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Propagation of Uncertainty and Comparison of Interpolation Schemes

  • D. R. WhiteEmail author
TEMPMEKO 2016
Part of the following topical collections:
  1. TEMPMEKO 2016: Selected Papers of the 13th International Symposium on Temperature, Humidity, Moisture and Thermal Measurements in Industry and Science

Abstract

The numerical information in the calibration reports for indicating instruments is typically sparse, often comprising a simple table of corrections or indicated values against a small number of reference values. Users are left to interpolate between the tabulated values using one of several well-known interpolation algorithms, including straight-line, spline, Lagrange, and least-squares interpolation. Although these algorithms are well known, there has apparently been no comparison of their performance in respect of uncertainty propagation. This paper provides an overview of the advantages and disadvantages of the most common interpolation algorithms with respect to uncertainty propagation, immunity to interpolation error, and sensitivity to data spacing. Secondly, the paper illustrates an unconventional method for the uncertainty analysis. The method exploits the linear dependence of the interpolations on measurements of the interpolated quantity, and is easily applied to any linear functional interpolation. In many respects, the best all-round interpolation scheme is a polynomial fitted by least-squares methods, which has a low propagated uncertainty, continuity to the chosen order of the fitted polynomial, and a good immunity to large gaps in the data.

Keywords

Interpolation Lagrange interpolation Least-squares Linear interpolation Measurement uncertainty Spline 

Notes

Acknowledgements

The author gratefully acknowledges members of the Metrology Group on LinkedIn whose questions prompted this study.

References

  1. 1.
    W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986)zbMATHGoogle Scholar
  2. 2.
    P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969)Google Scholar
  3. 3.
    P. Marcarino, P.P.M. Steur, A. Merlone, Interpolating equations for Industrial Platinum Resistance Thermometers, in Advanced Mathematical and Computational Tools in Metrology VI (World Scientific, Singapore, 2004), pp. 318–322Google Scholar
  4. 4.
    D.R. White, K. Hill, D. del Campo, C. Garcia Izquierdo, Guide on Secondary Thermometry: Thermistor Thermometry. (BIPM, Paris, 2014). http://www.bipm.org/utils/common/pdf/ITS-90/Guide-SecTh-Thermistor-thermometry.pdf
  5. 5.
    F. Sakuma, S. Hattori, Establishing a practical temperature scale standard by using a narrow-band radiation thermometer with a silicon detector, in Temperature: Its Measurement and Control in Science and Industry, vol. 5, ed. by J.F. Schooley (AIP, New York, 1982), pp. 421–427Google Scholar
  6. 6.
    P. Saunders, D.R. White, Physical basis of interpolation equations for radiation thermometry. Metrologia 40, 195–204 (2003)ADSCrossRefGoogle Scholar
  7. 7.
    E.W. Cheney, Introduction to Approximation Theory, 2nd edn. (Chelsea Publishing, Providence, 1982)zbMATHGoogle Scholar
  8. 8.
    T.J. Rivlin, An Introduction to the Approximation of Functions (Waltham, Blaisdell, 1969)zbMATHGoogle Scholar
  9. 9.
    R.W. Hamming, Numerical Methods for Scientists and Engineers, 2nd edn. (Dover, New York, 1973)zbMATHGoogle Scholar
  10. 10.
    M. Abramowitz, I.E. Stegun, Handbook of Mathematical Functions (US Gov. Printing Office, Washington DC, 1972)zbMATHGoogle Scholar
  11. 11.
    D.R. White, P. Saunders, The propagation of uncertainty with calibration equations. Meas. Sci. Technol. 18, 2157–2169 (2007)ADSCrossRefGoogle Scholar
  12. 12.
    BIPM, 2008, Evaluation of measurement data—guide to the expression of uncertainty in measurement JCGM 100:2008 (BIPM, Paris). http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf
  13. 13.
    K.D. Hill, A.G. Steele, The non-uniqueness of the ITS-90: 13.8033 K to 273.16 K, in Temperature: Its Measurement and Control in Science and Industry, vol. 7, ed. by D.C. Ripple (American Institute of Physics, New York, 2002), pp. 53–58Google Scholar
  14. 14.
    P. Saunders, D.R. White, Interpolation errors for radiation thermometry. Metrologia 41, 41–46 (2004)ADSCrossRefGoogle Scholar
  15. 15.
    J.L. Gardner, Correlations in primary spectral standards. Metrologia 40, S167–S171 (2003)ADSCrossRefGoogle Scholar
  16. 16.
    J.V. Beck, K.J. Arnold, Parameter Estimation in Science and Engineering (Wiley, New York, 1977)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Measurement Standards LaboratoryLower HuttNew Zealand

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