools in Metrology and Testing IX of Series on Advances in Mathematics for Applied Sciences
Chapter 16, page 126135
Publisher: World Scientific New Jersey,
84 edition
, 2012

• 
T. J. Esward, C. Matthews, S. Downes, A. Knott, S. Eichstädt and C. Elster
In F. Pavese, M. Bär, J.R. Filtz, A. B. Forbes, L. Pendrill, K. Shirono, editor, Volume Advanced Mathematical & Computational Tools in Metrology and Testing IX of Series on Advances in Mathematics for Applied Sciences
Chapter 19, page 143151
Publisher: World Scientific New Jersey,
84 edition
, 2012

• 
O. Bodnar, G. Wübbeler and C. Elster
Metrologia, 48(5),
333342,
2011.
[DOI: 10.1088/00261394/48/5/014]

• 
C. Elster and B. Toman
Metrologia, 48(5),
233240,
2011.
[DOI: 10.1088/00261394/48/5/001]

• 
G. Wübbeler, P. M. Harris, M. G. Cox and C. Elster
Metrologia, 47(3),
317324,
2010.
[DOI: 10.1088/00261394/47/3/023]

• 
S. Eichstädt, A. Link, T. Bruns and C. Elster
Measurement, 43(5),
708713,
2010.

• 
A. Link and C. Elster
Measurement Science and Technology, 20(5),
055104,
2009.

• 
C. Elster and B. Toman
Metrologia, 46(3),
261266,
2009.
[DOI: 10.1088/00261394/46/3/013]

• 
I. Lira, C. Elster, W. Wöger and M. G. Cox
In F. Pavese, M. Bär, J.M. Limares, C. Perruchet, N.F. Zhang, editor, Volume Advanced Mathematical & Computational Tools in Metrology VIIIof Series on Advances in Mathematics for Applied Sciences
Chapter 31, page 213
Publisher: World Scientific New Jersey,
78 edition
, 2009

• 
G. Wübbeler, A. Link, T. Bruns and C. Elster
In F. Pavese, M. Bär, J.M. Limares, C. Perruchet, N.F. Zhang, editor, Volume Advanced Mathematical & Computational Tools in Metrology VIIIof Series on Advances in Mathematics for Applied Sciences
Chapter 52, page 369374
Publisher: World Scientific New Jersey,
78 edition
, 2009

• 
C. Elster and A. Link
In F. Pavese, M. Bär, J.M. Limares, C. Perruchet, N.F. Zhang, editor, Volume Advanced Mathematical & Computational Tools in Metrology VIIIof Series on Advances in Mathematics for Applied Sciences
Chapter 13, page 8089
Publisher: World Scientific New Jersey,
78 edition
, 2009

• 
G. Wübbeler, M. Krystek and C. Elster
Measurement Science and Technology, 19(8),
084009,
2008.

• 
C. Elster and A. Link
Metrologia, 45(4),
464473,
2008.
[DOI: 10.1088/00261394/45/4/013]

• 
I. Lira, C. Elster and W. Wöger
Metrologia, 44(5),
379384,
2007.
[DOI: 10.1088/00261394/44/5/014]

• 
C. Elster, W. Wöger and M. G. Cox
Metrologia, 44(3),
L31L32,
2007.
[DOI: 10.1088/00261394/44/3/N03]

• 
C. Elster, A. Link and T. Bruns
Measurement Science and Technology, 18(12),
36823687,
2007.
[DOI: 10.1088/09570233/18/12/002]

• 
C. Elster
Metrologia, 44(2),
111116,
2007.
[DOI: 10.1088/00261394/44/2/002]

• 
C. Elster, F. Schubert, A. Link, M. Walzel, F. Seifert and H. Rinneberg
Magnetic resonance in medicine, 53(6),
128896,
2005.
[DOI: 10.1002/mrm.20500]
