Fractal dimension estimators for fractional Brownian motions

Abstract

Fractional Brownian Motions (fBm) are nonstationary and selfsimilar stochastic processes which extend ordinary Brownian motion and are of great importance for modeling processes with long-term dependencies, such as l/f-type processes. Identification of fBm amounts to estimate one single scalar parameter : the fractal dimension, related to the roughness of fBm's samples. According to the structural properties of fBm, different fractal dimension estimators can be considered. We have chosen five of them, which operate either in the frequency domain (identification of a spectral exponent via spectrum analysis), in the time domain (maximum likelihood on one hand, methods based on length measurements of fBm's samples at different observation scales on the other hand) or in a mixed time-scale domain (identification of a self-similarity parameter via the variance of wavelets coefficients). The relevance of these differents estimators is discussed and their performance is compared on simulated and real data.I -FRACTIONAL BROWNIAN MOTIONS I n a number of physical phenomena, strong long-term dependencies are involved and l/fp spectral behaviors are observed over a wide range of frequencies [ 11. A convenient modeling for processes of this kind has been proposed by Mandelbrot and Van Ness [2] and is referred to as fractional Brownian motion (fBm). By definition, fBm is a nonstationary process whose expression reads (S) .P + j ( t -s)H-l/*dB(s)); 0 < H < 1, where B ( t ) = B,/2(t) is ordinary Brownian motion. The increments of fBm are stationary and, moreover, they are statistically self-similar, which means that for any a > 0 (and with the convention BH(0) = 0), [ Btl(at)) and [ aHB&)) undergo the same probabilistic behavior.In the frequency domain, an average power spectrum density (PSD) of fBm can be defined [3] and it tums out that it is proportional to I/fp with p = 2H + 1. In the time domain, the selfsimilarity structure which underlies the definition of fBm allows to associate to it a fractal dimension D = 2 -H , which varies therefore between 1 and 2 [2]. This fractal dimension is related to the roughness of fBm's samples and it appears as a relevant parameter for identification and classification [4], motivating hence the development of efficient procedures aimed at its estimation. 0 * fomcrly with LTS-ICPI Lyon # formerly with EDFIDER'RISDM t also with G D R 134 CNRS "Trarlemenl du Signal el Images" I1 -FRACTAL DIMENSION ESTIMATORS According to the above properties, five different estimators of D have been considered, each based on some specific feature of fBm. A. Spectrum AnalysisThis exploits the fact that, because fBm has a llfp spectral behavior, its PSD is supposed to be a straight line in a log-log plot. The slope -b of this line can therefore be estimated by a least-squares fit on some log-log periodogram estimate ( Fig. 1) and D can be deduced from pas D = (5 -m/2.It is clear that such an estimator necessitates a large amount of data points for achieving a significant variance reduction. Another fundame...

Analytical formulation of the fractal dimension of filtered stochastic signals

“…However, the performance was similar to that obtained with estimators based on wavelet analysis (see [11]). Note that Flandrin [10] demonstrated the link between length measurement L BK (k) and variance of the wavelet coeffi-…”

“…The performance of such estimator in its average version, known as the Burlaga and Klein [2] estimator L BK (k), was excellent and outperformed the spectral estimators and the maximum likelihood estimators (see [11]). However, the performance was similar to that obtained with estimators based on wavelet analysis (see [11]).…”

“…By using equations (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25), it becomes:…”

“…Quantification of the impact of such linear filtering on the regularity of the processes studied is thus necessary. For these applications, the most commonly used fractal dimension estimators have been based on the box counting method (for example see [34]), the spectral slope (for example see [11]) and length measurement [2].…”

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“…The value of can be estimated as the negative gradient of the linear least squares fit to the -plot of the power spectrum [26]. However, to obtain accurate and stable estimates of the power law exponent, an ensemble average of power spectra over a long period of time is required, [27].…”