Parameter estimation of fractional Brownian motion

`HEST = wfbmesti(X)`

`HEST = wfbmesti(X)`

returns a one-by-three
vector `HEST`

which contains three estimates of the
fractal index `H`

of the input signal `X`

.
The signal `X`

is assumed to be a realization of
fractional Brownian motion with Hurst index `H`

.

The first two elements of the vector are estimates based on the second derivative with the second computed in the wavelet domain.

The third estimate is based on the linear regression in loglog plot, of the variance of detail versus level.

A fractional Brownian motion (`fBm`

) is a continuous-time
Gaussian process depending on the so-called 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

Var(fBm(t)-fBm(s)) = v |t-s|^(2H)

where `v`

is a positive constant.

This special form of the variance of the increments suggests
various ways to estimate the parameter `H`

. One can
find in Bardet et al. a survey of such methods. The `wfbmesti`

file provides three different
estimates. The first one, due to Istas and Lang, is based on the discrete
second-order derivative. The second one is a wavelet-based adaptation
and has similar properties. The third one, proposed by Flandrin, estimates `H`

using
the slope of the loglog plot of the detail variance versus the level.
A more recent extension can be found in Abry et al.

Abry, P.; P. Flandrin, M.S. Taqqu, D. Veitch (2003), "Self-similarity and long-range dependence through the wavelet lens," Theory and applications of long-range dependence, Birkhäuser, pp. 527–556.

Bardet, J.-M.; G. Lang, G. Oppenheim, A. Philippe, S. Stoev, M.S. Taqqu (2003), "Semi-parametric estimation of the long-range dependence parameter: a survey," Theory and applications of long-range dependence, Birkhäuser, pp. 557–577.

Flandrin, P. (1992), "Wavelet analysis and synthesis
of fractional Brownian motion," *IEEE Trans. on Inf.
Th.*, 38, pp. 910–917.

Istas, J.; G. Lang (1994), "Quadratic variations and
estimation of the local Hölder index of a Gaussian process," *Ann.
Inst. Poincaré*, 33, pp. 407–436.

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