Joint Committee for Guides in Metrology.
JCGM 102: Evaluation of Measurement Data - Supplement 2 to the "Guide to the Expression of Uncertainty in Measurement" - Extension to Any Number of Output Quantities.
Technical Report, JCGM, 2011.



The "Guide to the expression of uncertainty in measurement" (GUM) [JCGM 100:2008] is mainly concerned with univariate measurement models, namely models having a single scalar output quantity. However, mod- els with more than one output quantity arise across metrology. The GUM includes examples, from electrical metrology, with three output quantities [JCGM 100:2008 H.2], and thermal metrology, with two output quan- tities [JCGM 100:2008 H.3]. This Supplement to the GUM treats multivariate measurement models, namely models with any number of output quantities. Such quantities are generally mutually correlated because they depend on common input quantities. A generalization of the GUM uncertainty framework [JCGM 100:2008 5] is used to provide estimates of the output quantities, the standard uncertainties associated with the estimates, and covariances associated with pairs of estimates. The input or output quantities in the measurement model may be real or complex. Supplement 1 to the GUM [JCGM 101:2008] is concerned with the propagation of probability distributions [JCGM 101:2008 5] through a measurement model as a basis for the evaluation of measurement uncertainty, and its implementation by a Monte Carlo method [JCGM 101:2008 7]. Like the GUM, it is only concerned with models having a single scalar output quantity [JCGM 101:2008 1]. This Supplement describes a generalization of that Monte Carlo method to obtain a discrete representation of the joint probability distribution for the output quantities of a multivariate model. The discrete representation is then used to provide estimates of the output quantities, and standard uncertainties and covariances associated with those estimates. Appropriate use of the Monte Carlo method would be expected to provide valid results when the applicability of the GUM uncertainty framework is questionable, namely when (a) linearization of the model provides an inadequate representation, or (b) the probability distribution for the output quantity (or quantities) departs appreciably from a (multivariate) Gaussian distribution. Guidance is also given on the determination of a coverage region for the output quantities of a multivariate model, the counterpart of a coverage interval for a single scalar output quantity, corresponding to a stipulated coverage probability. The guidance includes the provision of coverage regions that take the form of hyper- ellipsoids and hyper-rectangles. A calculation procedure that uses results provided by the Monte Carlo method is also described for obtaining an approximation to the smallest coverage region.


@TechReport{     jcgm:2011:ANOQ,
  author = 	 {Joint Committee for Guides in Metrology},
  title = 	 {JCGM 102: Evaluation of Measurement Data -
                  Supplement 2 to the "Guide to the Expression of
                  Uncertainty in Measurement" - Extension to Any
                  Number of Output Quantities},
  institution =  {JCGM},
  year = 	 {2011},



[1] Barnett, V. The ordering of multivariate data. J. R. Statist. Soc. A 139 (1976), 318.
[2] Bich, W., Callegaro, L., and Pennecchi, F. Non-linear models and best estimates in the GUM. Metrologia 43 (2006), S196{S199.
[3] Bich, W., Cox, M. G., and Harris, P. M. Uncertainty modelling in mass comparisons. Metrologia 30 (1993), 1-12.
[4] Chayes, F. Petrographic Modal Analysis: An Elementary Statistical Appraisal. John Wiley & Sons, New York, 1956.
[5] Comtet, L. Bonferroni Inequalities { Advanced Combinatorics: The Art of Finite and Infinite Expansions.Reidel, Dordrecht, Netherlands, 1974.
[6] Cox, M. G., Esward, T. J., Harris, P. M., McCormick, N., Ridler, N. M., Salter, M. J., and Walton, J. P. R. B. Visualization of uncertainty associated with classes of electrical measurement. Tech. Rep. CMSC 44/04, National Physical Laboratory, Teddington, UK, 2004.
[7] Cox, M. G., and Harris, P. M. SSfM Best Practice Guide No. 6, Uncertainty evaluation. Tech. Rep. MS 6, National Physical Laboratory, Teddington, UK, 2010.
[8] Cox, M. G., Harris, P. M., and Smith, I. M. Software specifications for uncertainty evaluation. Tech. Rep.MS 7, National Physical Laboratory, Teddington, UK, 2010.
[9] EAL. Calibration of pressure balances. Tech. Rep. EAL-G26, European Cooperation for Accreditation of Laboratories, 1997.
[10] Engen, G. F. Microwave Circuit Theory and Foundations of Microwave Metrology. Peter Peregrinus, London, 1992.
[11] Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. Bayesian Data Analysis. Chapman & Hall/CRC, Boca Raton, Florida, 2004. Second edition.
[12] Gill, P. E., Murray, W., and Wright, M. H. Practical Optimization. Academic Press, London, 1981.
[13] Golub, G. H., and Van Loan, C. F. Matrix Computations. John Hopkins University Press, Baltimore, MD, USA, 1996. Third edition.
[14] Hall, B. D. On the propagation of uncertainty in complex-valued quantities. Metrologia 41 (2004), 173-177.
[15] Kacker, R., Toman, B., and Huang, D. Comparison of ISO-GUM, draft GUM Supplement 1 and Bayesian statistics using simple linear calibration. Metrologia 43 (2006), S167-S177.
[16] Kerns, D. M., and Beatty, R. W. Basic Theory of Waveguide Junctions and Introductory Microwave Network Analysis. Pergamon Press, London, 1967.
[17] Lewis, S., and Peggs, G. The Pressure Balance: A Practical Guide to its Use. Second edition. HMSO, London, 1991.
[18] Lira, I. Evaluating the Uncertainty of Measurement. Fundamentals and Practical Guidance. Institute of Physics, Bristol, UK, 2002.
[19] Mardia, K. V., Kent, J. T., and Bibby, J. M. Multivariate Analysis. Academic Press, London, UK, 1979.
[20] Possolo, A. Copulas for uncertainty analysis. Metrologia 47 (2010), 262{271.
[21] Rice, J. R. Mathematical Statistics and Data Analysis, second ed. Duxbury Press, Belmont, Ca., USA, 1995.
[22] Scott, D. W. Multivariate Density Estimation: Theory, Practice, and Visualization. John Wiley & Sons, New York, 1999.
[23] Scott, D. W., and Sain, S. R. Multi-dimensional density estimation. In Handbook of Statistics, Volume 23, Data Mining and Computational Statistics (Amsterdam, 2004), C. Rao and E. Wegman, Eds., Elsevier, pp. 229-261.
[24] Silverman, B. W. Density Estimation. Chapman and Hall, London, 1986.
[25] Smith, D. B. Granite, G{2. Certificate of Analysis. United States Geological Survey, Denver, Colorado, 1998.
[26] Somlo, P. I., and Hunter, J. D. Microwave Impedance Measurement. Peter Peregrinus, London, 1985.
[27] Strang, G., and Borre, K. Linear Algebra, Geodesy and GPS. Wiley, Wellesley-Cambridge Press, 1997.
[28] Wubbeler, G., Harris, P. M., Cox, M. G., and Elster, C. A two-stage procedure for determining the number of trials in the application of a Monte Carlo method for uncertainty evaluation. Metrologia 47, 3 (2010), 317.