Effective Number of Observations and Unbiased Estimators of Variance for Autocorrelated Data - an Overview

Journal title

Metrology and Measurement Systems




No 1



autocorrelated ; time series ; estimator ; unbiased ; variance ; effective number of observations

Divisions of PAS

Nauki Techniczne


Polish Academy of Sciences Committee on Metrology and Scientific Instrumentation




Artykuły / Articles


DOI: 10.2478/v10178-010-0001-0 ; ISSN 2080-9050, e-ISSN 2300-1941


Metrology and Measurement Systems; 2010; No 1




ISO/IEC. <i>Guide to the Expression of Uncertainty in Measurement.</i> (1995). Geneva. ; Zhang N. (2006), Calculation of the uncertainty of the mean of autocorrelated measurements, Metrologia, 43, ; Dorozhovets M. (2007), Upgrading calculating methods of the uncertainty of measurement results in practice, Przegląd Elektrotechniczny, 83, 1. ; Witt T. (2007), Using the autocorrelation function to characterize time series of voltage measurements, Metrologia, 44, 201. ; Kirkup L. (2006), An Introduction to the Uncertainty in Measurement. ; Freund R. (2006), Regression Analysis. Statistical Modeling of a Response Variable. ; Priestley M. (1981), Spectral Analysis and Time Series. ; Box G. (1944), Time Series Analysis: Forecasting and Control. ; Brockwell P. (1991), Time series: theory and methods. ; Anderson T. (1971), The Statistical Analysis of Time Series. ; Bendat J. (1971), Random data: Analysis and measurement procedures. ; Yaglom A. (1987), Correlation theory of stationary and related random processes. ; Bartels J. (1935), Zur Morphologie geophysikalischer Zeitfunktionen, Sitz.-Ber. Preuss. Akad. Wiss, 30, 504. ; Bayley G. (1946), The Effective Number of Independent Observations in an Autocorrelat-ed Time-Series, J. Roy. Stat. Soc. Suppl, 8, 184. ; Bagrov N. (1969), On the equivalent number of independent data, Tr. Gidrometeor. Cent, 44, 3. ; Lubman D. (1969), Spatial Averaging in a Diffuse Sound Field, J. Acoust. Soc. Am, 46, 532. ; Leith C. (1973), The standard error of time-averaged estimates of climatic means, J. Appl. Meteorol, 12, 1066. ; Taubenheim J. (1974), On the significance of the autocorrelation of statistical tests for averages, meansquare deviations and superposed epochs [geophysical data], Gerlands Beitr. Geophysik, 83, 121. ; Şen Z. (1998), Small sample estimation of the variance of time-averages in climatic time series, Int. J. Climatol, 18, 1725. ; Fortus M. (1999), Equivalent Number of Independent Observations: A Review, Izvestia AN. Fizika Atmosf. Okeana, 35, 725. ; That useful approximate form of Eq. (12) was introduced by Bealey and Hammersley [14]. However, because of an error by a factor of two, approximate formulae for <i>n<sub>eff</sub></i> and <i>V<sub>eff</sub></i> given at p. 185 of their paper are incorrect. ; Zięba A. (2008), Uncertainty of the mean of correlated observations, null, 15. ; Law A. (2000), Simulation Modelling and Analysis, 251. ; <a target="_blank" href=''></a> ; Incorrect version of Eq. (31) is given in [14]. See remark [21]. ; Barlett M. (null), On the theoretical specification and sampling properties of autocorrelated time-series, J. Roy. Stat. Soc. Suppl, 8, 27. ; Zięba, A., Ramza, P. In preparation. ; Cox M. (2006), The generalized weighted mean of correlated quantities, Metrologia, 43. ; Cliff A. (1975), The comparison of means when samples consist of spatially autocorrelated observations, Environment and Planning A, 7, 725. ; Bretherton C. (1999), The Effective Number of Spatial Degrees of Freedom of a Time-Varying Field, J. Climate, 12, 1990. ; Zhang N. (2008), Allan variance of time series models for measurement data, Metrologia, 45, 549.