Fractional Brownian motion synthesis
According to the value of the Hurst parameter
fBm exhibits for
H > 0.5, long-range dependence and for
H < 0.5, short or intermediate dependence. This example shows each situation using the
wfbm function, which generates a sample path of this process.
For purposes of reproducibility, set the random seed to the default value. Generate a fractional Brownian motion signal of length 1000 with the Hurst parameter of 0.3. Plot the signal.
rng default h = 0.3; l = 1000; fBm03 = wfbm(h,l,'plot');
Now generate a fractional Brownian motion signal of length 1000 with the Hurst parameter of 0.7. The signal clearly exhibits a stronger low-frequency component and has, locally, less irregular behavior than
h = 0.7; l = 1000; fBm07 = wfbm(h,l,'plot');
H— Hurst parameter
Hurst parameter, specified as a positive scalar strictly less than 1.
fBm = wfbm(0.4,1000) generates a fractional
Brownian motion of length
L = 1000 with Hurst parameter
H = 0.4.
L— Signal length
Signal length, specified as a positive integer strictly greater than 100.
fBm = wfbm(0.1,500) generates a fractional
Brownian motion of length
L = 500 with Hurst parameter
H = 0.1.
ns— Number of reconstruction steps
Number of reconstruction steps, specified as a positive integer greater than 1.
A fractional Brownian motion (
fBm) is a
continuous-time Gaussian process depending on the Hurst parameter
0 < H
< 1. It generalizes the ordinary Brownian motion corresponding to
H = 0.5 and whose derivative is the white noise. The
fBm is self-similar in distribution and the variance of the
increments is given by
Var(fBm(t)-fBm(s)) = v |t-s|^(2H)
v is a positive constant.
Starting from the expression of the
fBm process as a fractional
integral of the white noise process, the idea of the algorithm is to build a
biorthogonal wavelet depending on a given orthogonal one and adapted to the parameter
Then the generated sample path is obtained by the reconstruction using the new wavelet starting from a wavelet decomposition at a given level designed as follows: details coefficients are independent random Gaussian realizations and approximation coefficients come from a fractional ARIMA process.
This method was first proposed by Meyer and Sellan and implementation issues were examined by Abry and Sellan .
Nevertheless, the samples generated following this original scheme exhibit too many high-frequency components. To circumvent this undesirable behavior Bardet et al.  propose downsampling the obtained sample by a factor of 10.
Two internal parameters
delta = 10 (the downsampling factor) and a
prec = 1E-4, to evaluate series by truncated sums, can be
modified by the user for extreme values of
A complete overview of long-range dependence process generators is available in Bardet et al .
 Abry, Patrice, and Fabrice Sellan. “The Wavelet-Based Synthesis for Fractional Brownian Motion Proposed by F. Sellan and Y. Meyer: Remarks and Fast Implementation.” Applied and Computational Harmonic Analysis 3, no. 4 (October 1996): 377–83. https://doi.org/10.1006/acha.1996.0030.
 Bardet, Jean-Marc, Gabriel Lang, Georges Oppenheim, Anne Philippe, Stilian Stoev, and Murad S. Taqqu. “Generators of Long-Range Dependent Processes: A Survey.” In Theory and Applications of Long-Range Dependence, edited by Paul Doukhan, Georges Oppenheim, and Murad S. Taqqu, 579–623. Boston: Birkhauser, 2003.