Comparison of daubechies wavelets for hurst parameter estimation

Time scale dependence on the working nature of wavelet analysis makes it a valuable tool for Hurst parameter estimation. Similar to other wavelet-based signal processing applications, the selection of a particular wavelet type and vanishing moment in wavelet based Hurst estimation is a challenging problem. In this paper, we investigate the best Daubechies wavelet in wavelet based Hurst estimation for an exact self similar process, fractional Gaussian noise and how Daubechies vanishing moment affects the Hurst estimation accuracy. Daubechies wavelets are preferred in analysis because increasing vanishing moment does not cause excessive increase of time support of Daubechies wavelets. Thus, limited time support of wavelets reduces the border effects. Results show that Daubechies wavelets with one vanishing moment (Daubechies 1) gives the best estimation result for short range dependent fractional Gaussian noise. Daubechies 2 is the best preference for long range dependent fractional Gaussian noise.

Anahtar Kelimeler:

Daubechies wavelets, fractional gaussian noise, hurst estimation, vanishing moment.

Comparison of daubechies wavelets for hurst parameter estimation

Keywords:

Daubechies wavelets, fractional gaussian noise, hurst estimation, vanishing moment.,

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H. E. Hurst, “Long term storage of reservoirs,” Trans. Am. Soc. Civil Eng. 116, pp. 770–808, 1950.
J. Beran, “Statistics for Long-Memory Processes,” New York: Chapman & Hall, 1994.
D. Cox, P. Lewis, “The Statistical Analysis of Series of Events,” London: Chapman & Hall, 1966.
M. Taqqu, V. Teverovsky, W. Willinger, “Estimators for long-range dependence: an empirical study,” Fractals, Vol. 3, No. 4, pp. 785–788, 1995.
C.K. Peng, S. Buldyrev, S. Havlin, M. Simons, H. Stanley, A. Goldberger, “Mosaic organization of DNA nu- cleotides,” Physical Review E., Vol. 49, No. 2, pp.1685–1689,1994.
P. Robinson, “Gaussian semiparametric estimation of long-range dependence,” The Annals of Statistics, Vol. 23, No.5, pp. 1630–1661, 1995.
P. Flandrin, “On the spectrum of fractional Brownian motion,” IEEE Trans. Inform. Theory, Vol. 35, No.1, pp. 197–199, 1989. [8] E. Masry, “The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion,” IEEE Trans. Inform. Theory, Vol. 39, pp. 260–264, 1993.
G. Wornell, “A Karhunen Loe’ve like expansion for 1/f processes via wavelets,” IEEE Trans. Inform. Theory, Vol. 36, No. 4, pp. 859–861, 1990.
G. Wornell, A. V. Oppenheim, “Estimation of fractal signals from noisy measurements using wavelets,” IEEE Trans. Signal Processing, Vol. 40, No. 3, pp. 611–623, 1992.
D. Veitch, P. Abry, “A Wavelet Based Joint Estimator of the Parameters of Long-Range Dependence,” IEEE Transactions on Information Theory, Vol. 45, No. 3, pp. 878-897, 1999.
P. Abry, D. Veitch, “Wavelet Analysis of Long-Range-Dependent Traﬃc,” IEEE Transactions on Information Theory, Vol. 44, No.1, pp. 2–15, 1998.
H. J. Jeong, J.R. Lee, D. McNickle, K. Pawlikowski, “Comparison of various estimators in simulated FGN,” Simulation Modeling Practice and Theory, Vol. 15, No. 9, pp. 1173–1191, 2007.
H. J. Jeongy, D. McNicklez, K. Pawlikowski, “Fast Self-Similar Teletraﬃc Generation Based on FGN and Wavelets,” IEEE International Conference on Networks, Brisbane, Australia, pp. 75–82, 1999.
N. S. D. Brito, B.A. Souza, F.A.C. Pires, “Daubechies wavelets in Quality of Electrical Power,” 8thInternational Conference On Harmonics and Quality of Power Proceedings IEEE/PES and NTUX, Vol. 1, Athens, Greece, pp. 511-515, 1998. [16] S. Santoso, E. J. Powers and W. M. Grady, “Power quality assessment via wavelet transform analysis,” Proceedings of the IEEE PES Summer Meeting, Portland, pp. 924-930, 1995.
F. A. C. Pires, N.S.D. Brim, “The analysis of transient phenomena using the wavelet theory,” Proceedings of the International Conference on Power Systems Transients, Proceedings of IPST’97, Seattle, USA, pp. 171–176, 1997. [18] Z. Fan, P. Mars, “Self-similar traﬃc generation and parameter estimation using wavelet transform,” in Global Telecommunications Conference, GLOBECOM, IEEE, Vol. 3, pp. 1419–1423, 1997.
Y. Li, G. Liu, H. Li, X. Hou, “Wavelet-Based Analysis of Hurst Parameter Estimation For Self-Similar Traﬃc,” IEEE International Conference on Acoustics Speech and Signal Processing, Vol. 2, pp. 2061–2064, 2002.
B. B. Mandelbrot, J. W. Van Ness, ”Fractional Brownian motions, fractional noises. and applications,” SIAM Rev., Vol.10, No. 4, pp. 422–437, 1968.
A. Grossmann, J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Analm, Vol. 15, No. 4, pp. 277–283, 1984.
P. Flandrin, “Wavelet analysis and synthesis of fractional Brownian motion,” IEEE Trans. Vol. 38, No. 2, pp. 910–917, 1992. [23] P. Flandrin, “Some aspects of nonstationary signal processing with emphasis on time-frequency and time-scale methods” in Wavelets, Berlin: Springer, pp. 68-98, 1989.
S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Med. Imaging, Vol. 5, No.3, pp. 152–161, 1989.
P. Abry, F. Sellan, “The wavelet-based synthesis for the fractional Brownian motion proposed by F. Sellan and Y. Meyer: remarks and fast implementation,” Appl. Comp. Harmonic Anal. 3, pp. 377–383, 1996.