We estimate the anisotropic index of an anisotropic fractional brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. Finally we use simulated fields to test the proposed estimator in different anisotropic and isotropic cases. Results show that the estimator behaves similarly in all cases and is able to detect anisotropy quite accurately.
Mots-clés : anisotropic gaussian fields, identification, estimator, asymptotic normality, Radon transform
@article{PS_2008__12__30_0,
author = {Bierm\'e, Hermine and Richard, Fr\'ed\'eric},
title = {Estimation of anisotropic gaussian fields through {Radon} transform},
journal = {ESAIM: Probability and Statistics},
pages = {30--50},
publisher = {EDP-Sciences},
volume = {12},
year = {2008},
doi = {10.1051/ps:2007031},
mrnumber = {2367992},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps:2007031/}
}
TY - JOUR AU - Biermé, Hermine AU - Richard, Frédéric TI - Estimation of anisotropic gaussian fields through Radon transform JO - ESAIM: Probability and Statistics PY - 2008 SP - 30 EP - 50 VL - 12 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2007031/ DO - 10.1051/ps:2007031 LA - en ID - PS_2008__12__30_0 ER -
%0 Journal Article %A Biermé, Hermine %A Richard, Frédéric %T Estimation of anisotropic gaussian fields through Radon transform %J ESAIM: Probability and Statistics %D 2008 %P 30-50 %V 12 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps:2007031/ %R 10.1051/ps:2007031 %G en %F PS_2008__12__30_0
Biermé, Hermine; Richard, Frédéric. Estimation of anisotropic gaussian fields through Radon transform. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 30-50. doi : 10.1051/ps:2007031. http://archive.numdam.org/articles/10.1051/ps:2007031/
[1] and , The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: remarks and fast implementation. Appl. Comput. Harmon. Anal. 3 (1996) 377-383. | MR | Zbl
[2] , and , Identification and series decomposition of anisotropic Gaussian fields. Proceedings of the Catania ISAAC05 congress (2005).
[3] , , , , and , Semi-parametric estimation of the long-range dependence parameter: a survey. In Theory and applications of long-range dependence, Birkhäuser Boston (2003) 557-577. | MR | Zbl
[4] , Asymptotic development and central limit theorem for quadratic variations of gaussian processes. To appear in Bernoulli (2006). | MR | Zbl
[5] , , and , Identification of filtered white noises. Stochastic Process. Appl. 75 (1998) 31-49. | MR | Zbl
[6] , and , Elliptic Gaussian random processes. Rev. Mathem. Iberoamericana. 13 (1997) 19-89. | EuDML | MR | Zbl
[7] , Champs aléatoires : autosimilarité, anisotropie et étude directionnelle2005).
[8] and , Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9 (2003) 215-236. | MR | Zbl
[9] , An effective method for simulating Gaussian random fields, in Proceedings of the statistical Computing section, 133-138, www.stat.uiowa.edu/grchan/ (1999). Amerir. Statist.
[10] , Inférence statistique pour les mouvements browniens fractionnaires et multifractionnaires. PhD thesis, Université Joseph Fourier (2000).
[11] , Estimating the parameters of fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199-227. | MR | Zbl
[12] and , Probabilités et statistiques, Vol. 2. Masson (1983). | MR | Zbl
[13] and , Fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18 (1997) 1088-1107. | MR | Zbl
[14] , A simple construction of the fractional brownian motion. Stochastic Process. Appl. 109 (2004) 203-223. | MR | Zbl
[15] and , Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré, Prob. Stat. 33 (1997) 407-436. | Numdam | MR | Zbl
[16] , , , and , Analyse de champs browniens fractionnaires anisotropes2001) 99-102.
[17] and , An Improved Method for 2-d Self-Similar Image Synthesis. IEEE Trans. Image Process. 5 (1996) 754-761.
[18] and , Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. Roy. Statist. Soc. Ser. B 59 (1997) 679-699. | MR | Zbl
[19] and , Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates. Stat. Inference Stoch. Process. 4 (2001) 283-306. | MR | Zbl
[20] , Analyse stochastique de signaux multi-fractaux et estimations de paramètres. Ph.D. thesis, Université d'Orléans, http://www.univ-orleans.fr/mapmo/publications/leger/these.php (2000).
[21] and , Fractional Brownian motion, fractionnal noises and applications. Siam Review 10 (1968) 422-437. | MR | Zbl
[22] , and , Wavelets, Generalised White Noise and Fractional Integration: The Synthesis of Fractional Brownian Motion. J. Fourier Anal. Appl. 5 (1999) 465-494. | MR | Zbl
[23] and , Simulation of Fractional Brownian Motion with Conditionalized Random Midpoint Displacement. Technical report, Advances in Performance analysis, http://vtt.fi/tte/tte21:traffic/rmdmn.ps (1999).
[24] and , Multifractional Brownian motion: definition and preliminary results. Technical report, INRIA, http://www.inria.fr/rrrt/rr-2645.html (1996).
[25] , , , and , nth-order fractional Brownian motion and fractional Gaussian noises. IEEE Trans. Sign. Proc. 45 (2001) 1049-1059.
[26] , , and , Fast and Exact Synthesis for 1-D Fractional Brownian Motion and Fractional Gaussian Noises. IEEE Signal Processing Letters 9 (2002) 382-384.
[27] , Wavelet-based simulation of fractional Brownian motion revisited. Preprint, http://www.stat.unc.edu/faculty/pipiras (2004). | MR | Zbl
[28] and , The Radon Transform and Local Tomography. CRC Press (1996). | MR | Zbl
[29] , Fast and exact simulation of fractional Brownian surfaces. J. Comput. Graph. Statist. 11 (2002) 587-599. | MR
Cité par Sources :
