Estimation of anisotropic Gaussian fields through Radon transform

Hermine Biermé; Frédéric Richard

ESAIM: Probability and Statistics (2007)

  • Volume: 12, page 30-50
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Biermé, Hermine, and Richard, Frédéric. "Estimation of anisotropic Gaussian fields through Radon transform." ESAIM: Probability and Statistics 12 (2007): 30-50. <http://eudml.org/doc/104402>.

@article{Biermé2007,
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 quite accurately. },
author = {Biermé, Hermine, Richard, Frédéric},
journal = {ESAIM: Probability and Statistics},
keywords = {Anisotropic Gaussian fields; identification; estimator; asymptotic normality; Radon transform.; anisotropic Gaussian fields; Radon transform},
language = {eng},
month = {11},
pages = {30-50},
publisher = {EDP Sciences},
title = {Estimation of anisotropic Gaussian fields through Radon transform},
url = {http://eudml.org/doc/104402},
volume = {12},
year = {2007},
}

TY - JOUR
AU - Biermé, Hermine
AU - Richard, Frédéric
TI - Estimation of anisotropic Gaussian fields through Radon transform
JO - ESAIM: Probability and Statistics
DA - 2007/11//
PB - EDP Sciences
VL - 12
SP - 30
EP - 50
AB - 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.
LA - eng
KW - Anisotropic Gaussian fields; identification; estimator; asymptotic normality; Radon transform.; anisotropic Gaussian fields; Radon transform
UR - http://eudml.org/doc/104402
ER -

References

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  1. P. Abry and F. Sellan, 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.  Zbl0862.60036
  2. A. Ayache, A. Bonami and A. Estrade, Identification and series decomposition of anisotropic Gaussian fields. Proceedings of the Catania ISAAC05 congress (2005).  Zbl1181.60056
  3. 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.  Zbl1032.62077
  4. A. Begyn, Asymptotic development and central limit theorem for quadratic variations of gaussian processes. To appear in Bernoulli (2006).  Zbl1143.60030
  5. A. Benassi, S. Cohen, J. Istas and S. Jaffard, Identification of filtered white noises. Stochastic Process. Appl.75 (1998) 31–49.  Zbl0932.60037
  6. A. Benassi, S. Jaffard and D. Roux, Elliptic Gaussian random processes. Rev. Mathem. Iberoamericana. 13 (1997) 19–89.  Zbl0880.60053
  7. H. Biermé, Champs aléatoires : autosimilarité, anisotropie et étude directionnelle. PhD thesis, Université d'Orléans, www.math-info.univ-paris5.fr/~bierme (2005).  
  8. A. Bonami and A. Estrade, Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl.9 (2003) 215–236.  Zbl1034.60038
  9. 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.  
  10. J.F. Coeurjolly, Inférence statistique pour les mouvements browniens fractionnaires et multifractionnaires. PhD thesis, Université Joseph Fourier (2000).  
  11. J.F. Coeurjolly, Estimating the parameters of fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process.4 (2001) 199–227.  Zbl0984.62058
  12. D. Dacunha-Castelle and M. Duflo, Probabilités et statistiques, Vol. 2. Masson (1983).  Zbl0535.62004
  13. C.R. Dietrich and G.N. Newsam, Fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput.18 (1997) 1088–1107.  Zbl0890.65149
  14. N. Enriquez, A simple construction of the fractional brownian motion. Stochastic Process. Appl.109 (2004) 203–223.  Zbl1075.60019
  15. 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.  Zbl0882.60032
  16. R. Jennane, R. Harba, E. Perrin, A. Bonami and A. Estrade, Analyse de champs browniens fractionnaires anisotropes. 18e colloque du GRETSI (2001) 99–102.  
  17. L.M. Kaplan and C.C.J. Kuo, An Improved Method for 2-d Self-Similar Image Synthesis. IEEE Trans. Image Process.5 (1996) 754–761.  
  18. J.T. Kent and A.T.A. Wood, Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. Roy. Statist. Soc. Ser. B59 (1997) 679–699.  Zbl0889.62072
  19. G. Lang and F. Roueff, Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates. Stat. Inference Stoch. Process.4 (2001) 283–306.  Zbl1008.62081
  20. S. Leger, Analyse stochastique de signaux multi-fractaux et estimations de paramètres. Ph.D. thesis, Université d'Orléans, (2000).  URIhttp://www.univ-orleans.fr/mapmo/publications/leger/these.php
  21. B.B. Mandelbrot and J. Van Ness, Fractional Brownian motion, fractionnal noises and applications. Siam Review10 (1968) 422–437.  Zbl0179.47801
  22. Y. Meyer, F. Sellan and M.S. Taqqu, Wavelets, Generalised White Noise and Fractional Integration: The Synthesis of Fractional Brownian Motion. J. Fourier Anal. Appl.5 (1999) 465–494.  Zbl0948.60026
  23. 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).  
  24. R.F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results. Technical report, INRIA, (1996).  URIhttp://www.inria.fr/rrrt/rr-2645.html
  25. E. Perrin, R. Harba, C. Berzin-Joseph, I. Iribarren and A. Bonami, nth-order fractional Brownian motion and fractional Gaussian noises. IEEE Trans. Sign. Proc.45 (2001) 1049–1059.  
  26. E. Perrin, R. Harba, R. Jennane and I. Iribarren, Fast and Exact Synthesis for 1-D Fractional Brownian Motion and Fractional Gaussian Noises. IEEE Signal Processing Letters9 (2002) 382–384.  
  27. V. Pipiras, Wavelet-based simulation of fractional Brownian motion revisited. Preprint, (2004).  Zbl1074.60048URIhttp://www.stat.unc.edu/faculty/pipiras
  28. A.G. Ramm and A.I. Katsevich, The Radon Transform and Local Tomography. CRC Press (1996).  Zbl0863.44001
  29. M.L. Stein, Fast and exact simulation of fractional Brownian surfaces. J. Comput. Graph. Statist.11 (2002) 587–599.  

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