Maximum Likelihood Estimation, Interpolation and Prediction for Fractional Brownian Motion

  • Rachid Harba
  • Hassan Douzi
  • Mohamed El Hajji
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7340)


The maximum likelihood (ML) estimation approach for fractional Brownian motion (fBm) is explored in this communication. First, a ML based estimation of the H parameter is implemented on the signal itself. This approach on the signal itself can easily be applied on non-uniformly sampled data or directly useful in the case of incomplete data. Secondly, the method is extended to provide a ML prediction and a ML interpolation for fBm which could be of interest in many domains. Results also help to explain errors in other interpolating methods such as the midpoint displacement algorithm used to synthesize fBm data.


Fractional Brownian Motion Synthetic Signal Irregular Sampling Fractional Gaussian Noise IEEE Signal Processing Letter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian Motion, Fractional Noises and Applications. SIAM 10(4), 422–438 (1968)zbMATHCrossRefGoogle Scholar
  2. 2.
    Gache, N., Flandrin, P., Garreau, D.: Fractal Dimension Estimators for Fractional Brownian Motion. In: Proceedings of the ICASSP, vol. 5, pp. 3557–3560 (1991)Google Scholar
  3. 3.
    Jennane, R., Harba, R., Jacquet, G.: Quality of Synthesis and Analysis Methods for Fractional Brownian Motion. In: Proceedings of the IEEE Workshop on Digital Signal Processing, pp. 307–310 (1996)Google Scholar
  4. 4.
    Lundahl, T., Ohley, W.J., Kay, S.M., Siffert, R.: Fractional Brownian Motion: A Maximum Likelihood Estimator and its Application to Image Texture. IEEE Transactions on Medical Imaging 5(3), 152–161 (1986)CrossRefGoogle Scholar
  5. 5.
    Dahlhaus, R.: Efficient parameter estimation for self-similar processes. The Annals of Statistics 17, 1749–1766 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Hoeffer, S., Kumaresan, R., Pandit, M., Ohley, W.J.: Estimation of the Fractal Dimension of a Stochastic Fractal from Noise Corrupted Data. Archiv fuer Electronic und Übertragungstechnick 46(1), 13–21 (1992)Google Scholar
  7. 7.
    Flandrin, P.: On the Spectrum of Fractional Brownian Motions. IEEE Trans. on Info. Theory 35, 197–199 (1989)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Beran, J.: Statistics for Long-Memory Processes. Chapman & Hall (1994)Google Scholar
  9. 9.
    Perrin, E., Harba, R., Jennane, R., Iribaren, I.: Fast and exact synthesis for 1D fractional Brownian motion and fractional Gaussian noises. IEEE Signal Processing Letters 9(11), 382–384 (2002)CrossRefGoogle Scholar
  10. 10.
    Gripenberg, G., Norros, I.: On the prediction for fractional brownian motion. Journal of Applied Probabilities 33, 400–410 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Peitgen, H.O., Saupe, D. (eds.): The Science of Fractal Images. Springer, New York (1988)zbMATHGoogle Scholar
  12. 12.
    Mandelbrot, B.B.: Comment on Computer Rendering of Fractal Stochastic Models. Communications of the ACM 25, 581–583 (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Rachid Harba
    • 1
  • Hassan Douzi
    • 2
  • Mohamed El Hajji
    • 2
  1. 1.Laboratoire PRISME, Polytech’OrléansUniversité d’OrléansOrléansFrance
  2. 2.Laboratoire IRF-SIC, Faculté des sciencesUniversité Ibn ZohrAgadirMorocco

Personalised recommendations