## Abstract

A preliminary essential procedure in time series analysis is the separation of the deterministic component from the random one. If the signal is the result of superposing a noise over a deterministic trend, then the first one must estimate and remove the trend from the signal to obtain an estimation of the stationary random component. The errors accompanying the estimated trend are conveyed as well to the estimated noise, taking the form of detrending errors. Therefore the statistical errors of the estimators of the noise parameters obtained after detrending are larger than the statistical errors characteristic to the noise considered separately. In this paper we study the detrending errors by means of a Monte Carlo method based on automatic numerical algorithms for nonmonotonic trends generation and for construction of estimated polynomial trends alike to those obtained by subjective methods. For a first order autoregressive noise we show that in average the detrending errors of the noise parameters evaluated by means of the autocovariance and autocorrelation function are almost uncorrelated to the statistical errors intrinsic to the noise and they have comparable magnitude. For a real time series with significant trend we discuss a recursive method for computing the errors of the estimated parameters after detrending and we show that the detrending error is larger than the half of the total error.

## Authors

Călin **Vamoş
**Tiberiu Popoviciu, Institutue of Numerical Analysis

Maria **Crăciun
**Tiberiu Popoviciu, Institutue of Numerical Analysis

## Keywords

## Paper coordinates

C. Vamoş, M. Crăciun, *Serial correlation of detrended time series*, Physical Review E, Vol. 78 (2008) article id. 036707,

doi: 10.1103/PhysRevE.78.036707

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[1] G. M. Viswanathan, S. V. Buldyrev, E. K. Garger, V. A. Kashpur, L. S. Lucena, A. Shlyakhter, H. E. Stanley, and J. Tschiersch, Phys. Rev. E 62, 4389 2000.

[2] W. Knospe, L. Santen, A. Schadschneider, and M. Schreckenberg, Phys. Rev. E 65, 056133 2002.

[3] K. Kiyono, Z. R. Struzik, N. Aoyagi, S. Sakata, J. Hayano, and Y. Yamamoto, Phys. Rev. Lett. 93, 178103 2004; K. Kiyono, Z. R. Struzik, N. Aoyagi, F. Togo, and Y. Yamamoto, ibid. 95, 058101 2005.

[4] W. M. Macek, R. Bruno, and G. Consolini, Phys. Rev. E 72, 017202 2005.

[5] K. Kiyono, Z. R. Struzik, and Y. Yamamoto, Phys. Rev. Lett. 96, 068701 2006.

[6] H.-D. Xi, Q. Zhou, and K.-Q. Xia, Phys. Rev. E 73, 056312 2006.

[7] P. Weber, F. Wang, I. Vodenska-Chitkushev, S. Havlin, and H. E. Stanley, Phys. Rev. E 76, 016109 2007.

[8] K. Hu, P. Ch. Ivanov, Z. Chen, P. Carpena, and H. E. Stanley, Phys. Rev. E 64, 011114 2001.

[9] Z. Chen, K. Hu, P. Carpena, P. Bernaola-Galvan, H. E. Stanley, and P. Ch. Ivanov, Phys. Rev. E 71, 011104 2005.

[10] J. W. Kantelhardt, E. Koscielny-Bunde, H. H. A. Rego, S. Havlin, and A. Bunde, Physica A 295, 441 2001.

[11] P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods Springer Verlag, New York, 1996.

[12] P. J. Brockwell, R. A. Davis, Introduction to Time Series and Forecasting Springer Verlag, New York, 2003.

[13] J. D. Hamilton, Time Series Analysis Princeton University Press, Princeton, NJ, 1994.

[14] D. Maraun, H. W. Rust, and J. Timmer, Nonlinear Processes Geophys. 11, 495 2004.

[15] E.-J. Wagenmakers, S. Farrell, and R. Ratcliff, Psychon. Bull. Rev. 11, 579 2004; T. L. Thorton and D. L. Gilden, ibid. 12, 409 2005.

[16] J. Timmer, U. Schwarz, H. U. Voss, I. Wardinski, T. Belloni, G. Hasinger, M. van der Klis, and J. Kurths, Phys. Rev. E 61, 1342 2000.

[17] S. Yue and P. Pilon, Water Resour. Res. 39, 1077 2003.

[18] C. Stărică and C. Granger, Rev. Econ. Stat. 87, 495 2005.

[19] J. Gao, J. Hu, W.-W. Tung, Y. Cao, N. Sarshar, and V. P. Roychowdhury, Phys. Rev. E 73, 016117 2006.

[20] C. Vamoş, Phys. Rev. E 75, 036705 2007.

[21] A. Carbone, G. Castelli, and H. E. Stanley, Phys. Rev. E 69, 026105 2004.

[22] A. Carbone and H. E. Stanley, Physica A 384, 21 2007.

[23] C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 1994.

[24] G. Box, G. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. Prentice-Hall, Upper Saddle River, NJ, 1994.

[25] C. Vamoş, Ş. M. Şoltuz, and M. Crăciun, e-print arXiv:0709.2963.

[26] The observational data, not yet published, was kindly provided by V. V. Morariu.

[27] F. Brochard and J. F. Lennon, J. Phys. Paris 36, 1035 1975.

[28] H. Strey, M. Peterson, and E. Sackmann, Biophys. J. 69, 478 1995.

[29] H.-G. Döbereiner, G. Gompper, C. K. Haluska, D. M. Kroll, P. G. Petrov, and K. A. Riske, Phys. Rev. Lett. 91, 048301 2003.

[30] S. Zhao and G. W. Wei, Comput. Stat. Data Anal. 42, 219 2007.

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