Abstract
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.
Chapter PDF
Similar content being viewed by others
Keywords
- 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.
References
Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian Motion, Fractional Noises and Applications. SIAM 10(4), 422–438 (1968)
Gache, N., Flandrin, P., Garreau, D.: Fractal Dimension Estimators for Fractional Brownian Motion. In: Proceedings of the ICASSP, vol. 5, pp. 3557–3560 (1991)
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)
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)
Dahlhaus, R.: Efficient parameter estimation for self-similar processes. The Annals of Statistics 17, 1749–1766 (1989)
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)
Flandrin, P.: On the Spectrum of Fractional Brownian Motions. IEEE Trans. on Info. Theory 35, 197–199 (1989)
Beran, J.: Statistics for Long-Memory Processes. Chapman & Hall (1994)
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)
Gripenberg, G., Norros, I.: On the prediction for fractional brownian motion. Journal of Applied Probabilities 33, 400–410 (1996)
Peitgen, H.O., Saupe, D. (eds.): The Science of Fractal Images. Springer, New York (1988)
Mandelbrot, B.B.: Comment on Computer Rendering of Fractal Stochastic Models. Communications of the ACM 25, 581–583 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Harba, R., Douzi, H., El Hajji, M. (2012). Maximum Likelihood Estimation, Interpolation and Prediction for Fractional Brownian Motion. In: Elmoataz, A., Mammass, D., Lezoray, O., Nouboud, F., Aboutajdine, D. (eds) Image and Signal Processing. ICISP 2012. Lecture Notes in Computer Science, vol 7340. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31254-0_37
Download citation
DOI: https://doi.org/10.1007/978-3-642-31254-0_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31253-3
Online ISBN: 978-3-642-31254-0
eBook Packages: Computer ScienceComputer Science (R0)