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Identification of the multiscale fractional Brownian motion with biomechanical applications. (English) Zbl 1164.62034

A generalized fractional Brownian motion \(X(t)\) is considered where the Hurst parameter \(H\) can be different on different frequencies. More precisely, \[ X(t)=2\sum_{j=0}^K\int_{\omega_j}^{\omega_{j+1}}\sigma_j(e^{it\xi}-1)(| \xi| ^{H_j+1/2)})^{-1}\overline{\hat W}(dt), \] where \(\hat W\) is the Fourier transform of a standard Brownian motion, \(H_j\) is the Hurst parameter of \(X\) at the frequency domains \([\omega_j,\omega_{j+1}]\), \(0=\omega_0<\omega_1<\dots<\omega_K<\omega_{K+1}=\infty\), and \(K\) is the number of changes in \(H\). (The case \(K=0\) corresponds to the usual fractional Brownian motion). A wavelet-based technique is described for the estimation of \(\omega_j\) and \(H_j\). Asymptotic normality of the \(H_j\) estimates and consistency of the \(\omega_j\) estimates are demonstrated. A goodness-of-fit test is described. An empirical procedure for the estimation of \(K\) is proposed. Results of simulations and an application to biomechanical data are presented.

MSC:

62M05 Markov processes: estimation; hidden Markov models
60J65 Brownian motion
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
92C10 Biomechanics
62P10 Applications of statistics to biology and medical sciences; meta analysis
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References:

[1] Abry P., Long-range Dependence: Theory and Applications pp 527– (2002)
[2] Amemiya T., Advanced Econometrics (1985)
[3] Ayache A., Fractal Theory an Application in Engineering (1999)
[4] Bai J., Journal of Time Series Analysis 5 pp 453– (1998)
[5] DOI: 10.2307/2998540 · Zbl 1056.62523 · doi:10.2307/2998540
[6] DOI: 10.1111/1467-9892.00195 · Zbl 0972.62070 · doi:10.1111/1467-9892.00195
[7] DOI: 10.1109/18.992817 · Zbl 1061.60036 · doi:10.1109/18.992817
[8] J. M. Bardet, and P. Bertrand(2003 ) Definition, properties and wavelet analysis of multiscale fractional Brownian motion . Preprint LSP, Toulouse III. · Zbl 1142.60329
[9] DOI: 10.1023/A:1009953000763 · Zbl 1054.62579 · doi:10.1023/A:1009953000763
[10] A. Benassi, and S. Deguy(1999 ) Multi-scale fractional Brownian motion: definition and identification . Preprint LAIC. Universite B. Pascal, Clermont-Ferrand, France.
[11] Benassi A., Rev. Mathematica Iberoamericana 13 pp 19– (1997) · Zbl 0880.60053 · doi:10.4171/RMI/217
[12] DOI: 10.1016/S0167-7152(98)00078-9 · Zbl 0931.60022 · doi:10.1016/S0167-7152(98)00078-9
[13] Bertrand P., IEEE Engineering in Medicine and Biology Society 2 pp 1163– (2001)
[14] DOI: 10.1007/s007800300101 · Zbl 1035.60036 · doi:10.1007/s007800300101
[15] Collins J. J., Experimental Brain Research 9 pp 308– (1993) · doi:10.1007/BF00229788
[16] Dahlhaus R., Annals of Statistics 17 pp 1749– (1989)
[17] DOI: 10.1109/18.119751 · Zbl 0743.60078 · doi:10.1109/18.119751
[18] Fox R., Annals of Statistics 14 pp 517– (1986)
[19] DOI: 10.1007/BF01207515 · Zbl 0717.62015 · doi:10.1007/BF01207515
[20] Ikeda N., Stochastic Differential Equations and Diffusion Processes (1989) · Zbl 0684.60040
[21] DOI: 10.1016/S0246-0203(97)80099-4 · Zbl 0882.60032 · doi:10.1016/S0246-0203(97)80099-4
[22] Jacod J., Limit Theorems for Stochastic Processes (1987) · doi:10.1007/978-3-662-02514-7
[23] Kolmogorov A. N., C. R. (Doklady) Acad. Sci. URSS 26 pp 115– (1940)
[24] DOI: 10.1016/S0304-4149(99)00023-X · Zbl 0991.62014 · doi:10.1016/S0304-4149(99)00023-X
[25] DOI: 10.1111/1467-9892.00172 · Zbl 0974.62070 · doi:10.1111/1467-9892.00172
[26] DOI: 10.1137/1010093 · Zbl 0179.47801 · doi:10.1137/1010093
[27] R. Peltier, and J. Levy Vehel(1995 ) Multifractional Brownian motion: definition and preliminary results , Technical Report 2645, INRIA, Le Chesnay, France.
[28] DOI: 10.1007/s004400200196 · Zbl 1007.60026 · doi:10.1007/s004400200196
[29] DOI: 10.1111/1467-9965.00025 · Zbl 0884.90045 · doi:10.1111/1467-9965.00025
[30] Samorodnitsky G., Stable Non-Gaussian Random Processes (1994) · Zbl 0925.60027
[31] Van der Vaart A., Asymptotic Statistics (1998) · Zbl 0910.62001 · doi:10.1017/CBO9780511802256
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.