Abstract
In this paper, we investigate the use of wavelet techniques in the study of the nth order fractional Brownian motion (n-fBm). First, we exploit the continuous wavelet transform’s capabilities in derivative calculation to construct a two-step estimator of the scaling exponent of the n-fBm process. We show, via simulation, that the proposed method improves the estimation performance of the n-fBm signals contaminated by large-scale noise. Second, we analyze the statistical properties of the n-fBm process in the time-scale plan. We demonstrate that, for a convenient choice of the wavelet basis, the discrete wavelet detail coefficients of the n-fBm process are stationary at each resolution level whereas their variance exhibits a power-law behavior. Using the latter property, we discuss a weighted least squares regression based-estimator for this class of stochastic process. Experiments carried out on simulated and real-world datasets prove the relevance of the proposed method.
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References
Abry P, Flandrin P, Taqqu MS, Veitch D (2000) Wavelets for the analysis, estimation, and synthesis of scaling data. In: Park K, Willinger W (eds) Self-similar network traffic and performance evaluation. Wiley, New York, pp 39–88
Abry P, Flandrin P, Taqqu MS, Veitch D (2003) Self-similarity and long-range dependence through the wavelet lens. In: Doukhan P, Oppenheim G, Taqqu MS (eds) Long-Range Dependence: Theory and Applications, Birkhäuser, Boston, pp 527–556
Bel L, Oppenheim G, Robbiano L, Viano MC (1998) Distribution processes with stationary fractional increments. In: Proceedings of the colloquium FDS’98: fractional differential systems: models, methods and applications ESAIM: Proceedings 5, pp 43–54
Beran J (1994) Statistics for long-memory processes, monographs on statistics and applied probability, vol 61. Chapman & Hall, New York
Biermé H (2005) Champs Aléatoires, Autosimilarité, Anisotropie et Etude Directionnelle. Ph.D. thesis, Université d’Orléans
Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2 edn. Springer, New York
Brouste A, Istas J, Lambert-Lacroix S (2007) On fractional gaussian random fields simulations. J Stat Softw 23: 1–23
Coeurjolly J-F (2001) Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat Inference Stoch Process 4: 199–227
Daubechies I (1992) Ten lectures on wavelets. CBMS-NSF regional conference series on applied mathematics, SIAM, Philadelphia, PA, p 61
Dickey DA, Fuller WA (1981) The likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49: 1057–1072
Dietrich CR, Newsam GN (1997) Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J Sci Comput 18: 1088–1107
Flandrin P (1992) Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans Inf Theory 38: 910–917
Gupta A, Joshi SD, Prasad S (2005) A new approach for estimation of statistically matched wavelet. IEEE Trans Signal Process 53: 1778–1793
Handy CR, Murenzi R (1998) Moment-wavelet quantization: a first principles analysis of quantum mechanics through continuous wavelet transform theory. Phys Lett A 248: 7–15
Hwang W-L (1999) Estimation of fractional Brownian motion embedded in a noisy environment using nonorthogonal wavelets. IEEE Trans Signal Process 47: 2211–2219
Istas J, Lang G (1997) Quadratic variations and estimation of the local Hölder index of a gaussian process. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques 33: 407–436
Kwiatkowski D, Phillips P, Schmidt P, Shin Y (1992) Testing the null hypothesis of stationary against the alternative of a unit root: how sure are we that economic time series have a unit root?. J Econom 54: 159–178
Liu H-F, Yang Y-Z, Dai Z-H, Yu Z-H (2003a) The largest Lyapunov exponent of chaotic dynamical system in scale space and its application. Chaos 13: 839–844
Liu H-F, Li W-F, Dai Z-H, Yu Z-H (2003b) The dimension of chaotic dynamical system in wavelet space and its application. Phys Lett A 316: 44–54
Loussot T, Harba R, Jacquet G, Benhamou CL, Lespesailles E, Julien A (1996) An oriented fractal analysis for the characterization of texture: application to bone radiographs. EUSIPCO Signal Process 1: 371–374
Mallat S (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Machine Intell 11: 674–693
Mallat S (1998) A wavelet tour of signal processing. Academic Press, New York
Mallat S, Hwang WH (1992) Singularity detection and processing with wavelets. IEEE Trans Inf Theory 38: 617–643
Mandelbrot BB (1999) Multifractals and 1/f noise: wild self-affinity in physics. Springer, New York
Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motion, fractional noises and applications. SIAM Rev 10: 422–438
Meyer Y (1992) Wavelets and operators (trans: DH Salinger). Cambridge University, Cambridge
Mielniczuk J, Wojdyllo P (2007) Estimation of Hurst exponent revisited. Comput Stat Data Anal 51: 4510–4525
Parra C, Iftekharuddin K, Rendon D (2003) Wavelet based estimation of the fractal dimension in fBm images. In: First International IEEE EMB Conference on Neural Engineering. Conference Proceedings, pp 533–536
Pérez DG, Zunino L, Garavaglia M, Rosso OA (2006) Wavelet entropy and fractional Brownian motion time series. Phys A 365: 282–288
Perrin E, Harba R, Berzin-Joseph C, Iribarren I, Bonami A (2001) Nth order fractional Brownian motion and fractional Gaussian noises. IEEE Trans Signal Process 49: 1049–1059
Perrin E, Harba R, Jennane R, Iribarren I (2002) Fast and exact synthesis for 1-D fractional Brownian motion and fractional gaussian noises. IEEE Signal Process Lett 9: 382–384
Pesquet-Popescu B, Larzabal P (1997) Higher order and lower order properties of the Wavelet decomposition of self similar process. IEEE signal processing workshop on higher-order statistics (SPW-HOS ’97), spwhos, pp 4–58
Phillips PCB, Perron P (1988) Testing for unit roots in time series regression. Biometrica 75: 335–346
Power GJ, Turvey GC (2010) Long-range dependence in the volatility of commodity futures prices: wavelet-based evidence. Phys A 389: 79–90
Sembiring J, Soemintapoera K, Kobayachi T, Akizuki K (2003) Diffusive representation of Nth order fractional brownian motion. In: Proceedings of the 13th IFAC symposium on system identification, Rotterdam, the Netherlands, 1, pp 181–186
Shao X, Ma C (2003) A general approach to derivative calculation using wavelet transform. Chemom Intell Lab Syst 69: 157–165
Stoev S, Taqqu MS (2005) Path properties of the linear multifractional stable motion. Fractals 13: 157–178
Tewfik AH, Kim M (1992) Correlation structure of the discrete wavelet coefficients of fractional Brownian motion. IEEE Trans Inf Theory 38: 904–909
Trimech A, Kortas H, Benammou S, Benammou S (2009) Multiscale Fama-French model: application to the French market. J Risk Finance 10: 179–192
Veitch D. (2001) MATLAB code for estimation of scaling exponents. Available at: http://www.cubinlab.ee.mu.oz.au/~darryl/secondordercode.html
Veitch D, Abry P (1999) A wavelet-based joint estimator of the parameters of long-range dependence. IEEE Trans Inf Theory 45: 878–897
Zhang L, Bao P, Wu X (2004) Wavelet estimation of fractional Brownian motion embedded in a noisy environment. IEEE Trans Inf Theory 50: 2194–2200
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Kortas, H., Dhifaoui, Z. & Ben Ammou, S. On wavelet analysis of the nth order fractional Brownian motion. Stat Methods Appl 21, 251–277 (2012). https://doi.org/10.1007/s10260-012-0187-2
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DOI: https://doi.org/10.1007/s10260-012-0187-2