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Metrology and Measurement Systems

The Journal of Committee on Metrology and Scientific Instrumentation of Polish Academy of Sciences

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Volume 18, Issue 4 (Jan 2011)

Issues

Standard Deviation of the Mean of Autocorrelated Observations Estimated with the Use of the Autocorrelation Function Estimated From the Data

Andrzej Zięba
  • Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Mickiewicza 30, 30-059 Krakow, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Piotr Ramza
  • Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Mickiewicza 30, 30-059 Krakow, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2011-12-25 | DOI: https://doi.org/10.2478/v10178-011-0052-x

Standard Deviation of the Mean of Autocorrelated Observations Estimated with the Use of the Autocorrelation Function Estimated From the Data

Prior knowledge of the autocorrelation function (ACF) enables an application of analytical formalism for the unbiased estimators of variance s2a and variance of the mean s2a(xmacr;). Both can be expressed with the use of so-called effective number of observations neff. We show how to adopt this formalism if only an estimate {rk} of the ACF derived from a sample is available. A novel method is introduced based on truncation of the {rk} function at the point of its first transit through zero (FTZ). It can be applied to non-negative ACFs with a correlation range smaller than the sample size. Contrary to the other methods described in literature, the FTZ method assures the finite range 1 < neffn for any data. The effect of replacement of the standard estimator of the ACF by three alternative estimators is also investigated. Monte Carlo simulations, concerning the bias and dispersion of resulting estimators sa and sa(×), suggest that the presented formalism can be effectively used to determine a measurement uncertainty. The described method is illustrated with the exemplary analysis of autocorrelated variations of the intensity of an X-ray beam diffracted from a powder sample, known as the particle statistics effect.

Keywords: autocorrelated data; time series; effective number of observations; estimators of variance; measurement uncertainty

  • Zięba, A. (2010). Effective number of observations and unbiased estimators of variance for autocorrelated data - an overview. Metrol. Meas. Syst., 17, 3-16.Web of ScienceGoogle Scholar

  • Chipman, J. S., Kadiyala, K. R., Madansky, A., Pratt, J. W. (1968). Efficiency of the sample mean when residuals follow a first-order stationary Markoff process. J. Amer. Statist. Assoc., 63, 1237-1246.Google Scholar

  • Pham, T. D., Tran, L. T. (1992). On the best unbiased estimate for the mean of a short autoregressive time series. Econometric Theory, 8, 120-126.Google Scholar

  • Bayley, G. V., Hammersley, J. M. (1946). The "effective" number of independent observations in an autocorrelated time series. J. R. Stat. Soc. Suppl., 8, 184-197.Google Scholar

  • Box, G. E. P., Jenkins, G. M., Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control 3rd ed. New Jersey: Prentice Hall, Englewood Cliffs.Google Scholar

  • Zhang, N. F. (2006). Calculation of the uncertainty of the mean of autocorrelated measurements. Metrology, 43, 276-281.Google Scholar

  • Percival, D. B. (1993). Three curious properties of the sample variance and autocovariance for stationary processes with unknown mean. The American Statistician, 47, 274-276.Google Scholar

  • Quenouille, M. H. (1949). Approximate tests of correlation in time-series. J. R. Statist. Soc. B, 11, 68-84.Google Scholar

  • Marriott, F. H. C. Pope, J. A. (1954). Bias in the estimation of autocorrelations. Biometrika, 41, 390-402.Google Scholar

  • Zieba, A., Ramza, P., to be published.Google Scholar

  • ISO/IEC. (1995). Guide to the Expression of Uncertainty in Measurement. Geneva: ISO.Google Scholar

  • Dinnebier, R. E., Billinge, S. J. L. Eds. (2008). Powder Diffraction: Theory and Practice. Cambridge: RSC Publishing.Web of ScienceGoogle Scholar

About the article


Published Online: 2011-12-25

Published in Print: 2011-01-01


Citation Information: Metrology and Measurement Systems, ISSN (Print) 0860-8229, DOI: https://doi.org/10.2478/v10178-011-0052-x.

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