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Fractional Brownian motion synthesis

`FBM = wfbm(H,L)`

FBM = wfbm(H,L,'plot')

FBM = wfbm(H,L,NS,W)

FBM =
wfbm(H,L,W,NS)

wfbm(H,L,'plot',NS)

wfbm(H,L,'plot',W)

wfbm(H,L,'plot',NS,W)

wfbm(H,L,'plot',W,NS)

`FBM = wfbm(H,L)`

returns a fractional Brownian
motion signal `FBM`

of the Hurst parameter `H`

(```
0
< H < 1
```

) and length `L`

, following
the algorithm proposed by Abry and Sellan.

`FBM = wfbm(H,L,'plot')`

generates and plots
the `FBM`

signal.

`FBM = wfbm(H,L,NS,W)`

or ```
FBM =
wfbm(H,L,W,NS)
```

returns the `FBM`

using `NS`

reconstruction
steps and the sufficiently regular orthogonal wavelet `W`

.

`wfbm(H,L,'plot',NS)`

or `wfbm(H,L,'plot',W)`

or `wfbm(H,L,'plot',NS,W)`

or `wfbm(H,L,'plot',W,NS)`

generates
and plots the `FBM`

signal.

`wfbm(H,L)`

is equivalent to `WFBM(H,L,6,'db10')`

.

`wfbm(H,L,NS)`

is equivalent to `WFBM(H,L,NS,'db10')`

.

`wfbm(H,L,W)`

is equivalent to `WFBM(H,L,W,6)`

.

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)

where `v`

is a positive constant.

According to the value of `H`

, the `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`

file, which
generates a sample path of this process.

% Generate fBm for H = 0.3 and H = 0.7 % Set the parameter H and the sample length H = 0.3; lg = 1000; % Generate and plot wavelet-based fBm for H = 0.3 fBm03 = wfbm(H,lg,'plot');

H = 0.7; % Generate and plot wavelet-based fBm for H = 0.7 fBm07 = wfbm(H,lg,'plot'); % The last step is equivalent to % Define wavelet and level of decomposition % w = ' db10'; ns = 6; % Generate % fBm07 = wfbm(H,lg,'plot',w,ns);

`fBm07`

clearly exhibits a stronger low-frequency
component and has, locally, less irregular behavior.

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 `H`

.

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 10.

Two internal parameters `delta = 10`

(the downsampling
factor) and a threshold` prec = 1E-4`

, to evaluate
series by truncated sums, can be modified by the user for extreme
values of `H`

.

A complete overview of long-range dependence process generators is available in Bardet et al.

Abry, P.; F. Sellan (1996), “The wavelet-based synthesis
for the fractional Brownian motion proposed by F. Sellan and Y. Meyer:
Remarks and fast implementation,” *Appl. and Comp.
Harmonic Anal.*, 3(4), pp. 377–383.

Bardet, J.-M.; G. Lang, G. Oppenheim, A. Philippe, S. Stoev, M.S. Taqqu (2003), “Generators of long-range dependence processes: a survey,” Theory and applications of long-range dependence, Birkhäuser, pp. 579–623.

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