NPG Nonlinear Processes in Geophysics NPG Nonlin. Processes Geophys. 1607-7946 Copernicus Publications Göttingen, Germany 10.5194/npg-18-441-2011 Bayesian estimation of the self-similarity exponent of the Nile River fluctuation Benmehdi S. 1 Makarava N. 2 Benhamidouche N. 3 Holschneider M. 2 Departement of Mathematics, University of Bourdj-Bouarreridj, Box 64, 34265 Bourdj-Bouarreridj, Algeria Institute for Mathematics, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany Departement of Mathematics, University of M'Sila, Box 166, Msila, Algeria 29 06 2011 18 3 441 446 Copyright: © 2011 S. Benmehdi et al. 2011 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://npg.copernicus.org/articles/18/441/2011/npg-18-441-2011.html The full text article is available as a PDF file from https://npg.copernicus.org/articles/18/441/2011/npg-18-441-2011.pdf

The aim of this paper is to estimate the Hurst parameter of Fractional Gaussian Noise (FGN) using Bayesian inference. We propose an estimation technique that takes into account the full correlation structure of this process. Instead of using the integrated time series and then applying an estimator for its Hurst exponent, we propose to use the noise signal directly. As an application we analyze the time series of the Nile River, where we find a posterior distribution which is compatible with previous findings. In addition, our technique provides natural error bars for the Hurst exponent.

 References Abry, P. and Pipiras, V.: Wavelet-based synthesis of the Rosenblatt process, Signal Process., 86, 2326–2339, <a href="http://dx.doi.org/10.1016/j.sigpro.2005.10.021">https://doi.org/10.1016/j.sigpro.2005.10.021</a>, 2006. Abry, P. and Veitch, D.: Wavelet Analysis of Long-Range-Dependent Traffic, IEEE T. Inform., 44, 2–15, 1998. Dieker, T.: Simulation of Fractional Brownian motion, Ph.D. thesis, Twente University, 2004. Geweke, K. and Porter-Hudak, S.: The estimation and application of long memory time series models, Time Ser. Anal., 4, 221–238, 1983. Goldberger, A.&nbsp;L., Amaral, L. A.&nbsp;N., Glass, L., Hausdorff, J., Ivanov, P., Mark, R., Mietus, J., Moody, G., Peng, C., and Stanley, H.: Components of a New Research Resource for Complex Physiologic Signals, PhysioBank, PhysioToolkit, and Physionet, Circulation, 101(23), e215–e220, 2000. Kang, D.&nbsp;S.: Simulation of The Fractional Brownian Motion, Ph.D. thesis, J.&nbsp;W.&nbsp;Goethe University, 2008. Liu, Y., Liu, Y., Wang, K., Yang, L., and Jiang, T.: Modified periodogram method for estimating the Hurst exponent of fractional Gaussian noise, Phys. Rev. E, 80(6), 066207–0662014, 2009. Makarava, N., Holschneider, M., and Benmehdi, S.: Bayesian estimation of self-similarity exponent, Phys. Rev. E, submitted, 2011. Mandelbrot, B.&nbsp;B. and Wallis, J.&nbsp;R.: Computer experiments with fractional Gaussian noises. Part 2, rescaled ranges and spectra, Water Resour. Res., 5(1), 242–259, 1969. McCoy, E. and Walden, A.: Wavelet Analysis and Synthesis of Stationary Long-Memory Processes, Comput. Graph. Stat, 5, 26–56, 1996. Millen, S. and Beard, R.: Estimation of the Hurst Exponent for the Burdekin River using the Hurs-Mandelbrot Rescaled Range Statistic, in: Proceedings of the First Queensland Statistics Conference, 2003. Peng, C.-K., Buldyrev, S., Havlin, S., Simons, M., Stanley, H.&nbsp;E., and Goldberger, A.: Mosaic organization of DNA nucleotides, Phys. Rev. E, 49, 1685–1689, 1994. Robinson, P.: Log-Periodogram Regression of Time Series with Long Range Dependence, Ann. Stat., 23, 1048–1072, 1995. Rosenblatt, M.: Independence and dependence, In Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability, 2, 431–443, 1961. Taqqu, M., Teverovsky, V., and Willinger, W.: Estimators for long range dependence:an empiracl study, Fractals, 3, 785–798, 1995. Veitch, D. and Abry, P.: A Wavelet Based Joint Estimator of the Parameters of Long-Range Dependence, IEEE Trans. Info. Theory, special issue "Multiscale Statistical Signal Analysis and its Applications", 45, 878–897, 1999. Veitch, D. and Abry, P.: LDestimate, available at: <a href="http://www.cubinlab.ee.unimelb.edu.au/&nbsp;darryl/">http://www.cubinlab.ee.unimelb.edu.au/&nbsp;darryl/</a>, last access date: 28. June 2011, 2002. Whittle, P.: Estimation and information in stationary time series, Ark. Mat., 2, 423–434, 1953.