, Volume 34, Issue 3, pp 387–402 | Cite as

Analysis and Comparison of Hyper-Ellipsoidal and Smallest Coverage Regions for Multivariate Monte Carlo Measurement Uncertainty Analysis Simulation Datasets

  • Vishal RamnathEmail author
Original Paper


Traditionally metrology systems have been analysed for measurement uncertainties in terms of the frequency statistics-based Guide to the Uncertainty in Measurement (GUM); however, a key challenge in the application of the GUM has been in terms of its inherent limitations and internal inconsistencies with Type A and Type B uncertainties in adequately and accurately determining appropriate coverage intervals and regions for measurement uncertainty results. Subsequently in order to address these particular issues the Bayesian statistical-based GUM supplements for univariate and multivariate models were developed that supersede the original GUM and which resolve these challenges. In this paper, a GUM supplement 2 uncertainty analysis for a multivariate oil pressure balance model is numerically implemented using an experimental dataset, and then the multivariate Monte Carlo method simulation results are processed in order to construct and study the corresponding optimal hyper-ellipsoidal and smallest coverage regions for bivariate and trivariate distributions with new proposed numerical algorithms for specified probability levels. The results are then further investigated in order to study the accuracy, validity limits and potential confidence region implications for measurement models that exhibit non-Gaussian joint probability density function distributions.


Uncertainty analysis Monte Carlo method GUM supplement 2 Multivariate confidence region Pressure metrology 



This work was performed with funds provided by the Department of Higher Education and Training (DHET) on behalf of the South African government for research by public universities.


  1. [1]
    BIPM, IEC, IFCC, ILAC, ISO, IUPAP, OIML, Evaluation of measurement data—guide to the expression of uncertainty in measurement, Tech. Rep., JCGM/WG1 GUM (2008), Revised 1st edition—
  2. [2]
    I. Lira, The GUM revision: the Bayesian view toward the expression of measurement uncertainty, Eur. J. Phys., 37 (2016) 025803 (16 pp). Scholar
  3. [3]
    BIPM, IEC, IFCC, ILAC, ISO, IUPAP, OIML, Evaluation of measurement data—Supplement 1 to the “Guide to the expression of uncertainty in measurement”—Propogation of distributions using a Monte Carlo method, Tech. Rep., JCGM/WG1 GUM Supplement 1 (2008), 1st edition—
  4. [4]
    BIPM, IEC, IFCC, ILAC, ISO, IUPAP, OIML, Evaluation of measurement data—Supplement 2 to the “Guide to the expression of uncertainty in measurement”—Propogation of distributions using a Monte Carlo method, Tech. Rep., JCGM/WG1 GUM Supplement 2 (2011), 1st edition—
  5. [5]
    W. Bich, M. Cox, C. Michotte, Towards a new GUM – an update, Metrologia, 53 (2016) S149–S159. Scholar
  6. [6]
    R. Willink, Representating Monte Carlo output distributions for transfereability in uncertainty analysis: modelling with quantile functions, Metrologia, 46 (2009) 154–166. Scholar
  7. [7]
    CGPM. Resolution 1 of the 26th CGPM—On the revision of the international system of units (SI) (2018).
  8. [8]
    V. Ramnath, Numerical analysis of the accuracy of bivariate quantile distributions utilizing copulas compared to the GUM supplement 2 for oil pressure balance uncertainties, Int. J. Metrol. Qual. Eng., 8 (2017) 29. Scholar
  9. [9]
    R.S. Dadson, S.L. Lewis, G.N. Peggs, The Pressure Balance: Theory and Practice (HMSO: London, 1982). ISBN 0114800480.Google Scholar
  10. [10]
    K. Jousten, J. Hendricks, D. Barker, K. Douglas, S. Eckel, P. Egan, J. Fedchak, J. Flugge, C. Gaiser, D. Olson, J. Ricker, T. Rubin, W. Sabuga, J. Scherschligt, R. Schodel, U. Sterr, J. Stone, G. Strouse, Perspectives for a new realization of the pascal by optical methods, Metrologia, 54 (2017) S146–S161. Scholar
  11. [11]
    W.J. Bowers, D.A. Olson, A capacitive probe for measuring the clearance between the piston and the cylinder of a gas piston gauge, Rev. Sci. Instrum., 81(035102) (2010) 7. Scholar
  12. [12]
    B. Blagojevic, I. Bajic, in Metrology in the 3rd Millennium, XVII. IMEKO World Congress, (IMEKO, Dubrovnik, Croatia, 2003), pp. 1982–1985. ISBN: 953-7124-00-2.Google Scholar
  13. [13]
    S. Yadav, V.K. Gupta, A.K. Bandyopadhyay, Standardisation of pressure measurement using pressure balance as transfer standard, MAPAN - J. Metrol. Soc. India, 26(2) (2011) 133–151.Google Scholar
  14. [14]
    P.M. Harris, C.E. Matthews, M.G. Cox, Summarizing the output of a Monte Carlo method for uncertainty evaluation, Metrologia, 51 (2014) 243–252. Scholar
  15. [15]
    V. Ramnath, Application of quantile functions for the analysis and comparison of gas pressure balance uncertainties, Int. J. Metrol. Qual. Eng., 8 (2017) 4 (18 pp). Scholar
  16. [16]
    G. Carlier, V. Chernozhukov, A. Galichon, Vector quantile regression: an optimal transport approach, Ann. Stat., 44(3) (2016) 1165–1192. Scholar
  17. [17]
    P.M. Harris, M.G. Cox, On a Monte Carlo method for measurement uncertainty evaluation and its implementation, Metrologia, 51 (2014) S176–S182. Scholar
  18. [18]
    A. Possolo, Copulas for uncertainty analysis, Metrologia, 47 (2010) 262–271. Scholar
  19. [19]
    J. Segers, M. Sibuya, H. Tsukahara, The empirical beta copula, J. Multivar. Anal., 155 (2017) 35–51. Scholar
  20. [20]
    D.W. Scott, S.R. Sain, in Handbook of statistics—Data mining and data visualization, ed. by C.R. Rao, E.J. Wegman, J.L. Solka (Elsevier, Oxford, 2005), chap. 9, pp. 229–261. Scholar
  21. [21]
    T.A. O’Brien, K. Kashinath, N.R. Cavanaugh, W.D. Collins, J.P. O’Brien, A fast and objective multidimensional kernel density estimation method: fastKDE, Comput. Stat. Data Anal., 101 (2016) 148–160. Scholar
  22. [22]
    T. Nagler, C. Czado, Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas, J. Multivar. Anal., 151 (2016) 69–89. Scholar
  23. [23]
    Y.C. Chen, A tutorial on kernal density estimation and recent advances, Biostat. Epidemiol., 1(1) (2017) 161–187. Scholar
  24. [24]
    S.N. Lee, M.H. Shih, A volume problem for an n-dimensional ellipsoid intersecting with a hyperplane, Linear Algebra Appl. 132 (1990) 90–102. Scholar
  25. [25]
    M. Friendly, G. Monette, J. Fox, Elliptical insights: understanding statistical methods through elliptical geometry, Stat. Sci., 28(1) (2013) 1–39. Scholar

Copyright information

© Metrology Society of India 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringUniversity of South AfricaFloridaSouth Africa

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