Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-11T01:09:15.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of anisotropic Gaussian fields through Radon transform

Published online by Cambridge University Press:  13 November 2007

Hermine Biermé  and
Frédéric Richard
Hermine Biermé
Affiliation:
MAP5-UMR 8145, Université René Descartes45, rue des Saints-Pères, 75270 Paris cedex 06 France, hermine.bierme@math-info.univ-paris5.fr; frederic.richard@math-info.univ-paris5.fr
Frédéric Richard
Affiliation:
MAP5-UMR 8145, Université René Descartes45, rue des Saints-Pères, 75270 Paris cedex 06 France, hermine.bierme@math-info.univ-paris5.fr; frederic.richard@math-info.univ-paris5.fr
Get access

Abstract

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 quiteaccurately.

Keywords

Anisotropic Gaussian fieldsidentificationestimatorasymptotic normalityRadon transform.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 30 - 50
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abry, P. and Sellan, F., 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. CrossRef
A. Ayache, A. Bonami and A. Estrade, Identification and series decomposition of anisotropic Gaussian fields. Proceedings of the Catania ISAAC05 congress (2005).
J.M. Bardet, G. Lang, G. Oppenheim, A. Philippe, S. Stoev and M.S. Taqqu, Semi-parametric estimation of the long-range dependence parameter: a survey. In Theory and applications of long-range dependence, Birkhäuser Boston (2003) 557–577.
A. Begyn, Asymptotic development and central limit theorem for quadratic variations of gaussian processes. To appear in Bernoulli (2006).
Benassi, A., Cohen, S., Istas, J. and Jaffard, S., Identification of filtered white noises. Stochastic Process. Appl. 75 (1998) 31–49. CrossRef
A. Benassi, S. Jaffard and D. Roux, Elliptic Gaussian random processes. Rev. Mathem. Iberoamericana. 13 (1997) 19–89.
H. Biermé, Champs aléatoires : autosimilarité, anisotropie et étude directionnelle. PhD thesis, Université d'Orléans, www.math-info.univ-paris5.fr/~bierme (2005).
Bonami, A. and Estrade, A., Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9 (2003) 215–236. CrossRef
G. Chan, 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.
J.F. Coeurjolly, Inférence statistique pour les mouvements browniens fractionnaires et multifractionnaires. PhD thesis, Université Joseph Fourier (2000).
Coeurjolly, J.F., Estimating the parameters of fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199–227. CrossRef
D. Dacunha-Castelle and M. Duflo, Probabilités et statistiques, Vol. 2. Masson (1983).
Dietrich, C.R. and Newsam, G.N., Fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18 (1997) 1088–1107. CrossRef
Enriquez, N., A simple construction of the fractional brownian motion. Stochastic Process. Appl. 109 (2004) 203–223. CrossRef
J. Istas and G. Lang, Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré, Prob. Stat. 33 (1997) 407–436.
R. Jennane, R. Harba, E. Perrin, A. Bonami and A. Estrade, Analyse de champs browniens fractionnaires anisotropes. 18 e colloque du GRETSI (2001) 99–102.
Kaplan, L.M. and Kuo, C.C.J., Improved Method, An for 2-d Self-Similar Image Synthesis. IEEE Trans. Image Process. 5 (1996) 754–761. CrossRef
Kent, J.T. and Wood, A.T.A., Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. Roy. Statist. Soc. Ser. B 59 (1997) 679–699. CrossRef
Lang, G. and Roueff, F., Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates. Stat. Inference Stoch. Process. 4 (2001) 283–306. CrossRef
S. Leger, 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).
Mandelbrot, B.B. and Van Ness, J., Fractional Brownian motion, fractionnal noises and applications. Siam Review 10 (1968) 422–437. CrossRef
Meyer, Y., Sellan, F. and Taqqu, M.S., Wavelets, Generalised White Noise and Fractional Integration: The Synthesis of Fractional Brownian Motion. J. Fourier Anal. Appl. 5 (1999) 465–494. CrossRef
I. Norros and P. Mannersalo, 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).
R.F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results. Technical report, INRIA, http://www.inria.fr/rrrt/rr-2645.html (1996).
Perrin, E., Harba, R., Berzin-Joseph, C., Iribarren, I. and Bonami, A., nth-order fractional Brownian motion and fractional Gaussian noises. IEEE Trans. Sign. Proc. 45 (2001) 1049–1059. CrossRef
Perrin, E., Harba, R., Jennane, R. and Iribarren, I., Fast and Exact Synthesis for 1-D Fractional Brownian Motion and Fractional Gaussian Noises. IEEE Signal Processing Letters 9 (2002) 382–384. CrossRef
V. Pipiras, Wavelet-based simulation of fractional Brownian motion revisited. Preprint, http://www.stat.unc.edu/faculty/pipiras (2004).
A.G. Ramm and A.I. Katsevich, The Radon Transform and Local Tomography. CRC Press (1996).
Stein, M.L., Fast and exact simulation of fractional Brownian surfaces. J. Comput. Graph. Statist. 11 (2002) 587–599. CrossRef