Experiments in Fluids

, Volume 44, Issue 4, pp 591–596 | Cite as

Confidence estimation using dependent circular block bootstrapping: application to the statistical analysis of PIV measurements

  • R. TheunissenEmail author
  • A. Di Sante
  • M. L. Riethmuller
  • R. A. Van den Braembussche
Research Article


In this paper the authors show an original application to time-resolved PIV of an existing method for confidence level and error determination, called dependent bootstrapping. Due to the high sampling frequencies the measured velocity samples are no longer uncorrelated making classical statistical procedures not applicable. Examples show that the various ways in calculating the number of independent samples based on the autocorrelation function question the reliability of the rarely, if ever, mentioned confidence levels in literature. Instead, the dependent bootstrapping technique reports consistent results of confidence estimates for both correlated and uncorrelated PIV velocity samples making this technique robust and general for further applications. The step-by-step description of the dependent circular block bootstrap implementation is given. The practical application and viability of the method are illustrated by two experimental cases.


Autocorrelation Function Block Length Fractional Error Sample Record Block Bootstrap 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


block length


number of bootstrap replications


step height of the backward facing step


Least Squares


number of recorded samples


number of recorded independent samples (effective sample size)


Probability Density Function


Particle image velocimetry


autocorrelation function


Standard Deviation


recording time


Student’s t value


integral time-scale


observed variable


sampling time


mean of the ensemble of bootstrap replications


autocorrelation coefficient


variance of the ensemble of bootstrap replications


maximum time-lag in the calculation of the autocorrelation function


desired statistical parameter


statistical estimator of ‘ξ



The authors would like to acknowledge the support from the Instituut voor de aanmoediging van innovatie door Wetenschap & Technologie in Vlaanderen (IWT, SBO Project no. 040092 and 030288).


  1. Bendat JS, Piersol AG (1966) Measurement and analysis of random data. Wiley, New YorkzbMATHGoogle Scholar
  2. Benedict LH, Gould RD (1996) Towards better uncertainty estimates for turbulence statistics. Exp Fluids 22:129–136CrossRefGoogle Scholar
  3. Bruun HH (1995) Hot-wire anemometry: principles and signal analysis. Oxford University Press, Oxford, pp 405–445Google Scholar
  4. Di Sante A, Gonzalez CJ, Van den Braembussche RA (2006) Time-resolved PIV in a rotating diverging channel. In: 13th international symposium application of laser techniques to fluid mechanics, LisbonGoogle Scholar
  5. Doebelin EO (1990) Measurement systems—application and design, 4th edn. McGraw Hill, New York, pp 169–182Google Scholar
  6. Garcia CM, Jackson PR, Garcia MH (2006) Confidence intervals in the determination of turbulence parameters. Exp Fluids 40:514–522CrossRefGoogle Scholar
  7. Künsch H (1989) The jackknife and the bootstrap for general stationary observations. Ann Stat 17:1217–1241zbMATHCrossRefGoogle Scholar
  8. O’ Neill PL, Nicolaides D, Honnery D, Soria J (2004) Autocorrelation functions and the determination of integral length with reference to experimental and numerical data. In: 15th Australasian fluid mechanics conference, Sydney, AustraliaGoogle Scholar
  9. Politis DN, White H (2004) Automatic block-length selection for the dependent bootstrap. Econ Rev 23(1):53–70zbMATHMathSciNetCrossRefGoogle Scholar
  10. Riethmuller ML, Lourenco L (1981) Measurement in particulate two-phase flows. In: Proceedings of laser velocimetry, VKI-LS-1981-3 February 1981Google Scholar
  11. Singh K (1981) On the asymptotic accuracy of Efron’s boostrap. Ann Stat 9:1187–1195zbMATHCrossRefGoogle Scholar
  12. Tennekes H, Lumley JL (1972) A first course in turbulence. The MIT Press, CambridgeGoogle Scholar
  13. Thiébaux HJ, Zwiers FW (1984) The interpretation and estimation of effective sample size. J Clim Appl Meteorol 23:800–811CrossRefGoogle Scholar
  14. Tritton DJ (1988) Physical fluid dynamics, 2nd edn. Oxford Science Publications, Oxford, pp 306–308Google Scholar
  15. Ullum U, Schmidt JJ, Larsen PS, McCluskey DR (1998) Statistical analysis and accuracy of PIV data. J Visualization 1(2):205–216CrossRefGoogle Scholar
  16. Young GA (1994) Bootstrap: more than a stab in the dark?. Stat Sci 9(3):382–415zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • R. Theunissen
    • 1
    Email author
  • A. Di Sante
    • 1
  • M. L. Riethmuller
    • 1
  • R. A. Van den Braembussche
    • 1
  1. 1.von Karman Institute for Fluid DynamicsSint-Genesius RodeBelgium

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