WAVELET-BASED ESTIMATOR FOR THE HURST PARAMETERS OF FRACTIONAL BROWNIAN SHEET

doi:10.1016/S0252-9602(16)30126-6

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (1): 205-222.doi: 10.1016/S0252-9602(16)30126-6

WAVELET-BASED ESTIMATOR FOR THE HURST PARAMETERS OF FRACTIONAL BROWNIAN SHEET

吴量^1,2, 丁义明^3,4

1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China;
3. Department of Mathematics, Wuhan Institute of Technology, Wuhan 430070, China;
4. Key Laboratory of Magnetic Resonance in Biological Systems, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

收稿日期:2015-09-22 修回日期:2016-05-12 出版日期:2017-02-25 发布日期:2017-02-25
通讯作者: Yiming DING,E-mail:ding@wipm.ac.cn E-mail:ding@wipm.ac.cn
作者简介:Liang WU,E-mail:wuliangshine@gmail.com
基金资助:
This work was supported in part by the National Basic Research Program of China (973 Program, 2013CB910200, and 2011CB707802).

WAVELET-BASED ESTIMATOR FOR THE HURST PARAMETERS OF FRACTIONAL BROWNIAN SHEET

Liang WU^1,2, Yiming DING^3,4

1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China;
3. Department of Mathematics, Wuhan Institute of Technology, Wuhan 430070, China;
4. Key Laboratory of Magnetic Resonance in Biological Systems, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

Received:2015-09-22 Revised:2016-05-12 Online:2017-02-25 Published:2017-02-25
Contact: Yiming DING,E-mail:ding@wipm.ac.cn E-mail:ding@wipm.ac.cn
About author:Liang WU,E-mail:wuliangshine@gmail.com
Supported by:
This work was supported in part by the National Basic Research Program of China (973 Program, 2013CB910200, and 2011CB707802).

摘要/Abstract

摘要：

It is proposed a class of statistical estimators ?=(?₁, …, ?_d) for the Hurst parameters H=(H₁,…,H_d) of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are asymptotically normal. These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals, which is important in texture classification and improvement of diffusion tensor imaging (DTI) of nuclear magnetic resonance (NMR). Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators. We find that when H_i≥1/2, the estimators are accurate, and when H_i<1/2, there are some bias.

关键词: detection of long-range dependence, self-similarity, Hurst parameters, wavelet analysis, fractional Brownian sheet

Abstract:

Key words: detection of long-range dependence, self-similarity, Hurst parameters, wavelet analysis, fractional Brownian sheet

吴量, 丁义明. WAVELET-BASED ESTIMATOR FOR THE HURST PARAMETERS OF FRACTIONAL BROWNIAN SHEET[J]. 数学物理学报(英文版), 2017, 37(1): 205-222.

Liang WU, Yiming DING. WAVELET-BASED ESTIMATOR FOR THE HURST PARAMETERS OF FRACTIONAL BROWNIAN SHEET[J]. Acta mathematica scientia,Series B, 2017, 37(1): 205-222.

