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Identification of the multiscale fractional Brownian motion with biomechanical applications

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  • Jean‐Marc Bardet
  • Pierre Bertrand

Abstract

. In certain applications, for instance, biomechanics, turbulence, finance or internet traffic, it seems suitable to model the data by a generalization of a fractional Brownian motion (FBM) for which the Hurst parameter H depends on the frequency as a piece‐wise constant function. These processes are called multiscale fractional Brownian motions. In this article, we provide a statistical study of the multiscale fractional Brownian motions. We developed a method based on wavelet analysis. By using this method, we calculated the frequency changes, estimated the different parameters, tested the goodness‐of‐fit and gave the numerical algorithm. Biomechanical data are then studied with these new tools.

Suggested Citation

  • Jean‐Marc Bardet & Pierre Bertrand, 2007. "Identification of the multiscale fractional Brownian motion with biomechanical applications," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(1), pages 1-52, January.
    • Handle: RePEc:bla:jtsera:v:28:y:2007:i:1:p:1-52
    DOI: 10.1111/j.1467-9892.2006.00494.x
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    Cited by:

    1. Pierre R. Bertrand & Abdelkader Hamdouni & Samia Khadhraoui, 2012. "Modelling NASDAQ Series by Sparse Multifractional Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 14(1), pages 107-124, March.
    2. Jean‐Marc Bardet & Pierre R. Bertrand, 2010. "A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(3), pages 458-476, September.
    3. Matthieu Garcin, 2019. "Hurst Exponents And Delampertized Fractional Brownian Motions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(05), pages 1-26, August.
    4. Jean-Marc Bardet & Imen Kammoun & Veronique Billat, 2012. "A new process for modeling heartbeat signals during exhaustive run with an adaptive estimator of its fractal parameters," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(6), pages 1331-1351, December.
    5. Xiao, Yimin, 2009. "A packing dimension theorem for Gaussian random fields," Statistics & Probability Letters, Elsevier, vol. 79(1), pages 88-97, January.
    6. Billat, Véronique L. & Mille-Hamard, Laurence & Meyer, Yves & Wesfreid, Eva, 2009. "Detection of changes in the fractal scaling of heart rate and speed in a marathon race," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(18), pages 3798-3808.
    7. Marco Dozzi & Yuliya Mishura & Georgiy Shevchenko, 2015. "Asymptotic behavior of mixed power variations and statistical estimation in mixed models," Statistical Inference for Stochastic Processes, Springer, vol. 18(2), pages 151-175, July.

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