参考文献

[1] Flandrin P. Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans Inform Theory, 1992, 38(2):910-917
[2] Abry P, Flandrin P, Taqqu M S, et al. Self-similarity and long-range dependence through the wavelet lens. Theory and Applications of Long-Range Dependence, 2003:527-556
[3] Abry P, Helgason H, Pipiras V. Wavelet-based analysis of non-Gaussian long-range dependent processes and estimation of the hurst parameter. Lithuanian Mathematical Journal, 2011, 51(3):287-302
[4] Kamont A. On the fractional anisotropic Wiener field. Probability and Mathematical Statistics-PWN, 1996, 16(1):85-98
[5] Pesquet-Popescu B, Vehel J L. Stochastic fractal models for image processing. IEEE Signal Processing Magazine, 2002, 19(5):48-62
[6] Biermé H, Meerschaert M M, Scheffler H P. Operator scaling stable random fields. Stoch Proc Appl, 2007, 117(3):312-332
[7] Bonami A, Estrade A. Anisotropic analysis of some Gaussian models. J Fourier Anal Appl, 2003, 9(3):215-236
[8] Doukhan P, Oppenheim G, Taqqu M S. Theory and Applications of Long-Range Dependence. Springer, 2003
[9] Wu D, Xiao Y. Geometric properties of fractional Brownian sheets. J Fourier Anal Appl, 2007, 13(1):1-37
[10] Atto A M, Berthoumieu Y, Bolon P. 2-D Wavelet packet spectrum for texture analysis. IEEE Trans Image Proc, 2013, 22(6):2495-2500
[11] Roux S G, Clausel M, Vedel B, et al. Self-similar anisotropic texture analysis:the hyperbolic wavelet transform contribution. IEEE Trans Image Proc, 2013, 22(11):4353-4363
[12] Biagini F, Hu Y, Øksendal B, et al. Stochastic Calculus for Fractional Brownian Motion and Applications. London:Springer, 2008
[13] Moseley M. Diffusion tensor imaging and aging a review. NMR in Biomedicine, 2002, 15(7/8):553-560
[14] Amblard P O, Coeurjolly J F. Identification of the multivariate fractional Brownian motion. IEEE Trans Signal Proc, 2011, 59(11):5152-5168
[15] Coeurjolly J F, Amblard P O, Achard S. Wavelet analysis of the multivariate fractional Brownian motion. ESAIM:Prob Stat, 2013, 17:592-604
[16] Wu L, Ding Y. Estimation of self-similar Gaussian fields using wavelet transform. International Journal of Wavelets, Multiresolution and Information Processing, 2015, 13(6):1550044
[17] Bardet J M. Statistical study of the wavelet analysis of fractional Brownian motion. IEEE Trans Inform Theory, 2002, 48(4):991-999
[18] Morales C J. Wavelet-Based Multifractal Spectra Estimation:Statistical Aspects and Applications[D]. Boston University, 2002
[19] Bardet J M. Testing for the presence of self-similarity of Gaussian time series having stationary increments. J Time Series Anal, 2000, 21(5):497-515
[20] Mandelbrot B, Van Ness J. Fractional Brownian motions, fractional noises and applications. SIAM Review, 1968, 10(4):422-437
[21] Hu Y. Heat equations with fractional white noise potentials. Appl Math Opt, 2001, 43(3):221-243
[22] Brouste A, Istas J, Lambert-Lacroix S. On fractional Gaussian random fields simulations. J Stat Software, 2007, 23(1):1-23
[23] Basu A K. Measure Theory and Probability. PHI Learning Pvt Ltd, 2004
[24] Arcones M A. Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann Prob, 1994, 22(4):2242-2274
[25] Serfling R J. Approximation Theorems of Mathematical Statistics. John Wiley & Sons, 1980
[26] Van der Vaart A W, Wellner J A. Weak Convergence and Empirical Processes. New York:Springer, 1996
[27] Goswami J C, Chan A K. Fundamentals of Wavelets:Theory, Algorithms, and Applications. John Wiley & Sons, 2011
[28] Bardet J M, Lang G, Oppenheim G, et al. Semi-parametric estimation of the long-range dependence parameter:a survey. Theory and Applications of Long-Range Dependence, 2003:557-577
[29] Tewfik A H, Kim M. Correlation structure of the discrete wavelet coefficients of fractional Brownian motion. IEEE Trans Inform Theory, 1992, 38(2):904-909
[30] Dijkerman R W, Mazumdar R R. On the correlation structure of the wavelet coefficients of fractional Brownian motion. IEEE Trans Inform Theory, 1994, 40(5):1609-1612
[31] Aitken A C. On least squares and linear combinations of observations. Proc Royal Soc Edinburgh, 1935, 55:42-48
[32] Davies R B, Harte D. Tests for Hurst effect. Biometrika, 1987, 74(1):95-101
[33] Kroese D P, Botev Z I. Spatial process generation. arXiv preprint arXiv:1308.0399, 2013

WAVELET-BASED ESTIMATOR FOR THE HURST PARAMETERS OF FRACTIONAL BROWNIAN SHEET

WAVELET-BASED ESTIMATOR FOR THE HURST PARAMETERS OF FRACTIONAL BROWNIAN SHEET

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 3

Metrics

本文评价

推荐阅读 0

[1]	Ciprian A. TUDOR, Maria TUDOR. FRACTIONAL 2D-STOCHASTIC CURRENTS[J]. 数学物理学报(英文版), 2013, 33(6): 1507-1521.
[2]	陈振龙, 李慧琼. POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS[J]. 数学物理学报(英文版), 2010, 30(3): 857-872.
[3]	肖文辉, 陈莘萌. A DESIGN OF THE PARALLEL FILTER BASED ON WAVELETS[J]. 数学物理学报(英文版), 1995, 15(S1): 133-138.